Jurongville Prelim 2009 Em P1

JURONGVILLE SECONDARY SCHOOL PRELIMINARY EXAMINATION 2009

Elementary Mathematics (PAPER 1) Secondary 4 Express/ 5 Normal (Academic) 2nd September 2009 (Wednesday)

Duration Marks

Name

: ________________________________ [

: 2 hours (0745 - 0945) :

] 80

Class

: 4___ / 5N ___

Parent’s Signature: _______________________

INSTRUCTIONS: Write your name and index 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 staples, paper clips, highlighters, glue or correction fluid. Answer ALL questions. Write your answers on the question paper. 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  . 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 80.

*Observe our school values of Integrity and Excellence by not cheating and doing your best in this paper

DO NOT OPEN THE BOOKLET UNTIL YOU ARE TOLD TO DO SO Setter: Mdm Lucy Wu

This document consists of 18 printed pages and 2 blank pages. [Turn over

2

Mathematical Formulae

n

Compound interest

r   Total amount = P1    100 

Mensuration

Curved Surface area of a cone = rl Surface area of a sphere = 4r 2 Volume of a cone =

1 2 r h 3

Volume of a sphere =

4 3 r 3 1 ab sin C 2

Area of triangle ABC =

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

Trigonometry

1 2 r  , where  is in radians 2

a b c   sin A sin B sin C

a 2  b 2  c 2  2bc cos A

Statistics

Mean =

 fx f

Standard deviation =

 fx f

2

  fx    f   

2

3

1)

(a)

A new car is valued at $55 000. At the end of each year its value is reduced by 15% of its value at the start of the year. What will it be worth after 6 years?

(b) Throughout his life Mr Bean’s heart has beat at an average rate of 72 beats per minute. Mr Bean is 60 years old. How many times has his heart beat during his life? Give your answer in standard form correct to two significant figures.

2)

Ans: (a)

$__________________

[1]

(b)

___________________

[1]

6 men take 4 hours to dig a hole. Find the time taken if 8 men are available. If it takes 1 hour to dig the hole, how many men are there? 2

Ans: _________________________ h

[1]

______________________ men

[1]

[Turn over

4

3)

On a map of scale 1 : 20 000 the area of a forest is 50 cm2. On another map the area of the forest is 8 cm2. Find the scale of the second map.

Ans: ___________________________

4)

[2]

A group of NCC cadets are marching for their National Day Parade March Past. If they marched in pairs, one cadet is without a partner. If they marched in 3’s, 5’s and 7’s, there will be one cadet without a partner. Calculate the smallest number of pupils in the March Past contingent.

Ans: ___________________________

[2]

5

5)

p

q

r

Find a formula for the shaded part in terms of p, q and r.

Ans: ___________________________

6)

[2]

An intelligent fish lays black or white eggs and it likes to lay them in a certain pattern. Each black egg is surrounded by six white eggs.

Here are 3 black eggs and 14 white eggs. (a)

How many eggs does it lay altogether if it lays 200 black eggs?

(b) How many eggs does it lay altogether if it lays n black eggs?

Ans: (a)

___________________

[1]

(b)

___________________

[1]

[Turn over

6

7)

Each packet of Quaker Cereal carries a token and 5 tokens can be exchanged for a free packet. How many free packets will I receive if I bought 125 packets? Show clearly your working.

Ans: ___________________________

[2]

7

8)

2 (a) Simplify 12 p 8  16 p (b) Showing your working clearly, solve the equation 3  1 (3x  2) 2  64

Ans: (a)

___________________

[1]

(b)

x = ________________

[2]

7

9)

(a)

Channel U starts its transmission every evening at 17 30 and finishes at 03 45 the following morning. Calculate the duration of its transmission, giving your answer in hours. (b) If the minute hand of the clock moves through a duration of 36 minutes, what is the corresponding angle that the hour hand move?

Ans: (a)

_________________ h

[1]

(b)

_________________ 

[2]

Ans: (a)

___________________

[1]

(b)

___________________

[1]

(c)

___________________

[1]

10) If   { x: x is an integer such that 10  x  100} A = { x: x when divided 15 leaves a remainder 7} B = { x: x when divided by 17 leaves a remainder 14} (a)

Find the value of n (A).

(b) Find the value of n( A  B). (c)

State the largest member of B.

[Turn over

8

11) (a)

Write the number 438 pico in standard form. 1 1 (b) If it takes of a second to write a zero and sec to write an integer other than a 5 10 zero, how long does it take to write the number?

Ans: (a)

___________________

[1]

(b)

________________ sec

[2]

12) Given that 5  x  10 , y  k , where k is a constant and the smallest value of y is 2, find x3 (a)

the value of k,

(b) the largest value of y, (c) the largest value of x  y . xy

Ans: (a)

k = ________________

[1]

(b)

___________________

[1]

(c)

___________________

[1]

9

13) During the National Day Celebration, JVS ordered T-shirts of various sizes for the students. The matrices show the order of various sizes of T-shirts and the cost ($) for the various sizes.

XL Boys  220 Girls  50

(a)

Find 1

 220 1   50 

L

M

240

180

60

210

S XL  15    L 13.50  M  12    S  9.20 

85   135 

240

180

60

210

85   135 

(b) Explain what your answer (a) represent. (c)

Using matrix multiplication, find the total amount the school has to pay.

___________________

[1]

(b) __________________________________________________________________

[1]

Ans: (a)

(c)

$ __________________

[Turn over

[1]

10

14) In the diagram, ABCDEF is a regular hexagon and APQRB is a regular pentagon. PXR is a straight line. Calculate (a)

 BAP,

(b)

ABX .

(c)

Show that PR is parallel to AB.

[2] D

C R

Q E

B

X P F

A

Ans: (a)

__________________ 

[1]

(b)

__________________ 

[1]

15) John sells cylindrical tins of pet food in his provision shop. The height of a tin is 8 cm and the cost price is $3.20. He intends to sell another type which is geometrically similar to the ones he is selling but whose height is 4 cm. Calculate the price to the nearest one cent that he should sell this smaller tin so that he makes a profit of 35%.

Ans: $ _________________________

[4]

11

16) In the diagram, O is center and angle PQT = 28o and angle RPT = 40 o, find the bearing of (a)

P from T,

(b) R from T, (c)

P from R. N

P 40

Q

2 8

T

O

R

Ans: (a)

__________________ 

[1]

(b)

__________________ 

[2]

(c)

__________________ 

[1]

[Turn over

12

17) In a video game, a spot on the screen bounces off the four sides of a rectangular frame. The spot moves from A to B to C to D, as shown in the diagram below.

(a)

What can you say about the angle at which the spot bounces off each side, compared with the angle at which it approached the side?

(b)

3 The column vector describing the part AB of the movement is   . Write down 2 the column vector describing BC and CD.

(c)

From D the spot moves to E on the fourth side of the frame. Given that the spot continues in the same way, write down the vector DE.

(d) Given that the spot bounces off the fourth side at E and continues to move, describe its subsequent path.

Ans: (a)

___________________

[1]

(b)

___________________

[2]

(c)

___________________

[1]

(d) __________________________________________________________________

[1]

13

18) A racing car manufacturer makes two models, the Alfa and Beta which are alike in every aspect except body design and styling. To determine if the body design of the Alfa has less wind resistance, both cars were tested on the Bukit Speedway. The following table gives lap times in minutes on a 5 km track; Alfa Beta (a)

1.0 1.3

0.9 1.2

1.0 1.0

0.8 0.9

0.9 1.1

1.0 0.9

0.9 1.4

1.0 1.3

Find the mean lap times for the Alfa and the Beta.

(b) Find the standard deviation for each car. (c)

Comment on the performance of each car.

Ans: (a)

(c)

Alfa _______________ Beta _______________ (b) Alfa _______________ Beta _______________ _________________________________________________________________

[2] [2] [1]

19) Express y= x 2  10 x  31in the form ( x  a) 2  b where a and b are integers. Hence sketch the graph of y. State its minimum value and the line of symmetry. [1]

Sketch

Ans: ___________________________

[2]

Min value __________________

[1]

Line of symmetry ____________

[1]

[Turn over

14

20)

B

y

x

A

p

D

q

C

z

In the diagram, ABC  ADB  90 0 , AD = p and DC = q. (a)

Use similar triangles to show that x 2  pz .

(b) Find a similar expression for y 2 . (c)

Add the two expressions for x 2 and y 2 and hence prove the Pythagoras Theorem.

Ans: (a)

__________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________

[2]

(b) __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________

(c)

[2]

__________________________________________________________________ __________________________________________________________________ __________________________________________________________________

[1]

15

21) The diagram shows a grid of squares. A button is placed on one of the squares. The six faces of a fair die are marked 1, 2, 3, 4, 5and 6. The fair die is tossed. If ‘1’ is shown, the button is moved one square upwards followed by one square to the left. If ‘2’ or ‘3’ is shown, the button is moved one square downwards. If ‘4’ is shown, the button is moved one square to the right. If ’5’ or ‘6’ is shown, the button is moved two squares to the left. The table below illustrates the movement. Number shown

Direction of movement

P

1 U

G

I

2 or 3

D

4

E

5 or 6 (a)

R

The button is placed on the square I. The die is tossed once. Find the probability that the button finishes at the square U.

(b) On another occasion, the die is tossed once and the button is moved. When the die is tossed a second time, the button is moved again. (i)

If the button is placed at G, find the probability that the button finishes at square R.

(ii)

If the button is placed on the square R, find the probability that the button finishes at the square U.

Ans: (a)

___________________

[1]

(b)(i)

___________________

[2]

(b)(ii)

___________________

[2]

[Turn over

16

22)

Speed in ms 1

80

24

0

Time in sec 4

10

15

The diagram is the speed-time graph of an object during a period of 15 seconds. (a)

(i)

Calculate the retardation during the first 4 seconds.

(ii)

Calculate the distance travelled in the first 4 seconds.

(iii)

Given that the acceleration of the object between t = 10 and t = 15 is 20 ms 2 , calculate the speed of the object when t = 15.

_______________ ms-2

[1]

(a)(ii)

________________ m

[1]

(a)(iii)

_______________ ms-1

[1]

Ans: (a)(i)

17

(b) On the axes in the answer space, sketch the distance –time graph, indicating the distance travelled, for the journey. Distance in m

0

4

10

15

Time in sec

[Turn over

[2]

18

23) (a)

Using ruler and compasses, construct  XYZ in which XY = 10 cm, YZ = 9 cm and XZ = 8 cm.

(b) Measure, and write down, the size of the smallest angle in the triangle. Give a reason for your choice. (c)

A point P is 3.8 cm from X and also equidistant from Y and Z. On your diagram and using suitable constructions find and mark the point P. Constructions: (a) XYZ (c)

[2] [2]

Ans: (b)

__________________ 

[1]

Reason ___________________________________________________________

[1]

19

BLANK PAGE

[Turn over

20

No 1

Answer a) $20743.22 b) 2.3 X 109

No 13

2

3 hours 48 men

14

Answer a) 270 300 390 220 b) The total order for XL, L, M and S sized T-shirts respectively. c) $14 804 a) 1080 b) 60 0 QRF  36 0 c)

3 4

1 : 50 000 211

15 16

5

q+r-p

17

6

A0 802 b) 4n + 2

18

7

31

19

 PRB  72 0 PRB  RBA  180 0

Since they are interior angles, PR is parallel to AB. $0.54 a) 3320 b) 2200 c) 0120 a) Equally inclined to the horizontal or negative gradients to each other.

 6    9 b) BC    , CD      4   6   6 c)    4  d) E will move to A and continue the path in the same pattern. a) Mean Alfa: 0.9375 Beta:1.1375 b) Std deviation Alfa: 0.0696 Beta: 0.180 c) Alfa is a better performing car because it has consistently smaller lap times. y  ( x  5) 2  6 Min value = 6 Line of symmetry: x = 5.

8 a) 3 p b)



20

1 8

x4

2 3

a) Proof b) y 2  zq

21

c)

9

1 a) 10 h 4 b) 18 0

21

10

a) b) c) a)

22

11

b) 12

a) b) c)

6 1 99 4.38 x 10 -10 1 2 sec 10 k = 2000 16 0.8

23

x 2  y 2  qz  zp

 z( p  q)  z 2 This is the Pythagoras Theorem. 1 a) 6 1 b)i) 18 1 ii) 9 a) i) 14 ms-2 b) ii) 208 m c) 124 ms-1 b) 49 0  10 Reason: Angle opposite the shortest side would be the smallest angle.

[Turn over