Bowen 2009 Prelim Em P1 + Answers

Class

Full Name

Index Number

BOWEN SECONDARY SCHOOL MATHEMATICS Paper 1 Secondary 4 Exp / 5NA 2nd September 2009

2 hours

READ THESE INSTRUCTIONS FIRST 1. Write your name, class and register number in the spaces provided. 2. Answer all questions in this paper. Write all answers clearly in the space provided. 3. Use of electronic calculator is allowed in this paper. 4. If working is needed for any question, it must be neatly and clearly shown in the space below that question. Omission of essential working will result in loss of marks. 5. The number of marks is given in brackets [ ] at the end of each question or part question. 6. You should not spend too much time on any one question. 7. The total mark for this paper is 80.

DO NOT OPEN THIS PAPER UNTIL YOU ARE TOLD TO DO SO For Examiner’s

This document consists of 15 printed pages. Setter: Mrs Li

1

Mathematical Formulae

Compound interest r   Total amount = P1    100 

n

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 = Area of triangle ABC =

4 3 r 3

1 ab sin C 2

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

1 2 r  , where  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 =

2

 fx f

 fx f

2

  fx    f   

2

Answer ALL questions 1

(a)

Find the highest common factor of 180 and 156.

(b)

Find the smallest positive integer, k, such that 180k is a perfect cube.

Answer : (a) __________________ [1] (b) __________________ [1]

2

Amy sold a camera to Betty at a profit of 20%. Betty then sold it to Cindy at a loss of 30%. If Cindy paid Betty $210 for the camera, find the price at which Amy paid for the camera.

Answer : $ __________________ [2]

3

When an ink bottle is left uncapped, the ink evaporates at a rate of 0.2 microcubic metres per second. The bottle contains 45 cubic centimetres of ink. How long will it take for all the ink to evaporate? Give your answer in minutes and seconds.

Answer :

3

_________ mintues _________ seconds [2]

4

Express the following in the simplest form 2

(a)

 4y     3 

(b)

(3 p) 2  2 p 1 

3

64 p 6

Answer : (a) ____________________ [1] (b) ____________________ [2]

5

Solve the simultaneous equations. 8x = 7 + 4y 2y – 5x + 4 = 0

Answer : x = _____________ y = __________ [3]

4

6

(a) (b)

Solve the inequality

x3  4

2 

2 . 3

Hence write down the greatest and the smallest integers satisfying the inequalities.

Answer : (a) _______________________________ [2] (b) greatest integer = ________________ [1] smallest integer = ________________ [1]

7

(a)

Factorise completely 30 x  2 x 2  4 x 3

(b)(i)

Simplify v2 – ( v + 2w) (v – 2w)

(ii)

Using your answer in (b)(i) and showing clear workings, find the value of

4395 2  4399  4391

Answer : (a) ____________________________ [2] (b)(i) ___________________________ [2] (b) (ii) ___________________________ [1] 5

8

The scale of Map A is 1 : 2000 and the scale of Map B is 1 : 1000 (a)

If the length of a road is 8 cm on Map A, how long would the same road be on Map B?

(b)

If the area of a pond is 12 cm2 on Map A, what is the actual area of the pond in square metres?

Answer : (a) _________________ cm [1] (b) _________________ m2 [2] 9

Two pails are geometrically similar. The surface areas of the two pails are 28 cm2 and 63 cm2 respectively. (a)

What is the ratio of the height of the smaller pail to the height of the bigger pail?

(b)

Find the volume of the bigger pail, in cm3, if the volume of the smaller pail is 2.4 l ?

Answer : (a) ___________________ [1] (b) ________________ cm3 [2] 6

10

A polygon has n sides. Two of its interior angles are 110o and 100o and the remaining exterior angles are each 14o. Calculate n.

Answer : 11

n = ___________________ [2]

The diagram shows the sector OAB of a circle, centre O, and radius 8 cm. It is given that the length of arc AB is 10 cm.

(a)

Show that angle AOB is 1.25 radian.

(b)

Find the area of the shaded region.

[1]

Answer : (b) ________________ cm2 [2]

7

12

(a)

Lily invests $2500 in a bank that pays compound interest each year. At the end of 3 years, she receives $3327.50. What is the rate of compound interest per annum paid by the bank?

(b)

For the same interest rate as (a), how much is the difference in the amount of interest Lily receives if the bank is paying by simple interest rate instead.

Answer : (a) _________________ % [2] (b) $ __________________ [2] 13. (a)

y is proportional to xn.

(i)

Write down the value of n when y cm is the length of a cube with volume x cm3.

(ii)

y hours is the time taken to travel a fixed distance at a speed of x km/h.

(b)

The force, F , between two particles is inversely proportional to the square of the distance between them. The force is 48 units when the distance between the particles is p metres. Find the force when the distance is increased by 300%.

Answer : (a)(i) n = _______________ [1] (a)(ii) n = ________________ [1] (b) _______________________ units [2]

8

14

The universal set  and the sets A and B are given by:

 = { x : x is an integer, 6  x  15 }, A = { x : x is a multiple of 3} and B = { x : x is an integer, 4  (a)

1 x  7 }. 2

Draw a Venn diagram to illustrate the sets  , A and B. Indicate clearly the elements of each set.

(b)

List the members of A  B .

(c)

Find n ( A  B ).

(d)

A number y is chosen at random from  . Find the probability that y  ( A  B ) .

[2]

Answer : (b) A  B = ________________________________ [1] (c) n ( A  B ) = __________________ [1] (d) ______________________________ [1] 9

15

In the figure, PBR and PAQ are straight lines. It is given that PA = 3 cm, AQ = 7 cm, PB = 5 cm, BR = 1 cm and QR = 6 cm. Q 7 6

A 3 P 5

B

R

1

(a)

Prove that PQR is similar to PBA .

(b)

Hence find the value of AB.

(c)

Find the ratio of area of PQR : area of quadrilateral ABRQ.

[2]

Answer : (b) AB = ____________________ cm [2] (c) _____________________________ [1] 10

16

The graph shows the charges of company A for carrying out computer repairs. The total cost is made up of two parts. There is a fixed charge $a and a further charge of $b for each hour of work done.

Total Cost, $

No. of hours

(a)

Find the cost of a repair work which took 4 hours.

(b)

Find the values of a and b.

(c)

Another company B charges $30 per hour for carrying out repairs without a fixed charge.

(i)

Draw a line on the graph to represent the total cost of hiring company B.

(ii)

State with reason(s), which company will you engage to do a 3 hours repair work?

[1]

Answer : (a) $ ________________ [1] (b) a = ______________ [1] b=

_____________ [1]

(c) (ii) ________________________________________________________________ ______________________________________________________________ [1]

11

17

The speed-time graph below shows the motion of a moving object. The distance travelled during the first 6 seconds is 144m. Speed (m/s) 30

v

0

15

6

t

Time (s)

(a)

Find the initial speed, v, of the object.

(b)

Given that the rate at which the object slows down after 15 seconds is equal to half the rate at which it accelerates during the first 6 seconds, find the time t, at which the object comes to rest.

Answer : (a) __________________ m/s [2] (b) ____________________ s [2] (c)

Draw a clearly labelled sketch of the acceleration-time graph for the object on the given axes below. [2] Acceleration (m/s2)

0

6

15

Time (s)

12

18

(a)

Sketch the graph of y = ( x + 5 ) ( 3 – x ). Indicate on the graph the coordinates of the points that cross the x and y axes. y

x

0

[2] (b)

Write down the equation of the line of symmetry of y = ( x + 5 ) ( 3 – x ).

(c)

Write down the coordinates of the turning point of the curve.

Answer : (b) ________________________ [1] (c)

19

(a)(i)

The first 4 terms of a sequence are 1, 27, 125, 343, … Write down the next term of the sequence.

(ii)

Express, in terms of n , the nth term of the sequence.

(b)(i)

(ii)

(

,

)

[1]

In another sequence, the first 4 terms are 4, 30, 128, 346, … Write down the next term of the sequence. By comparing this sequence with your answer in (a)(ii), write down, without simplifying, the nth term of the sequence.

Answer : (a)(i) ____________________________ [1] (a)(ii) ___________________________ [1] (b)(i) ___________________________ [1] (b)(ii) ___________________________ [1]

13

20

Given that

 5  3  1 3  and B    . A    2 1    4 2

(a)

Evaluate A2 – 3B.

(b)

 x 9  Find the values of x and y if AB    2 y

Answer : (a) ____________________________ [2] (b) x = __________________________ y = _______________________ [2] 21

In a class of 40 pupils, 16 are girls and 5 of them are short-sighted. Among the 24 boys, 9 of them are short-sighted. If 2 pupils are selected at random, what is the probability that (a)

the first pupil selected is short-sighted,

(b)

2 short-sighted boys are selected

Answer : (a) ________________ [1] (b) ________________ [2]

14

22

North A

C

B

The diagram above is a plan of a triangular field ABC, drawn to a scale of 1 cm to 200 m.

(a)

Measure and write down the bearing of C from B.

(b)(i)

A tree, planted in the field, is equidistant from AC and BC. By constructing the angle bisector of  ACB, find the possible positions of the tree. [1]

(b)(ii)

Given that the tree is to be planted at a distance of 1.6 km from B. Indicate clearly, the position of the tree, labeled it as T. [1]

(c)

Use the diagram to find the distance, in metres, from T to A.

Answer : (a) ______________________ o [1] (c) _____________________ m [1]

  

End of Paper 1

15

  

Answer Key 1a

12

1b

150

2

$250

3

3 mins 45 seconds

4a

9 16 y 2 1 x= 2

4b

9 2p

6b

greatest integer = 5 , smallest integer = –5

5 6a

,

y= 

5  x < 5

3 4

2 3

7a

 2 x 2 x  5  x  3

7b

4w2 , 16

8a

16

8b

4 800

9a

2 : 3

9b

8 100

10

17

11a

10 = 8 θ θ = 1.25 rad (proved)

11b

9.63

12a

10 %

12b

$ 77.50

13b

3

14b

{ 9 , 12 }

14c

8

14d

3 10

PQ 10  2 PB 5  QPR =  BPA , PR 6  2 PA 3

15b

3

15c

4 : 3

16a, b

$100 , a = 40 , b = 15

16cii

Company A as it is cheaper.

17a

18

17b

45

18b

x = –1

18c

( –1 , 16 )

19a

729 , ( 2n – 1 )3

19b

732 , ( 2n – 1 )3 + 3

20a

 34  27    1  0 7 20

20b

x= 7 y = 4

21b

3 65

073o

22c

560 m

(i) ⅓

13a

(ii) –1

14

A 7 8 14

15a

21a 22a

B

6 15

9 12

10 11 13

16