Fairfield Methodist_11 Prelims_E Maths Paper 2

FAIRFIELD METHODIST SCHOOL (SECONDARY)

SECONDARY 4 Express / 5 Normal Academic

Preliminary Examination

MATHEMATICS 4016/02 Paper 2 15 August 2011

Additional materials: Foolscap Paper

Electronic calculator

Geometrical instruments

Graph paper

TIME 2 hours 30 minutes

READ THESE INSTRUCTIONS FIRST

Write your answers and working on the separate Answer Booklet/Paper provided.

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.

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.

Show all your working on the same page as the rest of the answer.

Omission of essential working will result in loss of marks.

The total of the marks for this paper is 100.

You are expected to use an electronic calculator to evaluate explicit numerical

expressions. You may use mathematical tables as well if necessary.

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

Setters: Mr Ganesan and Mdm Toh Siew Lan

This question paper consists of 13 printed pages.

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

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 = 22

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


1. The diagram shows a circle of center A with radius 5 cm. Given that ABCD is a

rectangle with AB = 3 cm, AD = 9 cm and P and Q are points of intersection of

the rectangle with the circle, calculate

(a) PAQ in radians, [2]

(b) the perimeter of the shaded region PQDC. [3]

2. (a) Make y as the subject of the formula

n y = m

ny 4. [3]

(b) Express as a single fraction in its simplest form 24

2

2

3

x

x

x

. [2]

(c) Factorise completely accbab 3962 . [2]

(d) Consider the pattern.

(i) Write down the next line. [1]

(ii) Find the value of 22 155156 . [1]

(iii) Without using calculator, use the number pattern above to evaluate

2222222 16151413121110 . [2]

22 23 = 3 + 2

22 34 = 4 + 3

22 45 = 5 + 4

22 56 = 6 + 5

=

P B

A Q

C

D


3. During the recent Mathematics Olympiad competition, the number of gold,

silver and bronze medals won by the various levels of an international school

are given in the table below.

Sec 1 Sec 2 Sec 3 Sec 4

Gold

9

8 7 9

Silver

5

10 9 2

Bronze

11

12 5 15

To reward the pupils for the excellent performance, the school decided to

award book vouchers of $12, $8 and $5 to each of the gold, silver and bronze

awardees.

(a) Write down a 43 matrix P to represent the information given in the

table above. [1]

(b) Given that Q = 5812 , evaluate QP. [1]

(c) What do the elements in QP represent? [1]

(d) Write down a matrix R such that QPR will give you the total amount

of money the school will use in rewarding the pupils. [1]

(e) Write down matrices S and T, such that ST will give you the total

number of medals received by the Secondary 2 pupils. [1]

(f) The total number of medals obtained by the same school last year is

represented by the matrix 5

4P. Describe the trend of the results over

the two years. [1]


4. (a) The diagram shows two concentric circles with their centre at O. AB is the

diameter of the bigger circle and ATC is a tangent to the smaller circle at T.

Name a pair of similar triangles, giving your reasons clearly. [3]

(b) In the diagram, AB is the diameter of a circle centre O.

P and Q are points on the circle. The lines AQ and PB when produced,

meet at R. Given that BAP = 28 and ARP = 43, calculate, stating

your reasons clearly,

(i) BOP, [1]

(ii) AQP, [1]

(iii) QPR, [1]

(iv) PBQ. [1]

B A O

T

C

A

O

Q

R

B

P

43

28


5. In the figure below, B is a point on CE produced. Such that CE: CB is 4 : 5.

D is the midpoint of AC. It is given that AC = x and AB = y.

(a) Express, in terms of x and y,

(i) BC , [1]

(ii) BE , [1]

(iii) DB , [1]

(iv) AE . [1]

(b) Given that CG = x + 4y, show that AECG // . [1]

(c) Given that AF : AE = 3 : 4, find the value of

(i) ACEofArea

ACGEofArea

, [1]

(ii) ABCofArea

BEFofArea

. [2]

(Diagram is not drawn to scale)

B

A

C

D

E

F

G


6. A motorised toy car has 2 pairs of wheels. The radius of one pair of wheels is

x m while the radius of the other pair of wheels is 03.0x m.

The toy car can travel 200 metres in 5 minutes.

(Take,7

22 )

(a) Write down an expression, in terms of x and , for the number of

revolutions made by the smaller pair of wheels per minute. [2]

(b) Write down an expression, in terms of x and , for the number of

revolutions made by the larger pair of wheels per minute. [1]

(c) If the smaller pair of wheels makes 245 revolutions more than the

bigger pair of wheels in one minute, form an equation in x and show

that it reduces to .062317700 2 xx [3]

(d) Solve the equation and state, in cm, the diameter of the larger wheel. [4]


7. The diagram shows a triangular field XYZ, which is surrounded by fences.

It is given that XZ = 98 m, YZ = 166 m. The bearing of Z from X and Z from Y

is 280 and 305 respectively.

(a) (i) Find the bearing of X from Z. [1]

(ii) Show that angle XZY = 25. [1]

(iii) Find the distance XY. [3]

A vertical tower is standing at X. The angle of depression of Z when viewed

from the top of the tower is 19.

(b) Calculate

(i) the height of the tower, [2]

(ii) the area of the field XYZ. [2]

(c) A boy walked along the fence YZ, calculate the greatest angle of

elevation of the top of the tower viewed from path YZ. [3]

98 m

166 m

N


8. The marks obtained by a group of 40 pupils from Secondary 4Y in a recent

Mathematics Test are represented in the cumulative frequency curve below.

(a) Use your graph to find the

(i) the median mark, [1]

(ii) the lower quartile, [1]

(iii) the interquartile range, [1]

(iv) the value of y if 20% of the pupils scored more than y marks.

[1]

Marks

10

20

30

40

0 20 40 60 80 100

Cum

ulativ

e Freq

uen

cy


(b) Copy and complete the group frequency table of the marks of the

pupils. [2]

(c) Use the above group frequency table to calculate an estimate of

(i) the mean mark of the pupils, [1]

(ii) the standard deviation. [2]

9. The drama team from Helix Secondary School constructed some props for

their performance in the schools 130th anniversary celebrations. Each prop is

made up of 5 identical pieces of solid E that formed the shape of a hemisphere

of radius 1.2 m and centre O. It stood flat on the stage and 20 such identical

props were made for the performance.

(a) Calculate,

(i) the volume, in m3 ,of solid E, [2]

(ii) the total surface area, in m2 ,of solid E. [3]

Marks 200 x 4020 x 6040 x 8060 x 10080 x

Frequency

Solid E

O

Prop

O


After the production was over, the school sold all the hemispherical props to a

hardware manufacturer. The manufacturer melted the props and made them

into a number of solid cones of radius 50 cm and a height 90 cm.

(b) (i) Calculate the maximum possible number of solid cones that

can be obtained. [3]

(ii) If the cost of producing a solid cone is $3.65, find the cost of

producing another geometrically similar cone with radius

150 cm. [2]

10. In 2007 three partners, Angel, Maggie and Nicole started a cake shop. They

invested money in the ratio 6 : 2 : 7 and agreed to share the profit in the same

ratio as their investments.

(a) In 2008, Nicole received $11 050 more profit than Maggie. Calculate

the total profit for that year. [2]

(b) The cash price of a 46 Full HD LED TV is $3199. Angel used her

profit earned to buy the television. This price included the 7% GST.

Calculate, correct to the nearest cent, the GST paid for the TV. [2]

(c) Maggie bought the same television using a hire purchase scheme. She

paid 15% of the cash price as deposit, followed by 24 monthly

installments of $125. Calculate the annual rate of simple interest

charged on the television under this scheme. [3]

50 cm

90 cm


(d) Nicole decided to invest $288 000. She exchanged $288 000 for

American dollars at the rate of US$1 = S$1.675.

She placed the American dollars at a US bank that pays 5.5% per

annum compound interest compounded half yearly.

A year later, she withdrew the whole sum of money including the

interest earned from the bank and exchanged it back to Singapore

dollars at a rate of US$1 = S$1.5423. Calculate the total amount of

money, in Singapore dollars, she received. [4]

(e) The total profit earned in 2009 was 18% more than that of 2008 and

the total profit earned in 2010 was 25% less than made of 2009.

Express the total profit earned in 2010 as a percentage of the total

profit earned in 2008. [2]


11. Answer the whole question on a piece of graph paper.

The cost of producing a plush toy, $y, when x plush toys are produced, is

given by the formula 22280

x

y , 0x .

The table below show some corresponding values of x and y.

x 10 20 30 40 50 60 70

y 50 36 s 29 27.6 26.7 t

(a) Find the value of s and of t. [1]

(b) Using a scale of 2 cm to represent 10 units on the horizontal axis and 2

cm to represent $5 on the vertical axis, draw the graph of 22280

x

y .

[3]

(c) Use your graph to estimate the number of push toys that have to be

produced in order for the cost of producing each plush toy to be $35.

[1]

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

x = 15. [2]

(e) The selling price of each plush toy is x4.045$ .

(i) On the same axis draw the graph of .4.045 xy [1]

(ii) Estimate the profit earned, in dollars, if 25 plush toys are to be

produced and all the 25 plush toys were sold. [1]

End of paper


Sec 4 & 5 Preliminary Exam 2011

Mathematics Paper 2 Answers

1a 0.644 rad 5c(i) 6

5c(ii) 1/20

1b 15.2 cm 5a(i) -y+x

5a(ii) 1/5(x-y)

2a 5a(iii) -1/2 x +y

5a(iv) 1/5x+4/5y

2b 6a

6b

2c

2d(i) 6d x = 0.0167 or -0.0467 m

2d(ii) 311 9.34 cm

2d(iii) 181 7a(i) 100o

7a(iii) 87.6m

3a 7b(i) 33.7 m

7b(ii) 3440 m2

7c 39.2o

8a(i) 46 marks

3b 8a(ii) 36 marks

8a(iii) 20 marks

3c Each element represents the total 8a(iv) 60

amount of prize money to be

given to each level.

8c(i) Mean = 47.5

Std D = 19.59

3d

9a(i) 0.724 m3

9a(ii) 4.98 m2

9b(i) 307

3e 9b(ii) $98.55

or S = 10a $33,150

T = 10b $209.28

10c 5.16% p.a.

10d $279,969.16

10e 88.50%

3f There is an increase of 25% in 11a s= 31.3 t=26

the total number of medals this year 11c 22

compared to last year. 11d -1.2

m

nmny

4

2282

22

28

xx

xor

xx

x

cba 323

6767 22

1551211

29105

9789

p

199181236203

1

1

1

1

R

12108S

1

1

1

T

x

20

)03.0(

20

x

111

12

10

8


11e $37.50

4a Triangles CAB and TAO.

4b(i) 56o

4b(ii) 62o

4b(iii) 19o

4b(iv) 133o

Documents

Fairfield Methodist_11 Prelims_E Maths Paper 2