Dunearn Prelim 2009 Em P2

2 Mathematical Formulae Compound interest

r   Total amount = P1    100 

n

Mensuration Curved surface area of a cone = πrl Surface area of a sphere = 4πr2 Volume of a cone =

1 2 πr h 3

Volume of a sphere =

Area of triangle ABC =

4 3 πr 3

1 ab sin C 2

Arc length = rθ, where θ is in radians Sector area =

1 2 r θ, where θ is in radians 2

Trigonometry a b c   sin A sin B sin C

a 2  b 2  c 2  2bc cos A Statistics Mean =

Standard deviation =

 fx f

 fx f

2

  fx    f   

2

3

Answer all the questions. 1 R

O

74

Q

S

36 P

A

O is the centre of the circle PQRS. The straight line AR cuts the circle at Q and C. AP is a tangent to the circle at P. Given that POQ = 74 and PAQ = 36, calculate, stating your reasons clearly

2

(a)

QRP,

[1]

(b)

QSP,

[1]

(c)

QPA,

[2]

(d)

PSR,

[2]

(e)

RPO.

[2]

A certain mass of aluminium costs $28. An alloy, which is 1 kg lighter, also costs $28. For a sculpture, John paid $55 for 10 kg of aluminium and 5 kg of alloy. Given that x kg of aluminium costs $28, find, in terms of x, expressions for (a) (i)

(b) (c) (d)

cost of 1 kg of aluminium,

(ii) cost of 1 kg of alloy. Form an equation in x and show that it reduces to 11x 2  95 x  56  0 . Solve the equation 11x 2  95 x  56  0 . Hence find the cost of 1 kg of aluminium.

[1] [1] [3] [2] [1]

4

3

(a)

The utilities company structures its charges as follows. Volume used per month

Rate (cents per m3)

20 m3 or below

75 3

90

3

105

On the next 20 m

On the next 60 m

Mr Lee’s family pays $43.08 in a certain month for water, calculate the volume of water used. (b)

[2]

Peter earned $55 800 from his employment and $750 from other sources such as investments. His total relief amounts to $3 635. [ Chargeable income = Total income − Total relief ] (i)

Calculate his chargeable income.

(ii)

Chargeable income is taxed at a rate of 2.25% for the first $40 000 and

[1]

8.5% on the amount above $40 000. Calculate the amount of income tax that Peter has to pay. (c)

[2]

Mr Lim put $5 800 in a 2-year Australian dollars fixed deposit account which gives a simple interest rate of 5.5% per annum. (i)

Calculate his initial deposit in Australian dollars, if the exchange rate is S$ 1.16 = A$ 1.

(ii)

[1]

If he withdraws all his money at the end of 2 years when the exchange rate is S$ 1.09 = A$ 1, calculate his gain in Singapore dollars.

[3]

5 The cumulative frequency curve below illustrates the weekly wages of 760 workers in Supermarket A.

Cumulative Frequency

4

weekly wages ($) (a)

(b)

Use the graph to find (i)

the median weekly wage,

[1]

(ii)

the lower quartile,

[1]

(iii)

the interquartile range,

[2]

(iv)

the percentage of workers earning $350 or less a week.

[2]

If one of the 760 workers is chosen at random, calculate, as a fraction in its simplest form, the probability that the worker earns more than $300 a week.

[1]

6 4

(c)

The weekly wages of another group of 760 workers in Supermarket B is illustrated by the box and whisker diagram below where

weekly wages ($) (i)

Compare the weekly wages of the two groups of workers in two different ways.

(ii)

[2]

If you are a worker seeking employment in either Supermarket A or Supermarket B, which supermarket will you choose to work for? Give a simple reason for your answer.

[1]

5

25 cm

The figure below shows a solid hemisphere attached to a solid cylinder. Part of the solid, in the shape of a cone, is removed from the base of the cylinder. The base area of the hemisphere is 162 cm2 and the height of the cylinder is 22 25 cm. (Take   ). 7

7 5

6

(a)

Given that the volume of the cone removed is 432 cm3, calculate

(i)

the height of the cone,

[1]

(ii)

the slant height of the cone.

[2]

(b)

Calculate the total volume of the solid figure.

[3]

(c)

Calculate the total surface area of the solid figure.

[4]

(a)

(b)

Factorise completely 8a 2  50 , (i) 3 xy  9by  15b  5 x . (ii) Two vertical poles of identical height stand on horizontal ground. The distance

[2] [2]

between the poles is A metres. When a cable of length D is suspended between the poles, the cable sags in the middle. The sag G is given by the formula G

(c)

3 AD  A 7

(i)

Find G when A = 40 and D = 48

[1]

(ii)

Express D in terms of G and A.

[3]

The ratio of an interior angle to an exterior angle of a regular polygon is 13:2. Find the size of the exterior angle and hence, the number of sides of the polygon.

[3]

8 7

T

S

8.4 m

78 7.2 m

R 9.5 m 6.4 m

P Q

In the diagram, P, Q, R and S are four markers on a horizontal ground and QRS is a straight line. PSR  78, PS = 7.2 m, PR = 9.5 m and QR = 6.4 m.

(a)

(b)

Calculate (i)

PRS ,

[2]

(ii)

the area of triangle PQR,

[2]

(iii)

the length of PQ,

[3]

(iv)

the shortest distance from R to the path PQ.

[2]

RT is a vertical flagpole at R such that RT = 8.4 m. Calculate the greatest angle of elevation of T as seen from the path PQ.

[2]

9 8 P A r B

O

1.5 radians

26 cm

Q

C R

In the figure, B is the centre of sector BPR. BP = BR = 26 cm. AQC is a circle of radius r, inscribed inside the sector and

PBR = 1.5 radians.

9

(a)

Find the arc length PQR.

[1]

(b)

Show that the radius, r = 10.5 cm correct to 3 significant figures.

[3]

(c)

Find the area of major sector AOCQ.

[3]

(d)

Find the area of the shaded region.

[4]

(a)

Sets C, P and M are defined by C = { teenagers who play computer games }, P = { teenagers who play games on the playstation }, M = { teenagers who play games on the M-Box }. Express the following statement in set notation. All teenagers who play games on the M-Box also play computer games.

[1]

(ii)

Describe which teenagers belong to the set M '  C  P',

[2]

(iii)

Describe which teenagers belong to the set (C  P)  M '.

[2]

(i)

10 9

(b)

The following table shows the number of tickets sold for two nightly performances to a musical at the Esplanade Theatre. Front Row 220 260

Saturday Sunday

Centre Row 410 410

Back Row 535 495

The price of tickets to the musical is as follows: Front Row: $80 Centre Row: $60 Back Row: $50

Circle Seat 335 355

Circle Seat: $35

The number of tickets sold for the two nightly performances can be  220 410 535 335  represented by the matrix A =  .  260 410 495 355  The price of tickets can be represented by a matrix B.

10

(i)

Write down the matrix B.

[1]

(ii)

Calculate AB.

[1]

(iii)

Describe what is represented by the elements of AB.

[1]

(iv)

1   1 Given that C =  , calculate D = AC. 1   1  

[1]

(v)

Describe what is represented by the elements of D.

[1]

(vi)

Given that E = 1 1, calculate F = ED and describe what it shows.

[1]

Answer the whole of this question on a sheet of graph paper. When x copies of a book are produced, the cost, $y, of each copy is given by the formula 3600 y  10  . x The table below gives some values of x and the corresponding values of y. (a) x y

100 46

200 28

300 22

400 19

600 16

900 14

1200 13

Using a horizontal scale of 2 cm to represent 200 books and a vertical scale of 2 cm to represent $5, draw a graph of y against x.

[3]

11

10

(b)

Use your graph to estimate the number of books to be printed if the cost of producing each book is $20.

(c)

(d)

[1]

(i)

By drawing a tangent, find the gradient of the curve when x = 400.

[2]

(ii)

State briefly what this gradient represents.

[1]

x   In order to sell x books, the selling price of each book must be $  30  . 50   x On the same axes, draw the graph of y  30  to represent the (i) 50 selling price of the books. (ii)

[2]

Using your graphs, find the range of the number of books that should be printed if no loss is to be made, assuming all the books will be sold.

END OF PAPER

[2]