Jurongville Prelim 2009 Am P2

Jurongville Secondary School Preliminary Year Examination 2009

ADDITIONAL MATHEMATICS (4038/2) SECONDARY 4 EXPRESS / 5 NORMAL 4 September 2009 (Friday)

TIME

2 h 30 min

ADDITIONAL MATERIALS: Answer paper

INSTRUCTIONS TO CANDIDATES Write your name, index number and class on all the work you hand in. Write in dark blue or black pen on both sides of the paper. Your 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. Write your answers on the separate Answer Paper provided. 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. The use of a scientific calculator is expected, where appropriate. You are reminded of the need for clear presentation in your answers. At the end of the examination, fasten your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question. The total of the marks for this paper is 100.

Setter:

Mrs Neo LY

This question paper consists of 7 printed pages. JVS/Sep 09/4038/2

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2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation ax 2  bx  c  0, x

 b  b 2  4ac 2a

Binomial expansion n  n  n (a  b) n  a n   a n1b   a n 2 b 2  ...   a n r b r  ...  b n , 1  2 r n n(n  1) ... (n  r  1) n! where n is a positive integer and     r!  r  r!(n  r )!

2. TRIGONOMETRY Identities

sin 2 A  cos 2 A  1 sec 2 A  1  tan 2 A cos ec 2 A  1  cot 2 A sin( A  B)  sin A cos B  cos A sin B cos( A  B)  cos A cos B  sin A sin B tan A  tan B 1  tan A tan B sin 2 A  2 sin A cos A

tan( A  B) 

cos 2 A  cos 2 A  sin 2 A  2 cos 2 A  1  1  2 sin 2 A 2 tan A 1  tan 2 A 1 1 sin A  sin B  2 sin ( A  B) cos ( A  B) 2 2 1 1 sin A  sin B  2 cos ( A  B) sin ( A  B) 2 2 1 1 cos A  cos B  2 cos ( A  B) cos ( A  B) 2 2 1 1 cos A  cos B  2 sin ( A  B) sin ( A  B) 2 2 tan 2 A 

Formulae for ABC a b c   sin A sin B sin C a 2  b 2  c 2  2bc cos A 

1 ab sin C 2

JVS/SEP 09/4038/2

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3

1

(i)

The equation 2 x 2  5 x  3  0 has roots  and . Find a quadratic equation with integer coefficients whose roots are

(ii)

1 1 and 2 . 2  

[4]

Given that  is one of the roots of the equation x 2  2 x  5  0, show that

 3  9  10.

2

[2]

The mass, m grams, of a radioactive substances, present at time t days after first being measured, is given by the formula m  100e 0.008t . (i)

Find the mass of the substance when t = 80.

[1]

(ii)

Find the value of t when the initial mass of the substance has been reduced by 20%. [2]

3

4

(iii)

Find the rate at which the mass is decreasing when t = 150.

[2]

(iv)

Sketch the graph of m against t.

[2]

(i)

If log 4 y  4  5 log y 4 , find the values of y.

[3]

(ii)

Solve the equation lg 5  5 x 

(i)

Given that y  a cos bx  c , for 0≤x ≤360 and where a, b and c are positive integers.

1 1  lg(1  2 x ) . 2 2

[4]

If y has an amplitude of 2 with period 120, (a)

state the value of a and of b.

[2]

Given further that the minimum value of y is -1, (b) (ii)

state the value of c.

[1]

Sketch the graph of y  2 cos 2 x  1 for 0≤x≤π.

On the same axes, sketch an

additional graph required to find the number of solutions to the equation

 2 cos 2 x  1  2 x for 0≤x≤π. State the number of solutions for the equation in the given range.

[4]

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4 5

6

sin 3  sin  . 1  2 cos 2

(i)

Prove the identity

(ii)

Solve, for 0 <  < 360 , the equation 12 sin 2   8 cos 2   5cosec 2 .

[4] [4]

PQR is a triangle with vertices P(-4, 4), Q(5, 2) and R(4, 6). (i)

Find the equation of the circle whose centre is P and radius r = QR. Showing your method clearly, determine whether the points Q and R are inside or outside the circle. [4]

(ii)

Show that PQR is a right-angled triangle. Hence or otherwise, find the equation of the circle which passes through the points P, Q and R.

7

(i)

[4]

The expression 2 x 3  3 x 2  ax  2 is divisible by 2x – 1 but leaves a remainder 2b when divided by x + 2. Find the value of a and of b. Hence factorise the expression completely.

(ii)

[4]

Given that Ax 3  12 x 2  2 x  5  ( 2 x  1)(2 x  1)( x  B )  3 x  C

for all values of x, determine the values of A, B and C. Hence, state the remainder when Ax 3  12 x 2  2 x  5 is divided by 4x2 – 1.

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[4]

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5

8 A

B

R

Q

P

In the diagram, PR is a tangent to the circle at Q. AQ is the diameter and the chord AB produced meets the tangent at P. Show that, stating your reasons clearly, (i)

AQB = QPB,

[2]

(ii)

AB  PB  QB 2 .

[2]

Given further that AB = PB, find the value of

area of circle , leaving in your answer in area of AQP

terms of .

9

[4]

A particle moving in a straight line passes a fixed point O with a velocity of 10 m/s. Its acceleration, a m/s2 is given by a = 8 – 4t, where t is the time in seconds after passing O. Find (i)

the maximum speed attained by the particle in the original direction of the motion. [4]

(ii)

the value of t when the particle is momentarily at rest.

[2]

(iii)

the distance travelled in the first six seconds.

[3]

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10

(i)

Given that y  e 3 x cos 6 x , find the value of

dy when x = 2. dx

[2]

(ii)

S

The diagram shows part of the curve y 

10 . The tangent to the curve at P(1, 5) x 1

meets the y-axis at Q and the x-axis at R. The line RS is parallel to the y-axis.

11

(i)

(a)

Find the coordinates of Q and R.

[4]

(b)

Find the area of the shaded region.

[3]

The gradient function of a curve is given as

dy b  a  2 , where a and b are constants. dx x

2 Gradient of the tangent at R(1, 3) on the curve is 5, and S ( , 2) is a stationary point 3

on the curve. Find

(ii)

(a)

the value of a and of b,

[3]

(b)

the equation of the curve.

[3]

Given that y  ln[ x  16  x 2 ] , find

dy k and express it in the form , where dx 16  x 2

k is a constant to be determined. If x decreases at a constant rate of 2 units per second, find the rate of change of y when x = 3.

[5]

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12

(i)

The diagram shows a cyclic quadrilateral ABCD with AB = 7 cm, BC = 4 cm and BAD = , where  is a variable and 0° ≤ ≤ 90°. (a)

Express AD and CD in terms of . Hence show that the perimeter of ABCD, P cm, is given by P  11  11 sin   3 cos  .

(b)

(c) (ii)

[4]

Express P in the form k  R sin(   ) and hence find the value of  for which P = 20.

[3]

Find the maximum value of P and its corresponding value of .

[2]

Find the maximum and minimum values of

1 7  5 sin   2 cos 

.

[3]

END OF PAPER JVS/SEP 09/4038/2

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Answer

1(i)

9x2 – 13x + 4 = 0

2(i)

52.7 g

3(i)

0.25 or 1024

(ii)

-0.281

4(i)

a = 2, b = 3, c = 1

(ii)

5 solutions

5(ii)

45, 135, 225, 315

6(i)

( x  4) 2  ( y  4) 2  17 , Q and R are outside the circle

(ii)

4 x 2  4 y 2  4 x  24 y  48  0

7(i)

-3, -10, (2x – 1)(x – 2)(x + 1)

8

 2

(ii)

27.9 days

(ii)

(b)

10(i)

(ii)(a) Q(0, 7.5) , R(3, 0)

11(i)(a) a = 9, b = -4

12 (i)(b)

(b)

(c)

y  9x 

-0.241 g/day

-x

1 73 m 3

9(i)(a) 18 m/s

0.00171

5

(iii)

(b)

2.61 units2

4  10 (ii) k = 1, -0.4 units per second x

R  130 ,   15.3 ,   36.8

(c)

Max P = 22.4 cm,   74.7 

(ii) 0.25, 01

JVS/SEP 09/4038/2

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