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1

Candidate Name Class:

KRANJI SECONDARY SCHOOL Prel iminary Examination I Secondary 4 Express / 5 Normal Academic

ELEMENTARY MATHEMATICS 4016/1 PAPER 1

Thursday 29 April 2010 2 hours KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI

READ THESE INSTRUCTIONS FIRST

Write your name, class and register number in the spaces at the top of this page.

Write in dark blue or black pen 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 the questions.

The number of marks is given in brackets [ ] at the end of each question or part question.

If working is needed for any question it must be neatly and clearly shown in the space below the

question.

Omission of essential working will result in loss of marks.

The total of the marks for this paper is 80.

You are expected to use an electronic 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 .

Set by : Ms Sim Chin Chin

This question paper consists of 16 printed pages [Turn over

2

Mathematical Formulae

Compound Interest

Total amount =

nr

P

1001

Mensuration

Curved surface area of a cone = lr

Surface area of a sphere = 24 r

Volume of a cone = hr 2

3

1

Volume of a sphere = 3

3

4r

Area of triangle ABC = Cab sin2

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

f

fx

f

fx

3

Answer all the questions

For

Examiner’s

Use

1. Evaluate

(a) 7311213415

21112532

2

...

..

, give your answer correct to 2 significant figures.

(b) (7.21 10 – 24 ) – ( 4.35 10 – 25 ), give your exact answer in

standard form.

For

Examiner’s

Use

Answer: (a)__________________ [1]

(b)__________________ [1]

2. (a) A bell rings every 20 minutes while another bell rings every 35

minutes. The two bells rang together at 0800.

At what time do they next ring together again?

(b) On a typical day in a desert, the difference between the day and

night temperatures is 54C. If the day temperature is 47C, find

the night temperature.

Answer: (a)______________ [2]

(b)_______________ C [1]

3

(a) George spent $2 800 of his salary and saved 5

1 of it every month.

How much is his monthly salary?

(b) x grams of flour costs 40 cents. Find an expression for the number of

grams of flour that can be bought for y dollars.

Answer: (a) $ ___________ [1]

(b) ____________ g [1]

4

For

Examiner’s

Use

4. Sixteen workers can build a house in 25 days.

How many workers are needed if the house is to be built in 10 days?

For

Examiner’s

Use

Answer: ______________workers [1]

5.

p is indirectly proportional to the square of q. It is known that p = 12 for a

particular value of q. When q is increased by 100%, find

(a) the value of p,

(b) the percentage change in the value of p.

Answer: (a) p =_____________ [2]

(b) _______________ % [1]

6. (a) Simplify

3

2

x.

(b) Given that 81327 25 k, find the value of k.

Answer: (a) __ ______________ [1]

(b) k = ______________ [1]

5

For

Examiner’s

Use

7. In the diagram below, 90AOB , AC // OB and OA = 2 cm.

AB is an arc of a circle centre O and BC is an arc of a circle centre A.

Find the area of the shaded region.

For

Examiner’s

Use

Answer: _________________ cm 2 [3]

8. The line with equation 2y = 14ax passes through the point (3, 2) and

crosses the y-axis at the point A.

Find

(i) the coordinates of point A.

(ii) the gradient of the line.

Answer: (i)____ A (____,____)__ [1]

(ii) gradient = _________ [2]

A

B

C

O

2

6

For

Examiner’s

Use

9. The number 525 written as the product of its prime factors, is

753525 2 .

(a) If 525k is a perfect square, find the smallest integer value of k.

(b) Given that 525 is the lowest common multiple of 15, x and 35, find

two possible values of x.

For

Examiner’s

Use

Answer: (a) k =_____________ [1]

(b) x = _____, x=_______ [2]

10. (a) On the Venn diagram shown in the answer space, shade the set

'BA .

Answer (a)

(b) = { x : x is an integer and 1 ≤ x ≤ 20 }

P = { x : x is a multiple of 3}

Q = { x : x is a perfect square}

R = { x : x is a prime number}

Find

(i) QP ,

(ii) n )'( RP

[1]

Answer: (b) (i) QP = _____________ [1]

(ii) n )'( RP = __________ [1]

A B

7

For

Examiner’s

Use

11. The diagram below is the speed-time graph of a car. It travels at a constant

speed of U ms1 for the first 5 s. The car then retards uniformly at 1.5 ms2

from 5s to 15s. The car comes to rest after a further 20 s.

(a) Calculate the value of U in ms-1.

(b) At 15 s, a van passes the car. The van is travelling in the same

direction at a constant speed of 20 ms1. How far ahead is the

van when the car comes to rest?

For

Examiner’s

Use

Answer:(a) U =____________ [2]

(b) _____________ m [1]

20

U

15

Speed (m/s)

Time (s)

5 35 O

Speed (m/s)

8

For

Examiner’s

Use

12. In the diagram below, BCD is a straight line. Angle ABC = 90,

AD = 25 cm, CD = 4 cm and the area of triangle ACD = 14 cm².

Find, without evaluating any angle,

(a) the length of AB, and

(b) the length of BC.

For

Examiner’s

Use

Answer: (a) AB = ________ cm [1]

(b) BC = ________ cm [2]

13. (a) A regular polygon has interior angles of 1600.

Find the number of sides of the polygon.

(b) Four of the interior angles of a 10-sided polygon are each x and the

remaining interior angles are each 140 . Calculate the value of x.

Answer: (a) __________ sides [2]

(b) x = ______________ [2]

A B

C

D

25cm

4 cm

9

For

Examiner’s

Use

14. (a) Factorise completely 22 4123 yxyxyx .

(b) Factorise completely pp 502 3

For

Examiner’s

Use

Answer: (a)__________________ [2]

(b)_________________ [2]

15.

Two similar cups have surface areas of 60 cm2 and 135 cm2.

(a) Find, in its simplest integer form, the ratio of the height of the

smaller cup to the height of the larger cup.

(b) The capacity of the larger cup is 0.8 litres. Find the capacity of the

smaller cup. Give your answer in cubic centimetres.

Answer: (a) ______ : ______ [1]

(b)_____________ cm 3 [2]

10

For

Examiner’s

Use

16 In a test, each pupil of a group scores 5, 10 or 15 marks. The number of

pupils scoring each mark is shown in the table below.

Marks 5 10 15

No.of pupils 8 12 x

(i) If the mode is 10, write down the range of values of x.

(ii) If the median mark is 10, write down the largest possible

value of x.

For

Examiner’s

Use

Answer: (i) _________________ [1]

(ii) x = _____________ [1]

For

Examiner’s

Use

17 (a) A class of students took a test. Their marks are shown in the stem-and

leaf diagram.

(i) Write down the modal mark. (i

(ii) Find the median mark.

(b) The box-and-whisker diagram below shows the height of a number of

trees in a park. Find the interquartile range.

Answer: (a) (i) _____________

(ii)_____________

Answer

(b)__________ cm

[1]

[1]

[1]

For

Examiner’s

Use

1 2 4 5

2 0 7 9 9

3 1 1 2 5 5 5

4 0 0 1 2 2 7 8

key 3|1 means 31

100 120 110 130 140 150

Height in cm

11

For

Examiner’s

Use

18 In the figure below, ABCD and APQR are squares.

(a) Show that RABPAD

(b) Prove that triangles APD and ARB are congruent.

State the case for congruency

Answer: (a)

Answer: (b)

[2]

[2]

For

Examiner’s

Use

Q

R

A D

B C

P

12

For

Examiner’s

Use

19 (i) Solve the inequality 5(2p+1) < 67 .

(ii) Hence, write down the largest prime value of p which satisfies

the inequality 5(2p + 1) < 67

For

Examiner’s

Use

Answer: (i)_______________ [1]

(ii) p = ___________ [1]

20 Solve the simultaneous equations

2x + 7y = 11

3x + 1 = 7y

Answer: x = _____, y = ______ [3]

13

For

Examiner’s

Use

21 In the quadrilateral BXDC shown below, ABC = 94, BCD = 86,

CDX = a. E is a point on XD such that BAE = 94 and AED = a.

(i) Give a reason why AB is parallel to DC.

(ii) Calculate the value of a.

(iii) Show that AXE is an isosceles triangle.

For

Examiner’s

Use

Answer: (i) Reason: __________________________________________ [1]

(ii) a = _____________ [2]

(iii) ________________________________________________

________________________________________________ [2]

94

94

86

a a

A

B

C

D E X

Diagram NOT

drawn to scale

14

For

Examiner’s

Use

22 In the diagram, OA 3a and OB b. M is the point on AB such that

AM = 4

3AB. The lines OA and BC are parallel and OABC 43 .

(a) Find, in the form pa + qb,

(i)

AB ,

(ii)

OM ,

(iii)

AC .

(b) Use your answers to parts (a)(ii) and (a)(iii) to explain why OM is

parallel to AC.

Answer (b)

___________________________________________________________

___________________________________________________________

Answer:

(a)(i)

AB = _____________

(ii)

OM = ____________

(iii)

AC = ____________

[1]

[2]

[1]

[1]

For

Examiner’s

Use

3a O

A

B C

b M

15

For

Examiner’s

Use

23 (a) (i) Express 822 xx in the form bax 2

.

(ii) Sketch the graph of 822 xxy in the answer space below.

For

Examiner’s

Use

Answer: (a)(i)_________________ [2]

(a) (ii)

[3]

(b) Sketch the graph of 23 xxy , showing clearly all the axes

intercepts and turning points on your sketch.

Answer : (b)

[3]

y

x

x

y

16

For

Examiner’s

Use

24 A playground is in the shape of a triangle ABC.

Construct the model of the playground ABC such that AC = 12 cm

and BC = 5 cm using the line AB constructed for you in the answer

space below.

(a) Measure CAB

(b) In the triangle ABC, construct using only compasses and ruler

(i) the bisector of angle ABC.

(ii) the perpendicular bisector of the line AB.

(c) These two lines will intersect at a point P where a fountain

is to be constructed.

(i) Mark point P clearly.

(ii) Measure and write down the length of AP.

[1]

[2]

[1]

For

Examiner’s

Use

Answer:

(a) CAB = __________ [1]

(c) (ii) AP = ________ cm [1]

End of Paper

A B

17

Kranji Secondary School

Elementary Mathematics 2010

Sec 4 Express Preliminary Exam Paper 1 (Answer Key)

1) (a) 1.1

(b) 6.775 x 10-24

13) (a) 18

(b) 150

2) (a) 1020

(b) -7

14) (a) (3x-y)(x-4y)

(b) 2p(p+5)(p-5)

3) (a) 3500

(b) xy2

5

15) (a) 2:3

(b) 27

1237

4) 40

16) (i) 0 ≤ x < 12

(ii) 19

5) (a) 3

(b) 75

17) (a) (i) 35

(a) (ii) 33.5

(b) 35

6) (a)

x

8

(b) 7

18) (a)

Let PAD = x°

DAR = 90° – x°

RAB = 90° – (90° – x°)

= 90 – 90 + x

= x°

PAD = RAB (Shown)

(b)

AB = AD

PAD = RAB

AR = AP

∆APD ≡ ∆ARB (SAS)

7) 2 19) (a) p < 6.2

(b) 5

8) (i) A(0, -7)

(ii) 3

20) x = 2, y = 1

18

9) (a) 21

(b) x = 25, x = 75 OR 175

21) (i) ABC + BCD = 94° + 86°

= 180° (int s, // lines)

AB is parallel to DC

(ii) a =133

(iii)

AEX = 180° – 133° ( s on a

str line)

= 47°

AXE = 94° – 47° (ext of a ∆)

= 47°

Since AEX = AXE = 47°

∆AXE is an isos ∆

10) (i) {1,3,4,6,9,12, 15, 16, 183}

(ii) 5

22) (a) (i) 3a + b

(a) (ii) ba4

3

4

3

(a) (iii) a + b

(b)

)(4

3baOM , baAC

ACOM4

3

OM is parallel to AC

11) (a) 35

(b) 50

23) (a) (i) (x – 1)2 – 9

12) (a) 7

(b) 20

24) (a) 82° (± 1°)

(c) (ii)7.8cm (± 1cm)

3

1. a) i) ABCD is a rectangle with AB = 12 cm.

If the ratio of AB : BC = 2 : 5, find the area of ABCD. [2]

ii)

In the triangle ABC, AB = 7x cm, AX = 4 cm, CX = 12 cm and

90ABC . BX is the perpendicular from B to AC. Find the

value of x. [2]

b) i) Solve for x,

1

1

1

21

2

xx. [2]

ii) Make z the subject of the formula

bz

azba

2

22

. [4]

2. Adeline, Farah and Natalie were each left $120000 in their uncle’s will.

a) Adeline invested her money in a financial institution at 5% simple interest

per annum for 10 years. How much money did Adeline have at the end of

10 years? [2]

b) Farah invested her money in a financial institution which paid compound

interest of 4% per annum, compounded half yearly, for 10 years. How

much money did Farah have at the end of 10 years? [2]

c) Natalie spent all her money on a sports car.

At the end of the first year the value of the car had fallen by 20%.

At the end of the second year the value of the car had fallen further by

15% of its value at the end of the first year.

i) What was the value of Natalie’s car at the end of the second year?

[3]

ii) Natalie sold her car at the end of the second year for $80000.

Compared to the value of the car at the end of the second year,

did Natalie make a profit or a loss?

Calculate the percentage profit/loss made by Natalie. [3]

A 7x cm

4 cm

12 cm

B

C

X

4

3. i) Copy and complete the number pattern:

Row Pattern Sum, S S + 2

1 2 2 4

2 2, 3, 2 7 9

3 2, 3, 4, 3, 2

4 2, 3, 4, 5, 4, 3, 2

[1]

ii) What sort of numbers do you have in the S + 2 column? [1]

iii) Find the sum of 2, 3, 4, 5, 6, 7, 8, 9, 8, 7, 6, 5, 4, 3, 2. [1]

iv) Find the sum, S, in terms of N for Row N. [1]

v) Find the sum of

– 2, – 3, – 4,…, – 66, – 67, – 66,…, – 4, – 3, – 2. [1]

vi) If 2 + 3 + 4 + 5 + … + (k – 1) + k + (k – 1) + … + 5 + 4 + 3 + 2 = 223,

find the value of k. [2]

4. AD is a diameter of the circle and B, C and E are points on the circumference.

The tangents to the circle at points B and D meet at T. O is the centre of the circle.

65EAD , 122DOB and the circle has a radius of 5 cm.

a) State briefly a reason why 90OBT . [1]

b) Find

i) EDT , [2]

ii) DCB , [2]

iii) the length of OT. [2]

122

A

B

C

D

E

T

O

65

5

5. A company runs an office, a shop and a warehouse. The type and number of

lights used in the three places are summarized in the following table.

Type of light Number of lights Power Consumption of

Each Type of Light Office Shop Warehouse

Table lamp 4 3 2 11 watts

Light bulb 5 0 15 75 watts

Fluorescent tube 8 11 5 40 watts

a) Given that matrix

40

75

11

B , write down matrix A such that the product of

AB will give the power consumption of the lighting in each place. [1]

b) Evaluate AB. [2]

c) The operating hours of the office, shop and warehouse are 8, 12 and 6

hours respectively. Express the operating hours by C as a 1 3 matrix. [1]

d) Evaluate the product of AB and C. [2]

e) What does your answer in part (d) represent? [1]

6

6. Six hundred anglers took part in a fishing competition. The lengths of time taken

by each angler to catch a fish is shown in the cumulative frequency curve below.

0

50

100

150

200

250

300

350

400

450

500

550

600

650

700

0 10 20 30 40 50 60 70 80

Time (t minutes)

Cum.

Freq.

a) Use the graph to estimate

i) the number of anglers who took more than 1 hour to catch a fish, [1]

ii) the median of the distribution, [1]

iii) the lower quartile of the distribution. [1]

b) Copy and complete the grouped frequency table of the lengths of time

taken by each angler to catch a fish. [2]

Time (t minutes) 3015 t 4530 t 6045 t 7560 t

Number of anglers

c) Using your grouped frequency table, calculate an estimate of

i) the mean time taken to catch a fish, [2]

ii) the standard deviation. [2]

d) An angler is chosen at random from the group. Another angler is chosen

at random from those remaining. Given that both anglers belong to the

same class interval, find the probability that the total time taken by both

anglers is more than 2 hours 10 minutes. [2]

7

7. A man drove from town A to town B and then returned to town A. The distance

between the two towns is 400 km.

a) During the journey from town A to B, the man drove at an average speed

of )2( x km/h. Write down an expression, in terms of x, for the time

taken for the journey. [1]

b) Due to a traffic jam, his return journey from town B to town A was

affected. His average speed was reduced by 18 km/h. Write down an

expression, in terms of x, for the time taken for the return journey. [1]

c) Given that the difference in time for the two journeys was 2 hours, form an

equation in x and show that it reduces to

03632142 xx . [3]

d) Solve the equation 03632142 xx , giving both answers correct to

two decimal places. [3]

e) The man set off at 2125 h on Saturday for the journey from town A to

town B. Find his arrival time at town B, giving your answer correct to the

nearest minute. [2]

f) The car used 0.5 litres of petrol for a 6.6 km journey. How much petrol

was required to complete the two journeys? [1]

g) The petrol costs $1.82 per litre. How much did he spend on petrol for the

two journeys, correct to the nearest 50 cents? [1]

8. In the diagram, ABC is a triangular field with P and Q on AB and AC respectively.

Given that AP = 62 m, QC = 49 m, PQ = 92 m, PB = 79 m and 58BAC .

Calculate, giving all answers correct to three significant figures,

a) AQP , [2]

b) the length of AQ, [3]

c) the area of the quadrilateral PQCB, [3]

d) the length of BC. [2]

Given that AT is a vertical pole of height 12 m standing at A, calculate the angle of

depression of P from T. [2]

A

B C

Q

P

62 m

79 m 49 m

92 m

58

8

9. An inverted cone of base diameter 20 cm has its vertex 24 cm vertically below the

centre of its horizontal top. A hemisphere is removed from the cone. A vertical

cross-section through the vertex V of the cone is as shown.

In the triangle VAB, VA = VB. C and D are points on AB such that AC = DB.

a) Given that the volume of the hemisphere removed is 3

183 cm3, calculate

i) the radius of the hemisphere, [2]

ii) the length of VA, [2]

iii) the volume of this solid, [2]

iv) the total surface area of the solid. [3]

b)

The solid is melted down completely and recast to form a prism whose

cross-section is a trapezium. The lengths of the parallel sides of the

trapezium are 0.4 m and 0.6 m. The height of the prism is 0.3 m.

Calculate the length of the prism. [3]

A B C D

V

20 cm

24 cm

0.6 m

0.4 m

0.3 m

9

10. Answer the whole of this question on a sheet of graph paper. The table below gives some values of x and the corresponding values of y, where

20192 23 xxxy .

x – 4 – 3 – 2 – 1 0 1 2 3 4 5

y 0 – 32 a – 36 – 20 0 18 b 24 0

a) Find the value of a and of b. [1]

b) Using a scale of 2 cm to represent 1 unit, draw a horizontal x-axis for

54 x .

Using a scale of 1 cm to represent 5 units, draw a vertical y-axis for

3050 y .

On your axes, plot the points given in the table and join them with a

smooth curve. [3]

c) Use your graph to find

i) the smallest value of 20192 23 xxx in the interval 54 x .

[1]

ii) the largest value of x for which 030192 23 xxx . [1]

d) From your graph, write down the range of values of x for which the gradient of

the curve is positive. [1]

e) Draw the graph of 1015 xy for 41 x . By drawing suitable

tangents to your curve, find the coordinates of the points at which the

gradient of the tangent is equal to 15 . [3]

f) The x coordinates of the points where the curve 1202 xxy intersects

the curve 20192 23 xxxy are the solutions of the equation

023 cbxaxx . Find the value of a, of b and of c. [2]

10

Answers

1.a.i. 360 cm2

ii. 7

11

1.b.i. x = 0 or x = 2

ii. ba

babaz

1

22

2.a. $180000

b. $178313.69

c.i. $81600

ii. Loss

% loss = %51

491 1.96%

3.i. Row Pattern Sum, S S + 2

1 2 2 4

2 2, 3, 2 7 9

3 2, 3, 4, 3, 2 14 16

4 2, 3, 4, 5, 4, 3, 2 23 25

ii. square numbers

iii. 79

iv. S = (N + 1)2 – 2

v. – 4487

vi. k = 15

4.a. tangent is perpendicular to radius

b.i. 115EDT

ii. 119DCB

iii. OT = 10.3 cm

5.a.

5152

1103

854

A

b.

1347

473

739

AB

c. 6128C

d. 19670

e. Total power consumption of the

company in one day

6.a.i. 30 anglers

ii. 38 minutes

iii. 31 minutes

b. Time

(t minutes) 3015 t

4530 t

6045 t

7560 t

Number of

anglers 130 320 120 30

c.i. mean = 38.75

ii. s.d. = 11.7

d. probability = 11980

29

7.a. 2

400

x h

b. 16

400

x h

c. 22

400

16

400

xx

d. x = 67.67 or x = – 53.67

e. 0309 h on Sunday

f. 60.6 litres

g. $110.50

8.a. 9.34AQP

b. AQ = 108 m

c. Area = 6560 m2

d. BC = 145 m

of depression = 0.11

4

9.a.i. radius = 5 cm

ii. VA = 26 cm

iii. volume = 2250 cm3

iv. total surface area = 1210 cm2

b. length = 1.50 cm

10.a. 42a , 28b

b. smallest value = – 42

c. largest x = 4.66

d. 25.32 x

e. (4, 24), (– 2.8, – 35.5)

f. a = 1, b = 39, c = – 21

1

Candidate Name Class:

KRANJI SECONDARY SCHOOL Prel iminary Examination 2 Secondary 4 Express / 5 Normal Academic

ELEMENTARY MATHEMATICS 4016/1 PAPER 1

Wednesday 15 Septemeber 2010 2 hours KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI

READ THESE INSTRUCTIONS FIRST

Write your name, class and register number in the spaces at the top of this page.

Write in dark blue or black pen 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.



If working is needed for any question it must be neatly and clearly shown in the space below the

question.

Omission of essential working will result in loss of marks.

The total of the marks for this paper is 80.

You are expected to use an electronic 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 .

Set by : Mdm Mah

This question paper consists of 22 printed pages [Turn over

2


Compound Interest

Total amount =

nr

P

1001

Mensuration

Curved surface area of a cone = lr

Surface area of a sphere = 24 r


3

1

Volume of a sphere = 3

3

4r

Area of triangle ABC = Cab sin2

1


Sector area = 2

2

1r , where is in radians

Trigonometry

C

c

B

b

A

a

sinsinsin

Abccba cos2222

Statistics

Mean =

f

fx


22

f

fx

f

fx

3

Answer all the questions

For

Examiner’s

Use

1.

(a) Express 80 m/s in km/h.

(b) Find the sum of 9 kilograms, 60 grams and 200 milligrams, giving your answer in grams.

Answer: (a) ____________km/h [1]

(b) _______________g

[1]

For

Examiner’s

Use

2.

Two inverted cone constructed from the same material are geometrically similar. The ratio of

their areas is 9: 64. If the height of the bigger cone is 32 cm, find the height of the smaller cone.

Answer: ________________ cm [2]

4

For

Examiner’s

Use

3.

Light travels 1 metre in 3.3 nanoseconds.

The distance from the Sun to the Earth is 11105.1 metres.

How many seconds does light take to travel from the Sun to the Earth?

Answer: _________________ s [2]

For

Examiner’s

Use

4.

It takes 28 workers to complete half of a certain job in 15 days. If the remaining job needs to be

completed in 12 days, how many more workers are needed to be deployed?

Answer: ___________________ [2]

5

For

Examiner’s

Use

5.

A map is drawn to a scale of 1:40 000. The area of a housing estate on the map is 16.4 cm2.

Calculate the actual area of the estate in square kilometres

Answer: _______________ km2 [2]

For

Examiner’s

Use

6.

The numbers 480, 576 and 792, written as product of their prime factors are

532480 5 , 26 32576 and 1132792 23 .

Find

(a) the largest integer which is a factor of 480, 576 and 792,

(b) the smallest positive integer value of n for which 480n is a multiple of 576.

Answer: (a) ________________ [1]

(b) n = ____________

[1]

6

For

Examiner’s

Use

7.

Given that y is inversely proportional to x2, find the percentage increase in y when the value

of x is halved.

Answer: _________________ % [2]

For

Examiner’s

Use

8.

Two angles of a pentagon are 110o each. The remaining three angles are in the ratio

1: 3: 4. Find the largest exterior angle of the pentagon.

Answer: ___________________ [2]

7

For

Examiner’s

Use

9. (a) Solve the inequality )15(4

1

2

1723

x

xx .

(b) Using the result in (a), write down the greatest possible rational number.

Answer: (a) ________________

(b) ________________

[3]

[1]

For

Examiner’s

Use

8

For

Examiner’s

Use

10.

The marks scored by 12 students in a Mathematics Quiz are as follows:

72, 27, 38, 85, 54, 32, 76, 46, 68, 56, 64, 95

A box-and whisker diagram is drawn to represent the data.

Find the values of p, q, x, y and z.

Answer: p= ________________

q =________________

x =________________

y =________________

z =________________ [2]

For

Examiner’s

Use

p x y z q

9

For

Examiner’s

Use

11. Pauline wishes to deposit $50 000 over a period of 4 years.

Bank A offers a 0.6% per annum simple interest rate.

Bank B offers a 0.6% per annum compound interest rate compounded monthly.

Explain, showing your working clearly, which bank do you think Pauline will earn the

best interest from the $50 000.

Answer : ______________________________________________________________

______________________________________________________________

_______________________________________________________________ [3]

For

Examiner’s

Use

10

For

Examiner’s

Use

12.

(a) Express 452 xx in the form bax 2

.

(b) Sketch the graph of 452 xxy in the answer space below

(b) [3]

Answer:(a) ________________ [2]

For

Examiner’s

Use

x

y

11

For

Examiner’s

Use

13.

Some children were asked to estimate the length of the string that they are given. The results

are given below in the table.

Calculate

(a) the mean,

(b) the standard deviation of the length of the strings.

Length 0 < x 10 10 < x 14 14 < x 16

Frequency 16 8 12

For

Examiner’s

Use

12

Answer: (a) ________________ [2]

(b) ________________

[2]

For

Examiner’s

Use

14.

A right circular cone is divided into 3 portions X, Y and Z by planes parallel to the base as

shown. The height of each portion is h units.

Find the

(a) ratio of the volume of X to that of Y,

(b) ratio of the sums of volume of X and Y to that of Z,

(c) ratio of the volume of X to that of Z.

For

Examiner’s

Use

X

Y

Z

h

h

hh

h

h

X

Y

Z

13

Answer: (a) ________________ [2]

(b) ________________

[2]

(c) ________________ [1]

For

Examiner’s

Use

15. (a) Given that 5:4:32 yyx , find the value of x

y.

(b) Given that 1312 p

p . Find the value of p.

For

Examiner’s

Use

14

Answer: (a) x

y = ____________ [2]

(b) p = _____________

[1]

For

Examiner’s

Use

16.

The four faces of a red tetrahedral die are marked 1, 2, 3 and 3.

The four faces of a blue tetrahedral die are marked 1, 3, 5, and 6.

When such a die is thrown, the score is the number on the face on which it lands.

The two dice are thrown together and their scores added. The possibility diagram in the

answer space shows some of the totals.

(a) (i) Complete the possibility diagram.

(ii) Find the probability that the total score is 7.

(b) The faces of a green tetrahedral die are marked 10, 20, 30 and 40.

All the three dice are thrown together and the scores added.

Find the probability that the total is more than 35 but less than 38.

Answer (a) (i) [1]

For

Examiner’s

Use

15

Red

+ 1 2 3 3

Blue 1

3 4

5 7

6 9

Answer: (a) (ii) _____________ [1]

(b)______________

[2]

16

For

Examiner’s

Use

17.

In triangle QRT, RS = 20 cm, ST = 10 cm, QR = 12 cm and QS = 16 cm. RST is a straight

line.

(a) Explain why triangle RQS is a right-angled triangle.

(b) Find, without evaluating any angles, the value of .sin QST

(c) Hence, find the area of triangle QST.

Answer: (a) __________________________________________________________

___________________________________________________________

___________________________________________________________

____________________________________________________________ [2]

Answer: (b) QSTsin = ______ [1]

(c) ____________ cm2

[1]

For

Examiner’s

Use

20 cm S R

Q

T

16 cm

12 cm

10 cm

17

A B

For

Examiner’s

Use

18.

= { x : x is an integer and 1 ≤ x < 16 }

A = { x : x is a prime number}

B = { x : x is an odd number}

(a) On the Venn diagram shown in the answer space below, fill in the elements of set A and

B.

(b) (i) Find n )( BA .

(ii) List the elements of the set BA' .

Answer : (a) [2]

Answer: (b)(i) n )( BA = ____ [1]

(ii) BA' =________

[1]

For

Examiner’s

Use

18

For

Examiner’s

Use

19.

The diagram above shows part of a circle with centre O, passes through B and C. The point

P is the foot of the perpendicular from C to OB.

Given that the radius of the circle is 10 cm and BOC = 1.29 radians.

Find the shaded area.

Answer: ________________cm2 [4]

For

Examiner’s

Use

O

C

B P

19

For

Examiner’s

Use

20.

A six-faced die was thrown 28 times.

The table shows the number of times that each possible score occurred.

(a) Write down the modal score.

(b) After the 27th throw the median score was 2.

What was the least possible score on the 28th throw?

(c) The die was then thrown twice more.

The mean score of all 30 throws was exactly 3.

What were the scores on the extra two throws?

Score 1 2 3 4 5 6

Frequency 8 6 6 2 4 2

For

Examiner’s

Use

20

Answer: (a) _________________ [1]

(b)________________

[1]

(c) _____and _______ [2]

For

Examiner’s

Use

21. In the triangle OXY, the point B on OY is such that OB = OY7

2.

A is the midpoint of OX. AY intersects BX at C such that 5YC = 3YA.

Given that

OX = x and

OY = y, express, as simply as possible, in terms of

x and / or y,

(a)

BX ,

(b)

AY ,

(c)

OC .

For

Examiner’s

Use

O

Y

X

B

A

C

21

C

F

A

G

H

E

D

B

Answer: (a)

BX =____________ [1]

(b)

AY =___________

[1]

(c)

OC =___________

[2]

For

Examiner’s

Use

22.

In the diagram above, DFHA and EGHB are straight lines. DE, FG and BA are parallel

lines. Given that DE = AB = 10 cm, EH = 14 cm, HC = 5 cm and GF= 6 cm.

(a) Show that triangles DEH and ABH are congruent.

(b) Find the ratio of

(i) DEH

FGH

of area

of area,

(ii)DEH

DHC

of area

of area.

For

Examiner’s

Use

22

A B

C

Answer:(a)___________________________________________________________

___________________________________________________________

___________________________________________________________

___________________________________________________________ [2]

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

(ii)______________

[1]

For

Examiner’s

Use

23.

The diagram is a plan of a triangular field ABC,.

(a) Construct

(i) the perpendicular bisector of AB, [1]

(ii) the angle bisector of ∠ABC. [1]

(b) The Point T is equidistant from the points A and B and is also equidistant from the

lines AB and BC. Mark and label the point T. [1]

(c) Use the diagram to measure the bearing of T from A.

For

Examiner’s

Use

North

23

Speed (m/s)

Time

(s)

36

12

0

4 12 0

Answer: (c) _________________ [1]

For

Examiner’s

Use

24.

The diagram represents the speed-time graph of a moving object.

(a) Calculate the speed of the object when t = 3.

(b) Calculate the distance travelled in the first 12 seconds.

(c) Given that the rate at which the object slows down after t = 12 is equal to

half the rate at which it accelerates, during the first 4 seconds, calculate

the time at which it stops.

For

Examiner’s

Use

24

Answer: (a) ______________m/s [2]

(b)_______________m [2]

(c) _______________ s [2]


Elementary Mathematics 2010


1) (a) 288 km/h

(b) 9060.2g

13) (a) 9.89

(b) 4.51

2) 12 cm

14) (a) 1: 7

(b ) 8 : 19

1 : 19

3) 495 s

15) (a)

11

10

(b) 1 or 3

1

4) 7

16) (aii)

8

1

End of paper

25

(b) 64

5

Red

+ 1 2 3 3

Blue 1 2 3 4 4

3 4 5 6 6

5 6 7 8 8

6 7 8 9 9

5) 2.624 km 2

17) (a)

2

2222

20 200

1612

QSRQ

By converse of Pythagoras’ theorem, triangle

RQS is a right-angled triangle.

(b) 5

3

(c) 48 cm 2

6) (a) 24

(b) n=6

18)

b(i) 5 b(ii) {1, 9, 15}

7) 300 % 19) 51.2 cm 2

8) 140 0

20) (a) 1 (b) 3

( c) 6 and 6

9) (a)

3

15 x

(b) 3

1

21) (a) x − y

7

2

(b) xy2

1

(c) yx5

2

10

3

10) p =27, q = 42, x = 60, y = 74, z = 95

22)

(AAS) ABHcongruent DEH

(given) cm 10 AB DE

lines) // s,(alt BAHEDH

s) opp (vert.

AHBDHE

b(i) 25

9 b(ii)

14

5

A

B 2 3 5 7 1 9

11 13 15

4 6 8 10 12 14

26

11) Bank B has $14.21 more interest

than bank A.

23) (c ) 037° ± 1

12) (a)

4

12

2

12

2

x

24 (a) 30m/s

(b) 384 m

(c) 24 s

A B

C

T

0 1 4

)4

12,

2

12(

4

y

x

Candidate Name: ______________________ Class: _____

KRANJI SECONDARY SCHOOL

Prel iminary Examination 2 Secondary 4 Express

ELEMENTARY

MATHEMATICS 4016/2

PAPER 2

Friday 17 September 2010 2 hours 30 minutes

KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY




INSTRUCTIONS TO CANDIDATES

Do not open this booklet until you are told to do so.

Write your name, class and register number in the spaces provided on the answer paper.


Write your answers on the separate answer paper provided.

If you use more than one sheet of paper, fasten the sheets together.

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.

INFORMATION FOR CANDIDATES


The total number of marks for this paper is 100.

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

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

Set by : Mr. Mike Teo

_______________________________________________________________________________________

This question paper consists of 10 printed pages. Turn over

Kranji Secondary School 2 Preliminary Examination 2010 Secondary 4 Express Mathematics / Paper 2


Compound Interest

Total amount =

nr

P

1001

Mensuration

Curved surface area of a cone = rl

Surface area of a sphere =24 r


3

1

Volume of a sphere =3

3

4r

Area of triangle ABC = Cabsin2

1


Sector area = 2

2

1r where is in radians

Trigonometry

Abccba

C

c

B

b

A

a

cos2

sinsinsin222

Statistics

Mean =

f

fx


22

f

fx

f

fx


1 (a) Express as a single fraction in its simplest form.

.273

9

3

2

3

12

x

x

xx [3]

(b) Given that x y = 13 and ( x + y )2 = 42, calculate the numerical value

of

2

2

yx. [3]

(c) It is given that

qr

rqp

6

4

1.

(i) Find p when 1q and 5.2r . [1]

(ii) Express r in terms of p and of q. [3]

2. (a) When driven in town, a lorry runs x kilometres on each litre of petrol.

(i) Find, in terms of x, the number of litres of petrol used when the

lorry is driven 100 km in town. [1]

(ii) When driven out of town, the lorry runs 6x kilometres on

each litre of petrol. It uses 8 litres less petrol to go 100 km out

of town than to go 100 km in town.

Use this information to write down an equation involving x, and

show that it simplifies to

07562 xx [3]

(b) Solve the equation 07562 xx , giving both answers correct to two

decimal places. [3]

(c) Calculate the total volume of petrol used when the lorry is driven 55 km

in town and then 175 km out of town. [2]


3. The diagram below shows each link of a chain which has an outside length

of 20 mm and the metal is 3 mm thick.

(a) Find the maximum length in millimetres, of a chain which is made up of

(i) 2 links [1]

(ii) 3 links [1]

(iii) 10 links [1]

(b) Find an expression for the maximum length of a chain which is made up

of n links. [2]

(c) Use your answer in (b) to find the smallest number of links required to

make a chain at least 2.4 metres long. [2]

4 The coordinates of A and B are 2,5 and 1,4 respectively.

(a) Find the gradient of the line AB. [1]

(b) Find the equation of of the line AB. [1]

(c) Find the coordinates of X which lie on the y-axis such that AX = XB. [4]

(Note : Points A, X and B do not lie on a straight line.)

(d) Given that the perpendicular distance from point X to the line AB is

2.5 units, find the area of triangle AXB. [3]

20mm

3mm


12

9

5.

A solid cone has a base radius of 9 cm and height 12 cm.

A solid hemisphere has a radius of 9 cm.

A solid toy is formed by joining the plane faces of the cone and the hemisphere.

(a) Show that the length of the slant edge of the cone is 15 cm. [1]

(b) Calculate

(i) the surface area of the toy, [3]

(ii) the volume of the toy. [3]

(c) A solid metal cylinder with a radius of 2.5 m and height 1.5 m was melted

down and all the metal was used to make a large number of these toys.

Calculate, to the nearest number of toys, the number of toys that were

made. [4]


F

E

O

DA

X

C B

DA

X

C B

6. The points A, B, C and D lie on a circle as shown on Diagram I. AC cuts BD at X. AD is

parallel to BC.

Diagram I

(a) Show that triangle AXD and triangle CXB are similar. [2]

(b) Given that angle 29CBX and angle 81ADC , calculate the

angle ABX. [2]

(c) Diagram II shows the circle in Diagram I and a second circle, centre O.

The two circles intersect at A and B. CB produced cuts the second circle

at E. DB produced cuts the second circle at F. Angle 115BFE .

Diagram II

Calculate

(i) angle CAF, [3]

(ii) angle AOB. [2]


7. (a) Given that

13

7

6

1

1

5

1

21

40

b

a, find the value of

a and of b. [3]

(b) A florist sells 4 different types of bouquets of flowers. They are Blossom,

Cherish, Sunshine and Friendship. Each type of bouquet comprises roses

of different colours. The table shows the number of roses of each colour

for the bouquets.

The cost price of each red rose, pink rose, blue rose and yellow rose is

$1.00, $1.20, $1.50 and $2.00 respectively.

(i) Given that

5053

1125

2233

4580

A , write down matrix B such that

T = AB where T shows the total cost of each type of bouquet.

Hence, calculate T. [3]

(ii) During a particular week, 5 Blossom, 14 Cherish, 10 Sunshine

and 8 Friendship bouquets were stolen from the florist.

If 810145C and S = CT, calculate S. [1]

(iii) Describe what S represents. [1]

Number of

Red Roses

Number of

Pink Roses

Number of

Blue Roses

Number of

Yellow Roses

Blossom 0 8 5 4

Cherish 3 3 2 2

Sunshine 5 2 1 1

Friendship 3 5 0 5


D30

70

26

C

BA

N

N

8. A, B, C and D are four markers in a field. B is 70 m due east of A. C is 30 m

on a bearing of 020 from B and A is 26 m on a bearing of 110 from D.

A line is painted from A to C.

(a) Find the bearing of D from A. [1]

(b) Calculate the length of AC. [3]

(c) Find the bearing of C from A. [3]

(d) Find the area of triangle ABC. [2]

(e) A pole of length 4.2 m is erected vertically at B. Calculate the greatest

angle of elevation of the top of the pole from the line AC. [4]


9. The cumulative frequency curve below illustrates the marks obtained, out of 100,

by 160 students in an examination.

(a) Copy and complete the following frequency table for the distribution.

Height of the plants (cm) Frequency

3010 x

5030 x

7050 x

9070 x

[2]

(b) Use your graph to find

(i) the median, [1]

(ii) the interquartile range, [2]

(iii) the seventieth percentile mark. [1]

Cumulative

Frequency


(iv) Given that 55% of the students scored more than x marks,

find the value of x. [2]

(c) A student would be awarded a Grade C if he scores more than 50 marks

but less than or equal to 60 marks. Find the number of students who

were awarded a Grade C. [2]

(d) If two students were chosen at random, find the probability that one had

scored less than or equal to 40 marks and the other had scored more than

70 marks. [2]

10. Answer the whole of this question on a sheet of graph paper.

The following is a table of values for the graph )43(4

1 3 xxy .

x 5.2 2 5.1 1 5.0 0 0.5 1 1.5

y 3 1.5 a 0.5 0.7 1 1.3 1.5 b

(a) Calculate the value of a and of b, giving your answers to 1 decimal place. [1]

(b) By taking 2 cm to represent 0.5 units on both axes, draw the graph of

)43(4

1 3 xxy for 5.15.2 x . [3]

(c) By drawing a tangent, find the gradient of the curve at the point where

5.1x . [2]

(d) From the graph, find the range of values of x for which

2)43(4

1 3 xx . [2]

(e) By drawing a suitable straight line on the graph, solve the equation

025 3 xx . [4]

- End of paper -


Sec 4 exp Prelim Exam 2010 P2 Answer Key

1a) 33

3

xx 7a) a = 10 , b = 2

b) -2.5 bi)

0.19

9.10

6.13

1.25

T

ci) 3

11 ii) 9.576S

ii) 2

2

46 p

qqpr

iii) S represents the total cost of all the various

kinds of bouquets stolen from the florist.

2ai) litresx

100

ii) 86

100100

xx 8a) 290

b) 6.17 or -12.17 b) 85.1

c) 23.3 litres c) 6.070

d) 987

3ai) 34mm e) 3.10

ii) 48mm

iii) 146mm

b) 14n + 6 9a) 16 28 76 40

c) 171 bi) 60

ii) 22

4a) 3

1 iii) 67 or 68

b) 13 xy iv) 58

c) 2,0X c) 36

d) 11.9 or 7.50 d) 159

14

5a) 15

bi) 933 10a) a = 0.7 , b = 1.3

ii) 2540 c) -0.938

c) 11574 d) 175.2x or 5.1175.2 x

e) 45.02 or

6a)

XDAXBC

XCBXAD

CXBAXD

(Any 2 reasons)

b) 70

ci) 88

cii) 68


Elecmentary Mathematics 2010


1) (a) $180000

(b) $177629.31

(c) (i) $81600

(c) (ii) 1.96%

6) (a) (i) 30 anglers

(a) (ii) 38 mins

(a) (iii) 31 mins

(b) Time

(t mins) 15<t≤30 30<t≤45 45<t≤60 60<t≤75

No. of

anglers 130 320 120 30

(c) (i) 38.75

(c) (ii) 11.7

(d) 11980

29

2) (a) (i) 360

(a) (ii) 7

11

(b) (i) x = 0 OR x = 2

(b) (ii) z = ba

ab ba

1

22

7) (a) h

x 2

400

(b) time = hx 16

400

(c) x2 – 14x – 3632 = 0

(d) x = 67.67 OR -53.67

(e) 0309 h on Sunday

(f) 60.6 litres

(g) $110.50

3) (i)

Row Pattern Sum, s s + 2

1 2 2 4

2 2,3,2 7 9

3 2,3,4,3,2 14 16

8) (a) 34.9°

(b) 108m

(c) 6560m2

4 2,3,4,5,4,3,2 23 25

(ii) Square numbers

(iii) 79

(iv) s = (N + 1)2 – 2

(d) BC = 145

(e) of depression = 11.0°

4) (a) tangent is perpendicular to radius

(b) (i)EDT = 115°

(b) (ii) DCB = 119°

(b) (iii) OT = 10.3 cm

9) (a) (i) r = 5cm

(a) (ii) VA = 26cm

(a) (iii) 2250cm3

(a) (iv) 1210cm2

(b) 1.50cm

5)

(a)

5152

1103

854

A

(b)

40

75

11

5152

1103

854

AB

1347

473

739

(c) C = (8 12 6)

(d) (19670)

(e) Total power consumption of the

company in one day.

10) (a) a = - 42 , b = 28

(b) smallest value = - 42

(c) largest x = 4.66

(d) -2 < x < 3.25

(e) (4, 24)

(-2.8, -35.5)

(f) a = 1, b = 39, c = -21

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