Ahmad Ibrahim Sec. Mid-Year 2012: E Math P1 iss Em My Exam 4e5na p1

7/28/2019 Ahmad Ibrahim Sec. Mid-Year 2012: E Math P1 iss Em My Exam 4e5na p1

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AHMAD IBRAHIM SECONDARY SCHOOLMID YEAR EXAMINATION 2012

MATHEMATICS

PAPER 1

(80 Marks)

LEVEL: Secondary 4 Express DATE : 10thMay 2012Secondary 5 Normal Academic DURATION : 2 hours

NAME: _________________________________ CLASS: _____________

INDEX NO: ______________

ADDITIONAL MATERIALS:

Nil

INSTRUCTIONS TO CANDIDATES:

Write your name, class and index number on the space provided above.Write in dark blue orblack pen. You may use a pencil for any diagrams or graphs.

Answerall questions.If working is required for any question, it must be shown with the answer in the space belowthe question. Omission of essential working will result in loss of marks.Write your answers in the spaces provided on the question paper.Calculators should be used where appropriate.

If the degree of accuracy is not specified in the question, and if the answer is not exact, giveyour answer to three significant figures. Give answer in degrees to one decimal place.

For, use either your calculator value or 3.142, unless the question requires the answer interms of.

INFORMATION FOR CANDIDATES

The number of marks is given in brackets [ ] at the end of each question or part question.The total marks for this paper is 80.

___________________________________________________________________________This question paper consists of15 printed pages.

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MATHEMATICAL FORMULAE

Compound interest

Total amount aftern years = 1100

ni

P +

Mensuration

Curved surface area of a cone = rl

Curved surface area of a sphere = 2

4 r

Volume of a cone =21

3r h

Volume of a sphere =34

3r

Area of a triangle ABC =1

sin2

bc A

Arc length = r , where is in radians

Sector area =21

2r , where is in radians

Trigonometry

sin sin sin

a 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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1.

In a survey, 1400 students were asked how they travelled to school. The results of the

survey are shown in the bar chart.

(a) What is the modal transport taken by the students?(b) The same information is to be shown in a pie chart.

Calculate the angle representing the students who travelled by bus.

Ans: (a) _________________ [1](b) _________________ [1]

2. (a) It is estimated that the storage capacity of a mainframe computer is

16

1.27 10bytes, which can be expressed as n terabytes. Find the value ofn.

(b) A light in a machine comes on for 685 microseconds.

(i) Express the length of time in seconds.

Give your answer in standard form.

(ii) It was also given that the light travels at a speed of 83 10 m/s.

Find the distance travelled in kilometres by the light.

Ans: (a) n = ______________ [1]

(bi) ________________ s [1]

(bii) _______________km [1]3. (a) 12 workers painted a building in 15 days. Given that the workers work at the

3

0

50

100

150

200

250

300350

400

450

500

550

600

650

700

Bus Car Walk Cycle

N u m be

r of

St

ud

ents


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same rate, find how long it would take 9 workers to paint the same building.

(b) The resistance of a wire of constant length varies inversely to the square of its

diameter. The resistance is 23 ohms when the diameter is dmm.

Find the resistance of the wire when the diameter is halved.

Ans: (a) _____________ days [1]

(b) _____________ ohms[2]

4. (a) Evaluate2.461

37.45 0.875. Give your answer correct to 3 decimal places.

(b) Siti invests $6050 in a bank that pays a compound interest at a rate of 4% per

annual compounded every 3 months. Find the amount that Siti has in the bank

after 4 years.

Ans: (a) _________________ [1]

(b) $________________ [2]

5. The temperature inside a hot flask is p C and the temperature outside is q C ,

wherep and q are positive integers. Write down an expression in terms ofp and q, for

(a) the difference in the two temperatures,

(b) the mean of the two temperatures.

Ans: (a) ______________ C [1]

(b) ______________ C [1]6. (a) The numbers 198 and 324, written as the products of their prime factors, are

11321982 = and 42 32324 = .

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Use these results to find

(i) the highest common factor of 198 and 324,

(ii) the smallest integer, k, such that 198kis a perfect square.

(iii) the smallest positive integer value ofn for which 198n is a

multiple of 324.

(b) During one of the road works, two light blinkers were used as traffic signals to

warn motorists. If the first blinker flashes every 12 seconds and the second

blinker flashes every 40 seconds, how long will it be before both blinkers flash

at the same time?

Ans: (ai) HCF = __________ [1]

(aii) k= ______________ [1]

(aiii) n = ______________ [1]

(b) ________________ s [1]

7. (a) Simplify and express

3

3

23

b

ain positive index form.

(b) Given that

0

1 2 3 18 32

2

x x =

, find the value ofx.

Ans: (a) _________________ [2](b) x = ______________ [2]

8. (a) Two interior angles of an octagon are 32 and 34. The remaining interior

angles are in the ratio of 1 : 2 : 2 : 4 : 5 : 7. Find the largest interior angle.

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(b) The interior angle of a regular polygon ofn sides is 5 times the exterior angle.

Find the value ofn.

Ans: (a) ________________ [2]

(b) n = ______________ [2]

9. Solve the simultaneous equations

1132 += xy ,3

52

x y+ = .

Ans: x = ______________

y = ______________ [3]

10. (a) Solve the equation 23 12 0p p = .

(b) Expand and simplify 22 )5()32( + xx .

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Ans: (a) p = _____ or ______ [2]

(b) _________________ [2]

11. (a) Solve the inequality xxx2

125354 < .

(b) Represent your answer on the given number line.

Ans: (a) _________________ [2]

(b)

[1]

12. The table below shows the test scores of Class 4E8 in a recent Mathematics test.

Total Scores 100


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Marks

Frequency 1 5 25 9

(a) Calculate(i) the mean,

(ii) the standard deviation.

(b) Information for the same test done by Class 4E9 is given as follow:

Mean Standard Deviation

29.3 10.7

Compare briefly the performance of the two classes for this Mathematics test.

Ans: (ai) Mean = __________ [1]

(aii) Standard Deviation = ___________ [2]

(b) ___________________________________________________________

___________________________________________________________

___________________________________________________________ [2]

13. (a) Given the sequence 50 , 47 , 44 , 41 , (i) Write down the next two terms.

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(ii) Write down the nth term in the above sequence.

(b) Given another sequence 3 , 8 , 15 , 24 , 35 ,

Write down an expression, in terms ofn, for the nth term of this sequence.

Ans: (ai) ________ , ________ [1]

(aii) _________________ [1]

(b) _________________ [1]

14. The two open troughs in the diagram below are geometrically similar.

The ratio of the areas of the bases is 4:9 .

(a) The area of the top of the bigger trough is 540 cm2.

Find the area of the top of the smaller trough.

(b) Write down the ratio of the depth of the bigger trough : depth of the smaller

trough.

(c) Both troughs are filled with sand to the brim. The mass of sand in the smaller

trough is 24 kg. Find the mass of sand in the larger trough.

Ans: (ai) ______________ cm2 [1]

(aii) ________ : ________ [1]

(b) _______________ kg [2]

15. The equation 5 12 60y x+ = is a straight line lthat crosses thex-axis atPand the y-

axis at Q. Find

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(a) the coordinates of the pointPand Q,

(b) the gradient of the line l,

(c) the length of PQ,

(d) the equation of a line which passes through the point (0,2) and is parallel to

line l.

Ans: (a) P( ______ , ______ )

Q ( ______ , ______ ) [2]

(b) _________________ [1]

(c) _____________ units[1]

(d) _________________ [1]16. A box contains 20 identical marbles, of which 8 are red and 12 are white. Two

marbles are picked randomly, one after the other and there is no replacement. The tree

diagram below shows the possible outcomes and some of their probabilities.

10

Red

Red

White

White

Red

White

First Marble Second Marble

p

19

12

q

r

s

5

2


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(a) Write down the values ofp, q, rands shown on the tree diagram.

(b) Calculate the probability that

(i) both marbles will be red,

(ii) one marble will be red and the other white,

(iii) at least one marble is white.

(c) A third marble is now picked out at random.

Calculate the probability that none of the marbles is red.

Ans: (a) p = ______ , q = ______ , r= ______ ,s = ______ [2]

(bi) _________________ [1]

(bii) _________________ [1]

(biii) _________________ [1]

(c) _________________ [1]

17. In the diagram, OA = a and OB = b .

(a) Mark and label the pointsPand Q such that + 2OP= a b and1

2OQ = b a

uuur

.

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(b) If 4

2OA

=

uuur

and3

5AB

=

uuur

,

(i) express OB as a column vector,

(ii) calculate AB ,

(iii) find the value ofkif

2

kis parallel to AB .

Ans:(a) [2]

(bi) _________________ [1]

(bii) _____________ units [1]

(biii) k= ______________ [1]18.

12

O

A

B

a

b

A B

CD

P

8 cm

6 cm


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In the quadrilateralABCD, the diagonalsACandBD intersect atP.

AB is parallel to CD.AP=BPandDP= CPwhereDP= 8 cm andPB = 6cm.

(a) Prove that trianglesADPandBCPare congruent.

(b) Write down the value ofArea of

Area of

BCP

BDC

.

Ans: (a) ___________________________________________________________

___________________________________________________________

___________________________________________________________

___________________________________________________________

___________________________________________________________ [3]

(b) _________________ [1]

19. In the diagram, ABC is a straight line. Given 9AB = cm, 5BD = cm and4

sin5

x = ,

find

(a) the length of BC,

(b) the value of ABDcos .

Ans: (a) BC= __________ cm [1]

(b) cos ABD = _______ [1]20. (a) The diagram shows the distance-time graph of a water boat.

13

4 6 10

800

1600

Distance (metres)

0

D

B

x

5 cm

9 cmA C


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Calculate, in metres per second,

(i) the speed during the first 3 minutes,

(ii) the average speed for the complete journey.

Ans: (ai) ______________ m/s [2]

(aii) ______________ m/s [2]

(b) The first two stages of another journey are shown in the distance-time graph in

the answer space. The water boat travels from A for 4 minutes. It then stops

for 2 minutes to pick up passengers before returning to A in a further 3

minutes. Complete the distance-time graph in the answer space.

Ans: (b) [1]

14

4 6 10

800

1600

Distance (metres)FromA

Time(minutes)0

Time(minutes)


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x

y

0

x

y

B

A

O C

21. (a) The curve ))(3( xkxy += , where kis a constant, cuts they-axis at point

A(0, 6) and thex-axis atB and Cas shown.

Find

(i) the value of k,

(ii) the coordinates of B,

(iii) the equation of the line of

symmetry

of the curve.

Ans: (ai) k= ______________ [1]

(aii) B ( ______ , ______ ) [1]

(aiii) _________________ [1]

(b) (i) Express 422 xx in the form qpx +2)( .

(ii) Hence, sketch the graph of 422 = xxy .

Show your intercepts clearly.

Ans: (bi) _________________ [1]

(bii)

[2]

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-----------------------------------------------END OF PAPER-------------------------------------------

SETTER: Mr Fredrick Yong

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x

y

(1, 5)4

3.241.24

ANSWER KEY

Qn Answers

1 (a) Walk (b) 115.7

2 (a) 12700 or 41.27 10(b)(i) 46.85 10 s

(b)(ii) 205.5 km

3 (a) 20 days(b) 92 ohms

4 (a) 0.430(b) $7094.10

5 (a) ( )p q C+

(b)2

p qC

6 (a)(i) HCF = 18(a)(ii) 22k =

(a)(iii) 18n =

(b) 2 minutes

7(a)

6

27

a b

(b) 12x =

8 (a) 338(b) 12n =

9 1=x , 4=y10 (a) 0p = or 4

(b) 16223 2 + xx

11(a)

13 5

11x<

(b) Indicate 3 and1

511

on the number

line, with a hollow dot above 3 and

black dot above1

511

. Draw a line

linking the 2 dots.12 (a) 25.5

(b) 6.69(c) The average score of Class 4E9 ishigher but the performance is lessconsistent than that of Class 4E8.

13 (a)(i) 38, 35(a)(ii) 533 + n

(b) nn 22 + or ( ) 112 +n

14 (a) 2240cm

(b) 3 : 2(c) 81 kg

15 (a) P(5, 0) Q (0, 12)

(b)12

5

(c) 13 units

(d)12

25

y x= or 5 12 10y x=

16(a)

3 7 8 11, , ,

5 19 19 19p q r s= = = =

(b)(i)14

95

(b)(ii)48

95

(b)(iii)81

95

(c)11

57

17 (a)

(b)(i)

3

1

(b)(ii) 34 units or 5.93 units

(b)(iii)6

5

18 (a)AP=BP(given)

DP= CP(given) APD = BPC(vert. opp. angles)

By SAS Property, trianglesADPandBCPare

congruent.

(b)3

7

19 (a) 3 cm

(b)3

5

20 (a)(i) 133

m/s

(a)(ii)2

23

m/s

(b) See Graph

21 (a)(i) k= 2

(a)(ii) )0,3(B

(a)(iii) x=2

1

(b)(i) 5)1( 2 x

(b)(ii) See Diagram22

1

O

A

B

a

b

PQ


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Ahmad Ibrahim Sec. Mid-Year 2012: E Math P1 iss Em My Exam 4e5na p1