__________________________________________________________________________ This question paper consists of 5 printed pages.

Candidate Name _______________________________________

FUHUA SECONDARY SCHOOL

Secondary Four Express/ Five Normal Academic

Preliminary Examination 2009 Fuhua Secondary Fuhua Secondary Fuhua Secondary Fuhua Secondary Fuhua Secondary Fuhua Secondary Fuhua Secondary Fuhua Secondary Fuhua Secondary Fuhua Secondary Fuhua Secondary Fuhua Secondary Fuhua Secondary Fuhua Secondary Fuhua Secondary Fuhua Secondary Fuhua Secondary Fuhua Secondary Fuhua Secondary Fuhua Secondary Fuhua Secondary

ADDITIONAL MATHEMATICS 4038/1

Additional Materials: Writing paper (6) Graph paper (1)

DATE 18 September 2009

TIME 07 50 - 09 50

DURATION 2 hours

INSTRUCTIONS TO CANDIDATES

Answer all questions.

Write your answers and working on the separate writing papers provided.

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.

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 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.

The use of an electronic calculator is expected, where appropriate.

You are reminded of the need for clear presentation in your answers.

PARENT'S SIGNATURE FOR EXAMINERS USE Highest mark:

Lowest mark:

Setter: Mr Chia Chun Teck

Class

Index Number

80

[Turn Over

2

FSS_4E5N_PrelimAM1_2009

3

FSS_4E5N_PrelimAM1_2009

1 Given that 321

2

can be expressed in the form cba

2

1, find the value of

a, of b and of c. [4]

2 Given that A =

43

52, find A

1 and hence solve the simultaneous equation

1352

xy

0934

yx

[5]

3 Differentiatex

xe21

with respect to x. Hence, evaluate xxex

d2

0

2

1

. [5]

4 The equation of a curve is2

32ln

x

xy . Find the gradient of the curve at the point

where it meets the x-axis. [5]

5 The line 3x + 2y = 10 intersects the curve 3y + 2x = 5xy at the points A and B. Find the

equation of the perpendicular bisector of AB. [6]

6 (a) Given that 2

32

3

xx

x=

123

x

C

x

BAx for all values of x, find the value

of A, of B and of C, given that A, B and C are integers. [4]

(b) Hence, or otherwise, find .d

2

32

3

xxx

x [2]

7 In the diagram, points A, C and D lie on the circumference

of the circle with AD = p units, 30DAB and qADB .

B is a point on AC such that AB : BC = 3 : 1. If AD is the

diameter of the circle, show that

(a) the length of BD = p8

19 units, [4]

(b)

19

57sin60 1q . [2]

q

A B C

D

p

30

[Turn Over

4

FSS_4E5N_PrelimAM1_2009

8 In the expansion of

n

xx

3

12 , the coefficients of the fourth term and the third term

are in the ratio of 4: 9. Find the value of n and hence find the coefficient of x6 in the

expansion of 32 913

1x

xx

n

. [6]

9 PQRS is a rectangular field with PQ = 1200 m and QR = 1000 m. Mr Pythagoras started

walking from P to R. He first walked along PQ with a speed of

2.5 ms1

. At a certain point K, he cut across the field, walking with a speed of 1.5 ms1

,

in a straight line from K to R.

(a) Show that the total time taken by Mr Pythagoras is given by

22 100012003

2

5

2 xxT , where x m is the distance PK. [3]

(b) Find the shortest time that Mr Pythagoras would take and the corresponding

value of x. [5]

(Proof that it is a minimum is not required.)

10 The velocity, v m/s, of a particle moving in a straight line is given by v = 8 + pt + qt 3,

where t is the time in seconds after the particle passes through a fixed point O. Given

that when t = 2, the distance of the particle from O is 26 m and its acceleration is

35 m/s2, calculate

(a) the value of p and of q, [4]

(b) the minimum velocity of the particle, [3]

(c) the distance travelled by the particle in the fourth second. [2]

P Q

R S

K

5

FSS_4E5N_PrelimAM1_2009

11 (a) Given that cos 20 cos 40 = sin , deduce the exact value of the acute angle ,

without the use of the calculator. [3]

(b) Solve, for 0 < x < 360, the equation 2 cos (2x 40) = 3 . [4]

(c) Given that y > 5, find the smallest value of y that satisfies the equation

y

ysin

2cos5 . [3]

12 Answer the whole of this question on a sheet of graph paper

Two variables x and y are known to be related by the equation y = en p

x, where p and n

are constants to be determined. Measured values of x and y are given in the table below.

(a) Draw a straight line graph of ln y against x, using a scale of 1 cm to represent

1 unit on the x-axis and 2 cm to represent 0.2 units on the ln y-axis. [3]

(b) Use your graph to estimate the value of n and of p. [3]

(c) Estimate the value of x when y = 350. [2]

(d) On the same diagram, draw the line representing y3 = e

15 + x, and hence find the

value of x for which xn

x

p5

3e . [2]

End of Paper

x 2 4 6 8 10

y 235 302 399 528 675

"You may be disappointed if you fail, but you are doomed if you don't try."

Beverly Sills

6

FSS_4E5N_PrelimAM1_2009

Solutions to A Maths Prelim 2009 Paper 1

1 a = 2, b = 6, c = 2

2

7

2

7

3

7

5

7

4

1A

x =3

1, y = 1

3 22

12

1

2

1

xexe

dx

d xx

4

2

0

2

1

dxxex

4 Gradient of curve = 14

1

5 Equation of perpendicular bisector of AB: 36y = 24x + 37

6 A = 3, B = 8, C = 1

cxxxxdx

xx

x|1|ln|2|ln83

2

3

2

3 22

3

7 Use Pythagoras theorem for proof.

8 n = 6, coefficient of x6 =

3

18

9 Shortest time taken = 3

11013 s, Corresponding value of x = 450

10(a) p = 1, q = 3

10(b) minimum velocity of particle = 9

77 m/s

10(c) Distance travelled by particle in the fourth second = 4

3135 m

11(a) acute angle = 10

11(b) x = 5, 35, 185 or 215

11(c) Smallest value of y = 6.75

7

FSS_4E5N_PrelimAM1_2009

Q12(a) Best fit line to be drawn.

Q12(b) n 5.2, p 1.14

Q12(c) x 5

Q12(d) Plot the straight line ln y = 53

1x

From graph, x 0.9