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7/29/2019 Other School 2 EM P1

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Class Candidate Name Register Number

O-LEVEL PRELIMINARY EXAMINATION

Mathematics

Paper 1

Candidates answer on the Question Paper

READ THESE INSTRUCTIONS FIRST

Write your name, class and register 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 f luid.

Answer allquestions.

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 numericalexpressions.If the degree of accuracy is not specified in the question, and if theanswer is not exact, give the answer to three significant figures. Give answers indegrees to one decimal place.

For , use either your calculator value or 3.142, unless the question requires theanswer 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 partquestion.

The total mark for this paper is 80.

4016/01

2 hours

This question paper consists of 16 printed pages.[ Turn over


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O-Level Preliminary Examination Mathematics Paper 1

2

Mathematical Formulae

Compound interest

Total amount = 1100

nrP +

Mensuration

Curved surface area of a cone = rl

Surface area of a sphere = 24 r

Volume of a cone = 21

3r h

Volume of a sphere =34

3 r

Area of triangle ABC =1

sin2

ab C

Arc length = r , where is in radians

Sector area = 21

2r , where is in radians

Trigonometry

sin sin sina b c

A B C= =

2 2 2 2 cosa b c bc= + A

Statistics

Mean =fx

f

Standard deviation =

22fx fx

f f


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3

Answerall the questions.

1 Given that 5 +6

x< 2

6

13 x ,

(a) solve the inequality,(b) write down the least integer value of x that satisfies the inequality.

Answer (a)_________________________ [2]

(b) x=______________________ [1]________________________________________________________________________________

2 Consider the sequence

, 4, , 16, , . , ....2 8 32

(a) Write down the next two terms in the sequence.

(b) Find an expression, in terms of , for the th term of the sequence.n n

Answer (a)____________, ____________ [1]

(b)_________________________ [1]________________________________________________________________________________

3 Solve the equation 4231 3927 = + xx

Answer x= _______________________ [3]


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4

0

4 A map is drawn to a scale of 1 : 400 000.

(a) Find the actual distance, in kilometres, represented by 8.5 cm on the map.

(b) A town covers an area of 600 square kilometres. Find, in square centimetres, thearea representing the town on the map.

Answer (a)______________________km [1]

(b)_____________________ cm2 [2]________________________________________________________________________________

5 (a) Factorise completely .26226 kkmkm +

(b) Solve the equation 7 8 .( 5) 1t t =

Answer (a)_________________________[2]

(b) t=______________________ [1]


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5

6 Joe works 80 hours in order to earn $440.

Using the same rate of pay, calculate

(a) the amount he will earn in 7 hours,

(b) the total number of hours he will have to work in order to earn $82.50 more.

Answer (a) $ _______________________ [1]

(b) ____________________hours [2]

_________________________________________________________________________________________________________________________

7 In 2009, the population of China is estimated to be 1.33 billion.

(a) Write 1.33 billion in standard form.

(b) It is said that the population of China makes up 20% of the worlds population.

Find the worlds population. Express your answer in standard form.

Answer (a)_________________________ [1]

(b)_________________________ [1]


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6

8 The number 168 and 324, written as the products of their prime factors, are168 = 23 3 7, 324 = 22 34.

Using only the above results and showing your working clearly, find

(a) 4324x

(b) the largest integer which is a factor of both 168 and 324.

(c) the smallest positive integer value ofn for which 168n is a multiple of 324.

Answer (a)________________________ [1]

(b)________________________[1]

(c) n= _____________________[1]

_________________________________________________________________________________________________________________________

9 If V is inversely proportional to ( )4W and V = 2 when W= 8,(a) express V in terms ofW,

(b) find the value ofV when W= 6.

Answer (a) V =_____________________ [2]

(b) V =_____________________ [1]


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7

10 Some adults were asked how many television programmes they had watched inthe past week. The table below shows the results.

Number of programmes watched 0 1 2 3

Number of adults 9 11 5 x

(a) Write down the largest possible value ofxgiven that the mode is 1.

(b) Write down the largest possible value ofxgiven that the median is 1.

(c) Calculate the value ofxgiven that the mean is 1.

Answer (a) x =_____________________ [1]

(b) x =_____________________ [1]

(c) x= _____________________ [2]

_________________________________________________________________________________________________________________________

11 In the diagram, QRSand QTRare right-angles. PQR is a straight line, QR = 8 cm and

SQ = 10 cm. Find

S(a) the area ofQRS,

(b) the length ofRT, T10 cm

(c) the value of cos PQS.

P Q R8 cm

Answer (a) ___________________________cm2 [2]

(b)___________________________cm [1]

(c)______________________________ [1]


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8

12 (a) The exterior angle of a regular polygon with 2nsides is 30 smaller than the exterior0

angle of a regular polygon with nsides. Find the value ofn.

(b) A, B, C and D are the vertices of a regular polygon with 12 sides.(not drawn to scale)

Find the value of .BDC

B C

DA

Answer (a) n= _____________________ [2]

(b) BDC =_________________0 [2]


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913 Two jars shown in the diagram are geometrically similar. Their heights are 25 cm and 10 cm

as shown.

25 cm

10 cm

(a) The diameter of the larger jar is 8 cm. Find the diameter of the smaller jar.

(b) Find the ratio of the area of the base of the bigger jar to that of the smallerjar. Give your answer in the form of n : 1.

(a) Given that the smaller jar can hold 72 cm3

of milk, calculate the volume of milk

that the bigger jar can hold.

Answer (a)______________________cm [2]

(b)________________________ [2]

(c)_____________________cm3

[2]


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10

14 The diagram shows a rectangle ABCD where AB = 4 cm and BC = 7 cm. A quadrantof a circle , centre B is drawn with AB as radius.

Calculate, leaving your answers in terms of ,

A D

B C

4

7 E

(a) arc length AE .(b) area of the shaded region.

Answer (a)______________________cm [1]

(b)_____________________ cm2

[2]

________________________________________________________________________________

15 (a) Express in the form ( .462 + xx bax + 2)

(b) Sketch the graph of 46 + x , clearly labelling the turning point and2= xy

y-intercept.

y

xO

Answer (b) [2]

Answer (a)_________________________ [1]


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11

16 Given = { x : x is an integer such that 81


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12

17 There are six red cards numbered 1, 2, 2, 2, 3 and 3 and four blue cards numbered2, 3, 5 and 7 in a box.

David draws two cards from the box at random and add up the scores on the two cards.

(a) Complete the possibility diagram as shown. [1]

+ 2 3 5 7

1 3 4 6 8

2 4 5 9

2 4 7 9

2 4 5 7 9

3 6 8 10

3 5 6 10

(b) Find the probability that

(i) the sum of the two scores is at least 7.

(ii) the sum of the two scores is a prime number.

(iii) the sum is a multiple of one of the numbers drawn.

(c) Joe draws one card at a time from the box at random with replacement, untilhe gets a blue card. Find the probability that he will be successful exactly

on the third draw.

Answer (b)(i)______________________ [1]

(ii)______________________[1]

(iii)_____________________ [1]

(c)________________________ [1]


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13

18 In the diagram RQ and BC are parallel. AB is a straight line and parallel to PQ.

It is given that AR = 2 cm, RB = 4 cm and BC = 15 cm.

15

A

2

QR

4

CB P

(a) Show that ABC is similar to QPC and state the case of similarity.

(b) Given that ARQ is similar to ABC, find the length of RQ.

(c) Given that area of triangle ABC is 36cm2, calculate the area ofARQ,

Answer (a)

[3]

Answer (b)_______________________cm [2]

(c)______________________cm 2 [2]


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14

19 In April 2007, a survey was carried out to find the number of days 700 patients stayedat a hospital.

The graph shows the cumulative frequency curve for the data collected.

Cumulative Frequency

0

50

100

150

200

250

300

350

400

450

500

550

600

650

700

0 5 10 15 20 25 30

Number of days

Cumulative

frequenc

Use the graph to estimate

(a) the median number of days,

(b) the interquartile range,

(c) the percentage of patients who stayed more than 20 days at the hospital.

Answer (a)_____________________days [1]

(c)_____________________days [2]

(d) ______________________% [2]


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15

20 The graph below shows the overseas call charges under Bistro Mobile.

25

20

15

10

5

020 40 60 80 100

Length of overseas

calls (xmins)

Cost ($)

(a) Find the cost of the bill when the length of overseas calls is 24 minutes?(b) Find the total length of calls in minutes when the cost of bill is $12.(c) Another company, Sun Mobile, charges a fixed cost of $5 for first hour and

any additional units at 47.5 cents per minute.

(i) Draw the graph on the grid above to represent the charge made by SunMobile.

(ii) State the inequality to represent the interval of the length of overseas callsmade so that it is cheaper to subscribe to Sun Mobile than Bistro Mobile.

Answer (a) $______________________ [1]

(b)____________________mins [1]

(c) (i) On Graph [1]

(ii)_____________________ [1]


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16

21 The figure shows a triangle ABC with A(1, 3), B(1, 1) and C(7, k).

The gradient ofAB is 2nand the length ofAC is 72 .

A(1, 3)

B(1, 1)x

y

O

C(7, k)

Find (a) the value of

(i) n ,

(ii) k .

(b) the equation of the line AB,

Answer (a)(i) n = ___________________ [2]

(ii) k= ____________________ [2]

(b)__________________________ [2]

~~~~~End of Paper 1~~~~~


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17

Essential Steps Marks Alternative Steps/Remarks

1(a)5 +

6

x< 2

6

13 x

30 19 12x x+ < 18 42x <

7

3x>

M1

A1

(b) x= 3 B1

2(a) 64 , -128 B1(b) ( )n2 B1

3 4231 3927 = + xx 3( 1) 2(3 2) 43 3x x + = 3

4

3( 1) 2(3 2) 43 3x x + + =

3 3 6 4x x + + = 1

3x=

M1

M1

A1

4(a) 1 : 400 000

=1 cm : 400 000cm=1cm : 4000 m=1cm : 4km

8.5 4 34 = km B1(b) 1cm2 : 16km2

area =600

16= 37.5 cm2

M1

A1

5(a)26226 kkmkm +

= 2 ( 3 1) 2 (1 3 )m k k k + = 2 ( 3 1) 2 (3 1)m k k k

= (3 1)(2 2 )k m k

= 2(3 1)( )k m k

M1

A1

(b) 7 8( 5) 10t t =

7 8 40 10t t + = 30t =

30t = B1

6(a) 440 780

$38.50

=

B1


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18


(b) 440 82.5080

80

95

+

=

M1

B1

7(a) 91.33 10 B1

(b) 9

9

1.33 10

0.2

6.65 10

=

B1

8(a) 2 22 3 18 2x x = B1

(b) 12322 = A1

(c) n=27 B19(a)

4

kV

W=

2 8 4

k=

8k =

8

4V

W=

M1

A1

(b) 8

6 4V =

V = 4 B1

10(a) x= 10 B1

(b) x= 14 B1

(c) 11 10 31

25

x

x

+ +=

+

x= 2

M1

A1

11(a) SR = 22 810 = 6

area = 24682

1= cm2

M1

A1

(b)2410

2

1= RT

8.4=RT cm A1

Alternative solution :

6

8 1

RT=

0

RT = 4.8 cm

(c)

5

4

10

8= B1

12(a) 360 36030

2n n =

720 360 = 60n

660

360

==n

M1

A1


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19

y

xO

-5(3, -5)



(b)interior angle =

4

15012

180)212(=

BDC = 152

150180=

M1

A1

Alternative :

Ext. angle =360

12= 30

BDC =30

15

2

=

13(a)

25

10

8=

d

2.3=d cm

M1

A1

(b) 225 25

10 4

6.25

1

6.25:1

=

=

=

M1

A1

(c) 3

1025

72

3

=V

cm

M1

1125=V

A1

14(a) 12 (4) 2

4 = cm

B1

(b)2128 (4 )

4

= 28 4 c

M1

m2 A1

1 = 5 B15(a) 462 + xx 2( 3)x

(b)

B2

ngpoint and y-intercept

1 mark correct shape1 mark correct turni

16 )

B1

(a

A B

6

4

1

37

2

5


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20

+2 3 5 7

1 3 4 6 8

2 4 5 7 9

2 4 5 7 9

2 4 5 7 9

3 5 6 8 10

3 5 6 8 10


BA = }3,2{(b)(i) B1

(ii) '' BA = }6,4,1{ B1

(c) n( B ) = 4 B117(a)

B1

(b)(i)

24

11B1

8

3

24

9= B1

(ii)

B1(iii)

8

3

24

9=

(c) 6 6 4 18

10 10 1 125 =

0B1

CAB = CQP (corr. )

ABC = QPC (corr.

)

C is a common angle

18(a)

ABC is similar to QPC (AAA similarity) B1

irs of anglesuffices.

e ofmilarity is not stated.

B2Any 2 pas

o mark awarded if casNsi

(b)

6

2

15=

RQ

cm

M1

5=RQ

A1

(c) 2

6

2

36

=

ARQarea

24=ARQarea cm

M1

A1

1 9(a) median = 13 days B1

(b)

= 7.5 days

7.7 0.2 daysinterquaertile range = 17 9.5 M1B1

(c)%100

700

620700

% ( 3 s.f.s)= 11.4

M1

A1

2 0(a) $5.50 B152 mins(b) B1

(c)B1

25

20

Cost ($)


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21


B1

(ii) 22 < x< 96 B1

21(a)(i)

1 3

1

=

2

1

n +

n=1

2

M1

A1

(ii) 2

9 (rejected)

M12 2( 3) (7 1) 7k+ + = 2( 3) 36k+ =

k= 3 , -

A1

(b)

c = y =- x -2

M1A1

y =- x+ c- 2

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