2010 vs em4 prelim2 p2

Class Register Number

Name

4016/02 10/4P2/EM/2

MATHEMATICS PAPER 2 Wednesday 18 August 2010 2 hours 30 minutes

VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL VICTORIA SCHOOL

VICTORIA SCHOOL

SECOND PRELIMINARY EXAMINATION

SECONDARY FOUR

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. You may use a pencil for any diagrams or graphs. Do not use 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. You are expected to use a scientific calculator to evaluate explicit numerical expressions. 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 brackets [ ] at the end of each question or part question. The total number of marks for this paper is 100.

This document is intended for internal circulation in Victoria School only. No part of this document may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying or otherwise, without the prior permission of the Victoria School Internal Exams Committee.

Paper 2 consists of 9 printed pages, including the cover page. [Turn over

2

Mathematical Formulae

Compound interest

Total amount = 1100

nrP +

Mensuration

Curved surface area of a cone = rl Surface area of a sphere = 24 r

Volume of a cone = 213

r h

Volume of a sphere = 343

r

Area of triangle = 1 sin2

ab C

Arc length = r , where is in radians Sector area = 21

2r , where is in radians

Trigonometry

sin sin sina b c

A B C= =

2 2 2 2 cosa b c bc= + A

Statistics

Mean = fxf

Standard deviation = 22fx fx

f f

Victoria School 2010 4016/02/EM4/P2/10

3 1 18 students sat for a History and a Geography test. The stem-and-leaf diagram shows the History marks for the 18 students.

Stem Leaf

5 5 6 7 8 9 6 1 5 6 8 7 0 1 2 2 2 7 8 8 0 1

Key : 5 | 5 means 55 marks

Find (a) the median mark, [1]

(b) the interquartile range. [2]

The box-and-whisker diagram illustrates Geography test marks obtained by the 18 students.

80 90 50 60 7040Marks

(c) A Victorian claimed that the History test was more difficult then the Geography test. Do you agree? Give a reason for your answer. [2] 2 Two stalls A and B sell the same types of beverage. The table below shows the number of

cups and the selling price of each type of beverage sold for the day.

Mocha Tea Coffee Stall A 35 15 40 Number of cups Stall B 26 20 38

Selling price per cup $2.00 $1.20 $1.50

The information for the days sale is represented by the matrix . The

selling price of each cup of the different types of beverage can be represented by a column matrix P.

35 15 4026 20 38 = H

(a) Write down the column matrix P. [1] (b) It is given that . Find S and describe what is represented by its elements. [2] =S HP (c) The two stalls decided to increase their selling price of all the three types of beverage. Stall A increased its selling price by 18% and stall B increased its selling price by 20%. Write down another matrix Q such that QS gives the total income of both stalls after the price increase. Evaluate QS. [3] Victoria School 2010 4016/02/EM4/P2/10

4 3

A

B

C D

E

F

20

A, B, D, E and F are points on a circle. AB is a diameter of the circle. C is a point on AD and FB produced. It is given that AB BC= and 20BAC = . (a) Show that CD is an isosceles triangle. [2] F (b) Find angle ABD . [2] (c) Find angle AED . [1] (d) Find reflex angle . [2] DBF 4 (a) Paul deposited S$10 000 in a bank account which pays 4% simple interest per annum. Find the least number of years for the deposit to exceed S$11700. [2]

(b) John calculated that he would need A$7 500 for a holiday trip to Australia with his family. On a particular day, the banks exchange rates are as follows:

Selling Buying Singapore Dollar to

1 Australian Dollar 1.1931 1.1651 How much Singapore dollars did John have to change at the bank? [1] (c) Jake and Tom visited a car showroom together and each decided to buy a car of the same model. The list price of the car in the showroom was $80 000. (i) Jake paid for the new car in cash and was given a discount. It is given that he paid $70 400. Calculate the percentage discount. [2] (ii) Tom traded in his old car for $30 000. He then paid a cash deposit of 40% of the net price and paid the balance in equal monthly instalments over 5 years at a flat rate of 3.5% per annum. Calculate the monthly instalment. [3] Victoria School 2010 4016/02/EM4/P2/10

5 5 Two taps A and B run water at different speed. Tap A runs water at x litres per minute. Tap B runs water at a rate of 5 litres per minute faster than tap A. A rectangular tank with dimensions of 300 cm by 250 cm by 120 cm is to be filled with water. It takes 5 hours longer to fill the tank with water using tap A as compared to using tap B. (a) Find the volume of the tank in litres. [1] (b) Write down an expression, in terms of x, the time taken to fill the tank by using (i) tap A, [1] (ii) tap B. [1] (c) Form an equation in x and show that it reduces to 2 5 150 0x x+ = . [2] (d) Solve the equation . Hence, find the time taken, in hours, to fill the rectangular tank if both taps A and B are turned on together. [3]

2 5 150 0x x+ = 6 (a) Copy the Venn diagram and shade the region representing ( ) ( )' 'A B A B .

A B

[2] (b) A universal set and its subsets A and B are given by { } : is an integer, is an odd number between 0 and 11x x x= , { }: is a prime numberA x x= , { }: 7 6 23B x x= . (i) List the elements contained in the set 'A . [1] (ii) Find ( )n B . [2] (c) There are 28 boys in a class. Of these, 17 boys sing in the choir and 15 boys play the piano. It is given that

{ }boys in the class ,= { }boys who sing in the choirS = , { }boys who play the pianoP = . (i) Find the smallest value of ( )n S P . [2] (ii) Express in set notation { }boys who neither sing in the choir nor play the piano . [1]


6 7 Answer the whole of this question on a sheet of graph paper.

The variables x and y are connected by the equation2 3 2

4xy

x= + . Some corresponding

values are given in the following table.

x 0.5 0.6 0.8 1.0 1.5 2.0 2.5 3.0 y 4.06 3.09 p 1.25 0.57 0.50 0.76 1.25

(a) Find the value of p. [1] (b) Using a scale of 4 cm to represent 1 unit on each axis, draw a horizontal x-axis for and a vertical y-axis for 0 40 x 3 .5y . On your axes, plot the points given in the table and join them with a smooth curve. [3] (c) Use your graph to find the range of values of x for 1y < . [2] (d) By drawing a suitable straight line on your graph, find the solution of

2 12 12 5x xx

+ = + . [3] (e) By drawing a tangent, find the value of x of for which the gradient is 0.5. [2]

Q

R

P North S

16.5 10

8530

15

8

The diagram shows the locations of four towns P, Q, R and S on level ground where S is due north of Q. It is given that 30 , 15SQR PQS = = , 85QRS = , km and km.

10RS =16.5QP =

(a) Calculate QS. [2] (b) Calculate PS. [2] (c) Calculate the bearing of R from S. [1] (d) Calculate the bearing of S from R. [2] (e) A skyscraper of height 650 m is situated at S. Find the greatest angle of elevation of the top of the skyscraper as a man walks along QR. [2]


(f) A car traveled from Q to S. It stopped at a point T where the area of the triangle QPT is . Find the distance traveled. [2]

225 km

7 9 Diagram I Diagram II

O

30

A B

O

18

A piece of aluminium sheet is in a shape of a sector AOB with center O as shown in Diagram I. It is given that OA is18 cm and the angle . is 30AOB (a) Calculate the perimeter of the sector AOB. [2] (b) The aluminium sheet is used to make an inverted hollow cone by joining the edges OA and OB as shown in Diagram II. Calculate, (i) the base radius of the cone, [1] (ii) the height of the cone, [1] (iii) the volume of the cone. [1] (c) The inverted cone is filled with water to a depth of half its vertical height. Find the area of the aluminium that is in contact with the water. [2] (d) Some ball bearings, each of diameter 0.4 cm, are dropped into the cone until the water overflows. Find the maximum number of ball bearings that can be dropped into the cone before the water overflows. [4]


8 10 (a) The pupils in two classes took the same English test. Information relating to the results is shown in the table below.

Class A

Marks, m 5 < m 10 10 < m 15 15 < m 20 20 < m 25

Number of pupils 2 8 11 3

Class B

Mean = 15.625 marks

Standard Deviation = 3.8 marks

(i) Calculate the mean and standard deviation of the marks scored for Class A. [4] (ii) Compare and comment the results for the two classes in 2 ways. [2] (b) The cumulative frequency curve shows the mass distribution of 80 oranges.

(i) Find the median mass. [1] (ii) Find the interquartile range. [2] (iii) It is given that of the oranges are more than x g, find the value of x. [2] 42.5% (iv) Two oranges are chosen at random, one after the other, without replacement. Find the probability that one of the oranges has a mass less than 80 g and the other has a mass greater than 110 g. [2]


9 11


B

9b E D

O A C 12a

In the diagram, 12a and 9b. It is given that =OA =OB DBOD21= and OAOC

31= .

(a) Express, as simply as possible, in terms of a and b,

(i) , [1]

BC

(ii) DA . [1]

(b) Given that 41

of area of area =

ODAODE , find the position vector of E in terms of a and b. [2]

(c) Calculate the numerical value of ECBE . [3]

(d) Show that the areas of and ODE OCE are equal. [3]

(e) Find area of area of

BDEBOC

. [2]

End of Paper

Secondary Four Second Preliminary Examination 2010 Victoria School

Secondary Four Second Preliminary Examination 2010 Answer Key

1a 69 marks bi ' {1,9}A = ( ) 3n B = b 13 marks bii ( )n 4S P = ci

ii ( ) (' or ' 'S P S P ) 7a p = 1.91

c No. I do not agree. Geography test is more difficult because the median mark is lower i.e. 64 marks as compared to the median mark of History i.e. 69 marks.

c 1.15 x< < 2.75 0.85x = d

e 2.25x = 8a 19.9 kmQS =

2a P =

5.12.10.2

b 5.84 kmPS = 115 c

d 295 b

S = HP = = 35 15 4026 20 38e 3.7

5.12.10.2

148133

Elements in S represent the total amount of money made by each of stall A and B from the sale of all 3 types of beverages. f 11.7 km

c Q = ( )1.18 1.2 , ( ) 334.24 9a 140 cm 3b 70ABD = bi 16.5 cm c 110AED = bii 7.19 cm d Reflex 250DBF = biii 32050 cm 4a 5 years c 2233 cm

b S$8948.25 d 53554

ci 12% 10ai Mean 15.625 marks = cii $587.50 ii SD 4.034= 5a 9000 l bi 99 g

bi x

9000 mins ii 14 g

102 g bii 5

9000+x mins

iii

d 15 or 10x x= = , 6 hours iv 15632

6a

A B

10

Secondary Four Second Preliminary Examination 2010 Victoria School

11ai 4a 9b ii 12a 3b b 93a b

4+

c 3BEEC

=

e area of 1area of 2

BDEBOC

=

11

18 August 2010

2010 vs em4 prelim2 p2

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