4E5N Math P2 Prelim 2009 With Ans

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MONTFORT SECONDARY SCHOOL

O LEVEL PRELIMINARY EXAMINATION 2009

________________________________________________________________

Secondary 4 Express / 5 Normal

MATHEMATICS 4016/02

PAPER 2

Wednesday 2 September 2009 1100 1330 2 h 30 min

Additional Materials: Answer Paper

Graph Paper (1 sheet)________________________________________________________________

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 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 notexact, give the answer to three significant figures. Give answers in degrees toone 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 part

question. The total number of marks for this paper is 100.

________________________________________________________________This document consists of 11 printed pages.

Setter: A. Low


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Mathematical Formulae

Compound interest

Total amount =

nr

P

+

1001

Mensuration

Curved surface area of cone = rl

Surface area of a sphere = 24 r

Volume of a cone = hr2

3

1

Volume of a sphere = 3

3

4r

Area of triangleABC= ab2

1sin C

Arc length = r, where is in radians

Sector area = 2

2

1r , where is in radians

Trigonometry

C

c

B

b

A

a

sinsinsin==

bccba 2222 += cosA

Statistics

Mean =

f

fx

Standard deviation =

22

f

fx

f

fx


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Answer all the questions

1. (a) Simplify2

2

9 6

2 18

x x

x

+

. [2]

(b) Express as a single fraction in its simplest form,

6 52

3 4

y z

y z

+

.

[2]

(c) (i) Express 2 7 9x x in the form ( )2

x a b . [1]

(ii) Hence, solve the equation 2 7 9 0x x = giving your answerscorrect to two significant figures. [2]

2. The diagram shows a tent. BCFEis the rectangular base of the tent. ABCandDEF are vertical ends of the tent.BC=EF= 1 m, BE = CF = 2.2 m andAB =AC=DE=DF= 1.3 m.

.

(a)Show that the height of the tent is 1.2 m. [2](b)Calculate the volume of the tent, [2](c)Calculate the total surface area of the tent. [2](d)The canvas used to make the tent costs 0.02 cents per cm2. Calculate the

cost of the canvas used to make the tent. [2]


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3. A boy buys a 500 ml packet of mango juice for $1.50 and a 1 litre packet oforange juice for $2.50.

(a) Mango juice costsx cents more per litre than orange juice. Find the valueofx. [2]

The two packets of juices are geometrically similar.

(b) Given that the height of the packet of orange juice is 25 cm, calculate theheight of the packet of mango juice. Give your answer correct to twodecimal places. [2]

The boy makes a bowl of fruit punch with the juices and water. He mixes water,mango juice and orange juice in the ratio 1 : 2 : 5.

(c) Find the largest volume, in litres, of fruit punch he can make. [2]

(d) Find the percentage of juice left unused. [2]

4. The diagram shows a rectangleABCD. Mis the midpoint ofAD and CMintersectsDNat P. BC= 3 cm, CD = 7 cm andDN= 5 cm.

(a) Write down the value of sin BND. [1]

(b) Calculate

(i) MCD, [2](ii) ADN, [2](iii) MPD. [2]

A B

CD

M

N

7 cm

3 cm

P

5 cm


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5. A boat travelled 200 m downstream (with the current) fromA toB. AB is parallelto the banks of the river. The speed of a boat in still water is v m/s and the speedof the current is 2 m/s.

(a) Find an expression, in terms ofv, for the time taken, in seconds, for theboat to travel downstream fromA toB. [1]

(b) The boat then travelled upstream (against the current) fromB toA.

Find an expression, in terms ofv, for the time taken, in seconds, for theboat to travel fromB toA. [1]

(c) The total time taken for the journey downstream and upstream is4 minutes.

Write down an equation to represent this information, and show that it

simplifies to 23 5 12 0v v = . [3]

(d) Solve the equation

2

3 5 12 0v v =

. [2]

(e) Find the time, in seconds, for the boat to travel fromB toA. [1]

A B200 m

current

DownstreamUpstream


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6. The diagram shows a circleABCE. The tangents at CandEmeet atD. AngleGCD = angle GED = 90 and angleAMB = 54.

(a) Explain why G is the centre of the circle. [2]

(b) Find

(i) angleACB, [1](ii) angle CAB, [2](iii) angle CDE. [3]

(c) Show that trianglesABG andECG are similar. [2]

(d) State where the centre of the circle CDEG lies. [1]


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7. The diagram shows a horizontal field PQRS.R is due east ofS.PQ = 245 m, SR = 290 m, angle PSR = 47, angle SPR = 70 andangleRPQ = 36.

(a) Calculate the bearing ofP fromR. [2]

(b) Show that PR = 225.7 m. [2]

(c) Calculate

(i) how farR is south ofP, [2](ii) the distance QR, [3](iii) the shortest distance from P to QR. [2]

(d) A bird is 50 m vertically above P. Calculate the largest angle ofelevation of the bird when viewed from any point along QR.[2]


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8. (a) The cumulative frequency curve shows the marks for Mathematics Paper 1of 80 candidates. The maximum mark of Paper 1 is 80.

(i) Use the graph to find

(a) the median mark, [1](b) the upper quartile, [1](c) the interquartile range, [1](d) the 35

thpercentile. [1]

(ii) To score a distinction in Paper 1, a candidate has to attain morethan 67 marks. Two candidates are chosen at random. Find theprobability both of them attained distinction. [2]


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(b) The same 80 candidates sat for Mathematics Paper 2. The box andwhiskers diagram illustrates the marks obtained. The maximum mark forPaper 2 is 100.

(i) Find the range of marks for Paper 2. [1]

(ii) A candidate is chosen at random. Write down the probability thatthis candidate attained at most 40 marks for Paper 2. [1]

(iii) Compare the marks for Papers 1 and 2 in two different ways.[2]

(iv) Giving a reason, state whether Paper 1 or Paper 2 was moredifficult. [1]


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9. The diagram shows two circles with radii 6 cm with centres at O and at P. Thecircle with centre P passes through O. The two circles intersect atA andB.

(a) Explain why angleAOB =2

3

rad. [2]

(b) Find the perimeter of the shaded region. [2]

(c) Find the area of the shaded region. [3]

10.The first five terms of a sequence and the difference between successive terms areshown below.

Sequence 4 13 26 43 64 a

Difference 9 13 17 21 b

(a) Write down the value ofa and ofb. [2]

(b) Find an expression, in terms ofn, for the n th difference. [1]

(c) Find the tenth difference. [1]

(d) Given that the n th term of the sequence is given by 2pn qn r + + , find the

value ofp, of q and ofr. [3]


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11.The variablesx andy are connected by the equationy =x

x2

2

1 . Some

corresponding values ofx andy are given in the following table.

x 0.5 1 1.5 2 2.5 3 3.5 4y 3.75 1.5 p 0 0.45 0.83 q 1.5

(a) Find the value ofp and ofq giving your answers correct to two decimal places.[2]

(b) Using a scale of4 cm to represent 1unit draw a horizontalx-axis for

0.5 x 4.

Using a scale of 2 cm to represent 1 unit on theyaxis, draw a verticaly-axis

for 4 y 2.

On your axes, plot the points given in the table and join them with a smoothcurve. [3]

(c) Use your graph to find the range of values ofx for which 2 2 4 0x x .[2]

(d) By drawing a tangent, find the gradient of the curve at the point (1, 1.5).[2]

(e) On the same axes, draw the graph of 4 3 6x y+ = . [1]

(f) (i) Write down thex coordinate of the point where the two graphs intersect.[1]

(ii) The value ofx is a solution of the equation 2 12 0Ax Bx+ = . Find thevalue ofA and the value ofB. [2]

End of Paper

Answers

1. (a)3

2( 3)

x

x

+(b)

13

4 3

z

z y

(c) (i) ( )2

3.5 21.25x (ii) x = 8.1, -1.1

2. (a) 1.2 m (b) 1.32 m3

(c) 9.12 m2

(d) $18.243. x = 50 (b) 19.84 cm (c) 1.6 L (d) 6.67 %4. (a) 0.6 (b) (i) 12.2 (ii) 53.1 (iii) 49.0

5. (a)200

2v +(b)

200

2v (d) v = 3 (e) 200 s


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6. (a) Since GE and DE are perpendicular and GC and DC are perpendicular,GE and GC are radii (rad perpendicular to tan). Radii of the same circle intersectat the center. Hence G is the center.(b) (i) 54 (ii) 36 (iii) 72(c) AG = EG and BG = CG (radii), angle AGB = angle EGB (vert opp)

By SAS, triangles ABG and ECG are congruent.(d) Midpoint of DG

7. (a) 333 (c) (i) 201 m (ii) 147 m (iii) 222 m (iv) 12.78. (a) 42 marks (b) 51 marks (c) 15 marks (d)39 marks

(ii)3

632

(b) (i) 78 marks (ii) 0.25(iii) P2 median is higher than P1 and it has a larger spread of marks.(iv) P1 is more difficult as it has a lower median that P2.

9. (a) Since PAO and PBO are equilateral triangles, angle AOB =2

3

.

(b) 37.7 cm (c) 68.9 cm2

10. (a) a = 89 b = 25 (b) 4n + 5 (c) 45 (d) p = 2, q = 3, r = -1

11. (a) p = -0.58, q = 1.18 (c) 3.4x (d) 2.5(f) (i) x = 1.75 (ii) A = 11, B = -12

4E5N Math P2 Prelim 2009 With Ans

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