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SINGAPORE CHINESE GIRLS SCHOOL

Preliminary Examination 2010

ADDITIONAL MATHEMATICS 4038/1 PAPER 1 Wednesday 4 August 2010 2 hours Additional materials: Writing Paper

Graph paper Cover Sheet

READ THESE INSTRUCTIONS FIRST Write your Centre number, index number and name 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. 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 an electronic calculator is expected, where appropriate. You are reminded of the need for clear presentation in your answers. At the end of the examination, fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 80.

SCGS Preliminary Examination 2010

The Question Paper consists of 6 printed pages.

[Turn over

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Mathematical Formulae

1. ALGEBRA Quadratic Equation

For the equation , 02 =++ cbxax

aacbbx

242 =

Binomial Theorem

( ) nrrnnnnn bbarn

ban

ban

aba ++

++

+

+=+ KK221

21

where n is a positive integer and !

)1()1()!(!

!r

rnnnrnr

nrn +==

K .

2. TRIGONOMETRY

Identities 1cossin 22 =+ AA AA 22 tan1sec += AA 22 cot1cosec +=

BABABA sincoscossin)sin( = BABABA sinsincoscos)cos( m=

BABABA

tantan1tantan)tan( m

= AAA cossin22sin =

AAAAA 2222 sin211cos2sincos2 cos ===

AAA 2tan1

tan22tan =

)(21cos)(

21sin2sinsin BABABA +=+

)(21sin)(

21 cos2sinsin BABABA +=

)(21cos)(

21 cos2 coscos BABABA +=+

)(21sin)(

21sin2 coscos BABABA +=

Formulae for ABC

Cc

Bb

Aa

sinsinsin==

Abccba cos2222 += Abc sin

21=

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1. Given that , find 1A and hence solve the simultaneous equations

= 9532

A

0123 =+ yx , 0459 =+ yx . [4]

2. Find the greatest prime value of p for which px is always positive for all values of x. [4]

xx 22562 ++

3. (a) Given that 223=p , express p

p 12 2 in the form of 2ba + , where a and b are integers. [3]

(b) Find the exact value of q if 5125125 =q . [3]

4. The spread of a highly contagious virus in a high school can be modelled by the equation

tey 8.010001

5000+= ,

where t is the number of days after the virus is identified in the school and y is the number of pupils who are infected by the virus in the first t days. Find (i) the number of pupils who had the virus when it was first identified, [1]

(ii) the value of t when the total number of pupils infected reached 3744. [3]

5. Solve the equation (a) )32(log112log 9

23 +=+ xx [4]

(b) [4] 01522 4 =+ + xx

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6. (a) Given that the coefficients of 3x and 6x in the expansion of 9

2 3

x

x are p and q

respectively, find the value of qp . [4]

(b) Find the first 3 terms in the expansion of 5

22

+ x , in ascending powers of x. Given that

the coefficient of in the expansion of 2x5

2

22)1(

+ xkx is 12 , find the values of the

constant k. [5]

7. (a) Given that 73

5

9. A vertical redwood tree with height, PQ, stands on horizontal ground at Q. An ecologist wishes to find the height of the tree in metres. From a point A, due East of Q, he finds that the angle of elevation of the top of the tree P is 60 . He then walks to another point B, m 12 due East of A.

P

24 m

A 60

Q B 21 m

(i) Find the value of PQ . [2]

(ii) Show that

= 30

1334cos angle 1BPA . [3]

10. Sketch the graph of 11 += xy for 44 x , indicating on your graph the coordinates of

the vertex and of the points where the graph meets the coordinate axes. [4] Given that the graph of 11 += xy meets the graph of 3+= kxy at . Find the value 2=x of k. [2]

11. A curve has the equation 25

102 += x

xy .

(i) Find an expression for dxdy and the coordinates of the stationary points. [4]

(ii) Determine whether y increases or decreases as x increases between the stationary points. [2]

Hence, (iii) find the rate of change of x when 6=x , given that y is decreasing at a constant rate of

0.2 unit per second. [2]

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12. In the diagram, a solid with a volume of 512 cm3 is made by fixing a circular cylinder on a cuboid. The cuboid has a square base of sides 4x cm and a height x cm. The cylinder has a diameter of 4x cm and a height y cm.

y cm

x cm

4x cm

4x cm

(i) Express y in terms of x and hence show that the total surface area A cm2, of the solid is

given by

x

xA 51232 2 += . [5] Given that x can vary,

(ii) find the value of x for which A has a stationary value. Determine whether this value of A is a minimum or a maximum. [5]

~ End of Paper 1 ~

SCGS Preliminary Examination 2010

SCGS Preliminary Examination 2010

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Paper 1 (Answer Key)

1 1=x , 1=y 8(i) 4=a , 1=c

2 7=p

3(a) 22631

3(b) 127=q

4(i) 5

8(ii)

4=b

4(ii) 0.10=t 9(i) m 312=PQ

5(a) 7 ,2=x

5(b) 4=x

6(a) 3=qp

10

, 21=k

6(b) 2 ,21=k 11(i) 22

2

)25()25(10

+=

xx

dxdy , (5, 1).and (5, 1)

7(a) x = 3.61, 3.93, 6.75 11(ii) y increases

7(b)(i) 43tan =A 11(iii) unit/s 77.6

7(b)(ii) 6563 12(ii) 2=x , minimum

Quadratic Equation