Final Form 4 2011 ( Soalan Dan Skema), Peperiksaan Akhir Tahun Tingkatan 4 2011

3472/2 3472/2 Form Four Additional Mathematics Paper 2 October 2011 2 ½ hours

SEKOLAH MENENGAH SAINS TELUK INTAN

PEPERIKSAAN DIAGNOSTIK TINGKATAN 4 2011 ADDITIONAL MATHEMATICS Form Four Paper 2 Two hours and thirty minutes

DO NOT OPEN THIS QUESTION PAPER UNTIL INSTRUCTED TO DO SO INFORMATION FOR CANDIDATES 1

This question paper consists of three sections: Section A, Section B and Section C.

2

Answer all questions in Section A, four questions from Section B and two questions from Section C.

3

For Section C, answer one question from 12 or 13 and one question from 14 or 15.

4

Give only one answer/solution to each question.

5

Show your working. It may help you to get marks.

6

The diagrams in the questions provided are not drawn to scale unless stated.

7

The marks allocated for each question and sub-part of a question are shown in brackets.

8

A list of formulae is provided on pages 3 and 4.

9

You may use a four-figure mathematical table.

10 You may use a non-programmable scientific calculator.

This question paper consists of 13 printed pages.

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CONFIDENTAL

2

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The following formulae may be helpful in answering the questions. The symbols given are the ones commonly used. ALGEBRA

 b  b 2  4ac 2a

5

log a mn  log a m  log a n

am x an = a m + n

6

log a

3

am  an = a m – n

7

log a mn = n log a m

4

( am ) n = a m n

8

1

x

2

m  log a m  log a n n

log a b 

log c b log c a

GEOMETRY

1

Distance =

x1  x2 

2

  y1  y 2 

2

3

A point dividing a segment of a line

x , y   nx1  mx2 , ny1  my2   mn

2 Mid point

x, y    x1  x2 , y1  y2  

2

2



mn 

4 Area of a triangle =

1 x1 y 2  x2 y3  x3 y1   x 2 y1  x3 y 2  x1 y3  2

STATISTICS 1

x

2

x

3



x N

 fx f x  x   N 2

 f x  x   4  f 2

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x 2 x2 N

 fx 2 x2 f

5

1   N F  C m  L  2  fm     

6

I 

Q1  100 Q0

7

I

Wi I i Wi

3

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TRIGONOMETRY

1

Arc length, s =r

2

Area of a sector, A 

1 2 r 2

3

a b c   sin A sin B sin C

4

a2  b2  c2  2bc cos A

5

Area of triangle =

1 ab sin C 2

CALCULUS

1

dy dv du y = uv , u v dx dx dx

3

dy dy du   dx du dx

2

u dy y ,  v dx

v

du dv u dx dx v2

[See overleaf] 3472/2

CONFIDENTAL

4

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Section A [40 marks] Answer all questions in this section.

1

Solve the simultaneous equations 2x – y – 3 = 0 and 2x² - 10x + y + 9 = 0. [5 marks]

2

(a)

Express f(x) = x2  5x + 2m + 2 in the form of f(x) = p(x + q)2 + r.

(b)

The graph f(x) = x  5x + 2m + 2 has a minimum point at (k , - ). Find the values of k and of m. Hence, sketch the graph for 0  x  3 and state the corresponding range for f(x). [6 marks]

[2 marks] 2

3

It is given that  and  are roots of the equation x2  7x + k = 0 and  + 3 and  + 3 are roots of the equation x2 - mx + 25 = 0. Find the values of k and m. [6 marks]

4

If

5

A set of data consists of fifteen numbers. The sum of the numbers x, is 450 while the sum of the squares  x 2 , is 13 635. (a)

and y = 103 as x = 2 and y = 736 as x = 3, find the values of k and n. [6 marks]

Find the mean and the variance of the numbers.

[4 marks] When each of the numbers in the set of data is multiplied with m, followed by subtracting n from it, the new mean and standard deviation are 83 and 9 respectively. Find the values of m and n. [4 marks] 4 Given that and y  p Find (b)

6

(a) (b) (c)

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the rate of change of p, when x increases at a rate of 0.5 unit per second, [2 marks] dy in terms of x, dx [3 marks] the approximate change in y when x changes from 5 to 4.85 . [2 marks]

5

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Section B [40 marks] Answer four questions from this section.

7

Diagram 1 represents part of the mapping of x onto y by the function f(x) = ax + b, where a and b are constants and part of the mapping of y onto z by the function g. f(x) x

g(y) y

z 4

1

-1

c

-2 DIAGRAM 1 Find (a)

the values of a and b,

(b)

f -1 (x),

[5 marks]

(c)

g(x), given that

(d)

the value of c.

3x  1 2

[2 marks]

[2 marks] [1 mark]

[See overleaf] 3472/2

CONFIDENTAL 8

6

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Solution to this question by scale drawing will not be accepted. A line passes through A(-3 , 2) and B(2 , 5). The perpendicular bisector of line AB meets the x-axis at P and the y-axis at Q.

Find (a) the equation of the perpendicular bisector of AB, [4 marks] (b) the area of triangle OPQ, [3 marks] (c) the equation of locus R such that  APR is always 90. [3 marks]

9

Table 1 shows the frequency distribution of the marks of a group of pupils in a test. Marks Number of pupils

1-20

21-40

41-60

61-80

81-100

7

10

13

6

4

TABLE 1 (a)

Draw a histogram for the above data and estimate the modal age. [3 marks]

(b)

Calculate the standard deviation of the distribution.

(c)

Without using an ogive, calculate the median age.

[4 marks] [3 marks]

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7 10

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Diagram 2 shows a circle centre O with radius p cm. The area of rectangle OCTQ is 108 cm². The length CT is three times the breadth OC.

Using  = 3142, find

DIAGRAM 2

(a)

the length of CT,

(b)

angle  COT in radians,

(c)

the area of sector OAB,

(d)

the perimeter of the shaded region.

[2 marks] [2 marks] [2 marks] [4 marks]

[See overleaf] 3472/2

CONFIDENTAL

11

(a)

8

The gradient of the curve values of a and b.

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at the point (1, - 9 ) is 27. Find the [5 marks]

(b)

Diagram 3 shows a closed cone with diameter of (18 – 2d) cm and height of (6d – 21) cm. The vertex of the cone is vertically above the center of its base.

(6d – 21) cm

(18 - 2d) cm

DIAGRAM 3 Express the volume of the cone, V in terms of  and d. Hence, calculate the maximum volume of the cone, in term of . [5 marks]

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9

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Section C [20 marks]

Answer two questions from this section, in which one question from 12 or 13 and one question from 14 or 15.

12

Diagram 4 shows triangle ABC with ∠BAC = 34, AB = 7 cm, AC = 9 cm and BC = t cm.

DIAGRAM 4 (a) (b)

Find the value of t. Calculate the value of  BCA.

[2 marks] [3 marks]

(c)

Find the area of triangle ABC. [2 marks]

(d)

If AC is extended to D such that the area of triangle ABD is twice the area of triangle ABC, find the length of BD. [3 marks]

[See overleaf] 3472/2

CONFIDENTAL

13

10

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In the Diagram 4, ABC and EDC are straight lines.

DIAGRAM 4

Given that AE = 10 cm, BD = 7 cm, BC = 5 cm, CD = 6 cm and DE = 2 cm. Calculate (a) (b)

BCD, AEC,

[2 marks] [3 marks]

(c)

the length of AC, [2 marks]

(d)

the area of triangle BDE. [3 marks]

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11

14

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A particular type of cake is made by using four ingredients K, L, M and N. Table 2 shows the price indices for the four ingredients and the percentages of usage of the ingredients in making the cake.

TABLE 2

(a)

(b)

Calculate (i)

the price per kg of M in the year 2001 if its price per kg in the year 2003 was RM 3.25,

(ii)

the price index of K in the year 2003 based on the year 1999 if its price index in the year 2001 based on the year 1999 was 109. [5 marks]

The composite index number for the cost of making the cake in the year 2003 based on the year 2001 was 112.5. Calculate (i)

the value of x,

(ii)

the cost of making a cake in the year 2001 if the corresponding cost in the year 2003 was RM 6.30. [5 marks]

[See overleaf] 3472/2

CONFIDENTAL

15

12

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Table 3 shows the price indices of five components A, B, C, D and E that are needed to make an electronic equipment for the year 2003 based on the year 2001. The changes in the price indices of the components from the year 2003 to the year 2004 and their weightages are shown.

Components A B C D E

Price index in 2003 based on 2001 140 130 125 110 x

Change in price from 2003 to 2004 Increase 10% No change No change Increase 20% Decrease 5%

Weightage 3 2 6 5 4

TABLE 3 (a)

If the price of component B was RM8 in the year 2001, calculate its price in 2003. [2 marks]

(b)

Given that the prices of component E in the year 2001 and 2003 were RM5.50 and RM6.60 respectively, find the value of x. [2 marks]

(c)

Calculate the composite index for the cost of making the electronic equipment in the year 2004 based on the year 2003. [3 marks]

(d)

(i)

If the cost of making the electronic equipment in the year 2003 is RM2986, calculate the cost of the electronic equipment in the year 2004.

(ii)

The cost of making the electronic equipment is expected to increase by 30% from the year 2004 to the year 2009. Find the expected composite index for the year 2009 based on the year 2003. [3 marks]

END OF THE QUESTION PAPER

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13

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3472/2 Form Four Additional Mathematics Paper 2 October 2011

SEKOLAH MENENGAH SAINS TELUK INTAN

PEPERIKSAAN DIAGNOSTIK TINGKATAN 4 2011 ADDITIONAL MATHEMATICS Form Four Paper 2 Marking Scheme

.

This marking scheme consists of 11 printed pages.

[See overleaf] 3472/2

CONFIDENTAL

14

No.

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Solution y = 2x – 3 2x² - 10x + (2x – 3) + 9 = 0 (x – 1)(x – 3) = 0 x=1,3 y = -1 , 3

1

2 (a)

Marks K1 K1 K1 N1 N1 5 K1 N1

(b)

=-¼ m=2

k= ,

K1 N1, N1

y

N1 Shape of graph

6 x 3 -

3

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N1 (0 , 6), ( ,- ) and (3,0)

Range :

N1 8

or   7   k (  +3)(  +3) = m or   3(   )  9  25   m  6   3(7)  16 k =  = -5 -5 = m – 6 m=1

K1 K1

K1, K1 N1, N1 6

15 4

or

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K1

N1

n=5 k (2) 5  96

K1

k=3 5 (a)

N1 6 K1

x = = 30

N1

13635 2  30 15 =9

2 

N1 and

(b)

K1, K1

m=3, n=7

N1, N1 8 K1

= - 1.5

N1

6 (a)

(b)

K1

dy  4   3 dx p 2 12  3(10  x)2

K1

K1 N1

(c)

K1 = -0.008

N1 7 [See overleaf]

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CONFIDENTAL 7 (a)

16

K1

f (1)  a(1)  b  4 f (1)  a(1)  b  2

 a = 3 dan b = 1 (b)

Let f 1 ( x)  y  x

x 1 3

y  3x  1  x 

N1 y 1 3

 y 1 3  1 3   g ( y)  2 g ( x) 

x 2

g (4)  2

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N1, N1

K1

3x  2 g 3x  1  2

(c)

K1

f ( y)  x

y 1 3

f 1 ( x) 

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K1

N1 N1

N1 10

17 8 (a)

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Midpoint of AB = =

K1

= K1

K1

N1

(b)

K1

K1 =

(c)

N1

m AP  

2 15

dan

8  y   2   5   1    15  x  0      y

15 8 x 2 5

m PR 

15 2

K1

K1 N1 10

[See overleaf] 3472/2

CONFIDENTAL 9 (a)

Uniform scale All bars are correctly drawn Mode = 46.5

(b)

18

3472/2 K1 K1 N1

K1

K1

K1 23.98 (c)

40.5

N1 P1

K1

45.12

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N1 10

19 10 (a)

3472/2 K1

CT = 18 cm (b)

N1 K1

= 71.57 = 1.249 rad

N1

(c) = 5.791 cm² (d)

K1 N1 K1

= 7.494 cm K1 = 18.9737 cm BT = 18.9737 – 6 = 12.9737 cm

K1

38.47 cm

N1

10

[See overleaf] 3472/2

CONFIDENTAL

20

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11 (a)

K1 K1 K1

N1 N1

(b)

K1

 



K1 K1

K1

 cm³

N1 10 

12 (a)

K1 N1



(b)

K1, P1 N1



(d)

K1 N1



(c)

K1





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K1 N1 10

21 13 (a)



 



 

(b)

3472/2 K1 N1



K1



K1



N1 

(c)

(d)

K1 N1



K1



K1

= 19.5957 cm²

= 14.6968 cm²

14 (a)(i)

N1 10 K1 N1

(ii)

K1, K1 N1

(b)(i)

K1, K1 N1

(ii)

K1 N1 10

[See overleaf] 3472/2

CONFIDENTAL

22

15 (a)

3472/2 K1 N1

(b)

K1 N1

(c)

K1, K1 = 105.5

N1

(d)(i)

N1 K1

(ii)

= 137.15

N1 10

END OF MARKING SCHEME

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