Fairfield Em 1 Prelim 2009

FAIRFIELD METHODIST SCHOOL (SECONDARY) PRELIMINARY EXAMINATION 2009 SECONDARY 4 EXPRESS/ 5 NORMAL ACADEMIC

MATHEMATICS

4016/01

Paper 1 Date: 18/08/09 Candidates answer on the Question Paper. 2 hours _____________________________________________________________ READ THESE INSTRUCTIONS FIRST Write your name, index number and class 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. 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 80. For Examiner’s Use Paper 1

/ 80

Paper 2

/ 100

Total

This question paper consists of 16 printed pages.

%

Name: _______________________________ (

)

Class: ______

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 a 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 =

 fx f

Standard deviation =

FMS(S) Sec 4 E xp / 5 N(A) Preliminary Examination 2009 Mathematics Paper 1

fx 2   fx    f   f 

2

2

Name: _______________________________ (

)

Class: ______

Answer all the questions. 1

(a)

Evaluate, correct to 1 significant figure, the value of

(b)

If a  4.6  10 3 and b  3  10 5 , calculate

19.93  0.5692 3

7950

.

a , giving your answer in standard b

form.

Answer (a)……..…………………… [1] (b)……..…………………… [1]

2

A length of 2 cm on a map represents an actual distance of 1 km. Calculate (a) the actual distance represented by 60.7 cm on the map giving your answer in kilometres, (b) the area of the map, in square centimetres, which represents an actual area of 20 km 2 .

Answer (a)……..………………… km [1] (b)……..……………….. cm 2 [1] FMS(S) Sec 4 E xp / 5 N(A) Preliminary Examination 2009 Mathematics Paper 1

3

Name: _______________________________ (

)

Class: ______

3

3

(a) (b)

 5  Simplify  2  . y  Given that 3 24  27  3 x , find the value of x .

Answer (a)……..…………………… [1] (b) x  ..…….……………… [1]

4

Peter participated in a 40-km marathon and took 3 hours and 40 minutes to complete it. (a)

The marathon started at 07 10. At what time did Peter complete the race?

(b)

Find Peter’s average speed.

Answer (a)……..…………………… [1] (b)……..……………… km/h [1]

5

In the diagram, ABC is a straight line. Given AB = 9 cm, BD = 5 cm and 4 sin x  , find 5 (a) BC, (b) the value of cos ABD . 5 cm A

9 cm

B

D

x C

Answer (a)……..……………….cm [1] (b)……..…………………... [1]

FMS(S) Sec 4 E xp / 5 N(A) Preliminary Examination 2009 Mathematics Paper 1

4

Name: _______________________________ (

6

)

Class: ______

The figure shows two semicircles of diameters a cm and 3a cm. Find the perimeter of the shaded region. Give your answer in the form x + y  cm.

a 3a

Answer……………..………… cm [2]

7

One thousand identical drops of AQUA liquid of density 0.7 g/cm3 are found to have a total mass 0.000 000 045 g. (a)

Write 0.000 000 045 in standard form.

(b)

Calculate the volume of one drop, in m3, giving your answer in standard form.

Answer (a)……..………………….. [1] (b)……..………………m 3 [2] FMS(S) Sec 4 E xp / 5 N(A) Preliminary Examination 2009 Mathematics Paper 1

5

Name: _______________________________ (

8

)

Class: ______

In the diagram, the midpoint of AB lies on the y-axis. B is (6, 4). y

.

B (6 , 4) x

─8

.

A

(a)

Write down the coordinates of the point A.

(b)

Find the equation of AB.

(c)

Another line parallel to the x-axis and passing through (8, 5) meets AB produced at C. Find the coordinates of C.

Answer (a)

( .……… , ……… ) [1]

(b)…….....………………… [2] (c)

9

( .……… , ……… ) [1]

When the cost of fuel rose by 10%, Mr Tan decreased his fuel consumption by 10%. Find the percentage decrease in Mr Tan’s expenditure on fuel consumption.

Answer ……..…………………% [3]

FMS(S) Sec 4 E xp / 5 N(A) Preliminary Examination 2009 Mathematics Paper 1

6

Name: _______________________________ (

10

)

Class: ______

Two provision shops in Dover neighbourhood sell the same type of biscuits and toilet paper. A box of biscuits costs $2.50 at Shop A while a pack of toilet paper costs $4.25 at Shop B. Cindy and May plan to buy the following quantities. Biscuits (in boxes)

Toilet Paper (in Pack)

Cindy

7

4

May

6

2

Cindy needs to pay $41.50 for her supply regardless of whether she buys from Shop A x   7 4  2.50  41.50 41.50   , S    and T   . or Shop B. It is given that R   4.25  29.50   6 2  y  27 (a) By forming an equation connecting R, S and T, find the values of x and y. (b) Explain the significance of x and y.

Answer (a) x  ...……and y  ………… [2] Answer (b)………………………………………………………………………… …………………………………………………………………….......

11

Given that a 

[1]

4b2  1 , express b in terms of a and c. 7c  b 2

Answer……..…………………… [3] FMS(S) Sec 4 E xp / 5 N(A) Preliminary Examination 2009 Mathematics Paper 1

7

Name: _______________________________ (

12

)

Class: ______

Let  = x: x is an integer,  3 ≤ x ≤ 15, M =  x: x is a prime number  and N =  x: x is a perfect square. (a) Represent the sets M and N in a Venn diagram. (b) (i) List the elements of M '  N . (ii) Find n ( M  N ' ) .

[2]

Answer (a)

Answer (b)(i) ...…………………... [1] (b)(ii)....…………………... [1]

13

During the investigation of a car accident, the traffic police calculated the length of the skid, s metres, which is directly proportional to the square of the speed of the car, v km/h. (a) Write down the formula for s in terms of v and a constant, k. (b) If the car skidded for 40 m while travelling at 120 km/h, find (i) the value of k, (ii) the speed of a car which skidded 75 m.

Answer (a) ……………..………… [1] (b)(i) k = ...……………….. [1] (b)(ii)....………………km/h [1]

FMS(S) Sec 4 E xp / 5 N(A) Preliminary Examination 2009 Mathematics Paper 1

8

Name: _______________________________ (

14

(a)

)

Class: ______

Tins of Milk Powder Produced

XXX Milk

XXX Milk

2008

2007

The diagram is taken from a report of a Milk Powder Factory. It claims to show that the number of tins produced decreased by 50% between 2007 and 2008. Explain briefly why the report might be considered misleading. Answer (a) ………………………………………………………………………. …………………………………………………………………………………… …………………………………………………………………………………[2] (b)

The number of toy cars that students in a class have is shown in the table below. Number of toys

0

1

2

3

4

5

Number of students

4

12

x

15

4

5

If the median is 3, write down an inequality satisfied by x.

Answer (b)……..………………….. [2]

15

(a)

Factorise completely 6 st  2tz  9 s  3 z .

Answer (a) ………………………… [2] FMS(S) Sec 4 E xp / 5 N(A) Preliminary Examination 2009 Mathematics Paper 1

9

Name: _______________________________ (

15

(b)

)

Class: ______

Simplify a 2  (a  1)2  30 .

Answer (b) ……..………………….. [2]

16

(a) (b)

If each interior angle of a n-sided regular polygon is 156 , find the value of n. ABCDE is a pentagon such that A  125 , B  146 and D  117 . If  C :  E = 3 : 5, find  E.

Answer (a) n = ……..…….……….. [2] (b)……..…………………. o [2] FMS(S) Sec 4 E xp / 5 N(A) Preliminary Examination 2009 Mathematics Paper 1

10

Name: _______________________________ (

17

)

Class: ______

(a)

Find the smallest integer value of n for which 2160n is a perfect square.

(b)

During the Formula-Two competition, 3 cars start to drive around a 3-km track at the same time. It is observed that car Farara completes one lap at every 120 seconds, car Toyoda completes one lap at every 560 seconds and car Mercury completes one lap at every 480 seconds. How long will it take for the 3 cars to be at the starting position again?

Answer (a) .………..…………….... [1] (b)…….…………………..s [2]

18

(a)

(i) (ii)

Solve the inequality 4 ≤ 5x ≤ 2x + 5. 2x  5 4 If x  and x  list the integer(s) which satisfy this 5 5 condition.

Answer (a)(i) ……….…………….. [2] (a)(ii) ...………………….. [1]

FMS(S) Sec 4 E xp / 5 N(A) Preliminary Examination 2009 Mathematics Paper 1

11

Name: _______________________________ (

18

(b)

)

Class: ______

Solve the simultaneous equations.

2y  8  x 3 x  4 y  19

Answer (b) x  .………………………

y  …………………... [3]

19

A cylinder B has a volume of 400 cm 3 . Calculate the volume of 1 (a) a cylinder C with base radius that of B and height 4 times that of B, 3 (b) a cylinder D similar to B but with a curved surface area 9 times that of B.

Answer (a)……..………………cm 3 [2] (b)……..……………….cm 3 [2] FMS(S) Sec 4 E xp / 5 N(A) Preliminary Examination 2009 Mathematics Paper 1

12

Name: _______________________________ (

20

)

Class: ______

The diagram is the speed-time graph of a car. Speed (m/s) 60 50 40 30 20 10 0 10 20 30 40 50 60 70 (a) (b) (c) (d)

80

Time (s)

Calculate the acceleration of the car at 55 seconds. Calculate the distance travelled by the car during the first 30 seconds. Calculate the speed of the car after 76 seconds. On the axes in the answer space, sketch the distance time graph for the first 50 seconds of the motion of the car.

Answer (d) Distance (m) 1600 1400 1200 1000 800 600 400

[2]

200 0 10 20 30 40 50 60 70

80

Time (s)

Answer (a)……..………………m/s 2 [1] (b)……..…………………..m [1] (c)……..…………………m/s [1] FMS(S) Sec 4 E xp / 5 N(A) Preliminary Examination 2009 Mathematics Paper 1

13

Name: _______________________________ (

21

)

Class: ______

8  6  Given that OA    , OB    and that M and N are midpoints of OA and OB 1   3 respectively. (a)

Calculate AB .

(b)

Express MN as a column vector.

Answer (a) ……………………units [2] (b) …….………………….. [2]

22

(a)

(i) (ii)

Sketch the graph of y  x (9  x) . Write down the equation of the line of symmetry of y  x (9  x) .

Answer (a) (i)

y

0

x

[2]

Answer (a)(ii) ………..…………… [1]

FMS(S) Sec 4 E xp / 5 N(A) Preliminary Examination 2009 Mathematics Paper 1

14

Name: _______________________________ (

22

(b)

)

Class: ______

From graphs below, select one which illustrates each of the following statements.

y

y

x 0

y

x

x

0

0

Figure 1

Figure 3

Figure 2

y

y

x 0

x 0

Figure 4

(i) (ii) (iii)

Figure 5

The surface area y, of a cone is proportional to the square of the radius, x. The speed y, of a car varies inversely as the time, x. The cost y, of the rental of a handphone consists of a fixed charge and the number of calls made, x.

Answer (b)(i) Figure …….……….. [1] (b)(ii) Figure …….……….. [1] (b)(iii) Figure …………….. [1] FMS(S) Sec 4 E xp / 5 N(A) Preliminary Examination 2009 Mathematics Paper 1

15

Name: _______________________________ (

23

)

Class: ______

The scale drawing in the answer space below shows the positions of the towns A, B and C. The map is drawn to a scale of 1 cm to 10 km and B is due east of A. (a)

Find the bearing of A from C.

Answer (a)……..…………………. o [1]

(b)

(c)

By using ruler and compasses, (i) construct the perpendicular bisector of AB, (ii) construct the angle bisector of ABC .

[1] [1]

The fourth Town D is 95 km from C and the bearing of Town D from Town A is 350 . Using ruler and compasses only, find and label the position of Town D on the scale drawing. [2]

Answer (b) and (c) North

C

A

B

End of Paper

FMS(S) Sec 4 E xp / 5 N(A) Preliminary Examination 2009 Mathematics Paper 1

16

Name: _______________________________ (

Qn 1a

Answer 0.6

1b 2a 2b 3a

1.53  10 8 30.35 km

3b 4a 4b

Paper 1 Qn 14a

80 cm 2 y6 125 21 10 50 10.9 or 10

10 km/h 11

4 5 x 5 3 1

3 5 2a +2a 

18aii

9 10a 10b

11 12 a

4.5  10 6.43  10 17 (6,20) y  2x  8 1 ( 6 ,5) 2 1% x  3.5 y  6 y is the cost of a pack of toilet paper in shop A and x is the cost of a box of biscuit in shop B. 7 ac  1 b 4a E N 0,1,4,9

Tins are not of the same size Size is not proportional to decrease in numbers - Diagram is not a pictogram

95 o 15 3360s

5b

cm

-

16b 17a 17b 18ai

8

Answer

0 x8 (2t  3)(3s  z ) 2a + 29 n  15

3 cm

6 7a 7b 8a 8b 8c

Class: ______

14b 15a 15b 16a

5a



)

18b 19a 19b 20a 20b 20c 20d

x  3 y  2.5 178 cm 3 10800 3 0 m/s 2 500 m 8 m/s Distance km/h

1500

1000

200

M 2,3,5,7, 11,13

20

40

50

Time/s

-3, -2, -1, 6, 8, 10, 12, 14, 15

12bi 12bii

{0, 1, 4, 9} 15

21a 21b

13a 13bi

s  kv 2 1 k 360

22aii 22b

13bii

164 km/h

231

FMS(S) Sec 4 E xp / 5 N(A) Preliminary Examination 2009 Mathematics Paper 1

4.47 units  1     2 x  4.5 i) Figure 5 ii) Figure 3 iii) Figure 2 232.5 o 17