Jurongville Prelim 2009 Am p2

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JVS/Sep 09/4038/2 [Turn over

Jurongville Secondary SchoolPreliminary Year Examination

2009

ADDITIONAL MATHEMATICS (4038/2)

SECONDARY 4EXPRESS/5NORMAL

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 notuse 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.


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JVS/SEP 09/4038/2 [Turn over

Mathematical Formulae

1. ALGEBRA

Quadratic Equation

For the equation ,02 cbxax

a

acbbx

2

42

Binomial expansion

,......21

)(221 nrrnnnnn bba

r

nba

nba

naba

where n is a positive integer and!

)1(...)1(

)!(!

!

r

rnnn

rnr

n

r

n

2. TRIGONOMETRY

Identities

AAec

AA

AA

22

22

22

cot1cos

tan1sec

1cossin

BA

BABA

BABABA

BABABA

tantan1

tantan)tan(

sinsincoscos)cos(

sincoscossin)sin(

AAA

AAAAA

AAA

2

2222

tan1tan22tan

sin211cos2sincos2cos

cossin22sin

)(2

1sin)(

2

1sin2coscos

)(2

1cos)(

2

1cos2coscos

)(2

1sin)(

2

1cos2sinsin

)(2

1cos)(

2

1sin2sinsin

BABABA

BABABA

BABABA

BABABA

Formulae for ABC

Cab

Abccba

C

c

B

b

A

a

sin2

1

cos2

sinsinsin222


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1 (i) The equation 0352 2 xx has roots and . Find a quadratic equation with

integer coefficients whose roots are2

1

and

2

1

. [4]

(ii) Given that is one of the roots of the equation ,0522 xx show that

.1093 [2]

2 The mass, m grams, of a radioactive substances, present at time t days after first being

measured, is given by the formula tem 008.0100 .

(i) Find the mass of the substance when t= 80. [1]

(ii) Find the value oftwhen the initial mass of the substance has been reduced by 20%.

[2]

(iii) Find the rate at which the mass is decreasing when t= 150. [2]

(iv) Sketch the graph ofm against t. [2]

3 (i) If 4log54log 4 yy , find the values ofy. [3]

(ii) Solve the equation )21lg(2

1

2

155lg xx . [4]

4 (i) Given that cbxay cos , for 0x360 and where a, b and c are positive integers.

Ify has an amplitude of 2 with period 120,

(a) state the value ofa and ofb. [2]

Given further that the minimum value ofy is -1,

(b) state the value ofc. [1]

(ii) Sketch the graph of 12cos2 xy for 0x. On the same axes, sketch an

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

xx 212cos2 for 0x. State the number of solutions for the equation in the

given range. [4]


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5 (i) Prove the identity

sin

2cos21

3sin

. [4]

(ii) Solve, for 0


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8

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) 2QBPBAB . [2]

Given further that AB = PB, find the value ofAQPofarea

circleofarea, leaving in your answer in

terms of. [4]

9 A particle moving in a straight line passes a fixed point O with a velocity of 10 m/s. Itsacceleration, 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 oftwhen the particle is momentarily at rest. [2]

(iii) the distance travelled in the first six seconds. [3]

R PQ

B

A


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10 (i) Given that xeyx 6cos3 , find the value of

dx

dywhenx = 2. [2]

(ii)

The diagram shows part of the curve1

10

xy . The tangent to the curve at P(1, 5)

meets they-axis at Q and the x-axis atR. The lineRS is parallel to they-axis.

(a) Find the coordinates ofQ andR. [4]

(b) Find the area of the shaded region. [3]

11 (i) The gradient function of a curve is given as2

x

ba

dx

dy , where a and b are constants.

Gradient of the tangent atR(1, 3) on the curve is 5, and )2,3

2(S is a stationary point

on the curve. Find

(a) the value ofa and ofb, [3]

(b) the equation of the curve. [3]

(ii) Given that ]16ln[2xxy , find

dxdy and express it in the form

216 x

k

, where

kis a constant to be determined.

Ifx decreases at a constant rate of 2 units per second, find the rate of change ofy

whenx = 3. [5]

S


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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) ExpressAD and CD in terms of. Hence show that the perimeter ofABCD, P

cm, is given by cos3sin1111 P . [4]

(b) Express P in the form )sin(Rk and hence find the value of for

which P = 20. [3]

(c) Find the maximum value ofP and its corresponding value of. [2]

(ii) Find the maximum and minimum values of .cos2sin57

1

[3]

END OF PAPER


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Answer

1(i) 9x2

13x + 4 = 0

2(i) 52.7 g (ii) 27.9 days (iii) -0.241 g/day

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) 17)4()4( 22 yx , Q and R are outside the circle

(ii) 04824444 22 yxyx

7(i) -3, -10, (2x 1)(x 2)(x + 1) (ii) - x

82

9(i)(a) 18 m/s (b) 5 (c)3173 m

10(i) 0.00171 (ii)(a) Q(0, 7.5) , R(3, 0) (b) 2.61 units2

11(i)(a) a = 9, b = -4 (b) 104

9 x

xy (ii) k = 1, -0.4 units per second

12 (i)(b)

8.36,3.15,130 R (c) Max P = 22.4 cm,

7.74

(ii) 0.25, 01

Jurongville Prelim 2009 Am p2

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