FAIRFIELD METHODIST SCHOOL (SECONDARY) SECONDARY 4 Express / 5 Normal Academic Preliminary Examination MATHEMATICS 4016/01 Paper 1 17 August 2011 2 hours

READ THESE INSTRUCTIONS FIRST Candidates answer on the Question Paper. Write your name, class and index number on all the work you hand in. Write in dark blue or black pen on both sides of the paper. 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 of the marks for this paper is 80.

For Examiners Use

Paper 1

/ 80

Paper 2

/ 100

Overall total

/ 100

This question paper consists of 17 printed pages.

Setters :- Mr Jason Lum and Mr Wilson Ho

Name :____________________________( ) Class : ________

FMS(S) Sec 4 Exp and Sec 5 N(A) Preliminary Examination 2011 2 Mathematics Paper 1

Mathematical Formulae

Compound interest

Total amount = P 1100

nr

Mensuration

Curved surface area of a cone = rl

Surface area of a sphere = 24 r

Volume of a cone = hr 2

3

1

Volume of a sphere = 3

3

4r

Area of a triangle ABC = Cabsin2

1

Arc length = r , where is in radians

Sector area = 2

2

1r , where is in radians

Trigonometry

C

c

B

b

A

a

sinsinsin

Abccba cos2222

Statistics

Mean =

f

fx

Standard deviation =

22

ffx

f

fx

Name :____________________________( ) Class : ________

FMS(S) Sec 4 Exp and Sec 5 N(A) Preliminary Examination 2011 3 Mathematics Paper 1

1 (a) Express 438

5% as a decimal.

(b) Find the fraction which is exactly halfway between 1

11 and

4

11.

Answer (a) ________________________________ [1]

(b) ________________________________ [1]

2 Evaluate 3.93 1.76

5.04 5.22(7.78)

.

Give your answer correct to 4 significant figures.

Answer ___________________________________ [2]

3 The answer to a question is exactly 13.345. David estimates the answer to 3 significant figures.

Calculate the difference between the two answers as a percentage of the exact answer.

Answer ________________________________ % [2]

Name :____________________________( ) Class : ________

FMS(S) Sec 4 Exp and Sec 5 N(A) Preliminary Examination 2011 4 Mathematics Paper 1

4 A piece of rope is cut into three pieces in the ratio 2 : 7 : 11. If the longest piece is 72 cm longer

than the shortest piece, find the length of the original rope.

Answer _________________________________cm [2]

5 Simplify 9

32

11

5 )(222

k

k

k

k . Leave your answer in positive index notation.

Answer ___________________________________ [2]

6 By selling a digital camera for $308.20, a shopkeeper made a loss of 18% of his cost price. How

much must he sell the digital camera for in order to make a profit of 5% of his cost price?

Answer $ _________________________________ [2]

Name :____________________________( ) Class : ________

FMS(S) Sec 4 Exp and Sec 5 N(A) Preliminary Examination 2011 5 Mathematics Paper 1

7 A motorist took x hours to travel from Town A to Town B which are y kilometres apart. On the

return journey, he took one hour less. Write down an expression, in terms of x and y, for his

average speed in metres per second for his entire journey. Leave your answer in its simplest

form.

Answer _________________________________m/s [2]

8 Solve the equation 7

153 2

x

x

.

Answer x = _______________________________ [2]

Name :____________________________( ) Class : ________

FMS(S) Sec 4 Exp and Sec 5 N(A) Preliminary Examination 2011 6 Mathematics Paper 1

9 Solve the inequality 21733 x+ - .

Show your solution on the number line below.

Answer

[2]

10 Written as the product of its prime factors,

36 = 22 32 , 150 = 2532 .

Use these results to find

(a) the highest common factor of 36 and 150,

(b) the smallest positive integer, k, such that 150k is a cube number,

(c) the smallest positive integer, l, such that 36l is both a square number and a cube number.

Answer (a) ________________________________ [1]

(b) k = _____________________________ [1]

(c) l = ______________________________ [1]

Name :____________________________( ) Class : ________

FMS(S) Sec 4 Exp and Sec 5 N(A) Preliminary Examination 2011 7 Mathematics Paper 1

11 Solve the simultaneous equations.

2y = 11 4x

5x + 3y 10 = 0

Answer x = ____________________ y = ________________________ [3]

12 (a) A scale drawing of a rectangular playground is drawn with a representative fraction of 1 : n.

The length of the model is 10 cm and its width is 3 cm. If the area of the actual playground is

192 000 m2, calculate the value of n.

(b) Two similar jugs have volumes of 125 cm3

and 729 cm3

. Find, in its simplest form, the ratio

of the height of the smaller jug to the height of the larger jug.

Answer (a) n = ____________________________ [2]

(b) ________________________________ [1]

Name :____________________________( ) Class : ________

FMS(S) Sec 4 Exp and Sec 5 N(A) Preliminary Examination 2011 8 Mathematics Paper 1

13 (a) Each exterior angle of a regular polygon is 15o

. Calculate the number of sides of the

polygon.

(b) The diagram shows a prism whose cross-section is a regular hexagon. The side of the

hexagon is 6 cm and the length of the prism is 21 cm. Calculate

(i) the area of the hexagon,

(ii) the volume of the prism.

Answer (a) ________________________________ [1]

(b) (i) _________________________ cm2 [2]

(ii) __________________________ cm3 [1]

21

6

Name :____________________________( ) Class : ________

FMS(S) Sec 4 Exp and Sec 5 N(A) Preliminary Examination 2011 9 Mathematics Paper 1

14 The diagram shows a semicircle with centre O of radius 29 cm. S lies on the arc RT such that

RS = 42 cm. TR is produced to Y such that YR : RT = 1 : 2.

(a) Calculate the area of triangle RST.

(b) Express as a fraction in its simplest form

(i) sin RTS,

(ii) cos YRS.

Answer (a) ______________________________cm2 [2]

(b) (i) sin RTS = ____________________ [1]

(ii) cos YRS = ___________________ [1]

15 (a) (i) Factorise 2x2 5x 18.

(ii) Hence, solve 2x2 5x 18 = 0.

(b) Simplify )6(5)23(22 yy .

Answer (a) (i) ______________________________ [2]

(ii) x = ____________ or ____________ [1]

(b) ________________________________ [2]

S

42

R T Y O 29 29

Name :____________________________( ) Class : ________

FMS(S) Sec 4 Exp and Sec 5 N(A) Preliminary Examination 2011 10 Mathematics Paper 1

16 (a) F is directly proportional to the square of g. When g is increased by 150%, find

(i) an expression for the new g,

(ii) the percentage increase in F.

(b) 10 men took 21 days to paint the school hall. How many more men must be employed if

the school hall is to be painted in 6 days?

Answer (a) (i) _____________________________ [1]

(ii) ___________________________% [2]

(b) ________________________________ [2]

Name :____________________________( ) Class : ________

FMS(S) Sec 4 Exp and Sec 5 N(A) Preliminary Examination 2011 11 Mathematics Paper 1

17 The diagram shows the speed time graph of an object moving in a straight line during a period

of 25 seconds. The object travelled 90 metres from t = 12 to t = 16.

(a) Find the maximum speed of the object.

Answer (a) ________________ m/s [2]

(b) Find the speed of the object at t = 6.

Answer (b) _________________ m/s [2]

(c) On the axes below, draw the distance time graph of the object for the entire 25 seconds.

Answer

[2]

Speed (m/s)

25

Time (s)

10

5

0 12 16

Time (s)

Distance (m)

16 25 12 0

50

100

150

200

250

350

300

Name :____________________________( ) Class : ________

FMS(S) Sec 4 Exp and Sec 5 N(A) Preliminary Examination 2011 12 Mathematics Paper 1

18 The coordinates of point A and B are (0, 3) and (15, 8) respectively.

(a) Find the distance between point A and B.

Answer (a) _____________________ units [1]

(b) Find the equation of the line AB.

Answer (b) __________________________ [2]

(c) Calculate the area of the triangle AOB, where O is the origin.

Answer (c)______________________units2 [1]

x

(15, 8)

A

B

y

(0, 3)

O

Name :____________________________( ) Class : ________

FMS(S) Sec 4 Exp and Sec 5 N(A) Preliminary Examination 2011 13 Mathematics Paper 1

19 A magician has two bags of identical balls. Bag A contains 12 balls, 4 of them red, 5 of them

green and the rest are yellow. Bag B contains 9 balls, 7 of them red and the rest are yellow.

The magician picks a ball at random from Bag A and places it in Bag B. After mixing the

balls in Bag B, he picks a ball randomly from it.

(a) Complete the probability tree diagram in the answer space. [2]

Bag A Bag B

Red

Red

Yellow

Red

Green Green

Yellow

Red

Yellow

Yellow

(b) Calculate the probability that

(i) at least one of the balls picked is green,

(ii) both balls picked are of the same colours.

(b) (i) ___________________________ [1]

(ii) ___________________________ [2]

1

4

1

3

1

5

1

10

1

5

Name :____________________________( ) Class : ________

FMS(S) Sec 4 Exp and Sec 5 N(A) Preliminary Examination 2011 14 Mathematics Paper 1

20 (a) Express x2 + 5x + 3 in the form of (x + a)

2 b.

Answer (a) _______________________ [2]

(b) Hence, sketch y = x2 + 5x + 3.

[2]

(c) Write down the coordinates of the minimum point.

Answer (c)________________________ [1]

y

x 0

Name :____________________________( ) Class : ________

FMS(S) Sec 4 Exp and Sec 5 N(A) Preliminary Examination 2011 15 Mathematics Paper 1

21 PQRS is a rhombus. PX is perpendicular to QR and intersects QS at M.

(a) Show that triangles PQM and RQM are congruent.

Answer (a) In triangles PQM and RQM ___________________________________

__________________________________________________________

__________________________________________________________

__________________________________________________________

__________________________________________________________

__________________________________________________________

_________________________________________________________ [2]

(b) Name two other triangles that are congruent.

Answer (b) Triangles ____________ and ____________ [1]

(c) Name two triangles that are similar, but not congruent.

Answer (c) Triangles ____________ and ____________ [1]

(d) Given that angle PSR = 80o , find angle SMR.

Answer (d) Angle SMR = ________________________0 [1]

P Q

R S

M

X

M

Name :____________________________( ) Class : ________

FMS(S) Sec 4 Exp and Sec 5 N(A) Preliminary Examination 2011 16 Mathematics Paper 1

22 The weights, in kilograms, of 20 girls in a class are shown in the stem-and-leaf diagram.

3 5 7 8 9

4 3 5 6 7 9

5 0 1 4 5 5

6 1 2 4 5 7 8

key 4 3 means 43

(a) Write down the modal weight.

(b) Find the mean weight.

(c) A box-and-whisker diagram is drawn to represent the data above.

Find the values of w, of x and of y.

Answer (a) ____________________________ kg [1]

(b) ____________________________ kg [2]

(c) w = ___________________________ [1]

x = ___________________________ [1]

y = ___________________________ [1]

35 68 w x y

Name :____________________________( ) Class : ________

FMS(S) Sec 4 Exp and Sec 5 N(A) Preliminary Examination 2011 17 Mathematics Paper 1

23 The diagram shows the diagonal AC of a square ABCD.

(a) On the space below, construct the square ABCD.

(b) Measure and write down the length of AB.

(c) A tree is to be planted 3 cm from AB and equidistant from point A and B. By constructing

perpendicular bisectors, find and label the position of the tree T.

Answers (a) and (c)

[3]

Answer (b) ____________________________ cm [1]

End of paper

A

C

Name :____________________________( ) Class : ________

FMS(S) Sec 4 Exp and Sec 5 N(A) Preliminary Examination 2011 18 Mathematics Paper 1

Sec 4 & 5 Preliminary Exam 2011

Mathematics Paper 1 Answers

1a

0.43625

15a(i) (2x-9)(x+2)

15a(ii) 4.5 or -2

1b 15b

2 -0.1052 16a(i) 2.5 g

16a(ii) 525%

3 0.34% or 0.337% 16b 25

4 160 cm 17a 35 m/s

17b 7.5 m/s

5

18a 15.8 units

6 $394.64 or 394.65 18b y=1/3 x+3

18c 22.5 units

7

19b(i)

8 1.22 or 19b(ii)

9

20a(i)

10a 6

10b 180 20a(iii) (-2.5, -3.25)

10c 1296

21(b) Triangles SRM and SPM

11 x = 6.5 Triangles SRQ and QPS

y = -7.5

21c Triangles MXQ and MPS

12a

8000

Triangles PQX and RMX

12b 5 9 Triangles QXM and SRM

21d 50

13a 24

13b(i) 93.5 22a 55 kg

13b(ii) 1960 22b 51.55 kg

22c w=44, x=50.5, y= 61.5

14a 840 cm2

14b(i) 23(b) 4.95 to 5.10 cm

14b(ii)

5

22

3

11

k

5

9(2 1)

y

x

45

37

4 14

3 3x

21

29

21

29

24 12 34y y

5

12

23

60

2( 2.5) 3.25x

: