Fairfield Em 2 Prelim 2009

FAIRFIELD METHODIST SCHOOL (SECONDARY) SECONDARY 4 Express / 5 Normal Academic Preliminary Examination MATHEMATICS Paper 2

4016/02 20 August 2009

Additional materials: Foolscap Paper Electronic calculator Geometrical instruments Graph paper TIME

2 hours 30 minutes

READ THESE INSTRUCTIONS FIRST Write your answers and working on the separate Answer Booklet/Paper provided. Write your name, class and index number on all the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use a pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer ALL questions. 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. Show all your working on the same page as the rest of the answer. Omission of essential working will result in loss of marks. The total of the marks for this paper is 100. You are expected to use an electronic calculator to evaluate explicit numerical expressions. You may use mathematical tables as well if necessary. 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 

This question paper consists of 13 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 4 3 r 3

Volume of a sphere =

Area of a triangle ABC =

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 =

 fx f fx 2   fx    f   f 

FMS(S) Sec 4 Exp and Sec 5 N(A) Preliminary Examination 2009 Mathematics Paper 2

2

2

Name :____________________________(

1.

2.

)

Class : ________

5y 2y  as a single fraction in its simplest form. 3x  2 1  4 x

(a)

Express

(b)

Simplify

(c)

Given that wy 2 

(a)

Express x 2  8 x  11 in the form ( x + h) 2 - k where h and k are constants.

9 x 2  18 x . 2 x 2  3x  14

1  rt . Express t in terms of r, w and y . t

[3]

[2]

[2]

[3]

3.

(b)

Hence solve the equation x 2  8 x  11 = 7.

(a)

Mrs Thomas deposits $9780 in a fund that pays compound interest of 3.75%

[2]

per annum, compounded half-yearly.

(i)

Calculate the total amount of money that Mrs Thomas will have in the fund at the end of six years.

[2]

At the end of six years, Mrs Thomas withdrew $8,500 from the fund and invested this amount in a bank which offered simple interest rate of 4.8% per annum.

(ii)

Find the minimum number of full years she had to leave the money in the bank in order for it to be more than $12,000.

FMS(S) Sec 4 Exp and Sec 5 N(A) Preliminary Examination 2009 Mathematics Paper 2

[3]

3

Name :____________________________(

(b)

)

Class : ________

The Utilities Company uses the following rate to charge households for electricity usage.

(i)

Usage per month

Rate (cents per kilowatt hour(kWh))

For the first 100 kWh

15

On the next 50 kWh

24

On the next 50 kWh

38

In excess of 200 kWh

57

Calculate the amount Mrs Thomas has to pay if her monthly electricity usage is 168 kWh.

(ii)

Mrs Thomas paid $52.84 in a certain month for electricity. Calculate the amount of electricity used.

4.

[2]

[3]

A bag of twenty marbles contains five marbles of each of the colours red, blue, black and purple. The five marbles of each colour are numbered 1, 2, 3, 4 and 5.

(a)

(b)

One marble is drawn at random. Find the probability that it is a

(i)

black marble labelled with an even number,

[1]

(ii)

red marble or a blue marble numbered 3.

[1]

The marble is put back into the bag. Two marbles are drawn at random. Find the probability that at least one marble is purple.

FMS(S) Sec 4 Exp and Sec 5 N(A) Preliminary Examination 2009 Mathematics Paper 2

[2]

4

Name :____________________________(

5.

)

Class : ________

In the diagram below, A, B, C and D are four points on a horizontal field. A path runs along the edge CD of the field. A is due north of B. It is given that the bearing of D from A is 058 and the bearing of C from B is 110. AD = AB = 76 m and BC = 104 m. N D 76 m

A 76 m

B 104 m C (a)

(b)

Calculate (i) BD,

[2]

(ii) CD.

[3]

Find the bearing of (i) B from C,

[1]

(ii) A from D.

[1]

(c)

Calculate the area of triangle BCD.

[2]

(d)

Find the shortest distance from B to CD.

[2]

A vertical tree of height 124 m is planted at B. A boy has climbed exactly half way up the tree. (e)

Calculate the greatest angle of depression of any point on the path CD when viewed by the boy.

FMS(S) Sec 4 Exp and Sec 5 N(A) Preliminary Examination 2009 Mathematics Paper 2

[2]

5

Name :____________________________(

6.

)

Class : ________

A container which is formed by joining a cone to a cylinder is used to store scented oil. The radius of the cylinder is 12 cm and 1584  cm3 of scented oil is needed to fill up the container. The ratio of the height of the cone to the height of the cylinder is 5 : 2.

(a)

Find the height of the container.

(b)

Calculate the total surface area, including the base, of the outside of the container.

(c)

[4]

[3]

The container is full of scented oil. The oil is drained from the container at a rate of 0.05 litres per second into a storage tank to be stored. Find, the time taken, in minutes and seconds, to empty the container. Give your answer correct to the nearest seconds.

FMS(S) Sec 4 Exp and Sec 5 N(A) Preliminary Examination 2009 Mathematics Paper 2

[2]

6

Name :____________________________(

7.

(a)

)

Class : ________

Study the pattern below.

nth term

Numbers

1

5

5 = 1 x 5 = 1 x (4 + 1)

2

12

12 = 2 x 6 = 2 x (4 + 2)

3

21

21 = 3 x 7 = 3 x (4 + 3)

4

32

32 = 4 x 8 = 4 x (4 + 4)

:

:

:

:

:

:

c

437

:

:

:

:

:

:

35

1365

1365 = 35 x a = b x (4 + b)

(i)

Write down the 100th term of the pattern.

[1]

(ii)

Find the values of a and b.

[1]

(iii)

Find the nth term of the sequence, 5, 12, 21, 32, …, … .

[1]

(iv)

Write down the value of c.

[1]

FMS(S) Sec 4 Exp and Sec 5 N(A) Preliminary Examination 2009 Mathematics Paper 2

7

Name :____________________________(

7.

(b)

)

Class : ________

In the diagram, a circle with centre O passes through points P, Q, R, S and T and UV is a tangent to the circle at S. Given that PRQ = 48, TPS = 18 and both TOQ and POR are straight lines.

(i)

Explain why OSV = 90.

(ii)

Calculate

[1]

(a)

QPR,

[1]

(b)

PTQ,

[1]

(c)

ORS,

[1]

(d)

SOT.

[1]

FMS(S) Sec 4 Exp and Sec 5 N(A) Preliminary Examination 2009 Mathematics Paper 2

8

Name :____________________________(

8.

)

Class : ________

    6 In the diagram, OA = 2b, OC = 8a, AB = 3a and CD =  5 a  b. 5 B C 3a A 8a 2b O

(a)

D

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

(i) (ii) (iii)

 BC ,  AC ,  AD .

[1] [1] [1]

(b)

  Write down two facts about OD and BC .

[2]

(c)

(i)

Area of OBC . Area of ODC

[1]

Find the value of

Given that the area of triangle OBC is 45 square units, find

(ii)

the area of trapezium OABC.

FMS(S) Sec 4 Exp and Sec 5 N(A) Preliminary Examination 2009 Mathematics Paper 2

[1]

9

Name :____________________________(

9.

(a)

)

Class : ________

The diagram shows a triangle ABC inside a circle with centre O and AC is the diameter. B is a point on the circumference of the circle. Given that AB = 56 cm and BC = 33 cm, find

B

A

O

C

(i)

the area of the circle,

[2]

(ii)

the angle BAC in radians,

[2]

(iii)

the perimeter of the shaded region,

[3]

(iv)

the area of sector OBC.

[3]

FMS(S) Sec 4 Exp and Sec 5 N(A) Preliminary Examination 2009 Mathematics Paper 2

10

Name :____________________________(

9.

(b)

)

Class : ________

In the diagram below, ABCD and EFGH are two identical parallelograms. The points E, F and M are mid-point of BC, AD and EH respectively. The points M and N lie on the lines CD and EF respectively and ECM  x o . The lines AM and GN intersect at point L. NM is parallel to EC.

B

A

N

F

E

L xo D

G

M

C

H

(i)

Prove that triangle ADL is similar to triangle MNL.

[2]

(ii)

Show that NEM  x o .

[1]

(iii)

Find GFN in terms of x o .

[1]

(iv)

Prove that triangle GFN is congruent to triangle ADM.

[2]

FMS(S) Sec 4 Exp and Sec 5 N(A) Preliminary Examination 2009 Mathematics Paper 2

11

Name :____________________________(

10.

)

Class : ________

Answer the whole of this question on a graph paper. The following is a table showing corresponding values of x and y which are related by the equation y  Ax 2  Bx where A and B are constants.

x

1

0.5

0.5

1

1.5

2.5

3

4.5

5

y

19

8.5

6.5

11

13.5

12.5

9

13.5

2.5

(a)

Using a scale of 2 cm to 1 unit on the x-axis for 1  x  5 and 2 cm to 5 units on the y-axis for 25  y  15 , plot the points given in the table and join them with a smooth curve.

(b)

[3]

Use your graph to find (i)

the maximum value of the curve,

[1]

(ii)

the values of x when y is  10 .

[2]

(c)

By drawing a suitable tangent, find the gradient of the curve at x = 4. [2]

(d)

A straight line cut the graph at the points (0.5, 6.5) and (2, p). Using the graph, find

(i)

the value of p,

[1]

(ii)

the equation of this line.

[2]

FMS(S) Sec 4 Exp and Sec 5 N(A) Preliminary Examination 2009 Mathematics Paper 2

12

Name :____________________________( 11.

(a)

(b)

)

Class : ________

The following box-and-whisker diagrams show the distribution of marks of Chinese, English and Mathematics tests of a class of students.

(i)

Find the median mark for Mathematics test.

[1]

(ii)

State the test which has the lowest interquartile range of marks. [1]

(iii)

Compare the performance in the English and Mathematics test in two ways. [2]

(iv)

Jennifer gets 70 marks in all three tests. Explain clearly in which test she performs the best. [1]

In a factory, machine A is used to fill up packs of rice weighing 5 kg. In order to check the accuracy of the machine, an engineer inspects the weights of 100 packs of rice with the results shown in the following table. Weight of packs (kg) 4.7  x  4.8 4.8  x  4.9 4.9  x  5.0 5.0  x  5.1 5.1  x  5.2

Frequency 15 16 29 38 2

(i)

Find the mean and standard deviation of the weights of the 100 packs of rice. [3]

(ii)

Another machine B is also used for the same purpose. The mean and standard deviation of the weights of the packs of rice filled up by machine B is 4.983 kg and 0.101 kg respectively. Which machine is more accurate? Explain your answer.

[1]

End of paper FMS(S) Sec 4 Exp and Sec 5 N(A) Preliminary Examination 2009 Mathematics Paper 2

13

Name :____________________________(

)

Class : ________

Answers 1(a)

1(b)

1(c)

7(a)(iii) n(4 + n)

y(9  26 x)  3x  2 1  4 x 

19

7(b)(i) OS is the radius and UV is the tangent. The angle OSV formed by the radius and tangent is 90 o.

9x  2x  7 

1 t 2 4 w y r 2

(iv)

7(b)(ii) (a)

42 o

(b)

48 o

7(b)(ii) (c)

60 o

(d)

36 o

(ii)

8a-2b

8(a)(i) 5a-2b

2(a)

 x  4

2(b)

x  1.83 or x  9.83

3(a)(i)

$12,222.23

3(a)(ii)

9 years

8(c)(i)

3(b)(i)

$33.84

9(a)(i) 3320 cm2

(ii)

0.533 radians

3(b)(ii)

212 kwh

9(a)(iii) 124 cm

(iv)

562 cm2

4(a)(i)

1 10

4(b)

17 38

 27

1 8(a)(iii) 3a- 3 b 5

8(b) OD is parallel to BC and OD is 0.6 times of BC.

(ii)

3 10

5 3

(ii)

61.875 square units

9(b)(iii) 180-x 10(b)(i) y = 13 to 15 (ii) x = -048 to -0.68 and 4.23 to 4.43 10(c)

5(a)(i)

133 m

(ii)

155 m

5(b)(i)

290o

(ii)

238 o

-15 to -19

10(d)(i) p = 13 to 15 (ii) y = 5x + 4 11(a)(i) 65 mark

(ii)

English test

2

5(c)

6830 m

5(d)

87.8 m

5(e)

35.2 o

6(a)

21 cm

6(b)

1180 m2

6(c)

1 min 40 sec

7(a)(i) 10400 = 100 x 104 = 100 x (4+100) 7(a)(ii) a = 39, b = 35

11(a)(iii) Mathematics test has the highest median while English test has a lower median. Mathematics test has the largest interquartile range while English test has the lowest interquartile range. 11(a)(iv) English test as the mark she obtained is in the top 25% of her class. 11(b)(i) Mean = 4.946kg Std D. = 0.11038 kg 11(b)(ii) Machine B is more accurate because it has a bigger mean which is nearer the required standard mean of 5 kg. Furthermore, its standard deviation is smaller which means it is more consistent in meeting the standard requirement of packing rice of 5 kg.

FMS(S) Sec 4 Exp and Sec 5 N(A) Preliminary Examination 2009 Mathematics Paper 2

14

Name :____________________________(

)

FMS(S) Sec 4 Exp and Sec 5 N(A) Preliminary Examination 2009 Mathematics Paper 2

Class : ________

15