css 4e5n prelim 2009 emath paper 1 - with answer key

Name : ( ) Class :

COMMONWEALTH SECONDARY SCHOOL

PRELIMINARY EXAMINATION 2009

SECONDARY FOUR EXPRESS / FIVE NORMAL ACADEMIC

MATHEMATICS

Paper 1

Date : 27 August 2009

Candidates answer on the Question Paper.

4016/01

Time : 2 hours

0800 – 1000

READ THESE INSTRUCTIONS FIRST

Write your name, class and index number on all the work you hand in.Write in dark blue or black pen.You may use a soft pencil for any diagrams or graphsDo 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 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 .

At the end of the examination, fasten all your work securely together.The number of marks is given in the brackets [ ] at the end of each question or part question.The total number of marks for this paper is 80.

This question paper consists of 14 printed pages including this page.

CSS/Prelim 2009/Sec 4E5N/EMath P1/Chua IL/pg1 of 14

80

Mathematical Formulae

Compound Interest

Total amount =

MensurationCurved surface area of a cone = πrl

Surface area of a sphere =

Volume of a cone =

Volume of a sphere =

Area of triangle ABC =

Arc length = rθ, where θ is in radians

Sector area = , where θ is in radians

Trigonometry

a2 = b2 + c2 - 2bc cos A

Statistics

Mean =

Standard deviation =

Answer all the questions.


1 Rearrange the formulae to express y in terms of x.

Answer : [2]

2 Solve the equation .

Answer : x = [2]

3 Factorize fully the expression .

Answer : [3] 4 Mercury orbits around the Sun in 88 days, Venus does the same in 225 days and Earth

takes 360 days. The last time an eclipse occurred (when the Sun, Mercury, Venus and CSS/Prelim 2009/Sec 4E5N/EMath P1/Chua IL/pg3 of 14

BA

Sun Mercury

Earth

Venus

Earth are set in a straight line) was in the year 1992.

By writing 88, 225 and 360 into the product of their prime factors, find the year in which

the next eclipse would occur on Earth.

Answer : [3]

5 Mrs Cheah drove at 60kmh-1 for the first 1hour 20 minutes and 90kmh-1 for the rest of her

journey. If the whole journey took 2hours, find the exact value of the average speed of

Mrs Cheah’s journey, leaving your answer in ms-1.

Answer : [3] 6(a) In the Venn diagram, shade the region

. [1]


(b) Given that , and

. If , find the largest and smallest possible

values of p.

Answer : largest p = , smallest p = [2]

7 Write out the largest prime number satisfying the inequality .

Answer : the largest prime number = [3] 8 In one particular month, Hafizah gives her parents 15% of her salary, spends 5% on food,

1/5 on entertainment and 1/4 on rent. She uses the rest of her salary to invest in a


structured deposit that pays compound interest of 2% per year. Her rent is $1600.

(i) Find Hafizah’s salary.

(ii) Calculate the total interest she will receive in three years from her investment.

Answer : (i) [1]

(ii) [3]

9 The diagram shows a section of a regular 12-sided polygon which is cut from a circular

piece of paper of radius 5cm. All the vertices of the polygon lie on the circumference of

the circle. Find

(i) one interior angle of the polygon,

(ii) the amount of paper discarded, leaving your answer in terms of .

Answer : (i) [1]

(ii) [3]

10(i) Write the expression into the form .


6cm

29cm

15cm

y cm

C B

A

D

(ii) Hence sketch the graph of , showing clearly the turning point and

the x and y intercepts. [3]

Answer : (i) [2]

11 In the diagram, , AC = 29cm,


BD = 15cm, DC = 6cm and AD = y cm. Calculate

(i) the value of y,

(ii) the value of tan ADC, without solving for any angles.

Answer : (i) [2]

(ii) [2]

12 The variables x, y and z are related. z varies directly as the square of x, y varies inversely

as the cube root of z, and when x = 1, y = 1 and z = 27.

(i) Find an expression for z in terms of x and y in terms of z.

(ii) Hence show that . [1]

Answer : (i) z = , y = [4] 13 A piece of land on level ground is in the shape of an isosceles triangle ABC with the sides


N

A

C

AB = AC. The diagram, drawn to a scale of 1cm : 2m, shows the side AC.

Given that the bearing of B from A is 160,

(i) draw the triangle ABC and write down the length of BC in metres. [1]

(ii) A tree T is to be planted so that it is equidistant from points A and C and equidistant from

lines AC and BC. Construct the perpendicular bisector of AC and the angle bisector of

angle ACB and mark clearly with the point T the position of the tree. [3]

Answer : (i) BC = [1] 14 The table below shows the scores obtained when a die is thrown a number of times.


Score 1 2 3 4 5 6No. of times 3 4 x 1 2 3

(i) Write down the maximum value of x if the modal score is 2.

(ii) Write down the minimum value of x if the median score is 3.

(iii) Find the median score if the mean score is 23/7.

Answer : (i) [1]

(ii) [1]

(iii) [3]

15 The equation of a line l is y – 2x + 6 = 0.

(i) Find the equation of the line parallel to line l and which passes through the point (1,-2) .

(ii) Line l cuts the y-axis at A and the x-axis at B and B is the midpoint of the line AC. Find

(a) the coordinates of the point C,

(b) the length of AC.

Answer : (i) [2]

(ii)a) [2]

(ii)b) [2] 16 The price of a ticket in each category at the Night Safari is given below:


(i) The number of tickets sold on one weekend is given as follows.

Adult Senior Citizen ChildSaturday 52 85 125Sunday 102 40 63

By putting the prices into a column matrix A and the number of tickets sold as matrix B,

find the matrix C given by C = BA and describe what is represented by the elements of C.

(ii) To improve the revenue during weekends, two plans are proposed :

Plan 1 : Increase the price on Sunday only by 30%.

Plan 2 : Increase the price by 15% on each day.

A 1x2 matrix P is such that PC gives the revenue for the weekend under Plan 1. Another

1x2 matrix Q is such that QC gives the revenue for the weekend under Plan 2.

(a) Evaluate PC and QC.

(b) State which plan would be more profitable.

Answer : (i) C = . C represents [2]

(ii)a) PC = QC = . [2]

(ii)b) [1] 17 Two open troughs X and Y are geometrically similar


,

12cm

8cm

d cm

5cm

X

Y

prisms with 2 trapeziums and 3 rectangles making

up their sides.

The ratio of the sides of trough X to the sides of

trough Y is 1 : 4. If the capacity of the trough Y

is 1200 cm3, calculate

(i) the ratio of the surface area of X to Y.

(ii) the capacity of the trough X.

(iii) the depth d cm of trough Y.

Answer : (i) [1]

(ii) [2]

(iii) [3]


0

speed (m/s)

time (s)

6

3

20 50 T

0

distance (m)

time (s)20 50 T

18 The diagram shows the speed time graph of

a cyclist over a period of T seconds.

The cyclist sees a stretch of wet road ahead

and slows down uniformly from 6m/s to 3m/s

in 20 seconds. He then progresses at constant

speed for 30 seconds, passing the stretch of

wet road, before gaining speed uniformly

to 6m/s at T seconds.

(i) Given that the cyclist’s speed is 3.6m/s at t = 60s, find the value of T.

(ii) Find the average speed of the particle for the first 50 seconds.

(iii) On the axes in the answer space, sketch the corresponding distance-time graph

for the period of T seconds, indicating the values of distance travelled clearly.

Answer : (i) [2]

(ii) [2]

(iii)

[2]


L

N

M

P Q

RO

a

b

19 In the diagram, OPQR is a parallelogram. M is the midpoint of OQ, N is the midpoint of

OM and L is the point on OR such that OL = 2LR.

(a) Given that a and b , express as simply as possible in terms of a and b,

(i)

(ii)

(iii)

(b) Explain why NP and LM are parallel.

(c) Find the following ratios.

(i)

(ii)

Answer : (a)(i) [1] (ii) [1] (iii) [1]

(b) [1]

(c)(i) [1] (ii) [1]

End of Paper


Answer Key

1) 2) x = -1

3) a(b+2)(b-2)(1+c)(1-c) 4) 2047

5)

6)a) 6)b) smallest p = 30 , largest p = 35

7) 5 8)i) $6400 ii) $137.11

9)i) 150 ii) 10)

11)i) y = 25 ii) 12)i)

13) BC = 17.6m 14)i) 3 ii) 2 iii) 2.5

15)i) y = 2x – 4 ii) (6,6) , 13.4 units

16)i) ii) PC = $7159.10 , QC = $7136.90 , Plan 1 is more profitable

17)i) 1/16 ii) 18.75 iii) 6

18)i) T = 100 ii) 3.6

19ai) ii) iii) b) ci) 1/3 cii)


css 4e5n prelim 2009 emath paper 1 - with answer key

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