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2010 PRELIM EXAMINATION 3 CHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHCHSCHSCHSCHSCHSCHSCHSCHSCHS

Subject : Additional Mathematics Paper (4038/1) Level : Secondary 4 Express Date : 16 September 2010 Duration : 2 hours

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 80.

This question paper consists of 5 printed pages [including this cover page]

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Mathematical Formulae

1. ALGEBRA

Quadratic Equation For the equation 2 0ax bx c+ + = ,

2 42

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

+ = + + + + + +

K K ,

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

n n n n n r

r r n r r

+= =

K

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 = m

tan tantan( )

1 tan tanA BA B

A B =

m

sin 2 2sin cosA A A= 2 2 2 2cos 2 cos sin 2cos 1 1 2sinA A A A A= = =

22 tan

tan 21 tan

AAA

=

( ) ( )1 1sin sin 2sin cos2 2

A B A B A B+ = +

( ) ( )1 1sin sin 2cos sin2 2

A B A B A B = +

( ) ( )1 1cos cos 2cos cos2 2

A B A B A B+ = +

( ) ( )1 1cos cos 2sin sin2 2

A B A B A B = +

Formulae for ABC

sin sin sina b c

A B C= =

2 2 2 2 cosa b c bc A= + 1

sin2

ABC ab C =

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1 The line 2 6x y+ = meets the curve 2 22 4 28x xy y+ + = at the points P and Q. Find the coordinates of the points P and Q. [5]

2 Find the inverse of the matrix 4 13 4

and use it to solve the simultaneous equations

4 7 03 4 3 0y xx y+ =

+ + =. [5]

3 (a) Solve the equation ( ) ( )2 2log 4 5 log 5 1 2x x+ = + . [3] (b) Given the simultaneous equations 2 4xy = and 2log log 2 5xxy + = ,

show that 142x = . [4]

4 Given that 1lny x xx

= for x > 0,

(a) find dydx

, [2]

(b) hence, show that y is an increasing function of x. [3]

5 Solve the simultaneous equations

( )8 (16) 19 3 3 3

y x

yx

=

=

[5]

6

ABC is a triangle in which 2 6AB = + m, 3

ABC pi = radians and the area of

the 3 3 6ABC = + m2. Find, without using a calculator, the length of BC in the

form 3 2a b+ , where a and b are integers. [5]

A

B C

2 6+

3pi

4

7 The function f is defined by ( ) sin( ) ,f x a bx c= + where ba, and c are integers. Given that the amplitude of f is 3, the period of f is pi4 and the minimum value of f is 4 , (a) state the values of a , b and c, [3] (b) sketch the graph of ( )y f x= for 0 2x pi . [2]

8 The equation of the curve is 2xy e= .

(a) Find dydx

. [1]

(b) Find the equation of the tangent to the curve at the point (1, e). [2] (c) The normal to the curve at the point (1, e) intersects the x-axis at the point M.

Find the coordinates of M, leaving your answer in terms of e. [3]

9 The cubic polynomial f(x) is such that the coefficient of 3x is 2 and the roots of the equation f(x) = 0 are 1, 3 and k. Given that f(x) has a remainder of 24 when divided by (x +1), find (a) the value of k, [3] (b) the remainder when f(x) is divided by ).4( x [2]

10 (a) Find ( )cos 2d x xdx

. [2]

(b) Hence, evaluate 0

sin 2x x dxpi

. [4]

11 (a) Find the term independent of x in the expansion of .239

2

+

xx [3]

(b) (i) Find the first three terms of the expansion of 5)32( x in ascending powers of x. [2]

(ii) Hence, find the value of n given that the coefficient of x in the expansion of nxx )21()32( 5 is 432. [3]

12 A curve has the equation 2 14 1xyx

+=

+.

(a) Express dydx

in the form 3(4 1)

kxx +

, where k is a constant. [4]

(b) Hence, find the rate of change of x when x = 2, given that y is changing at a constant rate of 3 units per second. [3]

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

PQRS is a rhombus in which P is (1, 3), R is (6, 4) and Q lies on the x-axis.

Find

(a) the coordinates of Q, [3]

(b) the equation of the line QS, [3]

(c) the coordinates of S, [3]

(d) the area of the rhombus PQRS. [2]

-- END OF PAPER --

xO

y

P(1, 3)

Q

R

S

(6, 4)