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2010 PRELIM EXAMINATION 3 CHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHSCHCHSCHSCHSCHSCHSCHSCHSCHSCHS

Subject : Additional Mathematics Paper (4038/2)

Level : Secondary 4 Express

Date : 17 September 2010

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 80.

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 na 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 tan

A 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= = =

2

2 tantan 2

1 tan

AA

A=

( ) ( )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 sin

a b c

A B C= =

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

sin2

ABC ab C =

3

1 (a) Express 2

7 4

(2 1)( 1)

x

x x

as partial fractions. [4]

(b) Hence or otherwise, find 2

7 4

(2 1)( 1)

xdx

x x

. [3]

2 (a) Given that is a root of the equation 22 3 4x x= , show that 34 12= . [3]

(b) If and are roots of the equation 22 3 4x x= . Find the quadratic

equation whose roots are 2 and 2 . [5]

3

The diagram shows a closed rectangular box with a square base ABFE. The diagonal

of the front face AC is fixed at 10 cm and CAB = .

(a) Show that the surface area of the box, A cm2, is given by

200 sin 2 100 cos 2 100A = + + . [3]

Given that can vary,

(b) Find the value of for which A = 123 cm2. [6]

(c) State the maximum surface area of the box and the corresponding value of . [3]

4 (a) Find the range of values of k for which the expression 2 8 6kx x k+ + is always positive for all values of x. [4]

(b) Find the value of p and of q for which { }13

4

5 (a) Show that 2cos 2cos3 cos5 4cos cos3 + + . [3] (b) Hence or otherwise, solve the equation cos cos3 cos5 0 + + =

for 0 radians. [5]

6 A particle starts from a fixed point A and moves along a straight line. Its velocity, v ms-1,

is given by 2 8v t t= + + , where t is the time in seconds after leaving A. Find

(a) the time when the particle is instantaneously at rest, [3] (b) the distance travelled during the first 8 seconds of its journey, and [4] (c) the acceleration of the particle when t = 6. [2]

7 (a)

Points A, B and C lie on a circle. CT is a tangent to the circle. ABT is a straight

line. By using similar triangles, prove that 2CT AT BT= . [4]

(b)

Points A, B, Q, C and P lie on a circle. CT is a tangent to the circle. ABT and PQT are straight lines. Given BT = QT, prove that PQ=AB. [3]

T C

A

B

C T

B

A

P

Q

5

8 In an experimental environment, the population of certain insects was observed. After t days, the number of insects N was given by the equation 500 5000 ktN e= + where k is a constant.

(a) Find the initial value of N. [1] (b) Given that after 3 days, the population of the insects decreased to 5000, find

the value of k. [3] (c) After a long period of time, the population of the insects approaches a particular

value A. Find the value of A. [1]

(d) Sketch the graph of 500 5000 ktN e= + . [2] 9

y

xO a b

The diagram shows part of the curve 3 3 8y x= , meeting the x-axis at x = a.

The tangent to the curve at x = 0 meets the curve at x = b. (a) Find the value of a. [1]

(b) Show that b = 24. [5]

(c) Find the total area of the shaded regions. [5]

10 (a)

A circle passing through the origin O and the point P (6, 8) . OP is the diameter of the circle. The circle cuts the x-axis at A. Find

(i) the equation of the circle, [3] (ii) the coordinates of A. [2]

y

A

P (6, 8)

O x

6

(b) Sketch the graph of 3 2 1 3y x= . Indicate the coordinates of the vertex

and the points where the graph meets the axes. [3]

11

The variables x and y are connected by the equation x a

yx b

+=+

, where a and b are

constants. Experimental values of x and y were obtained. The diagram shows the straight line graph, passing through the points (5.5, 6.5) and (12, 26), obtained by plotting (x xy) against y. Find (a) the value of a and of b. [4]

(b) Another straight line x xy y = is drawn onto the same diagram. The point of

intersection of the two straight lines is a solution of the equation x c x a

x d x b

+ +=+ +

.

Find the value of c and the value of d. [2]

12

The figure shows a rectangular field in which AB = 300 m and AD = 1100 m.

A runner ran across the field from A to P at a speed of 4 m/s and then from P to

C at a speed of 5 m/s.

(a) Given that BAP = and that the total time taken for the runner to run

from A to C is T seconds. Show that 75 60sin

220cos

T = +

. [4]

(b) Find the value of for which T is stationary. [4] (c) Determine the nature of T. [2]

-- END OF PAPER --

(5.5, 6.5)

(12, 26)

x xy

y B

A

C

D

P

300

1100