IAL Mathematics Sample Assessment Material

INTERNATIONAL ADVANCED LEVEL

Mathematics, Further Mathematics and Pure Mathematics SAMPLE ASSESSMENT MATERIALS Pearson Edexcel International Advanced Subsidiary in Mathematics (XMA01) Pearson Edexcel International Advanced Level in Mathematics (YMA01) Pearson Edexcel International Advanced Subsidiary in Further Mathematics (XFM01) Pearson Edexcel International Advanced Level in Further Mathematics (YFM01) Pearson Edexcel International Advanced Subsidiary in Pure Mathematics (XPM01) Pearson Edexcel International Advanced Level in Pure Mathematics (YPM01) For first teaching in September 2013 First examination January 2014

Contents WMA01/01: Core Mathematics C12 Question Paper

3

WMA01/01: Core Mathematics C12 Mark Scheme

47

WMA02/01: Core Mathematics C34 Question Paper

57

WMA02/01: Core Mathematics C34 Mark Scheme

101

WFM01/01: Further Pure Mathematics F1 Question Paper

111

WFM01/01: Further Pure Mathematics F1 Mark Scheme

131

WFM02/01: Further Pure Mathematics F2 Question Paper

135

WFM02/01: Further Pure Mathematics F2 Mark Scheme

159

WFM03/01: Further Pure Mathematics F3 Question Paper

167

WFM03/01: Further Pure Mathematics F3 Mark Scheme

195

WME01/01: Mechanics M1 Question Paper

203

WME01/01: Mechanics M1 Mark Scheme

231

WME02/01: Mechanics M2 Question Paper

235

WME02/01: Mechanics M2 Mark Scheme

267

WME03/01: Mechanics M3 Question Paper

275

WME03/01: Mechanics M3 Mark Scheme

303

WST01/01: Statistics S1 Question Paper

311

WST01/01: Statistics S1 Mark Scheme

339

WST02/01: Statistics S2 Question Paper

347

WST02/01: Statistics S2 Mark Scheme

371

WST03/01: Statistics S3 Question Paper

379

WST03/01: Statistics S3 Mark Scheme

403

WDM01/01: Decision Mathematics D1 Question Paper

411

WDM01/01: Decision Mathematics D1 Answer Booklet

423

WDM01/01: Decision Mathematics D1 Mark Scheme

443

Notes on Marking Principles for all Mark Schemes

454

Pearson Edexcel International Advanced Level in Mathematics

© Pearson Education Limited 2013

Sample Assessment Materials

1

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Pearson Edexcel International Advanced Level in Mathematics

© Pearson Education Limited 2013

Sample Assessment Materials

Write your name here Surname

Other names

Pearson Edexcel

Centre Number

Candidate Number

International Advanced Level

Core Mathematics C12 Advanced Subsidiary

Sample Assessment Material Time: 2 hours 30 minutes

Paper Reference

WMA01/01

You must have: Mathematical Formulae and Statistical Tables (Blue)

Total Marks

Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation and integration, or have retrievable mathematical formulae stored in them.

Instructions

black ink or ball-point pen. •• Use If pencil is used for diagrams/sketches/graphs it must be dark (HB or B). Coloured pencils and highlighter pens must not be used. in the boxes at the top of this page with your name, • Fill centre number and candidate number. Answer all questions and ensure that your answers to parts of questions are • clearly labelled. Answer the questions in the spaces provided • – there may be more space than you need. You should show sufficient working to make your methods clear. Answers • without working may not gain full credit. When a calculator • degree of accuracy.is used, the answer should be given to an appropriate

Information

total mark for this paper is 125. •• The The marks for each question are shown in brackets – use this as a guide as to how much time to spend on each question.

Advice

each question carefully before you start to answer it. •• Read Try to answer every question. • Check your answers if you have time at the end.

S44999A ©2013 Pearson Education Ltd.

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1.

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Simplify fully 1

(a) (25 x 4 ) 2 ,

(1)

3 − 4 2

(b) (25 x ) .

(2)

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Q1

(Total 3 marks) 2

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Find the first 3 terms, in ascending powers of x, of the binomial expansion of (3 – x)6 and simplify each term.

(4)

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Q2

(Total 4 marks) Pearson Edexcel International Advanced Level in Mathematics

*S44999A0344* © Pearson Education Limited 2013

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Answer this question without the use of a calculator and show all your working. (i) Show that (5 − 8 )(1 +

2) ≡ a + b 2

giving the values of the integers a and b.

(3)

(ii) Show that 80 +

30 ≡ c 5 , where c is an integer. 5

(3)

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Q3

___________________________________________________________________________ (Total 6 marks) Pearson Edexcel International Advanced Level in Mathematics

*S44999A0544* © Pearson Education Limited 2013

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4.

Given that y = 2x5 + 7 +

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1 , x ≠ 0, find, in their simplest form, x3

(a)

dy , dx

(3)

(b)

∫ y dx.

(4)

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Q4

___________________________________________________________________________ (Total 7 marks) Pearson Edexcel International Advanced Level in Mathematics

*S44999A0744* © Pearson Education Limited 2013

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5. y=

5 3x − 2 2

The table below gives values of y rounded to 3 decimal places where necessary. x

2

2.25

2.5

2.75

3

y

0.5

0.379

0.299

0.242

0.2

Use the trapezium rule, with all the values of y from the table above, to find an approximate value for

∫

3

2

5 dx 3x − 2 2

(4)

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Q5

(Total 4 marks) Pearson Edexcel International Advanced Level in Mathematics

*S44999A0944* © Pearson Education Limited 2013

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f(x) = x4 + x3 + 2x2 + ax + b, where a and b are constants. When f (x) is divided by (x – 1), the remainder is 7 (a) Show that a + b = 3

(2)

When f (x) is divided by (x + 2), the remainder is –8 (b) Find the value of a and the value of b.

(5)

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Q6

___________________________________________________________________________ (Total 7 marks) Pearson Edexcel International Advanced Level in Mathematics

*S44999A01144* © Pearson Education Limited 2013

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A sequence a1, a2, a3, ... is defined by a1 = 2 an+ 1 = 3an – c where c is a constant. (a) Find an expression for a2 in terms of c. Given that

3

∑a i =1

i

(1)

=0

(b) find the value of c.

(4)

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Q7

___________________________________________________________________________ (Total 5 marks) Pearson Edexcel International Advanced Level in Mathematics

*S44999A01344* © Pearson Education Limited 2013

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The equation (k + 3)x2 + 6x + k = 5, where k is a constant, has two distinct real solutions for x. (a) Show that k satisfies k2 – 2k – 24 < 0 (b) Hence find the set of possible values of k.

(4) (3)

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Q8

___________________________________________________________________________ (Total 7 marks) Pearson Edexcel International Advanced Level in Mathematics

*S44999A01544* © Pearson Education Limited 2013

Sample Assessment Materials

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Given that y = 3x2, (a) show that log3 y = 1 + 2 log3 x

(3)

(b) Hence, or otherwise, solve the equation 1 + 2 log3 x = log3 (28x – 9)

(3)

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Q9

___________________________________________________________________________ (Total 6 marks) Pearson Edexcel International Advanced Level in Mathematics

*S44999A01744* © Pearson Education Limited 2013

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10.

y (3, 27)

C

A

O

x

Figure 1 Figure 1 shows a sketch of the curve C with equation y = f(x), where f(x) = x2(9 – 2x). There is a minimum at the origin, a maximum at the point (3, 27) and C cuts the x-axis at the point A. (a) Write down the coordinates of the point A.

(1)

(b) On separate diagrams sketch the curve with equation (i) y = f(x + 3), (ii) y = f(3x). On each sketch you should indicate clearly the coordinates of the maximum point and any points where the curves cross or meet the coordinate axes. (6) The curve with equation y = f(x) + k, where k is a constant, has a maximum point at (3, 10). (c) Write down the value of k.

(1)

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Q10

___________________________________________________________________________ (Total 8 marks) Pearson Edexcel International Advanced Level in Mathematics

*S44999A02144* © Pearson Education Limited 2013

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11.

y

y = –x2 + 2x + 24 R

A

B

O

y=x+4 x

Figure 2 The straight line with equation y = x + 4 cuts the curve with equation y = –x2 + 2x + 24 at the points A and B, as shown in Figure 2. (a) Use algebra to find the coordinates of the points A and B.

(4)

The finite region R is bounded by the straight line and the curve and is shown shaded in Figure 2. (b) Use calculus to find the exact area of R.

(7)

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Q11

___________________________________________________________________________ (Total 11 marks) Pearson Edexcel International Advanced Level in Mathematics

*S44999A02344* © Pearson Education Limited 2013

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12. The circle C has centre A(2, 1) and passes through the point B(10, 7) (a) Find an equation for C.

(4)

The line l1 is the tangent to C at the point B. (b) Find an equation for l1

(4)

The line l2 is parallel to l1 and passes through the mid-point of AB. Given that l2 intersects C at the points P and Q, (c) find the length of PQ, giving your answer in its simplest surd form.

(3)

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Q12

(Total 11 marks) Pearson Edexcel International Advanced Level in Mathematics

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13. x y

Figure 3 Figure 3 shows a flowerbed. Its shape is a quarter of a circle of radius x metres with two equal rectangles attached to it along its radii. Each rectangle has length equal to x metres and width equal to y metres. Given that the area of the flowerbed is 4 m2, (a) show that

y=

16 − π x 2 8x (3)

(b) Hence show that the perimeter P metres of the flowerbed is given by the equation P=

8 + 2x x (3)

(c) Use calculus to find the minimum value of P.

(5)

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Question 13 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________

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Q13

(Total 11 marks) Pearson Edexcel International Advanced Level in Mathematics

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14. In this question you must show all stages of your working. (Solutions based entirely on graphical or numerical methods are not acceptable.) (a) Solve for 0  x < 360°, giving your answers in degrees to 1 decimal place, 3sin(x + 45°) = 2

(4)

(b) Find, for 0  x < 2π, all the solutions of 2sin2 x + 2 = 7cos x giving your answers in radians.

(6)

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Question 14 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________

Q14

___________________________________________________________________________ (Total 10 marks) Pearson Edexcel International Advanced Level in Mathematics

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35

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15.

X Z

m 4c

α 6 cm

W

9 cm

Y

Figure 4 The triangle XYZ in Figure 4 has XY = 6 cm, YZ = 9 cm, ZX = 4 cm and angle ZXY = α. The point W lies on the line XY. The circular arc ZW, in Figure 4 is a major arc of the circle with centre X and radius 4 cm. (a) Show that, to 3 significant figures, α = 2.22 radians. (b) Find the area, in cm2, of the major sector XZWX.

(2) (3)

The region enclosed by the major arc ZW of the circle and the lines WY and YZ is shown shaded in Figure 4. Calculate (c) the area of this shaded region,

(3)

(d) the perimeter ZWYZ of this shaded region.

(4)

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Question 15 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 38

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Question 15 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________

Q15

___________________________________________________________________________ (Total 12 marks) Pearson Edexcel International Advanced Level in Mathematics

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41

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16. Maria trains for a triathlon, which involves swimming, cycling and running. On the first day of training she swims 1.5 km and then she swims 1.5 km on each of the following days. (a) Find the total distance that Maria swims in the first 17 days of training.

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(1)

Maria also runs 1.5 km on the first day of training and on each of the following days she runs 0.25 km further than on the previous day. So she runs 1.75 km on the second day and 2 km on the third day and so on. (b) Find how far Maria runs on the 17th day of training.

(2)

Maria also cycles 1.5 km on the first day of training and on each of the following days she cycles 5% further than on the previous day. (c) Find the total distance that Maria cycles in the first 17 days of training.

(3)

(d) Find the total distance Maria travels by swimming, running and cycling in the first 17 days of training. (3) Maria needs to cycle 40 km in the triathlon. (e) On which day of training does Maria first cycle more than 40 km?

(4)

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Q16

(Total 13 marks) TOTAL FOR PAPER: 125 MARKS END 44

46

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Sample Assessment Materials

WMA01/01: Core Mathematics C12 Question Number 1. (a) (b)

Scheme

Marks B1

5x 2 (25 x 4 )

− 32

=

1 4

(25 x )

or (25 x 4 )

3 2

3 2

= 1 2 5 x 6 o r b e tte r

M1

1

A1

125 x6

(2) (3 marks)

6 2  × (− x)  2

( 3 − x )6 = 36 + 35 × 6 × (− x) + 34 × 

2.

= 729,

3. (i)

(1)

−1458x ,

M1

+1215x 2

B1, A1, A1 (4 marks)

(5 − 8 )(1 + 2 ) Multiplies out brackets correctly.

M1 8 = 2 2 , seen or implied at any point. B1 1 + 3 2 or a = 1 and b = 3 A1

=5+5 2 − 8 −4 =5+5 2 −2 2 −4 =1+ 3 2 (ii)

Method 1

Either

80 +

30   5 

Method 2 5  5 

= 4 5 + ...

= 4 5+6 5

 400 + 30  5  5   5

Or  

 20 + ..  .. =   ..  ..  50 5  =    5  = 10

80 +

900 5

= 80 + 180 M1

= 4 5 + ...

B1

= 4 5+6 5 5

As this is a “show that” question – all working should be shown.

Pearson Edexcel International Advanced Level in Mathematics

(3)

Method 3

© Pearson Education Limited 2013

A1 (3) (6 marks)

Sample Assessment Materials

47

WMA01/01: Core Mathematics C12 Question Number 4. (a)

(b)

Scheme

dy = 10 x 4 − 3 x − 4 dx

(  = ) 26x

6

+ 7x +

−2

or

10 x 4 −

6

−2

3 x4

(3) M1 A1 A1

+C

1 × 0.25 ; × 0.5 + 0.2 + 2 ( 0.379 + 0.299 + 0.242 )} 2

{

1 8

M1 A1 A1

x x x = + 7x − −2 3 2

5.

{=

Marks

B1

(4) (7 marks)

Outside brackets B1 aef 1 × 0.25 or 18 2 For structure of M1 {................} ; Correct A1 expression inside brackets which all must be multiplied by their “outside constant”. awrt 0.32 A1

(2.540) } = 0.3175 or 0.318

4 marks 6. (a)

f ( x) = x 4 + x3 + 2 x 2 + ax + b Attempting f (1) or f (−1). f (1) = 1 + 1 + 2 + a + b = 7 or 4 + a + b = 7  a + b = 3 (as required) AG

M1 A1 * cso (2)

(b)

Attempting f ( − 2) or f (2). f (− 2) = 16 − 8 + 8 − 2a + b = − 8

{ − 2a + b = − 24}

Solving both equations simultaneously to get as far as a = ... or b = ... Any one of a = 9 or b = − 6 Both a = 9 and b = − 6

M1 A1 dM1 A1 A1 cso (5) (7 marks)

48

Pearson Edexcel International Advanced Level in Mathematics

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Sample Assessment Materials

WMA01/01: Core Mathematics C12 Question Number 7. (a)

Scheme

( a2 = )

Marks

B1

6−c

(1) (b)

M1 A1ft A1 o.e

c = 5.2

So

8. (a)

M1

(= 18 - 4c) a3 = 3(their a2 ) − c a1 + a2 + a3 = 2 + "(6 − c)"+ "(18 − 4c)" “ 26 − 5c ” = 0

(4) (5 marks)

M1

c≠k

Attempts b2 − 4ac for a = (k + 3) , b = 6 and their c. b 2 − 4 a c = 6 2 − 4 ( k + 3)( k − 5 )

A1 B1

− 4k 2 + 8k + 96 2

2

2

As b − 4 a c > 0 , then − 4k + 8k + 96 > 0

and so, k − 2k − 24 < 0

A1 * (4)

(b)

Attempts to solve k 2 − 2 k − 24 = 0 to give k = (  Critical values, k = 6 , − 4. )

M1

k 2 − 2 k − 24 < 0 gives - 4 < k < 6

M1 A1 (3) (7 marks)

9. (a)

(b)

log3 3x 2 = log3 3 + log3 x2 or log y − log x 2 = log 3 or log y − log 3 = log x 2

B1

log3 x 2 = 2log3 x

B1

Using log 3 3 =1 and deduces answer.

B1

3x 2 = 28 x − 9

M1

Solves 3 x 2 − 28 x + 9 = 0

to give x =

(3)

1 3

or x = 9

M1 A1 (3) (6 marks)

Pearson Edexcel International Advanced Level in Mathematics

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Sample Assessment Materials

49

WMA01/01: Core Mathematics C12 Question Number 10. (a) (b) i

Scheme

Marks

{Coordinates of A are} ( 4.5, 0 )

B1

y

Horizontal translation M1

27

-3 and their ft 1.5 on positive A1 ft x-axis

-3 (b) ii

1.5

O

Maximum at 27 marked on the B1 y-axis x

(3)

y

Correct shape, minimum at (0, 0) and a maximum within M1 the first quadrant.

(1, 27)

1.5 on x-axis A1 ft

O

(c)

{k =}

1.5

Maximum at (1, 27)

B1

x

− 17

(3) B1 (1) (8 marks)

50

Pearson Edexcel International Advanced Level in Mathematics

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Sample Assessment Materials

WMA01/01: Core Mathematics C12 Question Number 11. (a)

Scheme

Marks

Curve: y = − x 2 + 2 x + 24 , Line: y = x + 4 {Curve = Line}  − x 2 + 2 x + 24 = x + 4 x 2 − x − 20 {= 0}  ( x − 5)( x + 4) {= 0}  x = .....

So, x = 5, − 4 So corresponding y-values are y = 9 and y = 0

Eliminating y correctly. B1 Attempt to solve a M1 resulting quadratic to give x = their values. Both x = 5 and A1 x = −4 B1ft (4)

(b)

{

}

(− x 2 + 2 x + 24)dx = −

M1: x n → x n + 1 for any one term. 1st A1 at least two out of M1 A1 A1 three terms correct. 2nd A1 for correct answer.

x3 2 x 2 + + 24 x {+ c} 3 2

 x3 2 x 2  + + 24 x  = (......) − (......) − 2  3  −4

Substitutes 5 and − 4 (or their limits from part (a)) into an dM1 “integrated function” and subtracts, either way round.

  125 1   64   + 25 + 120  −  + 16 − 96  =  103 − 3 3 3     

2     −  − 58  = 162  3    

5

Area of ∆ = 12 (9)(9) = 40.5

Uses correct method for finding area of triangle.

So area of R is 162 − 40.5 = 121.5

Pearson Edexcel International Advanced Level in Mathematics

Area under curve – Area of triangle. 121.5

© Pearson Education Limited 2013

M1 M1 A1 oe cao (7) (11 marks)

Sample Assessment Materials

51

WMA01/01: Core Mathematics C12 Question Number 12. (a)

(10 − 2) 2 + (7 − 1) 2 or

( x ± 2) 2 + ( y ± 1) 2 = k 2

Scheme

Marks

(10 − 2) 2 + (7 − 1) 2

M1 A1 M1

(k a positive value)

2

2

( x − 2) + ( y − 1) = 100 (Accept 10 for 100)

A1

(Answer only scores full marks) (b)

(c)

7 −1 6 = (or equiv.) Must be seen in part (b) 10 − 2 8 −4 Gradient of tangent = (Using perpendicular gradient method) 3 y − 7 = m( x − 10) −4 y−7 = ( x − 10) or equivalent (ft gradient of radius, dep. on both M marks) 3

(Gradient of radius =)

r r2 −  2

B1 M1 M1 A1ft (4)

2

Condone sign slip if there is evidence of correct use of Pythag.

M1

= 10 2 − 5 2 or numerically exact equivalent.

A1

PQ = 2 75 = 10 3 Simplest surd form 10 3 required for final mark.

A1

(

52

(4)

)

Pearson Edexcel International Advanced Level in Mathematics

(3) (11 marks)

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Sample Assessment Materials

WMA01/01: Core Mathematics C12 Question Number

Scheme

13. (a)

kr 2 + cxy = 4 1 4

or

Marks

kr 2 + c[( x + y ) 2 − x 2 − y 2 ] = 4

π x2 + 2xy = 4

y=

M1 A1

4 − 14 π x2 16 − π x2 = 2x 8x

B1 cso

*

(3) (b)

P = 2x + cy + k π r where c = 2 or 4 and k = ¼ or ½

M1

 4 − 14 π x 2   16 − π x 2  πx + 2x + 4  + 2x + 4  P=  or P =  o.e. 2 2  8x   2x 

A1

πx

P=

πx 2

+ 2x +

8 πx − x 2

so P =

8 + 2x x

*

A1 (3)

(c)

 dP  8 = − 2 + 2   dx  x

−

M1 A1

8 + 2 = 0  x 2 = .. x2

and so x = 2 o.e.

M1 (ignore extra answer x = –2)

P = 4 + 4 = 8 (m)

A1 B1 (5) (11 marks)

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53

WMA01/01: Core Mathematics C12 Question Number 14. (a)

Scheme 2 sin( x + 45° ) = , so ( x + 45° ) = 41.8103... 3

(α

So, x + 45° = {138.1897... , 401.8103...} and x = {93.1897... , 356.8103...}

Marks

2 sin −1   or 3 M1 awrt 41.8 or awrt 0.73c x + 45° = either "180 − their α " or M1 "360° + their α "

= 41.8103...)

Either awrt 93.2° or awrt 356.8°

A1

°

A1

°

Both awrt 93.2 and awrt 356.8

(4) (b)

2(1 − cos 2 x) + 2 = 7 cos x 2

2cos x + 7 cos x − 4 = 0

Applies sin 2 x = 1 − cos 2 x

Correct 3 term, 2cos x + 7 cos x − 4 { = 0} 2

(2cos x − 1)(cos x + 4) {= 0} , cos x = ... 1 , 2 π  β =  3  cos x =

x=

π

3 5π x= 3

54

{cos x = − 4}

or 1.04719...c or 5.23598...c

Pearson Edexcel International Advanced Level in Mathematics

A1 oe

Valid attempt at solving and cos x = ... M1 1 cos x = A1 cso 2

Either Either

M1

π 3

or awrt 1.05c

5π or awrt 5.24c or 2π − their β 3

© Pearson Education Limited 2013

B1 B1 ft (6) (10 marks)

Sample Assessment Materials

WMA01/01: Core Mathematics C12 Question Number 15. (a)

Scheme

Correct use of cosine rule M1 leading to a value for cos α

92 = 42 + 62 − 2 × 4 × 6cos α  cos α = ..... cos α =

(b)

Marks

42 + 62 − 92  29   = − = −0.604..  2× 4×6  48  α = 2.22 ∗ (NB α = 2.219516005)

cso (2.22 must be seen here) A1

2π − 2.22(= 4.06366......) 1 2

2π − 2.22 or awrt 4.06 B1 Correct method for major sector area. Allow π − 2.22 M1 for the major sector angle. awrt 32.5 A1

× 4 2 × "4.06"

32.5 Or (b)

Alternative method:

1 2

π ×4

2

Area of triangle =

Correct expression for circle area. B1 Correct method for M1 circle - minor sector area. awrt 32.5 A1

= 32.5

1 2

× 4 × 6 × sin 2.22 ( = 9.56 )

So area required = “9.56” + “32.5” = 42.1 cm2 or 42.0 cm2

(d)

Arc length = 4 × 4.06 ( = 16.24 ) Or 8π − 4 × 2.22 Perimeter = ZY + WY + Arc Length Perimeter = 27.2 or 27.3

Pearson Edexcel International Advanced Level in Mathematics

(3)

Circle – Minor sector

π × 42 − × 42 × 2.22 = 32.5

(c)

(2)

Correct expression for the B1 area of triangle XYZ Their Triangle XYZ (Not triangle ZXW) + (part (b) M1 answer or correct attempt at major sector) awrt 42.1 or 42.0 (Or just 42). A1

(3)

(3)

M1: 4 × their ( 2π − 2.22 )

M1 A1ft Or circumference – minor arc A1: Correct ft expression 9 + 2 + Any Arc M1 awrt 27.2 or awrt 27.3 A1

© Pearson Education Limited 2013

(4) (12 marks)

Sample Assessment Materials

55

WMA01/01: Core Mathematics C12 Question Number 16. (a) (b)

(c)

(d)

(e)

56

Scheme

Marks

17 ×1.5 = 25.5(km)

B1

Use l = a + (n − 1)d with a = 1.5, d = 0.25 and n = 17 So l = 5.5

A1

Use S =

a(1 − r n ) with a = 1.5, and n = 17 1− r

(1)

M1 (2)

M1

And r = 1.05 So S = 38.76(km)

A1 A1

Total distance running is S = n2 {2a + (n − 1)d} = 59.5(km) So total in three sports is 123.76(km)

M1

(3)

A1 B1

(3)

M1

> 40 so 1.5 × (1.05)n−1 > 40 with their r (1.05)n−1 > 26.7 so (n – 1)log1.05 > log 26.7

M1

n – 1 > 67.297 So 69th day of training.

M1 A1

Uses ar

n −1

Pearson Edexcel International Advanced Level in Mathematics

(4) (13 marks)

© Pearson Education Limited 2013

Sample Assessment Materials

Write your name here Surname

Other names

Pearson Edexcel

Centre Number

Candidate Number

International Advanced Level

Core Mathematics C34 Advanced

Sample Assessment Material Time: 2 hours 30 minutes

Paper Reference

WMA02/01

You must have: Mathematical Formulae and Statistical Tables (Blue)

Total Marks

Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation and integration, or have retrievable mathematical formulae stored in them.

Instructions

black ink or ball-point pen. •• Use If pencil is used for diagrams/sketches/graphs it must be dark (HB or B). Coloured pencils and highlighter pens must not be used. in the boxes at the top of this page with your name, • Fill centre number and candidate number. Answer all questions and ensure that your answers to parts of questions are • clearly labelled. Answer the questions in the spaces provided • – there may be more space than you need. You should show sufficient working to make your methods clear. Answers • without working may not gain full credit. When a calculator • degree of accuracy.is used, the answer should be given to an appropriate

Information

total mark for this paper is 125. •• The The marks for each question are shown in brackets – use this as a guide as to how much time to spend on each question.

Advice

each question carefully before you start to answer it. •• Read Try to answer every question. • Check your answers if you have time at the end.

S45000A ©2013 Pearson Education Ltd.

1/2/2/

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57

1.

(a) Express 5cos2θ – 12sin2θ in the form Rcos(2θ + α), where R > 0 and 0 < α < 90° Give the value of α to 2 decimal places.

Leave blank

(3)

(b) Hence solve, for 0  θ < 180°, the equation 5cos2θ – 12sin2θ = 10 giving your answers to 1 decimal place.

(5)

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*S45000A0244*

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Sample Assessment Materials

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Question 1 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________

Q1

(Total 8 marks) Pearson Edexcel International Advanced Level in Mathematics

*S45000A0344* © Pearson Education Limited 2013

Sample Assessment Materials

3

59

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Leave blank

2.

y Q

R O

π

x

Figure 1

Figure 1 shows a sketch of the curve with equation y = e x sin x , 0  x  π. The finite region R, shown shaded in Figure 1, is bounded by the curve and the x-axis.

π π (a) Complete the table below with the values of y corresponding to x = and x = , 4 2 giving your answers to 5 decimal places. x

0

y

0

π 4

π 2

3π 4

π

8.87207

0 (2)

(b) Use the trapezium rule, with all the values of y in the completed table, to obtain an estimate for the area of the region R. Give your answer to 4 decimal places. (3) The curve y = e x sin x , 0  x  π, has a maximum turning point at Q, shown in Figure 1. (c) Find the x coordinate of Q.

(6)

___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 4

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61

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Question 2 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 6

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Q2

___________________________________________________________________________ (Total 11 marks) Pearson Edexcel International Advanced Level in Mathematics

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63

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3.

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Using the substitution u = cos x + 1, or otherwise, show that π 2

∫e 0

(cos x + 1)

sin x dx = e(e – 1)

(6)

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Question 3 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________

Q3

___________________________________________________________________________ (Total 6 marks) Pearson Edexcel International Advanced Level in Mathematics

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65

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4.

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(a) Use the binomial theorem to expand (2 – 3x)–2,

½ x ½<

2 3

in ascending powers of x, up to and including the term in x3. Give each coefficient as a simplified fraction. (5) f(x) =

a + bx , (2 − 3x) 2

½ x ½<

2 , where a and b are constants. 3

In the binomial expansion of f(x), in ascending powers of x, the coefficient of x is 0 and 9 the coefficient of x2 is 16 Find (b) the value of a and the value of b, (c) the coefficient of x3, giving your answer as a simplified fraction.

(5) (3)

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Question 4 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________

Pearson Edexcel International Advanced Level in Mathematics

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Question 4 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 12

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Question 4 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________

Q4

___________________________________________________________________________ (Total 13 marks) Pearson Edexcel International Advanced Level in Mathematics

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69

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5.

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The functions f and g are defined by f : x  e–x + 2,

x∈

g : x  2 ln x,

x>0

(a) Find fg(x), giving your answer in its simplest form. (b) Find the exact value of x for which f(2x + 3) = 6 (c) Find f –1, stating its domain.

(3) (4) (3)

(d) On the same axes, sketch the curves with equation y = f(x) and y = f –1(x), giving the coordinates of all the points where the curves cross the axes. (4) ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 14

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Question 5 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________

Pearson Edexcel International Advanced Level in Mathematics

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Question 5 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 16

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Question 5 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________

Q5

___________________________________________________________________________ (Total 14 marks) Pearson Edexcel International Advanced Level in Mathematics

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The curve C has equation 16y3 + 9x2y − 54x = 0 (a) Find

dy in terms of x and y. dx

(5)

(b) Find the coordinates of the points on C where

dy =0 dx

(7)

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Question 6 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________

Pearson Edexcel International Advanced Level in Mathematics

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Question 6 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 20

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Question 6 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________

Q6

___________________________________________________________________________ (Total 12 marks) Pearson Edexcel International Advanced Level in Mathematics

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7.

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(a) Show that cot x – cot 2x ≡ cosec 2x,

x≠

nπ , n∈ 2

(5)

(b) Hence, or otherwise, solve for 0  θ  π

π π 1   cosec  3θ +  + cot  3θ +  =     3 3 3 You must show your working. (Solutions based entirely on graphical or numerical methods are not acceptable.)

(5)

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Question 7 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________

Pearson Edexcel International Advanced Level in Mathematics

*S45000A02344* © Pearson Education Limited 2013

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Question 7 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 24

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Question 7 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________

Q7

___________________________________________________________________________ (Total 10 marks) Pearson Edexcel International Advanced Level in Mathematics

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81

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8.

h( x) =

(a) Show that h( x) =

4 18 2 + 2 − 2 , x + 2 x + 5 ( x + 5)( x + 2)

x0

2x x +5 2

(4)

(b) Hence, or otherwise, find h′(x) in its simplest form.

(3)

y

y = h(x) O

Figure 2

x

Figure 2 shows a graph of the curve with equation y = h(x). (c) Calculate the range of h(x).

(5)

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Question 8 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________

Pearson Edexcel International Advanced Level in Mathematics

*S45000A02744* © Pearson Education Limited 2013

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83

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Question 8 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 28

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Question 8 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________

Q8

___________________________________________________________________________ (Total 12 marks) Pearson Edexcel International Advanced Level in Mathematics

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85

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9.

 2  1   The line l1 has equation r = 3 + λ  2 , where λ is a scalar parameter.      −4  1  0  5 The line l2 has equation r =  9 + μ  0 , where μ is a scalar parameter.      −3  2 Given that l1 and l2 meet at the point C, find (a) the coordinates of C.

(3)

The point A is the point on l1 where λ = 0 and the point B is the point on l2 where μ = –1 (b) Find the size of the angle ACB. Give your answer in degrees to 2 decimal places. (4) (c) Hence, or otherwise, find the area of the triangle ABC.

(5)

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Question 9 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________

Pearson Edexcel International Advanced Level in Mathematics

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Question 9 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 32

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Question 9 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________

Q9

___________________________________________________________________________ (Total 12 marks) Pearson Edexcel International Advanced Level in Mathematics

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89

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10. y l C

P

S Q O

√3

x

Figure 3 Figure 3 shows part of the curve C with parametric equations x = tanθ,

y = sinθ,

0θ

π 2

The point P lies on C and has coordinates  3 , 1 3   2  (a) Find the value of θ at the point P.

(2)

The line l is a normal to C at P. The normal cuts the x-axis at the point Q.

(

)

(b) Show that Q has coordinates k 3 , 0 , giving the value of the constant k.

(6)

The finite shaded region S shown in Figure 3 is bounded by the curve C, the line x = 3 and the x-axis. This shaded region is rotated through 2π radians about the x-axis to form a solid of revolution. (c) Find the volume of the solid of revolution, giving your answer in the form pπ 3 + qπ 2, where p and q are constants. (7) ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 34

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Question 10 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________

Pearson Edexcel International Advanced Level in Mathematics

*S45000A03544* © Pearson Education Limited 2013

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Question 10 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 36

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Question 10 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________

Pearson Edexcel International Advanced Level in Mathematics

*S45000A03744* © Pearson Education Limited 2013

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Question 10 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 38

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Question 10 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________

Q10

___________________________________________________________________________ (Total 15 marks) Pearson Edexcel International Advanced Level in Mathematics

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95

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11. A team of conservationists is studying the population of meerkats on a nature reserve. The population is modelled by the differential equation dP 1 = P(5 – P), dt 15

t0

where P, in thousands, is the population of meerkats and t is the time measured in years since the study began. Given that when t = 0, P = 1, (a) solve the differential equation, giving your answer in the form P=

a b + ce

1 − t 3

where a, b and c are integers.

(11)

(b) Hence show that the population cannot exceed 5000

(1)

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Question 11 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________

Pearson Edexcel International Advanced Level in Mathematics

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97

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Question 11 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 42

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Question 11 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________

Pearson Edexcel International Advanced Level in Mathematics

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Question 11 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________

Q11

(Total 12 marks) TOTAL FOR PAPER: 125 MARKS END 44

100

*S45000A04444*

Pearson Edexcel International Advanced Level in Mathematics

© Pearson Education Limited 2013

Sample Assessment Materials

WMA02/01: Core Mathematics C34 Question Number 1. (a)

Scheme R = 13

tan α =

(b)

Marks B1

12  α = 67.38° 5

M1 A1 (3)

13cos(2θ + 67.4°) = 10  cos(2θ + 67.4°) =

10 13

M1

2θ + 67.38° = 39.715°, ( 320.285°, 399.715°)

A1

θ = 126.5 °

A1

θ = 166.2 °

2. (a)

x y

π

π

2

3π 4

1.844321332

4.810477381

8.87207

4

M1 A1

(5) (8 marks)

π 0 awrt 1.84432 B1

awrt 4.81048 or 4.81047 B1 (2)

1 2

(b)

× π4

π

B1

or 8

1 π Area ≈ × ; × 0 + 2 (1.84432 + 4.81048 + 8.87207 ) + 0} 2 4

{

M1 12.1948

A1 (3)

(c)

Uses vu '+ uv '

1 1 − dy 1 = e x × (sin x) 2 (cos x) + e x (sin x) 2 dx 2

M1 A1 A1

1 1 − dy 1 x x 2 = 0  e × (sin x ) (cos x ) + e (sin x ) 2 = 0 dx 2

cos x = −2sin x 1 tan x = −  x = 2.68 2

M1 M1 A1 (6) (11 marks)

Pearson Edexcel International Advanced Level in Mathematics

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Sample Assessment Materials

101

WMA02/01: Core Mathematics C34 Question Number

Scheme

Marks

du = − sin x dx

3.

e

cos x +1

B1

sin xdx = −  eu du

M1 A1

=-eu (+c)

ft on sign error A1

= -e(cos x+1) (+c) π

 (cos x +1)  2 2  −e  = (−e) − (−e ) = e (e − 1)  0

4. (a)

 3x  (2 − 3x) − 2 = 2 − 2 1 −  2    3x  1 −  2  

−2

cso M1 A1* (6 marks)

−2

B1 2

3

 3 x  ( −2)( −3)  3 x  ( −2)( −3)(−4)  3 x  = 1 + ( −2)  −  + −  +  −  + ... M1 A1 2 ×1  2  3 × 2 ×1  2   2  = 1 + 3x +

(2 − 3 x ) − 2 =

27 2 27 3 x + x + .. 4 2

1 3 27 2 27 3 + x+ x + x + .. 4 4 16 8

M1 A1 (5)

(b)

27 2 27 3  1 3 f ( x ) = (a + bx)  + x + x + x + ..  16 8 4 4  Coefficient of x: Coefficient of x2:

3a b + =0 4 4

(3a + b = 0)

27 a 3b 9 + = 16 4 16

M1

(9a + 4b = 3)

M1

A1 (either correct) A1 Leading to a = − 1, b = 3 (c)

102

Coefficient of x3:

Pearson Edexcel International Advanced Level in Mathematics

27a 27b 27 27 + = × −1 + × 3 8 16 8 16 27  11  = = 1  16  16 

© Pearson Education Limited 2013

dM1 A1

(5)

M1 A1ft cao

A1 (3) (13 marks)

Sample Assessment Materials

WMA02/01: Core Mathematics C34 Question Number

Scheme

5. (a)

Marks

M1

fg( x) = e−2ln x + 2 , −2  1  = eln x + 2 = x −2 + 2 =  2 + 2  x  

(b)

(c)

M1 A1 M1 A1

e − ( 2 x + 3) + 2 = 6  e − ( 2 x + 3) = 4

 −(2 x + 3) = ln 4 −3 − ln 4 x= 2

Let y = e

−x

(3)

M1 A1 (4)

+ 2  y − 2 = e− x  ln( y − 2) = − x  x = − ln( y − 2)

M1

−1

f ( x) = − ln( x − 2), x > 2.

A1 B1 (3)

(d)

Shape for f (x) B1 (0, 3) B1 Shape for f

−1

(x) B1

(3, 0) B1

(4) (14 marks)

Pearson Edexcel International Advanced Level in Mathematics

© Pearson Education Limited 2013

Sample Assessment Materials

103

WMA02/01: Core Mathematics C34 Question Number

6. (a)

Scheme

Differentiating implicitly to obtain ± ay 2

48 y 2

Marks

dy dy and/or ±bx 2 dx dx

M1

dy + ... − 54 ... dx

A1

9x2 y → 9x2

dy + 18 xy dx

( 48 y

dy + 18 xy − 54 = 0 dx

2

+ 9x2 )

d y 54 − 18 xy = dx 48 y 2 + 9 x 2

or equivalent

B1 M1

 18 − 6 xy  = 2 2   16 y + 3 x 

A1 (5)

18 − 6 xy = 0

(b)

Using x =

M1

3 y

or

y=

3 x

2

3

3 3 3 3 16 y 3 + 9   y − 54   = 0 or 16   + 9 x 2   − 54 x = 0  x  x  y  y

M1

Leading to

16 y 4 + 81 −162 = 0

or

16 + x 4 − 2 x 4 = 0

81 16

or

x 4 = 16

3 3 ,− 2 2

or

x = 2, − 2

y4 = y=

M1

A1, A1

Subs either of their values into xy = 3 to obtain a value of other variable.

3  3   2,  ,  −2, −  2  2 

M1

both

A1 (7) (12 marks)

104

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Sample Assessment Materials

WMA02/01: Core Mathematics C34 Question Number

Scheme

cos x cos 2 x − sin x sin 2 x

B1

≡

sin 2 x cos x − cos 2 x sin x sin x sin 2 x

M1

≡

sin(2 x − x) sin x sin 2 x

M1

≡

sin x 1 ≡ ≡ cosec2 x sin x sin 2 x sin 2 x

M1 A1*

cot x − cot 2 x ≡

7. (a)

Marks

(5) (b)

2 x = 3θ +

π 3

 x = 1.5θ +

π

B1

6

π 1 π   cot 1.5θ +  =  tan 1.5θ +  = 3 6 6 3   π  π 4π  1.5θ +  = , 6 3 3  θ=

Pearson Edexcel International Advanced Level in Mathematics

π 7π 9

,

9

© Pearson Education Limited 2013

M1

M1 A1, A1 (5) (10 marks)

Sample Assessment Materials

105

WMA02/01: Core Mathematics C34 Question Number 8. (a)

Scheme

2 4 18 2( x 2 + 5) + 4( x + 2) − 18 + 2 − = x + 2 x + 5 ( x + 2)( x 2 + 5) ( x + 2)( x 2 + 5) =

2 x( x + 2) ( x + 2)( x 2 + 5)

M1

=

2x ( x + 5)

A1*

2

h '( x ) =

( x 2 + 5) × 2 − 2 x × 2 x ( x 2 + 5) 2

M1 A1

h '( x) =

10 − 2 x 2 ( x 2 + 5)2

A1 2

Maximum occurs when h '( x) = 0  10 − 2 x = 0  x = ..

x= 5 When x = 5  h( x) =

5 5

Range of h(x) is 0h( x )

106

M1 A1

(4)

(b)

(c)

Marks

Pearson Edexcel International Advanced Level in Mathematics

(3) M1 A1 M1 A1

5 5

© Pearson Education Limited 2013

A1ft (5) (12 marks)

Sample Assessment Materials

WMA02/01: Core Mathematics C34 Question Number 9. (a)

(b)

Scheme

Marks

Equate j components 3 + 2λ = 9  λ = 3 Leading to C = ( 5,9, − 1)

M1 A1 A1

  Choosing correct directions or finding AC and BC 1 5     2  .  0  = 5 + 2 = 6 × 29 × cos ∠ABC 1  2    ∠ACB = 57.95° A = (2, 3, − 4)

(c)

(3)

M1

Use of scalar product.

M1 A1 A1

(4)

B = ( −5, 9, − 5)

 3 10        AC =  6  AND BC =  0   3 4    

(

AC 2 = 32 + 62 + 32 = 3 6 Area triangle ABC =

)

(

BC 2 = 102 + 42 = 2 29

)

1 1 AC BC sin ∠ACB = × '3 6 ' × ' 2 29 ' × sin 57.95° 2 2 = 33.5

M1 A1 A1 M1 A1 (5) (12 marks)

Pearson Edexcel International Advanced Level in Mathematics

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Sample Assessment Materials

107

WMA02/01: Core Mathematics C34 Question Number

Scheme

3 2

tan θ = 3 or sin θ =

10. (a)

θ=

Marks

M1

π

awrt 1.05

3

A1 (2)

dx dy = sec 2 θ , = cos θ dθ dθ

(b)

d y cos θ = dx sec2 θ

( = cos θ ) 3

π  1 m = cos3   = 3 8

At P,

M1 A1 Can be implied.

Using mm′ = −1,

m′ = −8

For normal

y − 12 3 = −8 x − 3

At Q, y = 0

− 12 3 = −8 x − 3

)

leading to

x = 17 16 3

( k = 1716 )

A1 M1

(

(

)

dM1

1.0625

A1 (6)

(c)

y

2

 dx dx =  y 2 dθ =  sin 2 θ sec 2 θ dθ  dθ

M1 A1

=  tan 2θ dθ

A1

=  ( sec 2 θ − 1) dθ

dM1

= tan θ − θ π

( +C )

A1

(

V = π  y 2 dx = [ tan θ − θ ] 03 = π  0  2 1 1 = 3π − 3 π ( p = 1, q = − 3 ) 3

π

)

3 − π3 − ( 0 − 0 )  

dM1 A1 (7) (15 marks)

108

Pearson Edexcel International Advanced Level in Mathematics

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Sample Assessment Materials

WMA02/01: Core Mathematics C34 Question Number

Scheme



11. (a)

1 dP = P (5 − P )

giving Hence



Marks

1 dt 15

B1

1 = A(5 − P) + B P

M1

1 1 A= ,B= 5 5

A1

1 1 1 5 d P = +  P(5 − P)  P (5 −5 P)dP

A1

 P(5 − P) dP =  15 dt 1

1

1 1 1  ln P − ln(5 − P ) = t 5 5 15

{t = 0, P = 1 }

(+ c)

M1 A1ft

1 1 ln1 − ln(4) = 0 + c 5 5

1    c = − ln 4  5  

dM1

1  P  1 1 ln   = t − ln 4 5  5 − P  15 5

eg:

 4P  1 ln   = t 5 − P 3 Using any of the subtraction (or addition) laws for logarithms CORRECTLY.

eg:

1t 4P = e3 5−P

or

1t

gives

eg:

5−P −1t =e 3 4P

1t

1t

Eliminate ln’s correctly.

P =

1t

t  (÷ e 3 )   1t  3  (÷ e ) 

1t

1

5e 3

1t

(4 + e 3 ) 5 (1 + 4 e

− 13 t

)

or

P =

Make P the subject. 25 (5 + 20e

− 13 t

)

etc.

− 13 t

>1  P < 5.

So population cannot exceed 5000

dM1

A1

Note that the ‘dM’ marks are dependent upon the first two M marks. 1 + 4e

dM1

4 P = 5e 3 − Pe 3  P (4 + e 3 ) = 5e 3

P =

(b)

dM1

(11) B1 (1) (12 marks)

Pearson Edexcel International Advanced Level in Mathematics

© Pearson Education Limited 2013

Sample Assessment Materials

109

110

Pearson Edexcel International Advanced Level in Mathematics

© Pearson Education Limited 2013

Sample Assessment Materials

Write your name here Surname

Other names

Pearson Edexcel

Centre Number

Candidate Number

International Advanced Level

Further Pure Mathematics F1

Advanced/Advanced Subsidiary Sample Assessment Material Time: 1 hour 30 minutes

Paper Reference

WFM01/01

You must have: Mathematical Formulae and Statistical Tables (Blue)

Total Marks

Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation and integration, or have retrievable mathematical formulae stored in them.

Instructions

black ink or ball-point pen. •• Use If pencil is used for diagrams/sketches/graphs it must be dark (HB or B). Coloured pencils and highlighter pens must not be used. in the boxes at the top of this page with your name, • Fill centre number and candidate number. Answer all questions and ensure that your answers to parts of questions are • clearly labelled. Answer the questions in the spaces provided • – there may be more space than you need. You should show sufficient working to make your methods clear. Answers • without working may not gain full credit. When a calculator • degree of accuracy.is used, the answer should be given to an appropriate

Information

total mark for this paper is 75. •• The The marks for each question are shown in brackets – use this as a guide as to how much time to spend on each question.

Advice

each question carefully before you start to answer it. •• Read Try to answer every question. • Check your answers if you have time at the end.

S45001A ©2013 Pearson Education Ltd.

1/2/2/

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111

1.

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The complex numbers z1 and z2 are given by z1 = 2 + 8i

and

z2 = 1 – i

Find, showing your working, (a)

z1 in the form a + bi, where a and b are real, z2

(b) the value of

(3)

z1 , z2

(c) the value of arg

(2)

z1 , giving your answer in radians to 2 decimal places. z2

(2)

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Q1

(Total 7 marks) Pearson Edexcel International Advanced Level in Mathematics

*S45001A0320* © Pearson Education Limited 2013

Sample Assessment Materials

3

113

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2.

f(x) = 5x2 – 4x 2 – 6,

x0

The root α of the equation f(x) = 0 lies in the interval [1.6, 1.8] (a) Use linear interpolation once on the interval [1.6, 1.8] to find an approximation to α. Give your answer to 3 decimal places. (4) (b) Taking 1.7 as a first approximation to α, apply the Newton-Raphson process once to f (x) to obtain a second approximation to α. Give your answer to 3 decimal places. (6) ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 4

114

*S45001A0420*

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Question 2 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________

Q2

___________________________________________________________________________ (Total 10 marks) Pearson Edexcel International Advanced Level in Mathematics

*S45001A0520* © Pearson Education Limited 2013

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5

115

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3.

y P

O

C

S

x

Q Figure 1 Figure 1 shows a sketch of the parabola C with equation y2 = 8x. The point P lies on C, where y > 0, and the point Q lies on C, where y < 0 The line segment PQ is parallel to the y-axis. Given that the distance PQ is 12, (a) write down the y coordinate of P,

(1)

(b) find the x coordinate of P.

(2)

Figure 1 shows the point S which is the focus of C. The line l passes through the point P and the point S. (c) Find an equation for l in the form ax + by + c = 0, where a, b and c are integers.

(4)

___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 6

116

*S45001A0620*

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117

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Question 3 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 8

118

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Q3

___________________________________________________________________________ (Total 7 marks) Pearson Edexcel International Advanced Level in Mathematics

*S45001A0920* © Pearson Education Limited 2013

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9

119

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4.

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The quadratic equation 5x2 – 4x + 1 = 0 has roots α and β. (a) Write down the value of α + β and the value of αβ. (b) Show that

α β 6 + = β α 5

(2) (4)

(c) Find a quadratic equation with integer coefficients, which has roots

α+

1 1 and β + α β

(6)

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120

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Q4

___________________________________________________________________________ (Total 12 marks) Pearson Edexcel International Advanced Level in Mathematics

*S45001A01120* © Pearson Education Limited 2013

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11

121

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5.

Prove by induction that, for n ∈ +, f(n) = 5n + 8n + 3 is divisible by 4

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Q5

___________________________________________________________________________ (Total 6 marks) Pearson Edexcel International Advanced Level in Mathematics

*S45001A01320* © Pearson Education Limited 2013

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13

123

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6.

(a) Using the formulae for

n

∑r, r =1

n

∑r2 and r =1

n

∑r3, show that r =1

1

n

∑r(r + 1)(r + 3) = 12 n(n + 1)(n + 2)(3n + k) r =1

where k is a constant to be found. (b) Hence evaluate

(7)

40

∑ r(r + 1)(r + 3)

r = 21

(3)

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Question 6 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________

Q6

___________________________________________________________________________ (Total 10 marks) Pearson Edexcel International Advanced Level in Mathematics

*S45001A01520* © Pearson Education Limited 2013

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15

125

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7.

 6 The point P  6t ,  , t ≠ 0, lies on the rectangular hyperbola H with equation xy = 36  t (a) Show that an equation for the tangent to H at P is y= −

1 12 x+ 2 t t

(5)

The tangent to H at the point A and the tangent to H at the point B meet at the point (–9, 12). (b) Find the coordinates of A and B.

(7)

___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 16

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Question 7 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________

Q7

___________________________________________________________________________ (Total 12 marks) Pearson Edexcel International Advanced Level in Mathematics

*S45001A01720* © Pearson Education Limited 2013

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17

127

Turn over

8.

Leave blank

(i) The transformation U is represented by the matrix P where,  1 − 2 P=   3  2

3 2   1  −  2 

−

(a) Describe fully the transformation U.

(2)

The transformation V, represented by the matrix Q, is a stretch scale factor 3 parallel to the x-axis. (b) Write down the matrix Q.

(1)

Transformation U followed by transformation V is a transformation which is represented by matrix R. (c) Find the matrix R. (ii)

(3)  1 −3 S=   3 1

Given that the matrix S represents an enlargement, with a positive scale factor and centre (0, 0), followed by a rotation with centre (0, 0), (a) find the scale factor of the enlargement,

(2)

(b) find the angle and direction of rotation, giving your answer in degrees to 1 decimal place. (3) ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 18

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Question 8 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________

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129

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Question 8 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________

Q8

(Total 11 marks) TOTAL FOR PAPER: 75 MARKS END 20

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WFM01/01: Further Pure Mathematics F1 Question Number 1. (a)

(b)

(c)

2. (a)

Scheme

z1 2 + 8i 1 + i = × z2 1− i 1+ i 2 + 2i + 8i − 8 = = −3 + 5i 2

z1 = (−3) 2 + 52 = 34 z2

M1 A1 A1 (3)

(or awrt 5.83)

M1 A1ft (2)

5 5 tan α = − or 3 3 z arg 1 = π − 1.03... = 2.11 z2

f (1.6) = −1.29543081... f (1.8) = 0.5401863372 α − 1.6 1.8 − α = "1.29543081..." "0.5401863372"

α = 1.741143899 (b)

Marks

A1 (2) (7 marks)

awrt -1.30 awrt 0.54

1 2

© Pearson Education Limited 2013

B1 B1 M1

awrt 1.741

f ′( x) = 10 x − 6 x f (1.7) = −0.4161152711... f ′(1.7) = 9.176957114... f (1.7) α 2 = 1.7 − f ′(1.7) α 2 = 1.745

Pearson Edexcel International Advanced Level in Mathematics

M1

A1

(4)

M1 A1

awrt -0.42 awrt 9.18

B1 B1 M1

cao

A1 (6) (10 marks)

Sample Assessment Materials

131

WFM01/01: Further Pure Mathematics F1 Question Number 3. (a) (b)

(c)

Scheme

PQ = 12  By symmetry y p = 2

12 =6 2

Marks

B1 (1)

2

y = 8x  6 = 8x 36 9  x= = 8 2

M1 A1 (2)

Focus S (2 , 0)

B1

  6 − 0  6 − 0 12  Gradient PS = =  = 9 9 5  −2 −2  2  2 

M1

y−0 =

Either

12 5

( x − 2) or y − 6 =

Or y = 125 x + c and 0 =

l:

12 5

( 2) + c

12 5

( x − 92 )

M1

 c = − 245

12 x − 5 y − 24 = 0

A1 (4) (7 marks)

4. (a)

(b)

α +β=

4 1 , αβ = 5 5

B1, B1 (2)

α β α2 + β2 + = β α αβ

B1

α 2 + β 2 = (α + β ) − 2αβ 2

α β + = β α = (c)

132

(α + β )

2

− 2αβ

αβ

6 * 5

B1

16 2 − = 25 5 1 5

A1

α + β 24 = α β αβ 5 1  1 α β 1 1 6 32  = + +5 =  α +  β +  = αβ + + + α  β β α αβ 5 5 5  24 32 x2 − x + = 0  5 x 2 − 24 x + 32 = 0 5 5

α+

1

M1

+β +

1

=α +β +

Pearson Edexcel International Advanced Level in Mathematics

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(4) M1 A1 M1 A1 M1 A1 (6) (12 marks)

Sample Assessment Materials

WFM01/01: Further Pure Mathematics F1 Question Number 5.

Scheme

Marks

f (1) = 5 + 8 + 3 = 16 , (which is divisible by 4). (∴True for n = 1).

B1

Assume true for f(k) Using the formula to write down f(k + 1),

f ( k + 1) = 5 k +1 + 8( k + 1) + 3

f ( k + 1) − f ( k ) = 5 k +1 + 8( k + 1) + 3 − 5 k − 8k − 3 k

M1

k

k

= 5(5 ) + 8k + 8 + 3 − 5 − 8k − 3 = 4(5 ) + 8

6. (a)

A1

f ( k + 1) = 4(5 k + 2) + f ( k ) , which is divisible by 4

A1

∴True for n = k + 1 if true for n = k. True for n = 1, ∴true for all n.

A1

r (r + 1)(r + 3) = r 3 + 4r 2 + 3r , so use

r

3

+4

r

2

r

+3

1 2 2 1  1  n ( n + 1) + 4  n ( n + 1)( 2n + 1)  , + 3  n ( n + 1)  4 6  2 

=

1 1 n(n + 1){3n( n + 1) + 8(2n + 1) + 18} or = n {3n3 + 22n 2 + 45n + 26} 12 12

=

{

40

40

20

21

1

1

}

A1

M1 A1

M1

1 1 ( 40 × 41× 42 × 133) − ( 20 × 21× 22 × 73) , = 707210 12 12

Pearson Edexcel International Advanced Level in Mathematics

(k = 13)

M1

(7)

 = −  =

A1, A1

1 (n + 1) {3n3 + 19n 2 + 26n} 12

1 1 n( n + 1) 3n 2 + 19n + 26 = n(n + 1)(n + 2)(3n + 13) 12 12

(6 marks)

M1

=

or =

(b)

B1

© Pearson Education Limited 2013

A1, A1 (3) (10 marks)

Sample Assessment Materials

133

WFM01/01: Further Pure Mathematics F1 Question Number

7. (a)

Scheme

y=

Marks

36 dy  = −36 x −2 x dx

M1

c2 1  6  dy At  6t ,  , =− =− 2 2 t (6t )  t  dx

y−

(b)

6 1 = − 2 ( x − 6t ) t t

Substitute ( −9, 12) :

(2t − 3)(2t + 1) = 0 t=

(b)

(c)

(ii) (a)

1 12 x+ 2 t t

(*)

120° or

(5) M1

12t 2 − 12t − 9 = 0

A1



t=

3 1 t=− 2 2

M1 A1 M1 A1 A1 (7) (12 marks)

2π rotation about the origin, anticlockwise. 3

B1, B1 (2)

 3 0    0 1  1 −  3 0   2 R=  0 1 3   2

M1 A1cso

1 12 12 = − 2 ( −9 ) + t t

3 1 t = −  Points are ( 9, 4 ) and ( −3, − 12 ) 2 2

8. (i) (a)

 y=−

M1 A1

B1 (1) −

3  3 3√ 3  −   −  2   2 2  = 1 1   √3 −  −   2 2  2

M1 A1 A1 (-1 each error) (3)

det S = 1 × 1- 3 × -3 ( = 10 ) or 32 + 12 ( = 10 )  Enlargement scale factor = 10

M1 A1 (2)

(b)

134

tan θ =

3  θ = 71.6o , anticlockwise. 1

Pearson Edexcel International Advanced Level in Mathematics

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M1 A1, A1 (3) (11 marks)

Sample Assessment Materials

Write your name here Surname

Other names

Pearson Edexcel

Centre Number

Candidate Number

International Advanced Level

Further Pure Mathematics F2

Advanced/Advanced Subsidiary Sample Assessment Material Time: 1 hour 30 minutes

Paper Reference

WFM02/01

You must have: Mathematical Formulae and Statistical Tables (Blue)

Total Marks

Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation and integration, or have retrievable mathematical formulae stored in them.

Instructions

black ink or ball-point pen. •• Use If pencil is used for diagrams/sketches/graphs it must be dark (HB or B). Coloured pencils and highlighter pens must not be used. in the boxes at the top of this page with your name, • Fill centre number and candidate number. Answer all questions and ensure that your answers to parts of questions are • clearly labelled. Answer the questions in the spaces provided • – there may be more space than you need. You should show sufficient working to make your methods clear. Answers • without working may not gain full credit. When a calculator • degree of accuracy.is used, the answer should be given to an appropriate

Information

total mark for this paper is 75. •• The The marks for each question are shown in brackets – use this as a guide as to how much time to spend on each question.

Advice

each question carefully before you start to answer it. •• Read Try to answer every question. • Check your answers if you have time at the end.

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1/2/2/

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135

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1.

(a) Express

3 in partial fractions. (3r − 1)(3r + 2)

(2)

(b) Using your answer to part (a) and the method of differences, show that n

3

∑ (3r − 1)(3r + 2) r =1

(c) Evaluate

1000

=

3n 2(3n + 2)

(3)

3

∑ (3r − 1)(3r + 2) , giving your answer to 3 significant figures.

r =100

(2)

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Q1

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3

137

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2.

The displacement x metres of a particle at time t seconds is given by the differential equation

Leave blank

d2 x + x + cos x = 0 dt 2 When t = 0 , x = 0 and

dx 1 = . dt 2

Find a Taylor series solution for x in ascending powers of t, up to and including the term in t 3 . (5) ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 4

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Q2

___________________________________________________________________________ (Total 5 marks) Pearson Edexcel International Advanced Level in Mathematics

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5

139

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3.

Leave blank

(a) Find the set of values of x for which x+4>

2 x+3

(6)

(b) Deduce, or otherwise find, the values of x for which x+4>

2 x+3

(1)

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Q3

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7

141

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4.

z = −8 + (8√3)i (a) Find the modulus of z and the argument of z.

(3)

Using de Moivre’s theorem, (b) find z 3 ,

(2)

(c) find the values of w such that w4 = z , giving your answers in the form a + i b, where a, b ∈  . (5) ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 8

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Q4

___________________________________________________________________________ (Total 10 marks) Pearson Edexcel International Advanced Level in Mathematics

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9

143

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5.

θ=

π 2

Leave blank

r=2

S

r = 1.5 + sin 3θ

O

θ=0 Figure 1

Figure 1 shows the curves given by the polar equations

and

r = 2,

0q

,

r = 1.5 + sin 3q,

0q

.

(a) Find the coordinates of the points where the curves intersect.

(3)

The region S, between the curves, for which r >2 and for which r < (1.5 + sin 3q), is shown shaded in Figure 1. (b) Find, by integration, the area of the shaded region S, giving your answer in the form aπ + b√3, where a and b are simplified fractions. (7) ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 10

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Question 5 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 12

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Q5

___________________________________________________________________________ (Total 10 marks) Pearson Edexcel International Advanced Level in Mathematics

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147

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6.

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A complex number z is represented by the point P in the Argand diagram. (a) Given that z − 6 = z , sketch the locus of P.

(2)

(b) Find the complex numbers z which satisfy both z − 6 = z and z − 3 − 4i = 5 . The transformation T from the z-plane to the w-plane is given by w =

(3)

30 . z

(c) Show that T maps z − 6 = z onto a circle in the w-plane and give the cartesian equation of this circle.

14

148

(5)

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Question 6 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________

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Question 6 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 16

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Q6

___________________________________________________________________________ (Total 10 marks) Pearson Edexcel International Advanced Level in Mathematics

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151

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7.

1

(a) Show that the transformation z = y 2 transforms the differential equation 1 dy − 4 y tan x = 2 y 2 dx

(I)

into the differential equation dz − 2 z tan x = 1 dx

(II)

(b) Solve the differential equation (II) to find z as a function of x. (c) Hence obtain the general solution of the differential equation (I).

(5) (6) (1)

___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 18

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Question 7 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________

Pearson Edexcel International Advanced Level in Mathematics

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19

153

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Question 7 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 20

154

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Question 7 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________

Q7

___________________________________________________________________________ (Total 12 marks) Pearson Edexcel International Advanced Level in Mathematics

*S45002A02124* © Pearson Education Limited 2013

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155

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8.

(a) Find the value of λ for which y = λx sin 5x is a particular integral of the differential equation d2 y + 25 y = 3cos 5 x dx 2

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(4)

(b) Using your answer to part (a), find the general solution of the differential equation d2 y + 25 y = 3cos 5 x dx 2 Given that at x = 0 , y = 0 and

(3)

dy =5, dx

(c) find the particular solution of this differential equation, giving your solution in the form y = f(x). (5) (d) Sketch the curve with equation y = f(x) for 0  x  π.

(2)

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Q8

(Total 14 marks) TOTAL FOR PAPER: 75 MARKS END 24

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WFM02/01: Further Pure Mathematics F2 Question Number 1(a)

(b)

Scheme

1 1 − 3r − 1 3r + 2 n

3

 (3r − 1)(3r + 2) = r =1

M1 A1

1 1 1 1 1 1 1 1 − + − + − + ... − 2 5 5 8 8 11 3n − 1 3n + 2

1 1 3n = − = 2 3n + 2 2(3n + 2) (c)

Marks

*

Sum = f(1000) – f(99) 3000 297 − = 0.00301 6004 598

M1 A1ft

A1

or 3.01 × 10 −3

(2)

M1 A1

(3)

(2) 7

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159

WFM02/01: Further Pure Mathematics F2 Question Number 2

Scheme

f ′′(t ) = − x − cos x ,

f ′′(0) = -1

B1

dx , f ′′′(0) = −0.5 dt t2 t3 ′ ′′ f (t ) = f (0) + tf (0) + f (0) + f ′′′(0) + ... 2 3! 2 1 3 = 0.5t − 0.5t − 12 t + ...

f ′′′(t ) = (−1 + sin x)

160

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Marks

© Pearson Education Limited 2013

M1A1

M1 A1

5

Sample Assessment Materials

WFM02/01: Further Pure Mathematics F2 Question Number 3(a)

(b)

Scheme

Marks

( x + 4)( x + 3)2 − 2( x + 3) = 0 , ( x + 3)( x 2 + 7 x + 10) = 0 so ( x + 2)( x + 3)( x + 5) = 0 or alternative method including calculator

M1

Finds critical values –2 and -5

A1 A1

Establishes x > -2

A1ft

Finds and uses critical value –3 to give –5 < x < -3

M1A1

x > -2

B1ft

(6)

(1) 7

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161

WFM02/01: Further Pure Mathematics F2 Question Number 4(a)

Scheme

Modulus = 16

B1

Argument = arctan(− 3) =

(b)

(c)

Marks

z 3 = 163 (cos(

2π 3

M1 A1

2π 2π ) + i sin( ))3 = 163 (cos 2π + isin2π ) =4096 or 16 3 3 3

1

w = 16 4 (cos(

2π 2π 1 π  π  ) + i sin( )) 4 = 2(cos   + isin   ) = 3 + i 3 3 6 6

(

OR −1 + 3i OR − 3 − i OR 1 − 3i

M1 A1

)

(3)

(2)

M1 A1ft M1A2 (1,0) (5) 10

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WFM02/01: Further Pure Mathematics F2 Question Number 5(a)

Scheme

1.5 + sin 3θ = 2

and ∴θ =

→

sin 3θ = 0.5 ∴ 3θ =

π  5π  or 6  6

Marks

 , 

π 5π or 18 18

M1 A1, A1

 18  1 Area = 12   (1.5 + sin 3θ )2 dθ  , - π × 22  18π  9 5π  18  1 = 12   (2.25 + 3sin 3θ + 12 (1 − cos 6θ ))dθ  - π × 22  18π  9

(3)

5π

(b)

5π

1   18 1 = 12 (2.25θ − cos 3θ + 12 (θ − sin 6θ ))  - π × 22 6  π 9

M1, M1 M1

M1 A1

18

=

13 3 5π − 24 36

M1 A1

(7) 10

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163

WFM02/01: Further Pure Mathematics F2 Question Number 6(a)

Scheme

Marks

Imaginary Axis Re(z) = 3

0

Real axis

6

Vertical Straight line Through 3 on real axis

B1 B1 (2)

(b)

(c)

These are points where line x = 3 meets the circle centre (3, 4) with radius 5.

M1

The complex numbers are 3 + 9i and 3 – i.

A1 A1

z −6 = z 

30 w

−6 =

(3)

M1

30 w

∴ 30 − 6w = 30  ∴ 5 − w = 5

M1 A1

This is a circle with Cartesian equation (u − 5)2 + v 2 = 25

M1 A1

(5) 10

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WFM02/01: Further Pure Mathematics F2 Question Number 7(a)

Scheme

dy dy dz dy dy dz and = 2 z so = . = 2 z. dx dz dx dz dx dx

Substituting to get 2 z.

(b)

I.F. = e  ∴

−2 tan xdx

M1 M1 A1

dz dz − 4 z 2 tan x = 2 z and thus − 2 z tan x = 1 dx dx

= e 2 ln cos x = cos 2 x

d z cos 2 x ) = cos 2 x ∴ z cos 2 x =  cos 2 xdx ( dx ∴ z cos 2 x

=



1 2

(cos 2 x + 1) dx = 14 sin 2x + 12 x +c

∴ z = 12 tan x + 12 x sec 2 x + c sec2 x

(c)

Marks

∴ y = ( 12 tan x + 12 x sec 2 x + c sec 2 x ) 2

*

M1 A1

(5)

M1 A1

M1 M1 A1 A1 B1ft

(6) (1) 12

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165

WFM02/01: Further Pure Mathematics F2 Question Number 8(a)

Scheme

Marks

Differentiate twice and obtaining

d2 y dy = λ sin 5 x + 5λ x cos 5 x and 2 = 10λ cos 5 x − 25λ x sin 5 x dx dx Substitute to give λ =

(b)

3 10

M1 A1 5ix

Complementary function is y = A cos 5 x + B sin 5 x or Pe So general solution is y = A cos 5 x + B sin 5 x +

(c)

M1 A1

M1 A1

+ Qe−5ix

3 x sin 5 x or in exponential form 10

A1ft

y= 0 when x = 0 means A = 0

B1

dy 3 3 dy = 5 B cos 5 x + sin 5 x + x cos 5 x and at x = 0 = 5 and so 5 = 5A dx 10 2 dx

M1 M1

So B = 1

A1

So y = sin 5 x +

(4)

3 x sin 5 x 10

A1

(3)

(5)

(d) "Sinusoidal" through O amplitude becoming larger

B1

Crosses x axis at

π 2π 3π 4π , , , 5 5 5 5

B1 (2) 14

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Write your name here Surname

Other names

Pearson Edexcel

Centre Number

Candidate Number

International Advanced Level

Further Pure Mathematics F3

Advanced/Advanced Subsidiary Sample Assessment Material Time: 1 hour 30 minutes

Paper Reference

WFM03/01

You must have: Mathematical Formulae and Statistical Tables (Blue)

Total Marks

Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation and integration, or have retrievable mathematical formulae stored in them.

Instructions

black ink or ball-point pen. •• Use If pencil is used for diagrams/sketches/graphs it must be dark (HB or B). Coloured pencils and highlighter pens must not be used. in the boxes at the top of this page with your name, • Fill centre number and candidate number. Answer all questions and ensure that your answers to parts of questions are • clearly labelled. Answer the questions in the spaces provided • – there may be more space than you need. You should show sufficient working to make your methods clear. Answers • without working may not gain full credit. When a calculator • degree of accuracy.is used, the answer should be given to an appropriate

Information

total mark for this paper is 75. •• The The marks for each question are shown in brackets – use this as a guide as to how much time to spend on each question.

Advice

each question carefully before you start to answer it. •• Read Try to answer every question. • Check your answers if you have time at the end.

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1.

The line x = 8 is a directrix of the ellipse with equation x2 y 2 + = 1, a2 b2

a > 0, b > 0,

and the point (2, 0) is the corresponding focus. Find the value of a and the value of b.

(5)

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Q1

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3

169

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2.

Use calculus to find the exact value of

1

1 dx . −2 x + 4 x + 13

∫

2

(5)

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Q2

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3.

(a) Starting from the definitions of sinh x and cosh x in terms of exponentials, prove that cosh 2 x = 1 + 2sinh 2 x (3) (b) Solve the equation

cosh 2 x − 3sinh x = 15,

giving your answers as exact logarithms.

(5)

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Q3

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4.

In

a

∫ (a( a −– x)x) cos xddx,x, nn

0

a a> >0,0,n n 0 0

(a) Show that, for n  2 ,

(b) Hence evaluate

−π2 0

π 2

In = nan – 1 – n(n – 1)In – 2

−x

2

cosx cos xdx. dx.

(5) (3)

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Q4

(Total 8 marks) Pearson Edexcel International Advanced Level in Mathematics

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5.

Given that y = ( arcosh 3x ) , where 3 x > 1 , show that 2

2

 dy  (a) ( 9 x –−1) 1)   = 36 y ,  dx  (9x22

(b) (9x 1) ( 9 x22 –− 1)

(5)

d2y dy + 9x = 18 . 2 dx dx

(4)

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Q5

(Total 9 marks) Pearson Edexcel International Advanced Level in Mathematics

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 1 0 3   M =  0 −2 1  , where k is a constant.  k 0 1  

6.

6   Given that  1  is an eigenvector of M, 6   6   (a) find the eigenvalue of M corresponding to  1  , 6  

(2)

(b) show that k = 3 ,

(2)

(c) show that M has exactly two eigenvalues.

(4)

A transformation T : 3 → 3 is represented by M. The transformation T maps the line l1 , with cartesian equations the line l2 .

x − 2 y z +1 = = , onto 1 4 −3

(d) Taking k = 3 , find cartesian equations of l2 .

(5)

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Q6

(Total 13 marks) Pearson Edexcel International Advanced Level in Mathematics

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The plane Π has vector equation r = 3i + k + λ (–4i + j) + µ (6i – 2j + k) (a) Find an equation of Π in the form r.n = p , where n is a vector perpendicular to Π and p is a constant. (5) The point P has coordinates (6, 13, 5). The line l passes through P and is perpendicular to Π. The line l intersects Π at the point N. (b) Show that the coordinates of N are (3, 1, –1).

(4)

The point R lies on Π and has coordinates (1, 0, 2). (c) Find the perpendicular distance from N to the line PR. Give your answer to 3 significant figures.

(5)

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Q7

(Total 14 marks) Pearson Edexcel International Advanced Level in Mathematics

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8.

The hyperbola H has equation

x2 y 2 − = 1. 16 4

The line l1 is the tangent to H at the point P (4 sec t, 2 tan t). (a) Use calculus to show that an equation of l1 is 2 y sin t = x − 4 cos t

(5)

The line l2 passes through the origin and is perpendicular to l1 . The lines l1 and l2 intersect at the point Q. (b) Show that, as t varies, an equation of the locus of Q is

(x

2

+ y2

)

2

= 16 x 2 − 4 y 2

(8)

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Question 8 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________

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Question 8 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 26

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Question 8 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________

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Question 8 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________

Q8

(Total 13 marks) END 28

194

TOTAL FOR PAPER: 75 MARKS

*S45003A02828*

Pearson Edexcel International Advanced Level in Mathematics

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Sample Assessment Materials

WFM03/01: Further Pure Mathematics F3 Question Number

1.

Scheme

±

a = 8, ± ae = 2 e

Marks

B1, B1

a × ae = a 2 = 16 e a=4

B1

b2 = a2 (1 − e2 ) = a2 − a2e2  b 2 = 16 − 4 = 12

M1

 b = 12 = 2 3

A1

(5)

5

Pearson Edexcel International Advanced Level in Mathematics

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Sample Assessment Materials

195

WFM03/01: Further Pure Mathematics F3 Question Number

Scheme

2.

x2 + 4 x + 13 = ( x + 2)2 + 9 1

 ( x + 2)

2

1  x+2 dx = arctan   3 +9  3 

Marks

B1 M1 A1

1

1 1  x + 2   3 arctan  3   = 3 ( arctan1 − arctan 0 )    −2 

=

π 12

M1

A1

(5)

5

196

Pearson Edexcel International Advanced Level in Mathematics

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Sample Assessment Materials

WFM03/01: Further Pure Mathematics F3 Question Number

Scheme

3(a)

 e x − e− x  rhs = 1 + 2sinh 2 x = 1 + 2    2 

=

Marks 2

2 + e2 x − 2 + e−2 x 2

e 2 x + e −2 x = = cosh 2 x = lhs 2

(b)

M1

M1

*

A1

(3)

1 + 2 sinh 2 x − 3sinh x = 15 2 sinh 2 x − 3sinh x − 14 = 0

M1

( sinh x + 2)( 2sinh x − 7) = 0

M1

sinh x = −2,

(

7 2

) (

A1

)

M1

2 7   7 + 53  7  x = ln +   + 1)  = ln   2  2  2   

A1

x = ln −2 + ( −2) 2 + 1) = ln −2 + 5

(5) 8

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197

WFM03/01: Further Pure Mathematics F3 Question Number

4(a)

Scheme

 (a − x)

n

Marks

cos xdx = ( a − x ) sin x +  n ( a − x ) n

n −1

sin xdx

a

( a − x )n sin x  = 0  0 = −n ( a − x )

n −1

cos x −  n ( n − 1)( a − x )

I n = na n −1 − n(n − 1)I n − 2

(b)

M1A1 A1

n−2

cos xdx

*

π

 π I2 = 2   − 2 2 cos xdx 0  2 π

= π − 2 [sin x ]02 = π − 2

dM1 A1

(5)

M1 A1 A1

(3) 8

198

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WFM03/01: Further Pure Mathematics F3 Question Number

Scheme

5(a)

dy 3 = 2ar cosh ( 3 x ) × dx 9x2 −1

9x2 −1

Marks

M1A1A1

dy = 6ar cosh ( 3x ) dx 2

2  ( 9 x2 −1)  dy  = 36 ( ar cosh ( 3x ) ) dx  2

 ( 9 x − 1)  dy  = 36 y dx  2

(b)

*

  dy 2 dy d 2 y  dy 2 18 x   + ( 9 x − 1) × 2 × 2  = 36 dx dx  dx   dx 

( 9 x2 −1)

d2y dy + 9 x = 18 2 dx dx

*

dM1

A1

(5)

M1 {A1} A1

A1

(4) 9

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199

WFM03/01: Further Pure Mathematics F3 Question Number

Scheme

6(a)

Marks

 1 0 3  6  6       0 −2 1 1  = λ 1   k 0 1  6  6       24   6λ       4 = λ   6 k + 6   6λ      Uses the first or second row to obtain λ = 4

(b)

M1A1 (2)

Uses the third row and their λ = 4 to obtain

6k + 6 = 24  k = 3 (c)

1− λ

0

3

0

−2 − λ

1

3

0

1− λ

M1 A1 (2)

*

=0

 (1 − λ ) ( ( −2 − λ ) (1 − λ ) − 0) − 0 ( 0 (1 − λ ) − 3) + 3 ( 0 − 3 ( −2 − λ ) ) = 0

(

 (1 − λ ) ( −2 − λ )(1 − λ ) + 9 ( 2 + λ ) = ( 2 + λ ) 9 − (1 − λ )

2

) = 0 (λ

3

M1 A1

− 12λ − 16 = 0)

 ( λ + 2) ( λ 2 − 2λ − 8) = 0

(d)

 ( λ + 2)( λ + 2)( λ − 4) = 0

M1

λ = −2, 4

A1

Parametric form of l1 : ( t + 2, −3t,4t −1)  1 0 3  t + 2  13t − 1        0 −2 1  −3t  = 10t − 1  3 0 1  4t − 1  7t + 5      

Cartesian equations of l2 :

x +1 y +1 z − 5 = = 13 10 7

(4)

M1

M1 A1

ddM1A1(5) 13

200

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WFM03/01: Further Pure Mathematics F3 Question Number

7(a)

Scheme

i

j

Marks

k

1   −4 1 0 =  4  6 −2 1  2 

M1 A2(1,0)

 1  3      4•0 = 5  2 1     1   r • 4 = 5  2  

(b)

 6  1 Equation of l is r = 13  + t  4   5   2    

M1

 6 + t  1 At intersection 13 + 4t  •  4  = 5  5 + 2t   2     

M1

 6 + t + 4 (13 + 4t ) + 2 ( 5 + 2t ) = 5  t = −3

M1

N is ( 3,1, −1) (c)

M1A1 (5)

*

  PN  PR = ( −3i − 12 j − 6k )( −5i − 13j − 3k ) = 189

9 + 144 + 36 25 + 169 + 9 cos NPR = 189 NX = NP sin NPR = 189 sin NPR = 3.61

A1

(4)

M1 A1ft A1 M1A1 (5) 14

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201

WFM03/01: Further Pure Mathematics F3 Question Number

Scheme

Marks

8(a)

dx dy = 4sec t tan t = 2sec 2 t dt dt

B1 (both)

dy 2sec 2 t = dx 4sec t tan t

M1

y − 2 tan t =

2 y sin t − 2 y sin t = x − (b)

1   =   2 sin t 

1 ( x − 4sec t ) 2sin t

4sin 2 t 4 = x− cos t cos t 4 − 4sin 2 t = x − 4 cos t cos t

=

+y

y = −2 x sin t

A1

2 2

(2)

(1 + 4sin t ) 2

4

(1 + 4sin t )

256cos2 t 2

(1)

−8sin t cos t 1 + 4 sin 2 t

2

(1 + 4sin t )

Pearson Edexcel International Advanced Level in Mathematics

4 cos t 1 + 4sin 2 t

2

−

2

=

(5)

M1 A1

M1 A1

  16 cos 2 t 64sin 2 t cos 2 t  = +  (1 + 4 sin 2 t )2 (1 + 4 sin 2 t )2   

256 cos 4 t

16 x 2 − 4 y 2 =

202

)

A1 M1

y=

2

*

Gradient of l2 is − 2 sin t

2 ( −2 x sin t ) sin t = x − 4 cos t  x =

(x

M1 A1

2

256 cos 4 t

(1 + 4sin t ) 2

256sin 2 t cos 2 t

(1 + 4sin t ) 2

2

=

M1

2

A1

256cos 4 t

(1 + 4sin t )

© Pearson Education Limited 2013

2

2

(8) 13

Sample Assessment Materials

Write your name here Surname

Other names

Pearson Edexcel

Centre Number

Candidate Number

International Advanced Level

Mechanics M1

Advanced/Advanced Subsidiary Sample Assessment Material Time: 1 hour 30 minutes

Paper Reference

WME01/01

You must have: Mathematical Formulae and Statistical Tables (Blue)

Total Marks

Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation and integration, or have retrievable mathematical formulae stored in them.

Instructions

black ink or ball-point pen. •• Use If pencil is used for diagrams/sketches/graphs it must be dark (HB or B). Coloured pencils and highlighter pens must not be used. in the boxes at the top of this page with your name, • Fill centre number and candidate number. Answer all questions and ensure that your answers to parts of questions are • clearly labelled. Answer the questions in the spaces provided • – there may be more space than you need. should show sufficient working to make your methods clear. Answers • You without working may not gain full credit. Whenever a numerical value of g is required, take g = 9.8 m s , and give your • answer to either two significant figures or three significant figures. When a is used, the answer should be given to an appropriate • degree ofcalculator accuracy. –2

Information

total mark for this paper is 75. •• The The marks for each question are shown in brackets – use this as a guide as to how much time to spend on each question.

Advice

each question carefully before you start to answer it. •• Read Try to answer every question. • Check your answers if you have time at the end.

S45004A ©2013 Pearson Education Ltd.

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1.

A particle P is moving with constant velocity (−3i + 2 j) m s −1. At time t = 6 s P is at the point with position vector (−4i − 7 j) m. Find the distance of P from the origin at time t = 2 s. (5)

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Question 1 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________

Q1

(Total 5 marks) Pearson Edexcel International Advanced Level in Mathematics

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205

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2.

Particle P has mass m kg and particle Q has mass 3m kg. The particles are moving in opposite directions along a smooth horizontal plane when they collide directly. Immediately before the collision P has speed 4u m s–1 and Q has speed ku m s–1, where k is a constant. As a result of the collision the direction of motion of each particle is reversed and the speed of each particle is halved. (a) Find the value of k.

(4)

(b) Find, in terms of m and u, the magnitude of the impulse exerted on P by Q.

(3)

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Q2

(Total 7 marks) Pearson Edexcel International Advanced Level in Mathematics

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207

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3. 100 N 30° Figure 1 A small box is pushed along a floor. The floor is modelled as a rough horizontal plane and the box is modelled as a particle. The coefficient of friction between the box and the floor 1 is 2 . The box is pushed by a force of magnitude 100 N which acts at an angle of 30° with the floor, as shown in Figure 1. Given that the box moves with constant speed, find the mass of the box.

(7)

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Q3

(Total 7 marks) Pearson Edexcel International Advanced Level in Mathematics

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4.

A beam AB has length 6 m and weight 200 N. The beam rests in a horizontal position on two supports at the points C and D, where AC = 1 m and DB = 1 m. Two children, Sophie and Tom, each of weight 500 N , stand on the beam with Sophie standing twice as far from the end B as Tom. The beam remains horizontal and in equilibrium and the magnitude of the reaction at D is three times the magnitude of the reaction at C. By modelling the beam as a uniform rod and the two children as particles, find how far Tom is standing from the end B. (7)

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Question 4 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 10

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Q4

(Total 7 marks) Pearson Edexcel International Advanced Level in Mathematics

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5.

Two cars P and Q are moving in the same direction along the same straight horizontal road. Car P is moving with constant speed 25 m s −1. At time t = 0, P overtakes Q which is moving with constant speed 20 m s–1. From t = T seconds, P decelerates uniformly, coming to rest at a point X which is 800 m from the point where P overtook Q. From t = 25 s, Q decelerates uniformly, coming to rest at the same point X at the same instant as P.

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(a) Sketch, on the same axes, the speed-time graphs of the two cars for the period from t = 0 to the time when they both come to rest at the point X. (4) (b) Find the value of T.

(8)

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215

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Question 5 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 14

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Q5

(Total 12 marks) Pearson Edexcel International Advanced Level in Mathematics

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217

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6.

A ball is projected vertically upwards with a speed of 14.7 m s −1 from a point which is 49 m above horizontal ground. Modelling the ball as a particle moving freely under gravity, find (a) the greatest height, above the ground, reached by the ball,

(4)

(b) the speed with which the ball first strikes the ground,

(3)

(c) the total time from when the ball is projected to when it first strikes the ground.

(3)

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Q6

(Total 10 marks) Pearson Edexcel International Advanced Level in Mathematics

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7.

PN

α Figure 2 A particle of mass 0.4 kg is held at rest on a fixed rough plane by a horizontal force of magnitude P newtons. The force acts in the vertical plane containing the line of greatest slope of the inclined plane which passes through the particle. The plane is inclined to the 3 horizontal at an angle α, where tan tanαα = = ,, as shown in Figure 2. 4

The coefficient of friction between the particle and the plane is

1 . 3

Given that the particle is on the point of sliding up the plane, find (a) the magnitude of the normal reaction between the particle and the plane, (b) the value of P.

(5) (5)

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Q7

(Total 10 marks) Pearson Edexcel International Advanced Level in Mathematics

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8.

A

B

1m

1m

Figure 3 Two particles A and B have mass 0.4 kg and 0.3 kg respectively. The particles are attached to the ends of a light inextensible string. The string passes over a small smooth pulley which is fixed above a horizontal floor. Both particles are held, with the string taut, at a height of 1 m above the floor, as shown in Figure 3. The particles are released from rest and in the subsequent motion B does not reach the pulley. (a) Find the tension in the string immediately after the particles are released. (b) Find the acceleration of A immediately after the particles are released.

(6) (2)

When the particles have been moving for 0.5 s, the string breaks. (c) Find the further time that elapses until B hits the floor.

(9)

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Q8

(Total 17 marks) TOTAL FOR PAPER: 75 MARKS END 28

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WME01/01: Mechanics M1 Question Number

Scheme

Marks

(−4i − 7 j) = r + 4(−3i + 2 j) r = (8i − 15 j)

Q1

M1 A1 A1

r = 82 + (−15)2 = 17 m

M1 A1 ft [5]

Q2

(a)

ku

4u I

P

Q ku 2

2u

4mu − 3mku = −2mu + 3mk k= (b)

I

4 3

u 2

M1 A1 M1 A1cso (4)

For P, I = m (2u - -4u) = 6mu

ku OR For Q, I = 3m ( 2 - -ku)

M1 A1 A1

(3)

(M1A1) [7]

Q3

(→) 100cos30 = F F = 0.5 R seen

M1 A1 A1 (B1)

(↓) mg + 100cos60 = R m = 13 kg or 12.6 kg

M1 A1 DM1 A1 [7]

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231

WME01/01: Mechanics M1 Question Number

Scheme

R

Q4

500

200

Marks

S

500

M ( B ), 500x + 500.2x + 200x3 = Rx5 + Sx1 (or any valid moments equation)

M1 A1 A1

(↓) R + S = 500 + 500 + 200 = 1200 (or a moments equation)

M1 A1

solving for x; x = 1.2 m

M1 A1 cso [7]

Q5

Shape (both) Cross Meet on t-axis 25,20,T,25 Figures

(a) V 25 20

O (b)

T

25

t

 t + 25  For Q: 20   = 800  2  t = 55

232

DM1 A1 M1 A1 DM1 A1

T =9

Pearson Edexcel International Advanced Level in Mathematics

(4)

M1 A1

 T + 55  For P: 25   = 800  2 

solving for T:

B1 B1 B1 B1

© Pearson Education Limited 2013

(8) [12]

Sample Assessment Materials

WME01/01: Mechanics M1 Question Number Q6

(a)

(b)

Scheme

( ↑ )v 2 = u 2 + 2as 0 = 14.7 2 − 2x 9.8 x s s = 11.025 (or 11 or 11.0 or 11.03) m Height is 60 m or 60.0 m ft

( ↓ )v = u + at

34.3 = −14.7 + 9.8t t =5

Q7

(a)

(b)

M1A1 A1 A1ft

(4)

( ↓ )v 2 = u 2 + 2as M1 A1

v 2 = (−14.7) 2 + 2x 9.8 x 49 v = 34.3 or 34 m s -1 (c)

Marks

OR

A1

( ↓ )s = ut + 12 at 2

49 = −14.7t + 4.9t 2 t =5

M1 A1 A1

F = 13 R

B1

( ↑ ) R cos α − F sin α = 0.4 g R = 23 g = 6.53 or 6.5

M1 A1

(→)P − F cos α − R sin α = 0 P = 26 45 g = 5.66 or 5.7

M1 A2

Pearson Edexcel International Advanced Level in Mathematics

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(3)

M1 A1

M1 A1

(3) [10]

(5)

(5) [10]

Sample Assessment Materials

233

WME01/01: Mechanics M1 Question Number Q8

Scheme

(a)

Mark together

(c)

( ↓ )0.4 g − T = 0.4a

M1 A1

( ↑ )T − 0.3g = 0.3a solving for T T = 3.36 or 3.4 or 12g/35 (N)

M1 A1

0.4 g − 0.3 g = 0.7 a a = 1.4 m s -2 , g/7

(b)

( ↑ )v = u + at v = 0.5 x 1.4 = 0.7

DM1 A1

(6)

DM1 A1

(2)

M1 A1 ft on a

( ↑ )s = ut + 12 at 2

M1 A1 ft on a

s = 0.5 x 1.4 x 0.52 = 0.175 ( ↓ )s = ut + 12 at 2

DM1 A1 ft

1.175 = − 0.7t + 4.9t 2 4.9t 2 − 0.7t − 1.175 = 0

0.7 ± 0.7 2 + 19.6 x 1.175 9.8 = 0.5663..or − ...

DM1 A1 cao

Ans 0.57 or 0.566 s

A1 cao

t=

234

Marks

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(9) [17]

Sample Assessment Materials

Write your name here Surname

Other names

Pearson Edexcel

Centre Number

Candidate Number

International Advanced Level

Mechanics M2

Advanced/Advanced Subsidiary Sample Assessment Material Time: 1 hour 30 minutes

Paper Reference

WME02/01

You must have: Mathematical Formulae and Statistical Tables (Blue)

Total Marks

Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation and integration, or have retrievable mathematical formulae stored in them.

Instructions

black ink or ball-point pen. •• Use If pencil is used for diagrams/sketches/graphs it must be dark (HB or B). Coloured pencils and highlighter pens must not be used. in the boxes at the top of this page with your name, • Fill centre number and candidate number. Answer all questions and ensure that your answers to parts of questions are • clearly labelled. Answer the questions in the spaces provided • – there may be more space than you need. should show sufficient working to make your methods clear. Answers • You without working may not gain full credit. Whenever a numerical value of g is required, take g = 9.8 m s , and give your • answer to either two significant figures or three significant figures. When a is used, the answer should be given to an appropriate • degree ofcalculator accuracy. –2

Information

total mark for this paper is 75. •• The The marks for each question are shown in brackets – use this as a guide as to how much time to spend on each question.

Advice

each question carefully before you start to answer it. •• Read Try to answer every question. • Check your answers if you have time at the end.

S45005A ©2013 Pearson Education Ltd.

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1.

A particle P moves on the x-axis. The acceleration of P at time t seconds, t  0, 2 in the positive x-direction. When t = 0, the velocity of P is 2 m s–1 in the is (3t 5)mmss−–2 (3t + 5) positive x-direction. When t = T, the velocity of P is 6 m s–1 in the positive x-direction. Find the value of T. (6)

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Q1

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2.

A particle P of mass 0.6 kg is released from rest and slides down a line of greatest slope of a rough plane. The plane is inclined at 30° to the horizontal. When P has moved 12 m, its speed is 4 m s–1. Given that friction is the only non-gravitational resistive force acting on P, find (a) the work done against friction as the speed of P increases from 0 m s–1 to 4 m s–1, (b) the coefficient of friction between the particle and the plane.

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(4) (4)

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Q2

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3. A

10 cm

B

10 cm

12 cm

C

Figure 1 A triangular frame is formed by cutting a uniform rod into 3 pieces which are then joined to form a triangle ABC, where AB = AC = 10 cm and BC = 12 cm, as shown in Figure 1. (a) Find the distance of the centre of mass of the frame from BC.

(5)

The frame has total mass M. A particle of mass M is attached to the frame at the mid-point of BC. The frame is then freely suspended from B and hangs in equilibrium. (b) Find the size of the angle between BC and the vertical.

(4)

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Q3

___________________________________________________________________________ (Total 9 marks) Pearson Edexcel International Advanced Level in Mathematics

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4.

A car of mass 750 kg is moving up a straight road inclined at an angle θ to the horizontal, 1 where sin θθ = 15 . The resistance to motion of the car from non-gravitational forces has constant magnitude R newtons. The power developed by the car’s engine is 15 kW and the car is moving at a constant speed of 20 m s–1. (a) Show that R = 260.

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(4)

The power developed by the car’s engine is now increased to 18 kW. The magnitude of the resistance to motion from non-gravitational forces remains at 260 N. At the instant when the car is moving up the road at 20 m s–1 the car’s acceleration is a m s–2. (b) Find the value of a.

(4)

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Q4

___________________________________________________________________________ (Total 8 marks) Pearson Edexcel International Advanced Level in Mathematics

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5.

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[In this question i and j are perpendicular unit vectors in a horizontal plane.] A ball of mass 0.5 kg is moving with velocity (10i + 24j) m s–1 when it is struck by a bat. Immediately after the impact the ball is moving with velocity 20i m s–1. Find (a) the magnitude of the impulse of the bat on the ball,

(4)

(b) the size of the angle between the vector i and the impulse exerted by the bat on the ball, (2) (c) the kinetic energy lost by the ball in the impact.

(3)

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Q5

___________________________________________________________________________ (Total 9 marks) Pearson Edexcel International Advanced Level in Mathematics

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6. D

2a (m) B

C

A 3a

a

Figure 2 Figure 2 shows a uniform rod AB of mass m and length 4a. The end A of the rod is freely hinged to a point on a vertical wall. A particle of mass m is attached to the rod at B. One end of a light inextensible string is attached to the rod at C, where AC = 3a. The other end of the string is attached to the wall at D, where AD = 2a and D is vertically above A. The rod rests horizontally in equilibrium in a vertical plane perpendicular to the wall and the tension in the string is T. (a) Show that T = mg √13.

(5)

The particle of mass m at B is removed from the rod and replaced by a particle of mass M which is attached to the rod at B. The string breaks if the tension exceeds 2mg√13. Given that the string does not break, 5 2

(b) show that M  m .

(3)

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Q6

___________________________________________________________________________ (Total 8 marks) Pearson Edexcel International Advanced Level in Mathematics

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7.

Q 40 m s

−1

12 m P

θ°

36 m

O

R Figure 3

A ball is projected with speed 40 m s–1 from a point P on a cliff above horizontal ground. The point O on the ground is vertically below P and OP is 36 m. The ball is projected at an angle θ° to the horizontal. The point Q is the highest point of the path of the ball and is 12 m above the level of P. The ball moves freely under gravity and hits the ground at the point R, as shown in Figure 3. Find (a) the value of θ,

(3)

(b) the distance OR,

(6)

(c) the speed of the ball as it hits the ground at R.

(3)

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Q7

___________________________________________________________________________ (Total 12 marks) Pearson Edexcel International Advanced Level in Mathematics

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8.

A small ball A of mass 3m is moving with speed u in a straight line on a smooth horizontal table. The ball collides directly with another small ball B of mass m moving with speed u towards A along the same straight line. The coefficient of restitution between A and B 1 is . The balls have the same radius and can be modelled as particles.

Leave blank

2

(a) Find (i) the speed of A immediately after the collision, (ii) the speed of B immediately after the collision.

(7)

After the collision B hits a smooth vertical wall which is perpendicular to the direction of 2 motion of B. The coefficient of restitution between B and the wall is 5 . (b) Find the speed of B immediately after hitting the wall.

(2)

The first collision between A and B occurred at a distance 4a from the wall. The balls collide again T seconds after the first collision. (c) Show that T =

112a . 15u

(6)

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Q8

(Total 15 marks) TOTAL FOR PAPER: 75 MARKS END 32

266

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Sample Assessment Materials

WME02/01: Mechanics 2 Question Number

Scheme

Q1

Marks

3t + 5

dv = 3t + 5 dt v =  ( 3t + 5) dt

M1* A1

v = 32 t 2 + 5t ( + c) t =0 v=2  c=2 v = 32 t 2 + 5t + 2 t =T

B1

6 = 32 T 2 + 5T + 2

12 = 3T 2 + 10T + 4 3T 2 + 10T − 8 = 0 ( 3T − 2 )(T + 4 ) = 0

T=

2 3

(T = −4 )

∴T =

2 3

(or 0.67)

Pearson Edexcel International Advanced Level in Mathematics

DM1*

M1 A1 [6]

© Pearson Education Limited 2013

Sample Assessment Materials

267

WME02/01: Mechanics 2 Question Number

Scheme

Q2

Marks

0 m s-1 4 m s-1 R

F

12 m

30 0.6g (a)

(b)

K.E gained = 12 × 0.6 × 4 2 P.E. lost = 0.6 × g × (12 sin 30) Change in energy = P.E. lost - K.E. gained 1 = 0.6 × g × 12sin 30 − × 0.6 × 42 2 = 30.48 Work done against friction = 30 or 30.5 J R (↑ )

R = 0.6 g cos 30

30.48 12 F = µR 30.48 µ= 12 × 0.6 g cos 30 µ = 0.4987 µ = 0.499 or 0.50

Pearson Edexcel International Advanced Level in Mathematics

A1

(4)

B1

F=

268

M1 A1 A1

B1ft

M1 A1

© Pearson Education Limited 2013

(4) [8]

Sample Assessment Materials

WME02/01: Mechanics 2 Question Number

Scheme

Marks

A

Q3

10 cm

10 cm

B (a)

mass ratio dist. from BC

C

12 cm AB

AC

BC

frame

10 4

10 4

12 0

32 x

Moments about BC:

10 × 4 + 10 × 4 + 0 = 32x 80 x= 32 x = 2 12 (2.5)

(b)

B1 B1

M1 A1

A1

(5)

B D C

Moments about B:

Mg

θ

Mg

A

Mg × 6sin θ = Mg × ( x cos θ − 6sin θ ) 12sin θ = x cos θ x tan θ = 12 θ = 11.768..... = 11.8º

Alternative method : C of M of loaded frame at distance

x from D along DA 1 x tan θ = 2 6 θ = 11.768..... = 11.8º

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1 2

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M1 A1 A1

A1

(4)

B1 M1 A1 A1 [9]

Sample Assessment Materials

269

WME02/01: Mechanics 2 Question Number

Scheme

Q4

Marks

a m s-2 20 m s-1

R

T

θ 750g

(a)

15000 = 750 20 R(parallel to road) T = R + 750 g sin θ R = 750 − 750 × 9.8 × 151 R = 260 * T=

(b)

M1 M1 A1 A1

(4)

a m s-2 20 m s-1

260N

T’

θ 750g

18000 = 900 20 T ′ − 260 − 750 g × sin θ = 750a 900 − 260 − 750 × 9.8 × 151 a= 750 a = 0.2

T′ =

270

Pearson Edexcel International Advanced Level in Mathematics

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M1 M1 A1

A1

(4) [8]

Sample Assessment Materials

WME02/01: Mechanics 2 Question Number Q5 (a)

Scheme

Marks

I = mv − mu = 0.5 × 20i − 0.5 (10i + 24 j) = 5i − 12 j 5i − 12 j = 13 Ns

(b)

θ

M1 A1 M1 A1

(4)

5 12

12 5 θ = 67.38 θ = 67.4°

tan θ =

(c)

M1 A1

K.E.lost = 12 × 0.5 (102 + 242 ) − 12 × 0.5 × 202

= 69 J

Pearson Edexcel International Advanced Level in Mathematics

M1 A1 A1

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(2)

Sample Assessment Materials

(3) [9]

271

WME02/01: Mechanics 2 Question Number

Scheme

Q6

D

Marks

θ

2a

T

F A

R

2a

a

a

mg (a)

M(A)

mg M1 A1 A1

3a × T cos θ = 2amg + 4amg

cos θ =

2 = 9+4

2 13

B1

6 T = 6mg 13 T = mg 13 * (b)

3a × T × cos θ = 2amg + 4aMg (2mg + 4 Mg ) T = 13 ≤ 2mg 13 6 mg + 2 Mg ≤ 6mg 5 M ≤ m * 2

A1

(5)

M1 A1

cso

A1

(3) [8]

272

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Sample Assessment Materials

WME02/01: Mechanics 2 Question Number

Scheme

Marks

Q7 (a)

(b)

M1 A1

Vertical motion: v 2 = u 2 + 2as (40sin θ ) 2 = 2 × g × 12 2 × g × 12 (sin θ ) 2 = 402 θ = 22.54 = 22.5° (accept 23) Vert motion P → R : s = ut + 12 at 2

−36 = 40sin θt −

A1

g 2 t 2

(3)

M1

g 2 t − 40 sin θt − 36 = 0 2

A1 A1

40sin 22.54 ± (40sin 22.54) 2 + 4 × 4.9 × 36 9.8 t = 4.694... t =

Horizontal P to R: s = 40cos θt = 173 m

( or 170 m)

(c) Using Energy:

1 2 1 mv − m × 402 = m × g × 36 2 2 v 2 = 2 ( 9.8 × 36 + 12 × 40 2 )

v = 48.0….. v = 48 m s -1 (accept 48.0)

A1 M1 A1

(6)

M1 A1

A1

(3) [12]

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273

WME02/01: Mechanics 2 Question Number

Scheme

Q8

u

u

A

3m

Marks

m

B w

v

(a) (i) Con. of Mom: 3mu − mu = 3mv + mw

N.L.R:

1 2

(1) – (2)

2u = 3v + w (u + u) = w − v u = w−v u = 4v v = 14 u

(ii) In (2)

(b) B to wall: N.L.R:

e=

1 2

(1) (2)

u = w − 14 u w = 54 u 5 4

M1# A1 M1# A1 DM1# A1

A1

u × 52 = V

(7)

M1

V= u 1 2

A1ft

(2)

(c)

u A 1 4

1 2

u B

5 16a time = 4a ÷ u = 4 5u 1 16a 4 = a Dist. Travelled by A = u × 4 5u 5 1 1 In t secs, A travels ut , B travels ut 4 2 1 1 Collide when speed of approach = ut + ut , distance to cover = 2 4 4 4a − a 5 4a − 4 a 16a 4 64a ∴t= 3 5 = × = 5 3u 15u 4u 16a 64a 112a Total time = + = 5u 15u 15u *

B to wall:

274

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B1ft B1ft

M1$

DM1$ A1 A1

(6) 15

Sample Assessment Materials

Write your name here Surname

Other names

Pearson Edexcel

Centre Number

Candidate Number

International Advanced Level

Mechanics M3

Advanced/Advanced Subsidiary Sample Assessment Material Time: 1 hour 30 minutes

Paper Reference

WME03/01

You must have: Mathematical Formulae and Statistical Tables (Blue)

Total Marks

Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation and integration, or have retrievable mathematical formulae stored in them.

Instructions

black ink or ball-point pen. •• Use If pencil is used for diagrams/sketches/graphs it must be dark (HB or B). Coloured pencils and highlighter pens must not be used. in the boxes at the top of this page with your name, • Fill centre number and candidate number. Answer all questions and ensure that your answers to parts of questions are • clearly labelled. Answer the questions in the spaces provided • – there may be more space than you need. should show sufficient working to make your methods clear. Answers • You without working may not gain full credit. Whenever a numerical value of g is required, take g = 9.8 m s , and give your • answer to either two significant figures or three significant figures. When a is used, the answer should be given to an appropriate • degree ofcalculator accuracy. –2

Information

total mark for this paper is 75. •• The The marks for each question are shown in brackets – use this as a guide as to how much time to spend on each question.

Advice

each question carefully before you start to answer it. •• Read Try to answer every question. • Check your answers if you have time at the end.

S45006A ©2013 Pearson Education Ltd.

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1. A 13l C

5l

B (m)

Figure 1 A garden game is played with a small ball B of mass m attached to one end of a light inextensible string of length 13l. The other end of the string is fixed to a point A on a vertical pole as shown in Figure 1. The ball is hit and moves with constant speed in a horizontal circle of radius 5l and centre C, where C is vertically below A. Modelling the ball as a particle, find (a) the tension in the string,

(3)

(b) the speed of the ball.

(4)

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Q1

___________________________________________________________________________ (Total 7 marks) Pearson Edexcel International Advanced Level in Mathematics

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277

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2.

A particle P of mass m is above the surface of the Earth at distance x from the centre of the Earth. The Earth exerts a gravitational force on P. The magnitude of this force is inversely proportional to x2.

Leave blank

At the surface of the Earth the acceleration due to gravity is g. The Earth is modelled as a sphere of radius R. (a) Prove that the magnitude of the gravitational force on P is

mgR 2 2

.

(3)

A particle is fired vertically upwards from the surface of the Earth with initial speed 3U. At a height R above the surface of the Earth the speed of the particle is U. (b) Find U in terms of g and R.

(7)

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Q2

___________________________________________________________________________ (Total 10 marks) Pearson Edexcel International Advanced Level in Mathematics

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3.

1.5 m O

θ Figure 2 A particle of mass 0.5 kg is attached to one end of a light elastic spring of natural length 0.9 m and modulus of elasticity λ newtons. The other end of the spring is attached to a fixed point O on a rough plane which is inclined at an angle θ to the horizontal, where sinθθ = 3 . The coefficient of friction between the particle and the plane is 0.15. The particle sin 5

is held on the plane at a point which is 1.5 m down the line of greatest slope from O, as shown in Figure 2. The particle is released from rest and first comes to rest again after moving 0.7 m up the plane. Find the value of λ.

(9)

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Q3

___________________________________________________________________________ (Total 9 marks) Pearson Edexcel International Advanced Level in Mathematics

*S45006A01128* © Pearson Education Limited 2013

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11

285

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4. O

4l

6l

Figure 3 A container is formed by removing a right circular solid cone of height 4l from a uniform solid right circular cylinder of height 6l. The centre O of the plane face of the cone coincides with the centre of a plane face of the cylinder and the axis of the cone coincides with the axis of the cylinder, as shown in Figure 3. The cylinder has radius 2l and the base of the cone has radius l. (a) Find the distance of the centre of mass of the container from O.

(6)

θ° Figure 4 The container is placed on a plane which is inclined at an angle θ° to the horizontal. The open face is uppermost, as shown in Figure 4. The plane is sufficiently rough to prevent the container from sliding. The container is on the point of toppling. (b) Find the value of θ.

(4)

___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 12

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Pearson Edexcel International Advanced Level in Mathematics

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Q4

___________________________________________________________________________ (Total 10 marks) Pearson Edexcel International Advanced Level in Mathematics

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5. v P

T O

θ

(m)

a

Figure 5 A particle P of mass m is attached to one end of a light inextensible string of length a. The other end of the string is fixed at the point O. The particle is initially held with OP horizontal and the string taut. It is then projected vertically upwards with speed u, where u2 = 5ag. When OP has turned through an angle θ the speed of P is v and the tension in the string is T, as shown in Figure 5. (a) Find, in terms of a, g and θ, an expression for v 2 . (b) Find, in terms of m, g and θ, an expression for T. (c) Prove that P moves in a complete circle. (d) Find the maximum speed of P.

(3) (4) (3) (2)

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Q5

___________________________________________________________________________ (Total 12 marks) Pearson Edexcel International Advanced Level in Mathematics

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6.

At time t = 0, a particle P is at the origin O moving with speed 2 m s −1 along the x-axis in the positive x-direction. At time t seconds (t > 0), the acceleration of P has magnitude 3 m s −2 and is directed towards O. 2 (t + 1)  3  (a) Show that at time t seconds the velocity of P is  − 1 m s −1. t 1 +   (5) (b) Find, to 3 significant figures, the distance of P from O when P is instantaneously at rest. (7)

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Q6

___________________________________________________________________________ (Total 12 marks) Pearson Edexcel International Advanced Level in Mathematics

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7.

A light elastic string, of natural length 3a and modulus of elasticity 6mg, has one end attached to a fixed point A. A particle P of mass 2m is attached to the other end of the string and hangs in equilibrium at the point O, vertically below A. (a) Find the distance AO.

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(3)

The particle is now raised to point C vertically below A, where AC > 3a, and is released from rest.

a (b) Show that P moves with simple harmonic motion of period 2π  .  g

(5)

1 4

It is given that OC = a. (c) Find the greatest speed of P during the motion.

(3)

1

The point D is vertically above O and OD = 8 a . The string is cut as P passes through D, moving upwards. (d) Find the greatest height of P above O in the subsequent motion.

(4)

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Question 7 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________

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Question 7 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________

Q7

(Total 15 marks) TOTAL FOR PAPER 75 MARKS END 28

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Sample Assessment Materials

WME03/01: Mechanics M3 Question Number

Scheme

Q1

A

α

Marks

13l T B

5l

mg 12 13 T cos α = mg cosα =

(a)

R (↑ )

T×

(b)

12 = mg 13 13 T = mg 12

B1 M1

oe

v2 Eqn of motion T sin α = m 5l 13mg 5 v2 × =m 12 13 5l 25 gl v2 = 12 v=

5 2

gl 3

  gl 25 gl or or any other equiv)   accept 5 12 12  

A1

(3)

M1 A1 M1 dep

A1

(4) [7]

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Sample Assessment Materials

303

WME03/01: Mechanics M3 Question Number Q2

Scheme

k x2 k mg = ( − ) 2 R mgR 2 F= x2 F = (−)

(a)

M1

*

mgR2 x2 dv gR 2 v =− 2 dx x  gR 2  1 2 v =   − 2  dx 2  x 

x = R, v = 3U

x = 2R , v = U

304

M1

mx = −

(b)

1 2 gR 2 v = ( +c ) 2 x 9U 2 = gR + c 2 1 2 gR 2 9U 2 v = + − gR 2 x 2 1 2 gR 2 9U 2 U = + − gR 2 2R 2 gR U2 = 8 gR U= 8

Pearson Edexcel International Advanced Level in Mathematics

Marks

© Pearson Education Limited 2013

A1

(3)

M1 M1 M1 dep on 1st M mark A1 M1 dep on 3rd M mark

M1 dep on 3rd M mark

A1

(7) [10]

Sample Assessment Materials

WME03/01: Mechanics M3 Question Number

Scheme

Marks

Q3

R

F

θ

mg EPE lost = R (↑ )

λ × 0.62 2 × 0.9

−

λ × 0.12 

7  = λ 2 × 0.9  36 

R = mg cos θ

M1 A1 M1

4 = 0.4 g 5 F = µ R = 0.15 × 0.4 g P.E. gained = E.P.E. lost − work done against friction = 0.5 g ×

0.5g × 0.7sin θ =

λ × 0.62

λ × 0.12

− 0.15 × 0.4 g × 0.7 2 × 0.9 3 0.1944λ = 0.5 × 9.8 × 0.7 × + 0.15 × 0.4 × 9.8 × 0.7 5 λ = 12.70...... λ = 13 N or 12.7

Pearson Edexcel International Advanced Level in Mathematics

2 × 0.9

−

© Pearson Education Limited 2013

M1 A1 M1 A1 A1

A1 [9]

Sample Assessment Materials

305

WME03/01: Mechanics M3 Question Number Q4

(a)

Scheme

cone 4π l 3 mass ratio 3 4 dist from l O Moments:

container 68π l 3 3 68 x

cylinder 24π l 3 72 3l

4l + 68 x = 72 × 3l 212l 53 x= = l accept 3.12l 68 17

Marks

M1 A1

B1 M1 A1ft A1

(6)

(b)

G

X

θ GX = 6l − x

seen

2l 6l − x 2 × 17 = 49 θ = 34.75... = 34.8 or 35

tan θ =

306

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M1 M1 A1

A1

(4) [10]

Sample Assessment Materials

WME03/01: Mechanics M3 Question Number

Scheme

Q5

C

Marks

v

mg (a)

(b)

Energy:

1 1 m × 5ag − mv 2 2 2 2 v = 5ag − 2ag sin θ

mga sin θ =

mv 2 a m T = ( 5ag − 2ag sin θ ) − mg sin θ a T = mg ( 5 − 3sin θ )

(d)

A1

(3)

Eqn of motion along radius:

T + mg sin θ =

(c)

M1 A1

M1 A1 M1 A1

(4)

At C , θ = 90°

T = mg ( 5 − 3) = 2mg T > 0 ∴ P reaches C

M1 A1 A1

(3)

Max speed at lowest point

( θ = 270° ;

v 2 = 5ag − 2ag sin 270 ) v 2 = 5ag + 2ag

v =  ( 7ag )

M1 A1

(2) [12]

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307

WME03/01: Mechanics M3 Question Number

Q6

Scheme

d2 x 3 =− 2 2 dt ( t + 1)

(a)

dx −2 =  −3 ( t + 1) dt dt −1 = 3 ( t + 1) ( +c ) t = 0, v = 2 2 = 3 + c c = −1 dx 3 = −1 dt t + 1

t = 0, x = 0

*

 c′ = 0

x = 3ln ( t + 1) − t

v =0

t=2

3 =1 t +1

x = 3ln 3 − 2 = 1.295... = 1.30 m (Allow 1.3)

308

Pearson Edexcel International Advanced Level in Mathematics

M1

M1 A1 M1

 3  x= − 1  dt  t +1  = 3ln ( t + 1) − t (+c′)

(b)

Marks

© Pearson Education Limited 2013

A1

(5)

M1 A1 B1

M1 A1 M1 A1

(7) [12]

Sample Assessment Materials

WME03/01: Mechanics M3 Question Number

Scheme

Q7

Marks

A 3a

e

T O

R (↑ )

(a)

2mg T = 2mg

B1

6mge 3a 6mge 2mg = 3a e=a AO = 4a

Hooke's law: T =

M1 A1

(3)

(b)

T

4a

x x

H.L. Eqn. of motion

Pearson Edexcel International Advanced Level in Mathematics

2mg 6mg ( a − x ) 2mg ( a − x ) T= = 3a a -2mg + T = 2mx 2mg ( a − x ) -2mg + = 2mx a 2mgx − = 2mx a g  x=− x a a period 2π g

© Pearson Education Limited 2013

B1ft M1 M1

A1

*

A1

(5)

Sample Assessment Materials

309

WME03/01: Mechanics M3 Question Number

Scheme

v2 = ω 2 ( a2 − x2 )

(c)

(d)

Marks

a x=− 8

2  g  a     − 0  a  4   1 =  ( ga ) 4

vmax 2 =

M1 A1

vmax

A1

g  a2 a2  v =  −  a  16 64  2

3ag 64 2 2 v = u + 2as 3ag − 2 gh 0= 64 3a h= 128 a 3a 19a = Total height above O = + 8 128 128

(3)

M1

=

310

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M1 A1

A1

(4) [15]

Sample Assessment Materials

Write your name here Surname

Other names

Pearson Edexcel

Centre Number

Candidate Number

International Advanced Level

Statistics S1

Advanced/Advanced Subsidiary Sample Assessment Material Time: 1 hour 30 minutes

Paper Reference

WST01/01

You must have: Mathematical Formulae and Statistical Tables (Blue)

Total Marks

Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation and integration, or have retrievable mathematical formulae stored in them.

Instructions

black ink or ball-point pen. •• Use If pencil is used for diagrams/sketches/graphs it must be dark (HB or B). Coloured pencils and highlighter pens must not be used. in the boxes at the top of this page with your name, • Fill centre number and candidate number. Answer all questions and ensure that your answers to parts of questions are • clearly labelled. Answer the questions in the spaces provided • – there may be more space than you need. You should show sufficient working to make your methods clear. Answers • without working may not gain full credit. Values from the statistical should be quoted in full. When a calculator is • used, the answer should betables given to an appropriate degree of accuracy.

Information

total mark for this paper is 75. •• The The marks for each question are shown in brackets – use this as a guide as to how much time to spend on each question.

Advice

each question carefully before you start to answer it. •• Read Try to answer every question. • Check your answers if you have time at the end.

S45007A ©2013 Pearson Education Ltd.

1/2/2/

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311

1.

Gary compared the total attendance, x, at home matches and the total number of goals, y, scored at home during a season for each of 12 football teams playing in a league. He correctly calculated: S xx = 1022500

S yy = 130.9

Leave blank

S xy = 8825

(a) Calculate the product moment correlation coefficient for these data. (b) Interpret the value of the correlation coefficient.

(2) (1)

Helen was given the same data to analyse. In view of the large numbers involved she decided to divide the attendance figures by 100. She then calculated the product moment correlation coefficient between

x and y. 100

(c) Write down the value Helen should have obtained.

(1)

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Question 1 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________

Q1

___________________________________________________________________________ (Total 4 marks) Pearson Edexcel International Advanced Level in Mathematics

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3

313

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2.

An experiment consists of selecting a ball from a bag and spinning a coin. The bag contains 5 red balls and 7 blue balls. A ball is selected at random from the bag, its colour is noted and then the ball is returned to the bag. When a red ball is selected, a biased coin with probability

2 3

Leave blank

of landing heads is spun.

When a blue ball is selected a fair coin is spun. (a) Complete the tree diagram below to show the possible outcomes and associated probabilities. Ball

..............

..............

Coin ..............

Heads

..............

Tails

..............

Heads

..............

Tails

Red

Blue

(2) Shivani selects a ball and spins the appropriate coin. (b) Find the probability that she obtains a head.

(2)

Given that Tom selected a ball at random and obtained a head when he spun the appropriate coin, (c) find the probability that Tom selected a red ball.

(3)

Shivani and Tom each repeat this experiment. (d) Find the probability that the colour of the ball Shivani selects is the same as the colour of the ball Tom selects. (3) 4

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Question 2 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________

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315

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Question 2 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 6

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Question 2 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________

Q2

___________________________________________________________________________ (Total 10 marks) Pearson Edexcel International Advanced Level in Mathematics

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317

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3.

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The discrete random variable X has probability distribution given by x

−1

0

1

2

3

P( X = x)

1 5

a

1 10

a

1 5

where a is a constant. (a) Find the value of a.

(2)

(b) Write down E( X ).

(1)

(c) Find Var( X ).

(3)

The random variable Y = 6 − 2 X (d) Find Var(Y ).

(2)

(e) Calculate P( X  Y ).

(3)

___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 8

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Question 3 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________

Pearson Edexcel International Advanced Level in Mathematics

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319

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Question 3 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 10

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Question 3 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________

Q3

___________________________________________________________________________ (Total 11 marks) Pearson Edexcel International Advanced Level in Mathematics

*S45007A01128* © Pearson Education Limited 2013

Sample Assessment Materials

11

321

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4.

The Venn diagram in Figure 1 shows the number of students in a class who read any of 3 popular magazines A, B and C.

Leave blank

B

A 4

2

5 3 6

10

C

Figure 1 One of these students is selected at random. 1 6

(a) Show that the probability that the student reads more than one magazine is . (b) Find the probability that the student reads A or B (or both). (c) Write down the probability that the student reads both A and C.

(2) (2) (1)

Given that the student reads at least one of the magazines, (d) find the probability that the student reads C.

(2)

(e) Determine whether or not reading magazine B and reading magazine C are statistically independent. (3) ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 12

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Q4

___________________________________________________________________________ (Total 10 marks) Pearson Edexcel International Advanced Level in Mathematics

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Sample Assessment Materials

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325

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5.

A teacher selects a random sample of 56 students and records, to the nearest hour, the time spent watching television in a particular week. Hours

1 − 10

11 − 20

21 − 25

26 − 30

31 − 40

41 − 59

Frequency

6

15

11

13

8

3

Mid-point

5.5

15.5

28

(a) Find the mid-points of the 21 − 25 hour and 31 − 40 hour groups.

Leave blank

50 (2)

A histogram was drawn to represent these data. The 11 − 20 group was represented by a bar of width 4 cm and height 6 cm. (b) Find the width and height of the 26 − 30 group.

(3)

(c) Estimate the mean and standard deviation of the time spent watching television by these students. (5) (d) Use linear interpolation to estimate the median length of time spent watching television by these students. (2) The teacher estimated the lower quartile and the upper quartile of the time spent watching television to be 15.8 and 29.3 respectively. (e) State, giving a reason, the skewness of these data.

(2)

___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 16

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Q5

___________________________________________________________________________ (Total 14 marks) Pearson Edexcel International Advanced Level in Mathematics

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329

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6.

A travel agent sells flights to different destinations from Beerow airport. The distance d, measured in 100 km, of the destination from the airport and the fare £f are recorded for a random sample of 6 destinations. Destination

A

B

C

D

E

F

d

2.2

4.0

6.0

2.5

8.0

5.0

f

18

20

25

23

32

28

[You may use

∑d

2

= 152.09

∑f

2

= 3686

Leave blank

∑ fd = 723.1 ]

(a) Using the axes below, complete a scatter diagram to illustrate this information.

(2)

(b) Explain why a linear regression model may be appropriate to describe the relationship between f and d. (1) (c) Calculate S dd and S fd

(4)

(d) Calculate the equation of the regression line of f on d giving your answer in the form f = a + bd . (4) (e) Give an interpretation of the value of b.

(1)

Jane is planning her holiday and wishes to fly from Beerow airport to a destination t km away. A rival travel agent charges 5p per km. (f) Find the range of values of t for which the first travel agent is cheaper than the rival. (2)

£f 40

30

20

10

0

0 20

330

1

2

3

4

5

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8 d (100 km)

Sample Assessment Materials

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Q6

___________________________________________________________________________ (Total 14 marks) Pearson Edexcel International Advanced Level in Mathematics

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7.

The distances travelled to work, D km, by the employees at a large company are normally distributed with D ∼ N(30 , 82 ).

Leave blank

(a) Find the probability that a randomly selected employee has a journey to work of more than 20 km. (3) (b) Find the upper quartile, Q3 , of D.

(3)

(c) Write down the lower quartile, Q1 , of D.

(1)

An outlier is defined as any value of D such that D < h or D > k where h = Q1 − 1.5 × (Q3 − Q1 )

and

k = Q3 + 1.5 × (Q3 − Q1 )

(d) Find the value of h and the value of k.

(2)

An employee is selected at random. (e) Find the probability that the distance travelled to work by this employee is an outlier. (3) ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 24

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Q7

(Total 12 marks) TOTAL FOR PAPER: 75 MARKS END Pearson Edexcel International Advanced Level in Mathematics

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BLANK PAGE

28

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Sample Assessment Materials

WST01/01: Statistics S1 Question Number Q1

(a)

Scheme

r=

8825 , 1022500 ×130.9

Marks M1 A1

= awrt 0.763

(b) Teams with high attendance scored more goals (oe, statement in context)

B1

(c) 0.76(3)

B1ft

(2) (1) (1) Total 4

(a) M1 for a correct expression, square root required Correct answer award 2/2 (b) Context required (attendance and goals). Condone causality. B0 for ‘strong positive correlation between attendance and goals’ on its own oe (c) Value required. Must be a correlation coefficient between -1 and +1 inclusive. B1ft for 0.76 or better or same answer as their value from part (a) to at least 2 d.p.

Pearson Edexcel International Advanced Level in Mathematics

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Sample Assessment Materials

339

WST01/01: Statistics S1 Question Number Q2

Scheme

(a) R

5/12 7/12

(b)

P(H) =

(c) P(R|H) =

B

H

2/3

P(R) and P(B)

T

1/3 1/2

2nd set of probabilities

H

1/2

B1

5 ×2 12 3 41 " " 72

,=

B1

T

(2)

5 2 7 1 41 or awrt 0.569 × + × ,= 12 3 12 2 72

2

(d)

Marks

M1 A1 (2)

20 or awrt 0.488 41

M1 A1ft A1 (3)

2

5 7   +   12   12  25 49 74 37 or awrt 0.514 = + = or 144 144 144 72

M1 A1ft A1

(3) Total 10

(a) 1st B1

for the probabilities on the first 2 branches. Accept 0.416 and 0.583

2nd B1 for probabilities on the second set of branches. Accept 0.6 , 0.3 , 0.5 and Allow exact decimal equivalents using clear recurring notation if required.

1.5 3

(b) M1 for an expression for P(H) that follows through their sum of two products of probabilities from their tree diagram (c)

5 P( R ∩ H ) P( R ∩ H ) 12 with denominator their (b) substituted e.g. = award M1. Formula M1 for P( H ) P( H ) (their (b) ) seen 5

2

× Formula probability × probability 12 3. but M0 if fraction repeated e.g. not seen M1 for

their b

1st A1ft for a fully correct expression or correct follow through 2nd A1 for 20 o.e. 41 2

(d) M1

2 3

2

5 7 for   or   can follow through their equivalent values from tree diagram  12   12 

1st A1 for both values correct or follow through from their original tree and + 2nd A1 for a correct answer Special Case

340

5 4 7 6 × or × seen award M1A0A0 12 11 12 11

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Sample Assessment Materials

WST01/01: Statistics S1 Question Number Q3

Scheme

(a)

2a + 52 + 101 = 1 a=

Marks M1

(or equivalent)

1 or 0.25 4

A1

(b) E(X) = 1

(2)

B1

(c) E( X 2 ) = 1× 1 + 1× 1 + 4 × 1 + 9 × 1 5 10 4 5 Var(X) = 3.1 − 12 ,

2

(d) Var(Y) = ( −2 ) Var(X) ,

= 8.4 or

(e) X > Y when X = 3 or 2,

M1

(= 3.1) = 2.1 or

21 oe 10

M1 A1

42 oe 5

M1 A1

so probability = " 14 "+ 15 =

9 20

(1)

(3)

(2)

M1 A1ft

oe

A1

(3) Total 11

(a) M1for a clear attempt to use

 P( X = x) = 1

Correct answer only 2/2. NB Division by 5 in parts (b), (c) and (d) seen scores 0. Do not apply ISW. (b) B1 for 1 (c) 1st M1

for attempting

 x2P( X = x) at least two terms correct. Can follow through.

2nd M1 for attempting E( X 2 ) − [E( X )]2 or allow subtracting 1 from their attempt at E( X 2 ) provided no incorrect formula seen. Correct answer only 3/3. (d) M1 for (−2) 2 Var( X ) or 4Var(X) Condone missing brackets provided final answer correct for their Var(X) . Correct answer only 2/2. (e) Allow M1 for distribution of Y = 6 − 2 X and correct attempt at E( Y 2 ) − [E(Y )]2 M1 for identifying X = 2, 3 1st A1ft for attempting to find their P(X=2) + P(X = 3) nd 9 or 0.45 2 A1 for 20

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341

WST01/01: Statistics S1 Question Number Q4

Scheme

Marks

(a)

2+3 5 1 = = (** given answer**) their total their total 6

M1 A1cso

(b)

4+ 2+5+3 , total

M1 A1

(c)

P( A ∩ C ) = 0

(d)

P(C reads at least one magazine) =

(e) P(B) =

=

14 7 or or 0.46 30 15

B1

10 1 9 3 = , P(C ) = = , 30 3 30 10

6+3 20

=

9

M1 A1

3 1 = 30 10

or P(B|C) =

3 9

(2) (1)

20

P( B ∩ C ) =

(2)

(2)

M1

1 3 1 3 1 P( B ) × P(C ) = × = = P( B ∩ C ) or P(B|C) = = =P(B) 3 10 10 9 3

M1

So yes they are statistically independent

A1cso

(3)

Total 10 (a)

M1

for

5 2+3 or 30 their total

(b) M1 for adding at least 3 of “4, 2, 5, 3” and dividing by their total to give a probability Can be written as separate fractions substituted into the completely correct Addition Rule

(c) B1 for 0 or 0/30 (d)

M1 for a denominator of 20 or

9 only, 2/2 20

20 leading to an answer with denominator of 20 30

(e) 1st M1 for attempting all the required probabilities for a suitable test 2nd M1 for use of a correct test - must have attempted all the correct probabilities. Equality can be implied in line 2. A1 for fully correct test carried out with a comment

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Sample Assessment Materials

WST01/01: Statistics S1 Question Number Q5

(a)

Scheme 23,

Marks

35.5 (may be in the table)

B1 B1

(b) Width of 10 units is 4 cm so width of 5 units is 2 cm

B1

Height = 2.6 × 4 =10.4 cm

(c)

 fx = 1316.5  x =

M1 A1

1316.5 = 56

awrt 23.5

(d) (e)

37378.25 2 − x = awrt10.7 56

Q2 = (20.5) +

= 37378.25 can be implied

B1 M1 A1

awrt 23.7 or 23.9

11

[7.9 >5.6 so ]

2

allow s = 10.8

( 28 − 21) × 5 = 23.68…

Q3 − Q2 = 5.6, Q2 − Q1 = 7.9

(3)

M1 A1

 fx So σ =

(2)

M1 A1

(5)

(2)

M1

(or x < Q2 )

negative skew

A1

(2) Total 14

(b) M1 for their width x their height=20.8. Without labels assume width first, height second and award marks accordingly. (c) 1st M1

for reasonable attempt at

 x and /56

is required 2nd M1 for a method for σ or s , 2 (fx) = 354806.3 M0, f 2 x = 13922.5 M0 and ( Typical errors



Correct answers only, award full marks. (d) Use of

 f( x − x)

2



 fx)

2

= 1733172 M0

= awrt 6428.75 for B1

lcb can be 20, 20.5 or 21, width can be 4 or 5 and the fraction part of the formula correct for M1 - Allow 28.5 in fraction that gives awrt 23.9 for M1A1 (e) M1for attempting a test for skewness using quartiles or mean and median. Provided median greater than 22.55 and less than 29.3 award for M1 for Q3 − Q2 < Q2 − Q1 without values as a valid reason. SC Accept mean close to median and no skew oe for M1A1

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343

WST01/01: Statistics S1 Question Number Q6

Scheme

Marks

(a)

B1 B1

(2) (b) The points lie reasonably close to a straight line (o.e.) (c)

 d = 27.7, Sdd S fd

(d)

b= a=

(e)

 f = 146

2 27.7 ) ( = 152.09 −

B1

(both, may be implied)

= 24.208…..

6 27.7 ×146 = 49.06…. = 723.1 − 6 S fd Sdd

awrt 24.2 awrt 49.1

awrt 2.03

= 2.026….

146 27.7 = 14.97….. −b× 6 6

so f = 15.0 + 2.03d

A flight costs £2.03 (or about £2) for every extra 100km or about 2p per km.

(f) 15.0 + 2.03d < 5d

so d >

15.0 = 5.00 ~ 5.05 ( 5 − 2.03)

So t > 500~505

(1)

B1

M1 A1 A1

(4)

M1 A1 M1 A1 B1ft

(4) (1)

M1 A1

(2) Total 14

344

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Sample Assessment Materials

WST01/01: Statistics S1 WST01/01: Statistics S1 (a) 1st B1 for at least 4 points correct (allow + one 2mm square) st B1 for all points correctcorrect (allow(allow + one +2 one mm 2mm squaresquare) (a) 21nd B1 at least 4 points nd 2 B1 for all points correct (allow + one 2 mm square (b) Ignore extra points and lines reference points (b) Require Ignore extra pointstoand linesand line for B1. Require reference to points and line for B1. (c) M1 for a correct method seen for either - a correct expression st (c) 1M1 method seen for either - a correct expression A1 for aScorrect dd awrt 24.2 st for SS dd awrt awrt 49.1 24.2 21nd A1 A1 for fd

2nd A1 for S fd awrt 49.1

(d) 1st M1 for a correct expression for b - can follow through their answers from (c) st M1 for a correct method to find follow through theirtheir b and their means (d) 12nd M1 expression for ba -- can follow through answers from (c) in terms of to d and giventheir expressions. Accept 2nd A1 M1 for af =.... correct method findall a -values followawrt through b and their means15 as rounding from correct answer 2nd A1 only. for f =.... in terms of d and all values awrt given expressions. Accept 15 as rounding from correct answer only. (e) Context of cost and distance required. Follow through their value of b (e) Context of cost and distance required. Follow through their value of b (f) M1 for an attempt to find the intersection of the 2 lines. Value of t in range 500 to 505 seen award M1. Value in rangeto5 find to 5.05 M1. of the 2 lines. Value of t in range 500 to 505 seen award M1. (f) M1 forofandattempt the award intersection Acceptoft dgreater than 500 to 505 inclusive Value in range 5 to 5.05 award M1. to include graphical solution for M 1A1 Accept t greater than 500 to 505 inclusive to include graphical solution for M 1A1

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345

Question Number Q7

WST01/01: Statistics S1

(a)

Scheme

 

P(D > 20) = P  Z >

Marks M1

20 − 30   8 

= P(Z >- 1.25)

A1 awrt 0.894

= 0.8944

(b) P(D < Q3 ) = 0.75 so

Q3 − 30 = 0.67 8

(3)

M1 B1 A1

Q3 = awrt 35.4 (c) 35.4 - 30= 5.4 so Q1 = 30 − 5.4 = awrt 24.6 (d)

A1

(3)

B1ft

Q3 − Q1 = 10.8 so 1.5 ( Q3 − Q1 ) = 16.2 so Q1 − 16.2 = h or Q3 + 16.2 = k

M1

h=8.4 to 8.6 and k= 51.4 to 51.6

A1

both

(e) 2P(D > 51.6) = 2P(Z > 2.7)

(1)

(2)

M1

= 2[1 - 0.9965] = awrt 0.007

M1 A1

(3)

Total 12 (a) M1 for an attempt to standardise 20 or 40 using 30 and 8. 1st A1 for z = + 1.25 2nd A1 for awrt 0.894 (b)

M1

for

Q3 − 30 = to a z value 8

M0 for 0.7734 on RHS. B1 for (z value) between 0.67~0.675 seen. M1B0A1 for use of z = 0.68 in correct expression with awrt 35.4 (c) Follow through using their of quartile values. (d) M1 for an attempt to calculate 1.5(IQR) and attempt to add or subtract using one of the formulae given in the question - follow through their quartiles (e) 1st M1 for attempting 2P(D > their k) or ( P(D >their k)+ P(D < their h)) 2nd M1 for standardising their h or k (may have missed the 2) so allow for standardising P(D > 51.6) or P(D < 8.4) Require boths Ms to award A mark.

346

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Sample Assessment Materials

Write your name here Surname

Other names

Pearson Edexcel

Centre Number

Candidate Number

International Advanced Level

Statistics S2

Advanced/Advanced Subsidiary Sample Assessment Material Time: 1 hour 30 minutes

Paper Reference

WST02/01

You must have: Mathematical Formulae and Statistical Tables (Blue)

Total Marks

Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation and integration, or have retrievable mathematical formulae stored in them.

Instructions

black ink or ball-point pen. •• Use If pencil is used for diagrams/sketches/graphs it must be dark (HB or B). Coloured pencils and highlighter pens must not be used. in the boxes at the top of this page with your name, • Fill centre number and candidate number. Answer all questions and ensure that your answers to parts of questions are • clearly labelled. Answer the questions in the spaces provided • – there may be more space than you need. You should show sufficient working to make your methods clear. Answers • without working may not gain full credit. Values from the statistical should be quoted in full. When a calculator is • used, the answer should betables given to an appropriate degree of accuracy.

Information

total mark for this paper is 75. •• The The marks for each question are shown in brackets – use this as a guide as to how much time to spend on each question.

Advice

each question carefully before you start to answer it. •• Read Try to answer every question. • Check your answers if you have time at the end.

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Pearson Edexcel International Advanced Level in Mathematics

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1.

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Explain what you understand by (a) a population,

(1)

(b) a statistic.

(1)

A researcher took a sample of 100 voters from a certain town and asked them who they would vote for in an election. The proportion who said they would vote for Dr Smith was 35%. (c) State the population and the statistic in this case.

(2)

(d) Explain what you understand by the sampling distribution of this statistic.

(1)

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Q1

___________________________________________________________________________ (Total 5 marks) Pearson Edexcel International Advanced Level in Mathematics

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349

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2.

Bhim and Joe play each other at badminton and for each game, independently of all others, the probability that Bhim loses is 0.2

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Find the probability that, in 9 games, Bhim loses (a) exactly 3 of the games,

(3)

(b) fewer than half of the games.

(2)

Bhim attends coaching sessions for 2 months. After completing the coaching, the probability that he loses each game, independently of all others, is 0.05 Bhim and Joe agree to play a further 60 games. (c) Calculate the mean and variance for the number of these 60 games that Bhim loses. (2) (d) Using a suitable approximation calculate the probability that Bhim loses more than 4 games. (3) ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 4

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Q2

___________________________________________________________________________ (Total 10 marks) Pearson Edexcel International Advanced Level in Mathematics

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351

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3.

A rectangle has a perimeter of 20 cm. The length, X cm, of one side of this rectangle is uniformly distributed between 1 cm and 7 cm.

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Find the probability that the length of the longer side of the rectangle is more than 6 cm long. (5) ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 6

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Q3

___________________________________________________________________________ (Total 5 marks) Pearson Edexcel International Advanced Level in Mathematics

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353

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4.

The lifetime, X, in tens of hours, of a battery has a cumulative distribution function F(x) given by 0  4  F( x) =  ( x 2 + 2 x − 3) 9 1 

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x 1.5

(a) Find the median of X, giving your answer to 3 significant figures.

(3)

(b) Find, in full, the probability density function of the random variable X. (c) Find P(X  1.2)

(3) (2)

A camping lantern runs on 4 batteries, all of which must be working. Four new batteries are put into the lantern. (d) Find the probability that the lantern will still be working after 12 hours.

(2)

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Question 4 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________

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355

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Question 4 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 10

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Question 4 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________

Q4

___________________________________________________________________________ (Total 10 marks) Pearson Edexcel International Advanced Level in Mathematics

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357

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5.

A company has a large number of regular users logging onto its website. On average 4 users every hour fail to connect to the company’s website at their first attempt. (a) Explain why the Poisson distribution may be a suitable model in this case.

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(1)

Find the probability that, in a randomly chosen 2 hour period, (b) (i) all users connect at their first attempt, (ii) at least 4 users fail to connect at their first attempt.

(5)

The company suffered from a virus infecting its computer system. During this infection it was found that the number of users failing to connect at their first attempt, over a 12 hour period, was 60. (c) Using a suitable approximation, test whether or not the mean number of users per hour who failed to connect at their first attempt had increased. Use a 5% level of significance and state your hypotheses clearly. (9) ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 12

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Question 5 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________

Pearson Edexcel International Advanced Level in Mathematics

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359

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Question 5 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 14

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Question 5 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________

Q5

___________________________________________________________________________ (Total 15 marks) Pearson Edexcel International Advanced Level in Mathematics

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361

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6.

A company claims that a quarter of the bolts sent to them are faulty. To test this claim the number of faulty bolts in a random sample of 50 is recorded.

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(a) Give two reasons why a binomial distribution may be a suitable model for the number of faulty bolts in the sample. (2) (b) Using a 5% significance level, find the critical region for a two-tailed test of the hypothesis that the probability of a bolt being faulty is 1 . The probability of rejection 4 in either tail should be as close as possible to 0.025 (3) (c) Find the actual significance level of this test.

(2)

In the sample of 50 the actual number of faulty bolts was 8. (d) Comment on the company’s claim in the light of this value. Justify your answer.

(2)

The machine making the bolts was reset and another sample of 50 bolts was taken. Only 5 were found to be faulty. (e) Test at the 1% level of significance whether or not the probability of a faulty bolt has decreased. State your hypotheses clearly. (6) ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 16

362

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Question 6 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________

Pearson Edexcel International Advanced Level in Mathematics

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363

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Question 6 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 18

364

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Question 6 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________

Q6

___________________________________________________________________________ (Total 15 marks) Pearson Edexcel International Advanced Level in Mathematics

*S45008A01924* © Pearson Education Limited 2013

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19

365

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7.

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The random variable Y has probability density function f(y) given by  ky (a − y ) f ( y) =  0 

0 y3

otherwise

where k and a are positive constants. (a) (i) Explain why a  3 (ii) Show that k =

2 9(a − 2) (6)

Given that E(Y) = 1.75 (b) show that a = 4 and write down the value of k.

(6)

For these values of a and k, (c) sketch the probability density function, (d) write down the mode of Y.

(2) (1)

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366

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Question 7 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________

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Question 7 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 22

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Question 7 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________

Pearson Edexcel International Advanced Level in Mathematics

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Question 7 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________

Q7

(Total 15 marks) TOTAL FOR PAPER: 75 MARKS END 24

370

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Pearson Edexcel International Advanced Level in Mathematics

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Sample Assessment Materials

WST02/01: Statistics S2 Question Number Q1 (a)

Scheme

Marks

A population is collection of all items

B1

(b)

(A random variable) that is a function of the sample which contains no unknown quantities/parameters.

B1

(c)

The voters in the town

B1

Percentage/proportion voting for Dr Smith

B1

Probability Distribution of those voting for Dr Smith from all possible samples (of size 100)

B1

(d)

(1) (1)

(2) (1) [5]

Notes (a)

B1 – collection/group all items – need to have /imply all eg entire/complete/every

(b)

B1 – needs function/calculation(o.e.) of the sample/random variables/observations and no unknown quantities/parameters(o.e.) NB do not allow unknown variables e.g. “A calculation based solely on observations from a given sample.” B1 “A calculation based only on known data from a sample” B1 “A calculation based on known observations from a sample” B0

(c)

B1 – Voters

Solely/only imply no unknown quantities

Do not allow 100 voters. B1 – percentage/ proportion voting (for Dr Smith) the number of people voting (for Dr Smith) Allow 35% of people voting (for Dr Smith) Allow 35 people voting (for Dr Smith) Do not allow 35% or 35 alone (d)

B1 – answers must include all three of these features (i) All possible samples, (ii) their associated probabilities, (iii) context of voting for Dr Smith. e.g

“It is all possible values of the percentage and their associated probabilities.” B0 no context

Pearson Edexcel International Advanced Level in Mathematics

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Sample Assessment Materials

371

WST02/01: Statistics S2 Question Number Q2 (a)

Scheme

Let X be the random variable the number of games Bhim loses. X ~B(9, 0.2) P(X ≤ 3) - P(X ≤ 2) = 0.9144 – 0.7382 = 0.1762

(b) (c) (d)

Marks

or ( 0.2 ) ( 0.8) 3

6

9! 3!6!

= 0.1762

P(X ≤ 4) = 0.9804 Mean = 3

variance = 2.85,

Po(3)

B1

57 20

awrt 0.176

M1 A1

awrt 0.98

M1A1 (2)

(3)

B1 B1 (2)

poisson

P(X > 4) = 1 – P (X < 4)

M1 M1

= 1 – 0.8153 = 0.1847

A1

(3) [10]

Notes (a)

B1 – writing or use of B(9, 0.2)

M1 for writing/ using P(X ≤ 3) - P(X ≤ 2) or (p)3 (1 – p)6 A1 awrt 0.176

9! 3!6!

(b)

M1 for writing or using P(X ≤ 4) A1 awrt 0.98

(c)

B1 3 B1 2.85, or exact equivalent

(d)

M1 for using Poisson M1 for writing or using 1 – P (X < 4) NB P (X < 4) is 0.7254 Po(3.5) and 0.8912 Po(2.5) A1 awrt 0.185 Special case: Use of Po(1.8) in (a) and (b)

e -1.8 1.8 3 (a) can get B1 M1 A0 – B1 if written B(9, 0.2), M1 for or awrt to 0.161 3! If B(9, 0.2) is not seen then the only mark available for using Poisson is M1. (b) can get M1 A0 - M1 for writing or using P(X ≤ 4) or may be implied by awrt 0.964 Use of Normal in (d) Can get M0 M1 A0.- for M1 they must write 1 – P (X < 4) or get awrt 0.187

372

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Sample Assessment Materials

WST02/01: Statistics S2 Question Number Q3

Scheme

Method 1 P (X > 6) =

1 6

P( X < 4) =

1 2

total =

1 1 2 + = 6 2 3

Marks

Method 2 P(4 < X < 6) =

1 3

Method 3 P (X > 6) =

1 6

Y~U[3,9] P (Y > 6) = 1–

1 2 = 3 3

total =

B1 M1

1 2

1 1 2 + = 6 2 3

A1

M1dep B A1 (5) [5]

Notes Methods 1 and 2 B1 for 6 and 4 (allow if seen on a diagram on x-axis)

M1 for P( X > 6) or P( 6 < X < 7); or P(X < 4 ) or P( 1 < X < 4) ; or P(4 < X < 6) Allow ≤ and ≥ signs 1 1 1 A1 ; or ; must match the probability statement 6 2 3 M1 for adding their “ P( X > 6)” and their “P(X < 4 )” or 1 - their “P(4 < X < 6)” dep on getting first B mark 2 A1 cao 3 Method 3 Y~U[3, 9] B1 for 6 with U[1,7]and 6 with U[3,9] M1 for P( X > 6) or P( 6 < X < 7) or P( 6 < Y < 9) 1 1 A1 ; or ; must match the probability statement 6 2 M1 for adding their “ P( X > 6)” and their “P(Y >6 )” dep on getting first B mark 2 A1 cao 3

Pearson Edexcel International Advanced Level in Mathematics

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Sample Assessment Materials

373

WST02/01: Statistics S2 Question Number Q4 (a)

(b)

(c)

(d)

Scheme

Marks

4 2 m + 2m − 3 = 0.5 9 m2 + 2m – 4.125 = 0 −2 ± 4 + 16.5 m= 2 m =1.26, -3.264 (median =) 1.26

(

)

M1

M1 A1

4  d  x2 + 2x − 3  9  = 4 2x + 2 Differentiating  ( ) dx 9

(

)

M1 A1

8  ( x + 1) 1 ≤ x ≤ 1.5 f ( x) =  9  0 otherwise

B1ft

P(X > 1.2)

M1

= 1 – F(1.2) = 1 – 0.3733 47 = , 0.6267 75

(0.6267)4= 0.154

(3)

(3)

awrt 0.627

A1

awrt 0.154 or 0.155

M1 A1 (2)

(2)

[10]

Notes (a)

(b)

(c)

M1 putting F(x) = 0.5 M1 using correct quadratic formula. If use calc need to get 1.26 (384... ) A1 cao 1.26 must reject the other root. If they use Trial and improvement they have to get the correct answer to gain the second M mark. M1 attempt to differentiate. At least one x n → x n −1 A1 correct differentiation B1 must have both parts- follow through their F′ (x) Condone < 1.5 8 1.2 8 M1 finding/writing 1 – F(1.2) may use/write  ( x + 1)dx or 1 -  ( x + 1)dx 1.2 9 1 9 or



1.5

1.2

"their f ( x)" dx . Condone missing dx

A1 awrt 0.627 (d)

374

M1 (c)4 If expressions are not given you need to check the calculation is correct to 2sf. A1 awrt 0.154 or 0.155

Pearson Edexcel International Advanced Level in Mathematics

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Sample Assessment Materials

WST02/01: Statistics S2 Question Number Q5 (a) (b) (i) (ii)

(c)

Scheme

Marks

Connecting occurs at random/independently, singly or at a constant rate Po (8) P(X = 0) = 0.0003 P(X > 4) = 1 – P(X < 3) = 1 – 0.0424 = 0.9576 H0 : λ = 4 (48) H1 : λ > 4 (48) N(48,48) Method 1 Method 2 x − 0.5 − 48 59.5 − 48   =1.6449 P(X > 59.5) = P  Z ≥  48 48   = P ( Z > 1.66) = 1 – 0.9515 = 0.0485 x = 59.9 0.0485 < 0.05 Reject H0. Significant. 60 lies in the Critical region The number of failed connections at the first attempt has increased.

(a) (b) (i) (ii) (c)

B1

(1)

B1 M1A1 M1 A1 (5) B1 M1 A1 M1 M1 A1

A1

M1 A1 ft (9) [15]

Notes B1 Any one of randomly/independently/singly/constant rate. Must have context of connection/logging on/fail B1 Writing or using Po(8) in (i) or (ii) M1 for writing or finding P(X = 0) A1 awrt 0.0003 M1 for writing or finding 1 – P(X < 3) A1 awrt 0.958 B1 both hypotheses correct. Must use λ or µ M1 identifying normal A1 using or seeing mean and variance of 48 These first two marks may be given if the following are seen in the standardisation formula : 48 and 48 or awrt 6.93 M1 for attempting a continuity correction (Method 1: 60 + 0.5 / Method 2: x + 0.5 ) M1 for standardising using their mean and their standard deviation and using either Method 1 [59.5, 60 or 60.5. accept + z.] Method 2 [ (x+0.5) and equal to a +z value) 59.5 − 48 x − 0.5 − 48 A1 correct z value awrt +1.66 or + =1.6449 , or 48 48 A1 awrt 3 sig fig in range 0.0484 – 0.0485, awrt 59.9 M1 for “reject H0” or “significant” maybe implied by “correct contextual comment” If one tail hypotheses given follow through “their prob” and 0.05 , p < 0.5 If two tail hypotheses given follow through “their prob” with 0.025, p < 0.5 If one tail hypotheses given follow through “their prob” and 0.95 , p > 0.5 If two tail hypotheses given follow through “their prob” with 0.975, p > 0.5 If no H1 given they get M0 A1 ft correct contextual statement followed through from their prob and H1. need the words number of failed connections/log ons has increased o.e. Allow “there are more failed connections” NB A correct contextual statement alone followed through from their prob and H1 gets M1 A1

Pearson Edexcel International Advanced Level in Mathematics

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Sample Assessment Materials

375

WST02/01: Statistics S2 Question Number Q6 (a)

Marks

2 outcomes/faulty or not faulty/success or fail A constant probability Independence Fixed number of trials (fixed n)

B1 B1

X ~ B(50,0.25) P(X < 6) = 0.0194 P(X < 7) = 0.0453 P(X > 18) = 0.0551 P(X > 19) = 0.0287

M1

CR X < 6 and X > 19

A1 A1 (3)

(c)

0.0194 + 0.0287 = 0.0481

M1A1 (2)

(d)

8(It) is not in the Critical region or 8(It) is not significant or 0.0916 > 0.025; There is evidence that the probability of a faulty bolt is 0.25 or the company’s claim is correct.

M1; A1ft

H0 : p = 0.25 H1 : p < 0.25 P( X < 5) = 0.0070 or CR X < 5 0.007 < 0.01, 5 is in the critical region, reject H0, significant. There is evidence that the probability of faulty bolts has decreased

B1B1 M1A1

(b)

(e)

(a) (b)

(c) (d)

(e)

376

Scheme

(2)

M1 A1ft

(2)

(6) [15]

Notes B1 B1 one mark for each of any of the four statements. Give first B1 if only one correct statement given. No context needed. M1 for writing or using B(50,0.25) also may be implied by both CR being correct. Condone use of P in critical region for the method mark. DO NOT accept P(X < 6) A1 (X )< 6 o.e. [0,6] DO NOT accept P(X > 19) A1 (X) > 19 o.e. [19,50] M1 Adding two probabilities for two tails. Both probabilities must be less than 0.5 A1 awrt 0.0481 M1 one of the given statements followed through from their CR. A1 contextual comment followed through from their CR. NB A correct contextual comment alone followed through from their CR will get M1 A1 B1 for H0 must use p or π (pi) B1 for H1 must use p or π (pi) M1 for finding or writing P( X < 5) or attempting to find a critical region or a correct critical region A1 awrt 0.007/CR X < 5 M1 correct statement using their Probability and 0.01 if one tail test or a correct statement using their Probability and 0.005 if two tail test. The 0.01 or 0.005 needn’t be explicitly seen but implied by correct statement compatible with their H1. If no H1 given M0 A1 correct contextual statement follow through from their prob and H1. Need faulty bolts and decreased. NB A correct contextual statement alone followed through from their prob and H1 get M1 A1

Pearson Edexcel International Advanced Level in Mathematics

© Pearson Education Limited 2013

Sample Assessment Materials

WST02/01: Statistics S2 Question Number Q7 (ai) f(y) > 0 or f(3) > 0

Scheme

Marks M1

ky ( a − y ) > 0 or 3k(a – 3) > 0 or (a – y) > 0 or (a – 3) > 0 a>3

(ii)

3

 k (ay − y )dy = 1 2

A1 cso

integration

M1

answer correct

A1

answer = 1

M1

0

3

  ay 2 y 3   −  = 1 k  3  0   2  9a  k  − 9 = 1  2  9 a − 18   k  =1  2  2 * k= 9 ( a − 2) (b)



3 0

A1 cso (6)

Int  xf ( x )

k (ay 2 − y 3 )dy = 1.75 3

  ay 3 y 4  −  = 1.75 k  4  0   3 81   k  9a −  = 1 .75 4  81   2 9a −  = 15.75(a − 2) 4  81 2.25a = − 31.5 + 2 a=4 * 1 k= 9

Pearson Edexcel International Advanced Level in Mathematics

© Pearson Education Limited 2013

M1

Correct integration

A1

 xf ( x) = 1.75 and limits 0,3

M1dep

subst k

M1dep

A1cso B1

Sample Assessment Materials

(6)

377

WST02/01: Statistics S2 Question Number

Scheme

Marks

(c) 0 0

(d)

B1 B1

3

mode = 2

B1

(2) (1) [15]

(a) (i)

Notes M1 for putting f(y) > 0 or f(3) > 0 or ky ( a − y ) > 0 or 3k(a – 3) > 0 or (a – y) > 0 or (a – 3) > 0 or state in words the probability can not be negative o.e. A1 need one of ky ( a − y ) > 0 or 3k(a – 3) > 0 or (a – y) > 0 or (a – 3) > 0 and a > 3

(ii)

(b)

M1 attempting to integrate (at least one yn → yn+1) (ignore limits) A1 Correct integration. Limits not needed. And equals 1 not needed. M1 dependent on the previous M being awarded. Putting equal to 1 and have the correct limits. Limits do not need to be substituted. A1 cso M1 for attempting to find  yf ( y ) dy (at least one yn → yn+1) (ignore limits) A1 correct Integration M1  yf ( y ) = 1.75 and limits 0,3 dependent on previous M being awarded

(c)

M1 subst in for k. dependent on previous M being awarded A1 cso 4 B1 cao 1/9 B1 correct shape. No straight lines. No need for patios. B1 completely correct graph. Needs to go through origin and the curve ends at 3. Special case: If draw full parabola from 0 to 4 get B1 B0 Allow full marks if the portion between x = 3 and x = 4 is dotted and the rest of the curve solid.

0 0

(d)

378

4

B1 cao 2

Pearson Edexcel International Advanced Level in Mathematics

© Pearson Education Limited 2013

Sample Assessment Materials

Write your name here Surname

Other names

Pearson Edexcel

Centre Number

Candidate Number

International Advanced Level

Statistics S3

Advanced/Advanced Subsidiary Sample Assessment Material Time: 1 hour 30 minutes

Paper Reference

WST03/01

You must have: Mathematical Formulae and Statistical Tables (Blue)

Total Marks

Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation and integration, or have retrievable mathematical formulae stored in them.

Instructions

black ink or ball-point pen. •• Use If pencil is used for diagrams/sketches/graphs it must be dark (HB or B). Coloured pencils and highlighter pens must not be used. in the boxes at the top of this page with your name, • Fill centre number and candidate number. Answer all questions and ensure that your answers to parts of questions are • clearly labelled. Answer the questions in the spaces provided • – there may be more space than you need. You should show sufficient working to make your methods clear. Answers • without working may not gain full credit. Values from the statistical should be quoted in full. When a calculator is • used, the answer should betables given to an appropriate degree of accuracy.

Information

total mark for this paper is 75. •• The The marks for each question are shown in brackets – use this as a guide as to how much time to spend on each question.

Advice

each question carefully before you start to answer it. •• Read Try to answer every question. • Check your answers if you have time at the end.

S45009A ©2013 Pearson Education Ltd.

1/2/2/

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Sample Assessment Materials

379

1.

A report states that employees spend, on average, 80 minutes every working day on personal use of the Internet. A company takes a random sample of 100 employees and finds their mean personal Internet use is 83 minutes with a standard deviation of 15 minutes. The company’s managing director claims that his employees spend more time on average on personal use of the Internet than the report states.

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Test, at the 5% level of significance, the managing director’s claim. State your hypotheses clearly. (7) ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 2

380

*S45009A0224*

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Question 1 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________

Q1

___________________________________________________________________________ (Total 7 marks) Pearson Edexcel International Advanced Level in Mathematics

*S45009A0324* © Pearson Education Limited 2013

Sample Assessment Materials

3

381

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2.

Philip and James are racing car drivers. Philip’s lap times, in seconds, are normally distributed with mean 90 and variance 9. James’ lap times, in seconds, are normally distributed with mean 91 and variance 12. The lap times of Philip and James are independent. Before a race, they each take a qualifying lap. (a) Find the probability that James’ time for the qualifying lap is less than Philip’s.

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(4)

The race is made up of 60 laps. Assuming that they both start from the same starting line and lap times are independent, (b) find the probability that Philip beats James in the race by more than 2 minutes.

(5)

___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 4

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Q2

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5

383

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3.

A woodwork teacher measures the width, w mm, of a board. The measured width, X mm, is normally distributed with mean w mm and standard deviation 0.5 mm. (a) Find the probability that X is within 0.6 mm of w.

Leave blank

(2)

The same board is measured 16 times and the results are recorded. (b) Find the probability that the mean of these results is within 0.3 mm of w.

(4)

Given that the mean of these 16 measurements is 35.6 mm, (c) find a 98% confidence interval for w.

(4)

___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 6

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*S45009A0624*

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385

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Question 3 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 8

386

*S45009A0824*

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Q3

(Total 10 marks) Pearson Edexcel International Advanced Level in Mathematics

*S45009A0924* © Pearson Education Limited 2013

Sample Assessment Materials

9

387

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4.

A researcher claims that, at a river bend, the water gradually gets deeper as the distance from the inner bank increases. He measures the distance from the inner bank, b cm, and the depth of a river, s cm, at seven positions. The results are shown in the table below. Position

A

B

C

D

E

F

G

Distance from inner bank b cm

100

200

300

400

500

600

700

Depth s cm

60

75

85

76

110

120

104

(a) Calculate Spearman’s rank correlation coefficient between b and s.

Leave blank

(6)

(b) Stating your hypotheses clearly, test whether or not the data provides support for the researcher’s claim. Use a 1% level of significance. (4) ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 10

388

*S45009A01024*

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Q4

___________________________________________________________________________ (Total 10 marks) Pearson Edexcel International Advanced Level in Mathematics

*S45009A01124* © Pearson Education Limited 2013

Sample Assessment Materials

11

389

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5.

A random sample of 100 people were asked if their finances were worse, the same or better than this time last year. The sample was split according to their annual income and the results are shown in the table below. Finances

Worse

Same

Better

Under £15 000

14

11

9

£15 000 and above

17

20

29

Leave blank

Annual income

Test, at the 5% level of significance, whether or not the relative state of their finances is independent of their income range. State your hypotheses and show your working clearly. (10) ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 12

390

*S45009A01224*

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Question 5 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________

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391

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392

*S45009A01424*

Pearson Edexcel International Advanced Level in Mathematics

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Sample Assessment Materials

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Q5

___________________________________________________________________________ (Total 10 marks) Pearson Edexcel International Advanced Level in Mathematics

*S45009A01524* © Pearson Education Limited 2013

Sample Assessment Materials

15

393

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6.

A total of 228 items are collected from an archaeological site. The distance from the centre of the site is recorded for each item. The results are summarised in the table below. Distance from the centre of the site (m)

0–1

1–2

2–4

4–6

6–9

9–12

Number of items

22

15

44

37

52

58

Leave blank

Test, at the 5% level of significance, whether or not the data can be modelled by a continuous uniform distribution. State your hypotheses clearly. (12) ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 16

394

*S45009A01624*

Pearson Edexcel International Advanced Level in Mathematics

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395

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396

*S45009A01824*

Pearson Edexcel International Advanced Level in Mathematics

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Q6

___________________________________________________________________________ (Total 12 marks) Pearson Edexcel International Advanced Level in Mathematics

*S45009A01924* © Pearson Education Limited 2013

Sample Assessment Materials

19

397

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7.

Leave blank

A large company surveyed its staff to investigate the awareness of company policy. The company employs 6000 full-time staff and 4000 part-time staff. (a) Describe how a stratified sample of 200 staff could be taken.

(3)

(b) Explain an advantage of using a stratified sample rather than a simple random sample. (1) A random sample of 80 full-time staff and an independent random sample of 80 part-time staff were given a test of policy awareness. The results are summarised in the table below. Mean score ( x )

Variance of scores ( s2 )

Full-time staff

52

21

Part-time staff

50

19

(c) Stating your hypotheses clearly, test, at the 1% level of significance, whether or not the mean policy awareness scores for full-time and part-time staff are different. (7) (d) Explain the significance of the Central Limit Theorem to the test in part (c). (e) State an assumption you have made in carrying out the test in part (c).

(2) (1)

After all the staff had completed a training course the 80 full-time staff and the 80 part-time staff were given another test of policy awareness. The value of the test statistic z was 2.53 (f) Comment on the awareness of company policy for the full-time and part-time staff in light of this result. Use a 1% level of significance. (2) (g) Interpret your answers to part (c) and part (f).

(1)

___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 20

398

*S45009A02024*

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21

399

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*S45009A02224*

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23

401

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Question 7 continued ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________

Q7

(Total 17 marks) TOTAL FOR PAPER: 75 MARKS END 24

402

*S45009A02424*

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Sample Assessment Materials

WST03/01: Statistics S3

Hypothesis Tests (Final M1A1)

For an incorrect comparison (e.g. probability with z value) even with a correct statement and/or comment award M0A0 For a correct or no comparison with more than one statement one of which is false Award M0A0 (This is compatible with the principle above of contradictory statements being penalised) Apply these rules to all questions

Pearson Edexcel International Advanced Level in Mathematics

© Pearson Education Limited 2013

Sample Assessment Materials

403

WST03/01: Statistics S3 Question Number Q1

Scheme

H0: μ = 80, H1: μ > 80 z=

B1,B1 M1A1

83 − 80 =2 15

100 2 > 1.6449 Reject H0 or significant result or in the critical region Managing director’s claim is supported. st

1 B1 2nd B1 1st M1 1st A1 3rd B1 2nd M1

2nd A1 2nd M1A1

(accept 1.645 or better)

B1 M1 A1

for H0. They must use µ not x, p, λ or x etc for H1 (must be > 80). Same rules about µ . 15 . Can accept +. for attempt at standardising using 83, 80 and 100 May be implied by z = +2 for + 2 only

7

for +1.6449 seen (or probability of 0.0228 or better) for a correct statement about “significance” or rejecting H0 (or H1) based on their z value and their 1.6449 (provided it is a recognizable critical value from normal tables) or their probability (< 0.5) and significance level of 0.05. Condone their probability > 0.5 compared with 0.95 for the 2nd M1 for a correct contextualised comment. Must mention “director” and “claim” or “time” and “use of Internet”. No follow through.

If no comparison or statement is made but a correct contextualised comment is given the M1 can be implied. If a comparison is made it must be compatible with statement otherwise M0 e.g. comparing 0.0228 with 1.6449 is M0 or comparing probability 0.9772 with 0.05 is M0 comparing -2 with - 1.6449 is OK provided a correct statement accompanies it condone -2 >-1.6449 provided their statement correctly rejects H0.

Critical Region

They may find a critical region for X : X > 80 + 1st M1 3rd B1 1st A1

404

Marks

15 × (z value) 100 for 1.645 or better for awrt 82.5 The rest of the marks are as per the scheme.

15 ×1.6449 = awrt 82.5 100

for 80 +

Pearson Edexcel International Advanced Level in Mathematics

© Pearson Education Limited 2013

Sample Assessment Materials

WST03/01: Statistics S3 Question Number Q2 (a)

Scheme

Marks

[ P ~ N(90,9) and J ~ N(91,12)]

( J − P ) ∼ N(1, 21) P( J < P) = P( J − P < 0)

0 −1   = P Z <  21   = P( Z < −0.2182...) = 1 − 0.5871 = 0.4129 calculator (0.4136….) (b)

X = ( J1 + J 2 + .... + J 60 ) − ( P1 + P2 + .... + P60 ) E( X ) = 60 × 91 − 60 × 90 = 60 Var( X ) = 60 × 9 + 60 ×12 = 1260 P( X > 120)

M1, A1

dM1 awrt (0.413 ~ 0.414)

A1

(4)

M1 [stated as E(X) = 60 or X~N(60, …)]

120 − 60   = PZ >  1260   = P( Z > 1.69030...) = 1 − 0.9545 = 0.0455

B1 A1 M1

awrt (0.0455)

A1

(5) 9

(a) 1st M1

for attempting J - P and E(J - P) or P - J and E(P - J) for variance of 21 (Accept 9 + 12). Ignore any slip in µ here. 1st A1 nd 2 dM1 for attempting the correct probability and standardising with their mean and sd. This mark is dependent on previous M so if J - P ( or P - J) is not being used score M0 If their method is not crystal clear then they must be attempting P(Z< -ve value) or P(Z > +ve value) i.e. their probability after standardisation should lead to a prob. < 0.5 so e.g. P(J - P < 0) leading to 0.5871 is M0A0 unless the M1 is clearly earned. 2nd A1 for awrt 0.413 or 0.414 The first 3 marks may be implied by a correct answer

(b) 1st M1

B1 1st A1 2nd M1

Use of means

for a clear attempt to identify a correct form for X. This may be implied by correct variance of 1260 for E(X) = 60. Can be awarded even if they are using X = 60J - 60P. Allow P - J and -60 for a correct variance. If 1260 is given the M1 is scored by implication. for attempting a correct probability and standardising with 120 and their 60 and 1260 If the answer is incorrect a full expression must be seen following through their values 120 − their 60  −120 − − 60    for M1 e.g. P  Z >  . If using -60, should get P  Z <  their variance  their variance   

Attempt to use J − P for 1st M1, E( J − P ) = 1 for B1 and Var( J − P ) = 0.35 for A1 Then 2nd M1 for standardisation with 2, and their 1 and 0.35

Pearson Edexcel International Advanced Level in Mathematics

© Pearson Education Limited 2013

Sample Assessment Materials

405

WST03/01: Statistics S3 Question Number Q3

(a)

Scheme 2

E ∼ N(0,0.5 ) P( E < 0.6)

0.6   = P Z <  0.5   = P( Z < 1.2)

(

2

Marks

)

or

X ~ N w,0.5

or

0.6   P ( X − w < 0.6) = P  Z <  0.5  

= 2 × 0.8849 − 1 = 0.7698 (b)

 1  E ∼ N  0,   64 

(

P E < 0.3

)

 0.3  = P Z < 1  8  

awrt 0.770 2

or

 0.5 X ~ N  w,  16

or

 0.3  P X − w < 0.3 = P  Z < 1    8  

(

  

M1 A1

(2)

M1

)

M1, A1

= P( Z < 2.4)

= 2 × 0.9918 − 1 = 0.9836 (c)

1 35.6 ± 2.3263 × 8 (35.3, 35.9)

(a) 1st M1

1st A1

st

(b) 1 M1

2nd M1 1st A1 2nd A1 Sum of 16, not means

1st M1 2nd M1 1st A1

(c) M1

B1 1st A1 2nd A1

406

awrt 0.984

A1

(4)

M1 B1 A1,A1

(4) 10

for identifying a correct probability (they must have the 0.6) and attempting to standardise. Need | |. This mark can be given for 0.8849 - 0.1151 seen as final answer. for awrt 0.770. NB an answer of 0.3849 or 0.8849 scores M0A0 (since it implies no | |) M1 may be implied by a correct answer for a correct attempt to define E or X but must attempt

σ2 . n

Condone labelling as E or X This mark may be implied by standardisation in the next line. for identifying a correct probability statement using E or X . Must have 0.3 and | | for correct standardisation as printed or better for awrt 0.984 The M marks may be implied by a correct answer. for correct attempt at suitable sum distribution with correct variance ( = 16× 14 ) for identifying a correct probability. Must have 4.8 and | | 4.8 or better for correct standardisation i.e. need to see 4 0.5 for 35.6 + z × 16 for 2.3263 or better. Use of 2.33 will lose this mark but can still score ¾ for awrt 35.3 for awrt 35.9

Pearson Edexcel International Advanced Level in Mathematics

© Pearson Education Limited 2013

Sample Assessment Materials

WST03/01: Statistics S3 Question Number Q4

Scheme

Distance rank Depth rank d

(a)

d2

d

2

1

2

3

4

5

6

7

1

2

4

3

6

7

5

0

0

1

1

1

1

2

0

0

1

1

1

1

4

= 1− =

6×8 7 × 48

6   = 0.857142 7

awrt 0.857

rs < 0.8929 so not significant evidence to reject H 0 , The researcher's claim is not correct (at 1% level). or insufficient evidence for researcher’s claim or there is insufficient evidence that water gets deeper further from inner bank. or no (positive) correlation between depth of water and distance from inner bank (a) 1 M1

2nd M1 3rd M1 1st A1 4th M1 2nd A1

(b) 1st B1

2nd B1 M1 A1ft

M1

M1

H 0 : ρ = 0, H1 : ρ > 0 Critical value at 1% level is 0.8929

st

M1

M1A1

=8

rs

(b)

Marks

A1 B1 B1 M1 A1ft

(6)

(4) 10

for an attempt to rank the depths against the distances for attempting d for their ranks. Must be using ranks. for attempting  d 2 (must be using ranks)

for sum of 8 (or 104 for reverse ranking) for use of the correct formula with their  d 2 . If answer is not correct an expression is required. for awrt (+) 0.857. Sign should correspond to ranking (so use of 104 should get -0.857)

for both hypotheses in terms of ρ , H1 must be one tail and compatible with their ranking for cv of 0.8929 (accept +) for a correct statement relating their rs with their cv but cv must be such that |cv| 2.5758 so] significant evidence to reject H 0 There is evidence of a difference in policy awareness between full time and part time staff (d) Can use mean full time and mean part time ~ Normal (e) Have assumed s 2 = σ 2 or variance of sample = variance of population (f) 2.53 < 2.5758, not significant or do not reject H0

So there is insufficient evidence of a difference in mean awareness

(g) Training course has closed the gap between full time staff and part time staff’s mean

awareness of company policy.

A1 B1 dM1 A1ft B1 B1

(2) (1)

M1 A1ft

(2)

B1

for attempt at labelling full-time and part-time staff. One set of correct numbers. for mentioning use of random numbers for s.r.s. of 120 full-time and 80 part-time

(c) 1st M1

for attempt at s.e. - condone one number wrong . NB correct s.e. = for using their s.e. in correct formula for test statistic. Must be

(7)

B1

(a) 1st M1

2nd M1

(1)

M1,M1

= 2.828… (awrt 2.83) Two tailed critical value z = 2.5758 (or prob of awrt 0.002 (