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RAFFLES INSTITUTION 2013 Year 6 Preliminary Examination Higher 2

MATHEMATICS

9740/01

Paper 1

17 September 2013

3 hours Additional materials:

Answer Paper List of Formulae (MF15)

READ THESE INSTRUCTIONS FIRST Write your name and CT group on all the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use a soft pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer all the questions. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. You are expected to use a graphic calculator. Unsupported answers from a graphic calculator are allowed unless a question specifically states otherwise. Where unsupported answers from a graphic calculator are not allowed in a question, you are required to present the mathematical steps using mathematical notations and not calculator commands. You are reminded of the need for clear presentation in your answers. The number of marks is given in brackets [ ] at the end of each question or part question. At the end of the test, fasten all your work securely together.

This document consists of 4 printed pages. RAFFLES INSTITUTION

RI 2013

Math Department

[Turn over

2 1

By considering the expansion of

1 1 x2

, or otherwise, show that

sin 1 x x

x3 ax 5 ... 6

where a is a constant to be determined.

2

[5]

In the triangle PQR, angle PQR radians, angle PRQ radians and 6 6 QR = 3. Given that is sufficiently small, show that

PQ PR a sin sin b , 6 6 for constants a and b to be determined.

3

4

[5]

A sequence of positive integers u1 , u 2 , u 3 , is defined by u1 9 and u n 1 u n 2 n 3 for n 1 . (i)

Find u 2 , u3 and u 4 .

[1]

(ii)

By considering the value of u n 5 , make a conjecture for a formula for u n in terms of n. Prove your conjecture by induction. [5]

It is given that x and y satisfy the equation

y2 y 4 ln x 4 6 x 2 , y 0. 4

5

dy 2 xy ( x 2 3) . dx 2 y4 1

(i)

Show that

(ii)

Hence obtain the possible exact value(s) of

[3] dy when y 2 . dx

[3]

OABC is a trapezium such that OA is parallel to CB, and CB : OA = k : 1 , where k is a positive constant, and k 1 . Given that OA = a, OB = b, and X and Y are the midpoints of OB and AC respectively, find the following vectors in terms of k, a and b (i) OC , [1] [2] (ii) OY . Hence show that XY is parallel to OA.

[2]

It is given that OB and AC intersect at the point D. Find the ratio, in terms of k, between the area of the triangle XYD and the area of the triangle BCD. [2]

H2 MA 9740/ 2013 RI Year 6 Preliminary Examination/01

[Turn over

3

6

2 1 The line l has equation r = 1 1 , R and the plane has equation r . 3 1 (i)

2 1 7 . 1

Find the position vector of the point of intersection, A, of l and .

[3]

It is given that is the acute angle between l and . (ii)

Find the exact value of sin .

[2]

(iii) The point B has coordinates (2, 1, 3). Hence or otherwise, find the shortest distance from B to , giving your answer in the exact form. [2] 7

An entomologist is investigating the change in population size N of a certain species of insects at time t weeks. He suggests that N and t are related by the differential equation dN N kN 2 , dt where k is a positive constant. Show that (do not merely verify) the general solution to the differential equation is 1 , N k Ae t where A is an arbitrary constant.

[4]

Given that initially, there are 250 insects and after a very long time, the insect population is expected to approach a limit of 10, 000. Find the time required for the insect population to reach three times the initial population, giving your answer correct to the nearest number of days. [3]

8

1 Show that sin 4 A sin 2 A sin 2 2 A . 4 n 1 Given that S n r sin 4 2r x . r 0 4

(i)

[2]

(ii)

By using the result in (i), prove that S n sin 2 x

1 n 1

sin 2 2 n 1 x .

4 (iii) Hence give a reason why S n converges and state the sum to infinity.

9

[3] [2]

[Give all answers correct to the nearest dollar.] Mr Tan decides to set up a scholarship fund for worthy students. On 1 January 2013, he places this scholarship fund in a bank investment which guarantees an annual interest rate of 2.5%. This interest is added to the fund at the end of each year. The annual scholarship award of $2000 is first awarded on 1 January 2014. (i) To award the scholarship for year 2014, find the minimum amount of money $k that Mr Tan needs for the fund. If the annual scholarship is to be given out for years 2014 and 2015, show that in addition to $k , Mr Tan will need at least a further $1904, correct to the nearest dollar, for the fund. [3] (ii) Find the minimum amount Mr Tan needs for the scholarship fund if he wants the annual scholarship to be given out for 10 consecutive years. [3] (iii) Find the minimum amount Mr Tan needs for the scholarship fund if he intends to keep the scholarship going long into the future. [2] H2 MA 9740/ 2013 RI Year 6 Preliminary Examination/01

[Turn over

4 10

x5 2 . Using an algebraic method, solve the inequality f x . [3] 2x 3 3 Hence find the exact range of values of x for which 2 (i) f ln x , [3] 3

It is given that f x

1 f x 2

(ii)

11

2 3.

[2]

Sketch, on separate diagrams, the graph of x 6 x 16 2

y 1

(i) k 2 , (ii) k 0 , making clear the main relevant features of each curve.

2

k

0 for [3] [4]

(iii) State the equation of one line of symmetry of the curve in part (i) and describe fully a sequence of two transformations which would transform this curve onto the curve [3] 2 x 2 y 2 50 .

12

(a)

Find the exact value of

(b)

Find

9 3 2x x2

2

x 2 ln x dx .

[3]

1

dx .

Hence, find the exact value of the constant a for which 2 a 9 1 2 x dx. dx 12 0 3 2 x x2

13

The curve C has parametric equations x t 2 8,

y 2 t 2 10 t 16 ,

[7]

where t .

dy in terms of t. Hence find the coordinates of the minimum point on C, and state dx the coordinates of the point A on C whose tangent to the curve at A is a vertical line. [You do not need to show that the stationary point is indeed a minimum point.] [5]

(i)

Find

(ii)

Sketch the curve C.

[1]

It is given that the point P on the curve C has parameter p. (iii) Show that the equation of the tangent at P is py 2 p 5 x 5 p 2 40 .

[2]

(iv) It is given further that the tangent at P passes through the origin. Find the possible exact coordinates of P. [3] The set of points Q in an Argand diagram represents the complex number z that satisfies z t 2 8 i 2t 2 10t 16 , t . By using the results obtained in (iv) or otherwise, find

the range of values of arg z , giving your answers correct to three decimal places.

[3]

END OF PAPER H2 MA 9740/ 2013 RI Year 6 Preliminary Examination/01

[Turn over

View more...
MATHEMATICS

9740/01

Paper 1

17 September 2013

3 hours Additional materials:

Answer Paper List of Formulae (MF15)

READ THESE INSTRUCTIONS FIRST Write your name and CT group on all the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use a soft pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer all the questions. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. You are expected to use a graphic calculator. Unsupported answers from a graphic calculator are allowed unless a question specifically states otherwise. Where unsupported answers from a graphic calculator are not allowed in a question, you are required to present the mathematical steps using mathematical notations and not calculator commands. You are reminded of the need for clear presentation in your answers. The number of marks is given in brackets [ ] at the end of each question or part question. At the end of the test, fasten all your work securely together.

This document consists of 4 printed pages. RAFFLES INSTITUTION

RI 2013

Math Department

[Turn over

2 1

By considering the expansion of

1 1 x2

, or otherwise, show that

sin 1 x x

x3 ax 5 ... 6

where a is a constant to be determined.

2

[5]

In the triangle PQR, angle PQR radians, angle PRQ radians and 6 6 QR = 3. Given that is sufficiently small, show that

PQ PR a sin sin b , 6 6 for constants a and b to be determined.

3

4

[5]

A sequence of positive integers u1 , u 2 , u 3 , is defined by u1 9 and u n 1 u n 2 n 3 for n 1 . (i)

Find u 2 , u3 and u 4 .

[1]

(ii)

By considering the value of u n 5 , make a conjecture for a formula for u n in terms of n. Prove your conjecture by induction. [5]

It is given that x and y satisfy the equation

y2 y 4 ln x 4 6 x 2 , y 0. 4

5

dy 2 xy ( x 2 3) . dx 2 y4 1

(i)

Show that

(ii)

Hence obtain the possible exact value(s) of

[3] dy when y 2 . dx

[3]

OABC is a trapezium such that OA is parallel to CB, and CB : OA = k : 1 , where k is a positive constant, and k 1 . Given that OA = a, OB = b, and X and Y are the midpoints of OB and AC respectively, find the following vectors in terms of k, a and b (i) OC , [1] [2] (ii) OY . Hence show that XY is parallel to OA.

[2]

It is given that OB and AC intersect at the point D. Find the ratio, in terms of k, between the area of the triangle XYD and the area of the triangle BCD. [2]

H2 MA 9740/ 2013 RI Year 6 Preliminary Examination/01

[Turn over

3

6

2 1 The line l has equation r = 1 1 , R and the plane has equation r . 3 1 (i)

2 1 7 . 1

Find the position vector of the point of intersection, A, of l and .

[3]

It is given that is the acute angle between l and . (ii)

Find the exact value of sin .

[2]

(iii) The point B has coordinates (2, 1, 3). Hence or otherwise, find the shortest distance from B to , giving your answer in the exact form. [2] 7

An entomologist is investigating the change in population size N of a certain species of insects at time t weeks. He suggests that N and t are related by the differential equation dN N kN 2 , dt where k is a positive constant. Show that (do not merely verify) the general solution to the differential equation is 1 , N k Ae t where A is an arbitrary constant.

[4]

Given that initially, there are 250 insects and after a very long time, the insect population is expected to approach a limit of 10, 000. Find the time required for the insect population to reach three times the initial population, giving your answer correct to the nearest number of days. [3]

8

1 Show that sin 4 A sin 2 A sin 2 2 A . 4 n 1 Given that S n r sin 4 2r x . r 0 4

(i)

[2]

(ii)

By using the result in (i), prove that S n sin 2 x

1 n 1

sin 2 2 n 1 x .

4 (iii) Hence give a reason why S n converges and state the sum to infinity.

9

[3] [2]

[Give all answers correct to the nearest dollar.] Mr Tan decides to set up a scholarship fund for worthy students. On 1 January 2013, he places this scholarship fund in a bank investment which guarantees an annual interest rate of 2.5%. This interest is added to the fund at the end of each year. The annual scholarship award of $2000 is first awarded on 1 January 2014. (i) To award the scholarship for year 2014, find the minimum amount of money $k that Mr Tan needs for the fund. If the annual scholarship is to be given out for years 2014 and 2015, show that in addition to $k , Mr Tan will need at least a further $1904, correct to the nearest dollar, for the fund. [3] (ii) Find the minimum amount Mr Tan needs for the scholarship fund if he wants the annual scholarship to be given out for 10 consecutive years. [3] (iii) Find the minimum amount Mr Tan needs for the scholarship fund if he intends to keep the scholarship going long into the future. [2] H2 MA 9740/ 2013 RI Year 6 Preliminary Examination/01

[Turn over

4 10

x5 2 . Using an algebraic method, solve the inequality f x . [3] 2x 3 3 Hence find the exact range of values of x for which 2 (i) f ln x , [3] 3

It is given that f x

1 f x 2

(ii)

11

2 3.

[2]

Sketch, on separate diagrams, the graph of x 6 x 16 2

y 1

(i) k 2 , (ii) k 0 , making clear the main relevant features of each curve.

2

k

0 for [3] [4]

(iii) State the equation of one line of symmetry of the curve in part (i) and describe fully a sequence of two transformations which would transform this curve onto the curve [3] 2 x 2 y 2 50 .

12

(a)

Find the exact value of

(b)

Find

9 3 2x x2

2

x 2 ln x dx .

[3]

1

dx .

Hence, find the exact value of the constant a for which 2 a 9 1 2 x dx. dx 12 0 3 2 x x2

13

The curve C has parametric equations x t 2 8,

y 2 t 2 10 t 16 ,

[7]

where t .

dy in terms of t. Hence find the coordinates of the minimum point on C, and state dx the coordinates of the point A on C whose tangent to the curve at A is a vertical line. [You do not need to show that the stationary point is indeed a minimum point.] [5]

(i)

Find

(ii)

Sketch the curve C.

[1]

It is given that the point P on the curve C has parameter p. (iii) Show that the equation of the tangent at P is py 2 p 5 x 5 p 2 40 .

[2]

(iv) It is given further that the tangent at P passes through the origin. Find the possible exact coordinates of P. [3] The set of points Q in an Argand diagram represents the complex number z that satisfies z t 2 8 i 2t 2 10t 16 , t . By using the results obtained in (iv) or otherwise, find

the range of values of arg z , giving your answers correct to three decimal places.

[3]

END OF PAPER H2 MA 9740/ 2013 RI Year 6 Preliminary Examination/01

[Turn over

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