INDEX NO: _____________

Anglo-Chinese School (Independent)

PRELIMINARY EXAMINATION 2012

YEAR FOUR EXPRESS MATHEMATICS PAPER 1 4016/01

TUESDAY 31 July 2012 2 hours Candidates answer on the Question Paper. READ THESE INSTRUCTIONS FIRST Write your index number on all the work you hand in. Write in dark blue or black pen. You may use a pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer all questions. If working is needed for any question it must be shown with the answer. Omission of essential working will result in loss of marks. Calculators should be used where appropriate. If the degree of accuracy is not specified in the question, and if the answer is not exact, give the answer to three significant figures. Give answers in degrees to one decimal place. For , use either your calculator value or 3.142, unless the question requires the answer in terms of . 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 17 printed pages. [Turn over

80

2

Mathematical Formulae

Compound Interest

Total amount = ( ) 100r 1+ nP

Mensuration Curved surface area of a cone = rl

Surface area of a sphere = 4 2r

Volume of a cone = hr 231

Volume of a sphere = 334 r

Area of a triangle = 21

absin C

Arc length = r , where is in radians

Sector area = 221 r , where is in radians

Trigonometry

Cc

Bb

Aa

sin

sin

sin==

a 2 = b 2 + c 2 2bc cos A

Statistics

Mean =

ffx

Standard deviation = 22

f

fx

f

fx

3

Answer ALL the questions.

1 (a) Express 4130 as a percentage, correct to three significant figures.

(b) Express abba

c4)( 2

2

+ as a perfect square.

Answer: (a) .. [1]

(b) .. [1]

2 The diagram shows the graph of y = ax, a > 1. On a separate diagram, sketch the graph of .1 xay =

Answer: [2]

[Turn over

. . (1, a) (0, 1)

y

x O

O

x

y

4

3 The figure below shows a circle with centre O. The lines RS and TOU are parallel. The lines WU and VOS are parallel. Given that ,37=RSV find .VOW

Answer: .VOW ......[2]

4 It is given that = {2, 5, 10, 16, 17, 23, 25}, A = {prime numbers}, B = {odd numbers} and C = {multiples of 5}.

(a) Find ).'( CAn I

(b) List the elements in '.)' ( BCA UI

Answer: (a) ......[1]

(b) ..[2]

O

R

S

T

U

V W

37

5

5 Usain Bolt set the world record of 9.58 seconds in the 100-metre race at the 2009 World Athletic Championships Final.

(a) Express his speed in m/s.

(b) Express his speed in kilometre per micro second in standard form, correct to two decimal

places.

Answer: (a) .. [1]

(b) .. [2]

6 Simplify .2

4

32

51

34

52

ba

ba

+

Answer: ... [3]

[Turn over

6

7 (a) The diagram shows a line L such that the angle between L and the positive direction of the x-axis is 135. Write down the gradient of the line L.

(b) The diagram shows part of the graph of a quadratic function y = f(x). Write down the equation of the graph in the form y = k(x a)(x b).

Answer: (a) .. [1]

(b) .. [2]

8 The points A, B, C and D are consecutive vertices of a regular polygon of fifteen sides. Calculate

(a) ,ABC

(b) .CBD

Answer: (a) .. [2]

(b) .. [1]

1 4

y

x

12 .. .

135

y

x

L

A

B

C

D

7

9 In a friendly shooting match between Y and Z at a rifle range with a single target, the probability

that Y hits the target is 54 and the probability that Z does not hit the target is .

41 Y fires at the

target first, then Z fires at the target. Find the probability that

(a) only one of them hits the target,

(b) neither of them hits the target.

Answer: (a) ......[2]

(b) ..[1]

10 Triangle PQR is right-angled at P and QS is the bisector of angle PQR. PQ = 7 cm and RQ = 25 cm. Determine the value of QS.

Answer: QS =.......[3]

[Turn over

Q P

R

S

25 cm

7 cm x

x

8

11 A bungalow has a garden in the shape of a right-angled isosceles triangle. The owner of the bungalow wants to add a wooden decking in the garden as shown in the diagram.

(a) Find the exact value of AB.

(b) The breadth of the decking is x metres. Find an expression in terms of x for the area of the decking.

Answer: (a) AB =.......[1]

(b) ..[2]

Decking

Side Wall

10 m 10 m

x m

B A

9

12 The diagram shows the distance-time graph for a car which travels a distance of 24 km in a minutes at an average speed of v km/h. The car stops to rest for 15 minutes before it goes a further

24 km at an average speed of v23 km/h. The total time for the journey is 90 minutes. Find

(a) the ratio ,ba

(b) the value of a.

Answer: (a) ......[2]

(b) ..[2]

[Turn over

a b 15

24

48

km

min

10

O .

A

B

C

D

6 cm

18 cm

13 In the diagram, AB is a diameter of the circle, centre O. AD is a tangent to the circle and BD cuts the circle at C.

(a) Show that DAB and DCA are similar. (b) Given that CD = 6 cm and BC = 18 cm, calculate the

length of AD.

Answer: (a) ......[2]

..

(b) ..[2]

11

14 (a) Given that ba21

21 +

= kUW and ),(

21 ba +=UV show that

21 kVW = a.

(b) If 12=a and ,4= VW determine the values of k.

Answer: (a) ......[2]

(b) ..[2]

[Turn over

12

15 Given that

= 7001

G and .10

1

= kH

(a) Find GH. (b) Find .2 HG (c) Hence, or otherwise, write down the matrix for .HG n

Answer: (a) ......[2]

(b) ..[2]

(c) ..[1]

13

16 (a) Construct the triangle PQR in which PQ = 10 cm, PR = 11 cm and .70=PQR [2]

(b) Measure and write down the length of QR. [1] (c) Construct the perpendicular bisector of QR. [1]

(d) Point S lies within PQR. Mark point S on the perpendicular bisector of QR such that

the area of triangle PSR = 16.5 cm2. Indicate the length of the altitude from S to PR. [2]

[Turn over

10 cm P Q

14

17 The following table gives the number of e-books borrowed by 100 people, who have borrowed e-book readers from the National Library under a new scheme.

(a) State the sum of a + b.

(b) Given that the mean number of e-books borrowed per person is 4, show that .8832 =+ ba

(c) Hence, find the value of a.

(d) State the modal number of e-books borrowed.

Answer: (a) .. [1]

(b) .....[2]

..

..

(c) ...[2]

(d) ...[1]

No. of e-books borrowed 3 4 5 6 No. of people 48 a 16 b

15

18 Triangle ABC has vertex A on the x axis, as shown in the diagram. The line AD is perpendicular to BC. C and B are the points (4, 6) and (8, 2) respectively. The equation of AC is 6x 7y + 18 = 0.

(a) State the coordinates of A.

(b) Find the equation of AD. (c) Find the coordinates of D.

Answer: (a) .. [1]

(b) ...[2]

(c).....[3]

[Turn over

A

B

C

D

O

y

x

(4, 6)

(8, 2)

16

19 (a) Given that 3y is inversely proportional to 12 x and that 5=x when y = 4, find the value of 2x when .

32=y

(b) Solve the inequalities .4

32

33

11 xxx +

17

20 (a) Given that ,11

3

3

ttr

+= express t in terms of r.

(b) If 1322 =+ ba and ,3=ab calculate the value of .)( 4ba

(c) Express )5(11

15592

3224

2

xxxx

+ as a single fraction in its simplest form.

Answer: (a) ... [2]

(b) ...[3]

(c).....[3]

END OF PAPER

18

ANSWER

1(a) 73.2% (b)

2

+ bac

2 3(a) 37 4(a) 3 (b) 2, 10, 16, 17, 23

5(a) m/s

4795000

or m/s

47921010

(b) 1.04 104 km/micro s

6 32

51

2ba

7(a) 1 (b) 3(x 1)(x 4) 8(a) 156 (b) 12 9(a) 7/20 (b) 1/20

10 35/4 or 438

or 8.75

11(a) m 210

(b) )10(2 xx m2

12(a) 3/2 (b) 45 min 13(a) Since AB is perpendicular to AD and angle DCA = angle BCA (angle in semicircle)

We have, Angle DAB = 90 = Angle DCA (A)

Angle D is common (A)

It follows that,

Angle DBA = Angle DAC (Angle sum of ) (A)

Hence, by AAA, DAB and DCA are similar

1 a

(1, a2) (1, 1)

. .

O

x

y

19

13(b) By part (a), TS / TQ = TQ / RT

So, TS RT = TQ2

TQ 2 = 6 24

TQ = 12

14 (a)

=VW

a

=2

1 k

14 (b) 1/3 and 5/3

15(a)

= 701 k

GH (b)

= 22

)7(01 k

HG (c)

= nn kHG

)7(01

16 Diagram not drawn to scale 17(a) a + b = 36 (c) 20 (d) 3 19(a) A(3, 0) (b) 2y = x + 3 (c) D(5, 4)

20(a) 3

11

+=

rrt

or

31

11

+=

rrt

865 (b) 49 1/5 < x < 6

(c)

)12(113

2 +x

P Q

R

S

3 cm

9.1 cm

10 cm

11 cm

70