INDEX NO: _____________

Anglo-Chinese School (Independent)

PRELIMINARY EXAMINATION 2011

YEAR FOUR EXPRESS MATHEMATICS PAPER 1 4016/01

MONDAY 12 Sep 2011 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. You are expected to use a scientific calculator to evaluate explicit numerical expressions. 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 15 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 Solve 1

12

8113

=

xx .

Answer: ....[2] 2 When an ink bottle is left uncapped, the ink evaporates at a rate of 0.25 nano cubic metres per

second. The bottle contains 30 cubic centimetres of ink. How long will it take for all the ink to evaporate? Give your answer in standard form.

Answer: .. [2]

4

3 Simplify the following:

cbcbaabc

3

13122

4)3()2(

Answer: ....[2] 4 Sarah is organizing a tea party. She wishes to buy exactly the same number of bread rolls,

sausages and cheese. There are 30 bread rolls in a pack. There are 16 sausages in a pack. There are 12 slices of cheese in a pack. What is the minimum number of each pack she must buy?

Answer: ....[2]

5

5 Factorise cacbaba 27144 2 + completely.

Answer: ....[2] 6 (a) Express 3528 as a product of its prime factors in index form.

(b) Hence, find the smallest integer by which 3528 must be multiplied to obtain a perfect cube.

Answer: (a) .. [2]

(b) ...[1]

6

7 Express as a single fraction in its simplest form.

x

xxxx

x 15112

2

++

Answer: ....[3] 8 Given that 5 cm on map A represent 10 km on the ground, calculate

(a) the scale of the map in the form 1: n. (b) the actual area, in km2, of a region R which is represented by 9 cm2 on map A. (c) the area, in cm2, which represents the region R on a second map B, whose scale is

1:50 000.

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

(b) ..[1]

(c) ..[1]

7

9 Solve the inequality 2221553 +

8

11 On the axes in the answer space, sketch the graphs of y as a function of x.

(a) y = x3 (b) x + y - 2 = 0 (c) xy = 1 [3] Answer (a) (b) (c)

12 Given that y varies inversely as the square root of x and 2=y for a particular value of x , find, the value of y when this value of x is doubled.

Answer: .....[3] 13 AB, BC and CD are adjacent sides of a regular polygon. Given that = 18CAB , calculate (a) the exterior angle of the polygon,

(b) the number of sides of the polygon,

(c) ACD .

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

(b) ..[1]

(c) ..[1]

0 x

y

(1,1) 0 x

y

(1,1)

0 x

y

(1,1)

18 A B

C

D

9

14 There are 9 identical cards in a box, each numbered 1, 2, 3, 4, 5, 6, 7, 8 and 9 respectively. Two cards are drawn out at random, one after another, without replacement,

(a) Complete the tree diagram below. [2]

(b) Calculate the probability that

(i) the first card is odd, (ii) both cards are odd, (iii) the product of the two card is even.

Answer: (b) (i).. [1]

(ii)..... [1]

(iii).... [2]

Odd

Even

Odd

Odd

Even

Even

94

95

10

15 (a) (i) Express 342 += xxy in the form cbxay += 2)( .

(ii) Sketch the graph of 342 += xxy , showing the turning points clearly.

[1]

(b) (i) Sketch the graph of )3)(2( xxy += showing the x and y intercepts clearly.

[1] (ii) Write down the equation of the line of symmetry of )3)(2( xxy +=

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

(b)(ii) ...[1]

0

y

x

0

y

x

11

16 Given that n() = 100, n(A) = 55 and n(B) = 75, find

(a) the greatest )'( BAn , (b) the greatest )'( BAn .

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

(b) ..[2]

17 Soft drinks at a certain stall are served in paper cups in two sizes: regular and large. The cups are geometrically similar and the capacity of the regular and large cups are 270ml and 640ml respectively.

(a) The top of the regular cup has a diameter of 6 cm. Find the radius of the top of the large cup.

(b) The stall owner decided to add in another size which has radius double that of the regular cup but with the same height. Find the volume of the new cup.

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

(b) ..[2]

LargeRegular

12

18 The diagram shows a sector AOB of a circle, centre O, and radius 14 cm, in which angle AOB is

34 radians. AC represents an arc of another circle with centre at D, which is the mid-point of OA.

(a) Find angle ADC in radians. (b) Calculate the area of the shaded region.

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

(b) ..[3]

19 The table shows the number of passengers in a taxi during peak hours.

No. of passengers 0 1 2 3 4

No. of taxis 8 12 6 x 7

(a) Write down an inequality that must be satisfied by x if the mode is 3.

(b) Find the largest possible value of x if the median is 2.

(c) If x =17, find the mean number of passengers per taxi.

(d) Find the standard deviation.

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

(b) ..[2]

(c) ..[1]

(d) ..[2]

A D O C B

13

20

In the diagram, ABCD is a parallelogram. N is on AB such that AN : NB = 2 : 1.

(a) Prove that APN is similar to CPD.

(b) Prove that NDNP52= .

(c) Find the ratio of area of APD : area of PCBN. Answer (a).

.

...... [2] (b).

.

...... [2]

(c) [2]

A N B

C D

P

14

21 The diagram below shows a circle with centre O. The points A, B, C and D lie on the circumference of this circle such that AOB = 130, BCD = 110 and CDE = 81. ADE is a straight line. Find (a) DAB (b) ACD

(c) OBC

(d) ADO

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

(b) ..[2]

(c) ..[2]

(d) ..[1]

A

C

D

E 81

B

130

110 O

15

22 The diagram shows a speed-time graph of a car

Given that the average speed of the car in the 20 seconds is 21 m/s, calculate

(a) the maximum speed V m/s,

(b) the speed of the car at t = 18s

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

(b) ..[2]

(c) On the axes provided, complete the distance-time graph of the car for the period of 20 seconds.

[2]

END OF PAPER

distance travelled in m

4 12 20 time in seconds

Speed (m/s)

Time (s) 0 4 12 20

V

16

Answers

1. 65=x

2. 5102.1 3. 23

1c

4. 8 breads, 15 sausages and 20 cheese 5. )72()72(2 bacbaa 6(a) 223 732 (b) 21

7. )1(

432

++=

xxxx

8(a) 200000:1 (b) 22 36:`9 kmcm (c) 22 36:144 kmcm

9. 437

17

17(a) 4 (b) 1080ml

18(a) radADC38=

(b) 54.1 19(a) 12>x (b) 18 (c) 2.06 (d) 1.33

20(c) 116

21(a) 70 (b) 45 (c) 56 (d) 45 22(a) 30 (b) 7.5