1

2012 PRELIM EXAMINATION 3 CHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHCHSCHSCHSCHSCHSCHSCHSCHSCHS

Subject : Additional Mathematics Paper (4038/2)

Level : Secondary 4 Express

Date : 17 September 2012

Duration : 2 hours 30 minutes

CHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHCHSCHSCHSCHSCHSCHSCHSCHSCHS

NAME : ______________________( ) CLASS : _____________

READ THESE INSTRUCTIONS FIRST

Write your name, register number and class 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 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 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.

This question paper consists of 6 printed pages [including this cover page]

2

Mathematical Formulae

1. ALGEBRA

Quadratic Equation

For the equation 2 0ax bx c ,

2 4

2

b b acx

a

Binomial Theorem

1 2 2( )1 2

n n n n n r r nn n n

a b a a b a b a b br

,

where n is a positive integer and ! ( 1) ( 1)

!( )! !

n n n n n r

r r n r r

2. TRIGONOMETRY

Identities 2 2sin cos 1A A 2 2sec 1 tanA A

2 2cosec 1 cotA A

sin( ) sin cos cos sinA B A B A B

cos( ) cos cos sin sinA B A B A B

tan tantan( )

1 tan tan

A BA B

A B

sin 2 2sin cosA A A 2 2 2 2cos2 cos sin 2cos 1 1 2sinA A A A A

2

2 tantan 2

1 tan

AA

A

1 1

sin sin 2sin cos2 2

A B A B A B

1 1

sin sin 2cos sin2 2

A B A B A B

1 1

cos cos 2cos cos2 2

A B A B A B

1 1

cos cos 2sin sin2 2

A B A B A B

Formulae for ABC

sin sin sin

a b c

A B C

2 2 2 2 cosa b c bc A

1sin

2ABC ab C

3

4

1 The expression 3 22 3 x ax bx , where a and b are constants, has a factor of x +3

and leaves a remainder of 15 when divided by x 2.

(i) Calculate the value of a and of b. [4]

(ii) With these values of a and b, solve the equation 3 22 3 0 x ax bx . [4]

2

In the diagram, the curve 2y x and the line 2 8x y intersect at point P. Find

(i) the coordinates of point P, [3]

(ii) the area of shaded region. [5]

3 (i) Express 10

( 3) (3 2)

x

x x

in partial fractions. [3]

(ii) Hence evaluate 1

0

10d

( 3) (3 2)

xx

x x

, giving your answer correct to 2 decimal

places. [4]

O

y

x

2x + y = 8

y = x2

P

5

4 For 9x , a curve has gradient

3

29

k

x

, where k is a constant. Given that the

gradient of the tangent at the point 1

5, 2

is

1

16, find

(i) the value of k, [2]

(ii) the equation of the curve. [4]

5 (i) Given that the coefficient of 2x in the binomial expansion of

8

22

kx

x

is 112, find the value of the positive constant k. [3]

(ii) Using the value of k found in (i), show that there is no term in 2x in the

expansion of 8

3

21 4 2

kx x

x

. [4]

6 (a) Find the values of m for which the line 2 4y m x m is a tangent to the curve

212 3 0x xy . [4]

(b) Find the range of values of k for which 23 4k x x k is always positive

for all real values of x. [4]

7 (a) Given that

2

1

1 2 2 5 15 9

34

x

y x

y

pp q p q

q

, find the value of x and of y. [4]

(b) Solve the equation 2 2log 5 1 log 1 2x x . [5]

6

8

In the diagram, A, B, C, E and F are points on the circle with centre O.

The line AE and BF are diameters of the circle and the tangent at F meets AE and BC

produced at D.

(i) Show that triangle ABD is similar to triangle CED. [3]

(ii) Show that 2BF BD BC . [4]

(iii) Show that 2AE BD BC . [1]

9 Two positive numbers x and y vary in such a way that 2 2128 16 1 0x x y .

(i) Find the minimum value of x y . [6]

(ii) Show that this value of x y is indeed a minimum. [2]

10 (a) Given that 2 2 14 and that 7 , find the quadratic equation whose

roots are

and

. [3]

(b) The roots of the equation 23 13 2 2 0x k x k are negative and differ

by 2

3. Find the value of each of the roots. [5]

OA

B

C

D

E

F

7

11 Prove the identity cosec4 cot 4 cot 2 . [3]

Hence,

(i) find all the angles between 0 and 180 which satisfy the equation

1

cosec4 cot 43

, [3]

(ii) show that 2d 2

cosec4 cot 4d sin 2

and use it to find 2

1d

sin 2

. [5]

12 The diagram shows two intersecting circles, C1 and C2, with centres P and Q

respectively such that Q lies on C1 and P lies on C2. The point R is (7, 2) and lies on

both circles.

(i) Given that the equation of circle C1 is 2 2 6 4 19 0x y x y , find the

coordinates of P and the radius of C1. [2]

(ii) Find the equation of the perpendicular bisector of PR. [3]

(iii) Show that the coordinates of Q are 3, 3a b c d , where a, b, c and d are

integers. [5]

(iv) Calculate the area of triangle PQR. [2]

-- END OF PAPER

AM P2 (4038/2)

1. (i) 3 , 8a b (ii) 1

3 , & 12

x

O

C1

Q

R (7, 2)

P

x

y

C2

8

2. (i) (2, 4)P (ii) 226 units

3

3. (i) 1 4

3 3 2x x

(ii) 0.93

4. (i) 1

2k (ii)

11

9y

x

5. (i) 1

4k (ii)

2 2 3

2 2

2

28Term in 1 112 4

112 112

0

x x xx

x x

x

6. (a) 6m (b) 4k

7. (a) 1 , 8x y (b) 1

(rejected) or 34

x x

8. (i) To show: ABD is similar to CED

A D B C D E ( Common Angle)

o ( 1 8 0 ) ( a d j a c e n t a n g l e o n a s t r a i g h t li n e )

( o p p o s i t e a n g l e s o f c y c l i c q u a d r i l a t e r al)

BAD BCE

ECD

( a n g l e s u m o f t r i a n g l e ) A B C C E D

Thus, ABD is similar to CED. (AAA Similarity) (Shown)

(ii) 2 ( T a n g e n t - S e c a n t T h e o r e m )F D B D C D

2 2 2

2 2 2

2 2 2 2

2 2

2

2

90 (tangent perpendicular to radius)

(Pythagoras Theorem)

&

BFD

BD BF FD

BF BD FD

BF BD FD FD BD CD

BF BD BD CD

BF BD BD CD

BF BD BC

(iii) Since and are diameters, BF AE AE BF

2 2

2

AE BF

AE BD BC

9. (i) 3

Minimum value of 84

s

(ii)

42

21

2

3 16 0

8 2x

d s

dx

or

52

2

21

84

3 16 0

16 4y

d s

dy

10. (i) 2 2 1 0x x (ii) 1

1 & 23

9

11. (i) 30 , 120 (ii) 1 1

cot 2 or cos 4 cot 42 2

c ec c

12. (i) 3, 2P radius= 4 2 units /5.66 units (ii) 5y x

(iii) 5 2 3 , 2 3Q (iv) 28 3 / 13.9 units