2012_prelim_iii__maths_4016-1

CHS 2012 Prelim III Maths 4016/1 Page 1

CATHOLIC HIGH SCHOOL

PRELIMINARY EXAMINATION III

Subject : Mathematics 4016/1 Date : 12 September 2012

Level : : Secondary 4 Time : 11 00 13 00

Name : _________________________ ( ) Class : Sec 4 - ____

This document consists of 16 printed pages.

READ THESE INSTRUCTIONS FIRST

Write your name, class and index number in the spaces at the top of this page.

Write in dark blue or black pen. You may use a pencil for any diagrams or graphs.

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. The number of marks is given in brackets [ ] at the end of each question or part question. The total of the marks for this paper is 80.

For Examiner's Use

80


Mathematical Formulae

Compound interest

Total amount =

nr

P

1001

Mensuration

Curved surface area of a cone = lr

Surface area of a sphere = 2 4 r

Volume of a cone = hr 3

1 2

Volume of a sphere = 3 3

4r

Area of triangle ABC = Cba sin 2

1

Arc length = r , where is in radians

Sector area = 2 2

1r , where is in radians

Trigonometry

C

c

B

b

A

a

sin

sin

sin

Acbcba cos 2 222

Statistics

Mean =

f

xf

Standard deviation =

22

f

xf

f

xf


1 Calculate 8.96

(2.86 1.19)5.56 , giving your answer correct to 2 significant figures.

Answer ..[2]

2 SBS bus services P, Q and R leave the bus interchange at regular intervals. Bus service P

leaves every 6 minutes, bus service Q leaves every 8 minutes and bus service R leaves

every 13 minutes. All three bus services start their first service together at 06 05. At

what time do they next leave the bus interchange together?

Answer ..[2]

3 A computer operating system has a storage capacity of 0.34 terabytes.

(a) Express 0.34 terabytes in bytes, giving your answer in standard form.

(b) A 3-minute long video file requires 45 megabytes of storage capacity. Find the

number of video files that can be stored in the operating system.

Answer (a).[1]

(b)[1]


4 John was deciding between two geometrically similar bottles of orange juice in the supermarket. The ratio of their heights is 4 : 5. The cost of the smaller bottle is 90

cents and the bigger bottle is $1.70. Which size of bottle gives the better value?

You must show all your working clearly.

Answer ..bottle gives the better value because

[2]

5 Given that ={Quadrilaterals}, P ={Parallelograms}, Q = {Rectangles} and

R = {Rhombuses}.

(a) Give the name of the geometrical shape for the elements classified in the shaded

region in the Venn diagram.

(b) T is a set such that P T , draw on the same Venn diagram, the set T.

Answer : (a) ..[1]

(b) On the diagram [1]

6 (a) During the holidays, Daniel logged onto the Internet at 18 10 on Tuesday and

only logged off at 04 45 on Thursday. For how long was he connected to

the Internet?

(b) In the severe conditions of the desert, the difference between the day and night

temperatures is 54C. If the night temperature reaches 13 C, find the day temperature.

Answer (a)hoursmins[2]

(b)C.[1]


7 Benjamin bought a Samsung Galaxy for S$961, inclusive of 7% goods and services tax

(GST).

(a) Calculate the amount of GST paid.

(b) The exchange rate between US dollars (US$) and Singapore dollars (S$) was

US$1 = S$1.21. He sold the phone to a friend for US$691. Calculate his loss.

Answer (a) S$.....................................[1]

(b) S$.....................................[2]

8 (a) Simplify 41 3x .

(b) Solve the equation 4 2 3 3 2 2y y .

Answer (a) ...[1]

(b) y =......[2]

9 (a) y is directly proportional to 3x . It is known that 9y for a particular value of x.

Find the value of y when this value of x is halved.

(b) 10 men planned to complete painting a housing estate in 8 days. In actual, only a

quarter of the job was completed at the end of 3 days. Assumed that all men

worked at the same actual rate, how many more men were needed for the rest of

the job to be completed as scheduled?

Answer: (a)[1]

(b)....men [2]


10 A shop sells three types of sugar, White sugar, Fruit sugar and Brown sugar. Each variety

of sugar is sold in packets of two different sizes, small and large. The sale in two

consecutive months are given in the table below.

August September

Small Large Small Large

White sugar 20 15 22 17

Fruit sugar 15 22 11 18

Brown sugar 12 18 8 6

The information for Augusts sales can be represented by the matrix A =

20 15

15 22

12 18

.

The information for Septembers sales can be represented by the matrix B =

22 17

11 18

8 6

(a) Find C, where 2

A BC .

(b) Given that P =1

1

and that D = CP, find D.

(c) Hence, state what the elements of D represent.

Answer: (a) ..[1]

(b) [1]

(c)..

. [1]


11 (a) Solve the inequality 6 2 5 13x . (b) Write down the smallest integer which satisfy 6 2 5 13x .

Answer: (a) x .[2]

(b) ..[1]

12 Factorise 2 24 ( 4)x x x and hence solve the equation 2 24 ( 4)x x x .

Answer : . [2]

x = ,,[1]

13 The diagram shows the graph of2y ax bx c . Find the values of a ,b and c.

Answer: a =.[1]

b =.. .[1]

c =.... [1]


14 (a) Simplify 22( 3 ) (3 )

10

p q q p

r r

.

(b) Mr Samy bought y eggs at $x per dozen. He sold them for 15 cents each.

Find and express, as a single fraction in terms of x and y, for the profit, in cents,

that he made.

Answer (a) ........ [2]

(b) .cents [2]

15 The length of 40 leaves from a Frangipani tree were measured and recorded in the

frequency table shown below.

Length (x cm) 4 8x 8 12x 12 16x 16 20x 20 24x

Frequency 3 7 14 11 5

Find an estimate of

(a) the mean,

(b) the standard deviation.

Answer (a) .... cm [2]

(b) cm [2]


16 (a) Express 232 8x x in the form 2

q x p .

(b) Sketch the graph 232 8y x x on the given axes.

Answer: (a) ...[2]

(b)

[2]


17 The points P and Q are (2 , 7) and (8 , 5) respectively. The point R is such that

3

5QR

.

(a) Find the coordinates of R.

(b) It is given that 12

RSh

. Find the two possible values of h which will make

PQRS a trapezium. You may use the grid below to help you with your investigation.

Answer (a) , [2]

(b) h =..or .. [2]

P

Q


18 The diagram shows the speed-time graph of a particle after passing a point A.

(a) Find the average speed of the particle for the whole 35 seconds of its motion.

(b) On the diagram in the answer space, complete the corresponding acceleration-

time graph of the whole 35 seconds of its motion, indicating the values of the

acceleration on the vertical axis clearly.

Answer: (a) .m/s[2]

(b)

[2]

Speed (m/s)

Time (s) 20 30

5

50

35 0

Acceleration 2( / )m s

Time (s) 20 30 35


19 A frustum and a cone were obtained by slicing a conical container, height 2h, as shown in

Diagram I at the midway of the height. These figures were then attached to a cylinder, height

h, to form a new container as shown in Diagram II. Water was poured into the empty

container at a constant rate from the top and it took 66 minutes to fill to the brim.

Given that it took a minutes for the water to reach the container to a height of h and b minutes

to reach the height 2h.

(a) Find the value of a and of b.

(b) On the grid in the answer space, sketch the graph of the depth of water (d) against the

time (t).

`

Diagram I Diagram II

Answer: (a) a =..[1]

b = ..[1]

(b) [3]

depth (d)

3h

2h

h

a b 66 time (t) O


20 Box A contains 8 marbles numbered 1, 2, 3, 4, 5, 6, 7 and 8.

Box B contains 6 marbles numbered 2, 3, 5, 7, 8 and 9.

Box C contains 5 marbles numbered 1, 2, 3, 6 and 8.

A box is chosen at random and a marble is selected at random from the box.

Find the probability that

(a) Box B is chosen,

(b) Box C is chosen and the marble selected has an even number on it,

(c) Box A is chosen and the marble selected has a prime number on it,

(d) the marble selected has a prime number on it.

Answer: (a) [1]

(b) [1]

(c)..[1]

(d) [2]


21 The points P(2,6), Q(2,9) and R(14,1) are vertices of a triangle.

Find

(a) the gradient of PR, (b) the length of PR, (c) the area of triangle PQR,

(d) the value of cos QPR ,

(e) the coordinates of S such that PQRS is a parallelogram.

Answer: (a) [1]

(b) [1]

(c) [1]

(d) [1]

(e) , [1]

1 2 3 4 5 6 7 8 9 10 11 12 13 14

1

2

3

4

5

6

7

8

9

P

Q

R

x

y

O


22 The diagram shows a scale drawing of the positions of two submarines X and Y.

(a) Given that submarine Z is such that the bearing of Z from X is 210 and the

bearing of Z from Y is 150, construct and label clearly the submarine Z on the

diagram. [2]

(b) Construct (i) the angle bisector of angle YXZ, [1]

(ii) the perpendicular bisector of XY. [1]

(c) A ship is

I equidistant to XY and XZ,

II equidistant to X and Y.

Mark the position of the ship. Label this point R. [1]

X

Y

North


23

PQRST is a pentagon in which 104PQR QPT , 76QRS ,

RSX PTS x . QP and ST produced meet at X. (a) Calculate the value of x . (b) State, showing your reasoning, whether PQ is or is not parallel to SR. (c) Show that triangle PXT is an isosceles triangle.

Answer: (a) x = ..[2]

(b) PQ parallel to SR because ..

..

..[2]

(c)..

..

..[2]

~~~~~~~~~~~~~~~~~~End of Paper~~~~~~~~~~~~~~~~

X T S

P

Q

R 104

76

x x

104

2012_prelim_iii__maths_4016-1

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