scgs 2010 am prelims p2

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SINGAPORE CHINESE GIRLS SCHOOL Preliminary Examination 2010

ADDITIONAL MATHEMATICS 4038/2 PAPER 2 Thursday 5 August 2010 2 hours Additional materials: Writing 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 100.

SCGS Preliminary Examination 2010

The Question Paper consists of 6 printed pages.

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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. The roots of the quadratic equation 03 are 52 2 = xx and . Find the quadratic equation whose roots are 1

and 1 . [7]

2. (i) Find the values of a and b for which 5x is the solution set of bax . x >+22

[2] (ii) Show that, if , the line 5k 523 =+ yx meets the curve for all real values

of x. [4] kyx =+ 22 23

3. (i) Prove the identity

=+++ cot

2sin2cos12sin2cos1 . [3]

(ii) Hence or otherwise, find all the angles between 0 and 360 which satisfy the equation

=+++ cos4

2sin2cos12sin2cos1 . [4]

4. The cubic polynomial )(xf is such that the coefficient of 3x is 1 and the roots of 0)( =xf are

k , k2 and )1( k , where 0>k . It is given that )(xf has a remainder of 30 when divided by 1x ,

(i) show that , [2] 03032 23 =+ kkk (ii) find the number of real roots of the equation , [4] 03032 23 =+ kkk (iii) find the remainder when is divided by )(xf )12( +x . [2]

5. (a) (i) Express 583

32 +

xxx in partial fractions. [3]

(ii) Hence evaluate +3 2 2 583 26 dxxx x . [3]

(b) A curve has the equation , where . xxy ln)1( 2 = 0>x

(i) Find an expression for dxdy . [2]

(ii) Hence find dxxx ln . [3]


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6. (a) C is a circle which passes through the points P(1, 5), Q(3, 5) and R(3, 1).

(i) Show that the length of PQ is equal to the length of RQ. [1] (ii) Given that angle , find the equation of the circle, C. [3] = 90PQR

(b) In the diagram, O is the centre of the circle and the points A, B, C and E lie on the circle.

DEF is the tangent of the circle at the point E. ABCE is a trapezium where DCB is parallel to EA and . EBEA =

D

E

B C

O

F

A

(i) Find with explanation, an angle equal to angle BAE. [1] (ii) Prove that triangle AEB is similar to triangle CED. [3]

(iii) Prove that DCDBCE =

2

. [3]

7. The diagram shows part of the curve xxy sin2 += . The normal to the curve at the point

2 ,2

P cuts the y-axis at Q.

(i) Find coordinates of Q. [4]

(ii) Find the area of the shaded region. [4]


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8. Solutions to this question by accurate drawing will not be accepted.

Q

S

M

O

P (1, 2)

y

R(5,4)

x

The diagram, which is not drawn to scale, shows a rhombus PQRS in which P is (1, 2), Q is on the x-axis and R is (5, 4). The diagonals of the rhombus meet at M. (i) Find the equation of PR. [2] (ii) Find the coordinates of Q and of S. [4] Given that N is a point on QS produced such that 5:3: =MNQM , (iii) find the coordinates of N, [3] (iv) find the area of the quadrilateral PQRN. [2]

9. The velocity, v ms1, of a particle, P, travelling in a straight line, at time t s after leaving a fixed point, is given by 86 . 2 += ttv (i) Find the values of t when P is instantaneously at rest. [2] (ii) Find the average speed of P during the first 4 seconds. [5] (iii) Show that P will never return to its starting point. [3]


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10. The variables x and y are related by the equation a , where a and b are constants. The table below shows values of x and y.

yx b = 2

x 5 10 15 20 25 y 150 600 1350 2400 3750

(i) On the graph paper, plot against , using a scale of 4 cm to represent 0.5 unit on

the axis and 4 cm to represent 1 unit on the axis. Draw a straight line graph to represent [3]

ylg xlgxlg ylg

ayx b = 2 . (ii) Use your graph to estimate the value of a and of b. [4] (iii) On the same diagram, draw the line representing and hence find the value of x for

which . [3]

4xy =ax b = 24

11. Diagram I and II show two isosceles triangular tiles OXW and OYZ with cm 3== OXOW , cm 5= and the acute angle= OZOY =WOX .

Diagram III shows a pattern formed by using four pieces of each of these two types of tiles

with angle . = 90WOZ

I II III

Z O

WX Y

3 cm

O

W

X 3 cm

5 cm

5 cm Z

Y

O

(i) Show that the area of the pattern, A cm2, can be expressed in the form )cos25sin9( +k

where k is an integer. [3] (ii) Find the value of for which 48=A . [6] (iii) State the maximum value of A and the corresponding value of . [2]

~ End of Paper 2 ~



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Paper 2 (Answer Key)

1 049496 2 =++ xx 8(iii) )8 ,5.0(N

2(i) 7=a , 15=b 8(iv) 2unit 20

3(ii) = 14.5, 90, 165.5, 270 9(i) 4 ,2=t

4(ii) Only One real root. 9(ii) 1ms 2

4(iii) 8134

5(a)(i) 1

153

2 xx

5(a)(ii) 462.0

5(b)(i) x

xxxdxdy 1ln2 +=

5(b)(ii) Cxxxdxxx += 42lnln22

10(i)

6(a) 02622 =++ yxyx 10(ii) 62.3a , 2=b

7(i)

+

42 ,0

2Q 10(iii) 37.2x

7(ii) 2unit 29.3 11(i) )cos25sin9(2 +=A

8(i) 23

2+= xy 11(ii) = 2.45

8(ii) Q(4.5, 0) and S(1.5, 6) 11(iii) A = 53.1, 8.19

Quadratic Equation

scgs 2010 am prelims p2

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