Chij Prelim Em 2 2009

CHIJ SECONDARY (TOA PAYOH) PRELIMINARY EXAMINATION 2009 SECONDARY FOUR (SPECIAL/EXPRESS)

4016/02

MATHEMATICS Paper 2

4 September 2009 2 hour 30 minutes

Additional Materials:

Answer Paper Graph paper (1 sheet)

READ THESE INSTRUCTIONS FIRST Write your name, register number and class 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. 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 π , use either calculator value or 3.142, unless the question requires the answer in terms of π. At the end of the examination, fasten all your work securely together. The number of marks is given in the brackets [ ] at the end of each question or part question. The total number of marks for this paper is 100.

This document consists of 9 printed pages. [Turn over

2

Mathematical Formulae Compound interest r   Total amount = P 1 +   100 

n

Mensuration Curved surface area of a cone = π rl Surface area of a sphere = 4π r 2 Volume of a cone =

1 2 πr h 3

Volume of a sphere = Area of triangle ABC =

4 3 πr 3

1 ab sin C 2

Arc length = rθ , where θ is in radians Sector area =

1 2 r θ , where θ 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 ∑f

2

 ∑ fx  −  ∑ f   

2

Answer all the questions. chijsectp.4S/E.prelim.emath2.2009

3 1

2

2 x −1

(a)

. Simplify x +5 − 2 2 x −50

(b)

(i) (ii)

[2]

If x 2 − 6 x − 3 can be written in the form ( x + a ) 2 − b , find the values of a and of b.

[2]

Hence solve the equation x 2 − 6 x − 3 = 0 , giving your answers correct to 2 decimal places.

[2]

E

2

46.7 cm

A

36o

D

17.3 cm

In the figure,

B

C ABCD is a rectangle

where AB = 17.3 cm, DE = 46.7 cm and ∠EDA = 36 ° . Calculate

3

(a)

AE,

[1]

(b)

AD,

[1]

(c)

∠DEC ,

[2]

(d)

the area of ∆DEC .

[2]

Mr Ong, Mr Ali and Mr Kumar each bought a 1600 cc car from a car dealer. The cash price of the car is $65000 inclusive of COE and 6-months road tax. (a)

(b) 000

Mr Ong bought the car on hire purchase terms. He paid a deposit of 45% of the cash price followed by 72 equal monthly installments of $ 536.26. Calculate the difference in the price on hire purchase terms and the cash price. [2] Mr Ali bought the car on simple interest loan terms. He paid a down payment of $28 and the balance to be paid at the end of 5 years with a simple interest rate of 3.5% per annum. Calculate the amount that Mr Ali has to pay at the end of 5 years. [2]

(c)

Mr Kumar bought the car on compound interest loan terms. He paid a down payment

of $27 000 and the balance to be paid at the end of 5 years with a compound interest rate of 2.8% per annum compounded half-yearly. Calculate the amount that Mr Kumar has to pay at the end of 5 years. [2] (d)

In 2009 the price of one litre of engine oil was $5.56 which is 28% cheaper than the

price in 2008, find the price of one litre of engine oil in 2008. chijsectp.4S/E.prelim.emath2.2009

[2]

4 4.

Study the following pattern: 2

3

= 1

= 1

Row 2

1 3 + 23

= 9

= 32

Row 3

13 + 2 3+ 33

Row 4

1 3 + 2 3 + 33 + 4 3

Row 1

1

1×2  =    2 

2

= 36 = 62

2

= (1 +2 )2

2 ×3  =  

= ( 1 + 2 + 3 )2

3 ×4  =  

 2  2

 2  4 ×5    2 

= 100 = 102 = ( 1 + 2 + 3 + 4 )2 = 

2

…………………………………………………………………………………….. …………………………………………………………………………………….. Row n

5

13 + 23 + 33 + 43 + …… + n3 = p2

(a)

Write down Row 5.

[1]

(b)

By studying the pattern, (i) write down the sum of the numbers in row 10. (ii) express p in terms of n.

[1] [1]

(c)

Using your result in (b)(ii), find the value of n when p = 325.

[1]

(d)

A student found the sum of the numbers in a row as 500. Give a reason why we cannot accept this answer.

[1]

A box contains 12 coloured marbles. One half of the marbles are white and one third are yellow. The remaining marbles are blue. (a)

What is the probability of drawing a blue marble from the box?

[1]

(b)

Find the probability of drawing a marble that is not yellow.

[1]

(c)

Two marbles are randomly drawn from the box, one after the other without replacement. Draw a tree diagram to show the possible outcomes.

(d)

6

[2]

Use the tree diagram or otherwise to find the probability that, (i)

both marbles are yellow,

[1]

(ii)

one of the marbles is yellow and the other is red,

[1]

(iii) one of the marbles is white and the other is blue,

[2]

(iv) no white marbles will be drawn.

[2]

In 2008, the price of a litre of petrol in Singapore is $x. A man usually pumps $75 of petrol when he goes to a petrol station. (a)

Write down an expression in terms of x, for the number of litres of petrol that can chijsectp.4S/E.prelim.emath2.2009

5 be bought with $75.

[1]

In 2009 the price of petrol is 40 cents per litre cheaper. Write down an expression in terms of x for the number of litres of petrol that can be bought for $75.

[1]

If $75 can buy 9 extra litres of petrol at the reduced price, form an equation in x and show that it reduces to 15 x 2 − 6 x − 50 = 0 .

[3]

Solve the equation 15 x 2 − 6 x − 50 = 0 , giving your answers correct to 3 decimal places.

[3]

(e)

Write down the number of litres of petrol that can be bought with $60 in 2009.

[1]

(f)

In 2009, the price of petrol in Malaysia is RM2.52 per litre, How many more litres of petrol can one buy in Malaysia as compared to Singapore with S$60? Take the exchange rate in 2009 to be S$1=RM2.42.

[2]

(b) (c) (d)

7

In the diagram, AB is a tangent to the circle with centre O, ∠CAB = 56 ° and CDE is a straight line. (a)

(b)

Find, stating your reasons clearly, (i)

∠BDE

[2]

(ii)

obtuse ∠BOE

[2]

(iii) ∠EBO

[2]

(iv) ∠ABE

[2]

What can you say about the lines CA and BE? Give a reason for your answer.

[2]

E

A 56o

D

O

C

8 is

B

The diagram (not drawn to scale) shows three points P, Q and R on level ground such that Q due north of P and the bearing of R from P is 035 °. Given that PR =48 m and QR = 52 m, (a)

find the value of ∠PQR ,

[3]

(b)

find the bearing of R from Q.

[1] chijsectp.4S/E.prelim.emath2.2009

(c)

6 A vertical pole QT stands at Q and is 26 m tall. A man walks from P to R, find the greatest angle of elevation of T from the man as he walks from P to R. [2] T N

26 m Q

52 m 35o

P

9

48 m

R

The diagrams below (not drawn to scale) show three blocks of solid candle wax of different shapes and sizes. Diagram 2

Diagram 1

Diagram 3

2.7 cm 7.5 cm

x cm

h cm

4 cm 1.6 cm

Diagram 1 is a conical wax of height 7.5 cm sitting on top of a cylindrical wax of base radius 4 cm and height 1.6 cm. Diagram 2 is a wax cylinder of height h cm and radius 2.7 cm. Diagram 3 is a hemisphere of radius x cm. (a)

(b)

10

Calculate, giving your answers in terms of π, (i)

the total surface area of the candle in Diagram 1,

[3]

(ii)

the volume of the candle in Diagram 1.

[2]

The candle in Diagram 1 was melted down to form the candle in Diagram 2 of height h cm. Calculate the value of h.

[2]

(c)

15 blocks of candles in Diagram 2 was melted down to form 6 blocks of candles in Diagram 3. Calculate the value of x. [2]

(d)

If the cost of 1 block of Diagram 2 candle is $10.90, find the cost of 3 similar blocks of wax of height one-third that of Diagram 2. [1]

Answer the whole of this question on a sheet of graph paper. The variables x and y are connected by the equation y = x +

15 −7 . x

Some corresponding values of x and y, correct to 1 decimal place, are given in the table below. x

1

1.5

2

3

4

5

6

7

8

chijsectp.4S/E.prelim.emath2.2009

7 y

9

4.5

a

1

0.8

1

1.5

b

2.9

(a)

Calculate the value of a and of b.

(b)

Using a scale of 2 cm to represent 1 unit, draw a horizontal axis for 0 ≤ x ≤ 8 . Using a scale of 2 cm to represent 1 unit, draw a vertical axis for 0 ≤ y ≤10 . On these axes, draw the graph of y = x +

(c)

[1]

15 −7 . x

[3]

Use your graph to find the value of x when the gradient of the curve y=x+

15 − 7 is equal to zero. x

[1]

(d)

By drawing a tangent, find the gradient of the graph at x = 2.

(e)

The line y = kx touches the curve y = x +

15 − 7 at exactly one point. x

By drawing a suitable straight line on the same axes, use your graph to find the least possible value of k. (f)

[2]

[2]

By drawing a suitable straight line on the same axes, use your graph to find the solutions of the equation

3 15 x+ −11 = 0 . 2 x

[3]

chijsectp.4S/E.prelim.emath2.2009

40 0 35 0 30 0 25 0 20 0 15 0

Cumulative Frequency

11

8 The cumulative frequency curve below shows the distribution of the Mathematics marks obtained by 380 students from School P.

10 0 5 0 0 0

(a)

1 0

2 0

3 0 Mark s

4 0

5 0

6 0

7 0

Estimate from the graph, (i)

the median,

[1]

(ii)

the upper quartile,

[1]

(iii) the interquartile range,

[1]

(iv) the percentage of students who obtained more than 46 marks.

[1]

(b)

Given that 5 % of the students will be awarded a distinction grade, use the graph to find the lowest mark scored by this group of students. [1]

(c)

(i)

Copy and complete the following frequency distribution table. No of marks obtained 0 < x ≤ 10 10 < x ≤ 20 20 < x ≤ 30 30 < x ≤ 40 40 < x ≤ 50 50 < x ≤ 60 60 < x ≤ 70

(ii)

Frequency 20

10 Using the data from (c)(i), calculate the mean and standard deviation of the chijsectp.4S/E.prelim.emath2.2009

[2]

9 Mathematics marks scored by the 380 students, giving your answers correct to the nearest 0.1 mark. (d)

[3]

The marks obtained by 380 students from School Q who sat for the same examination paper are also noted. The results are summarised by a box-and-whisker plot shown

below.

(i) (ii)

State the median and the interquartile range. Find the range.

[2] [1]

(e)

Compare the marks obtained by students from the two schools in two different ways.[2]

(f)

Ryan said that the students in School P put in more effort in their study than students from School Q. Do you agree? Give a reason for your answer. [1]

--- End of Paper ---

CHIJ Sec TP Answers to E. Mathematics Paper 2 Prelim 2009

chijsectp.4S/E.prelim.emath2.2009

10 1(a)

2 x −19 2( x + 5)( x − 5)

1(bi) a = −3, b = 12

1(bii) x = 6.46 or −0.46

2(a)

27.4 cm

2(b) 37.8 cm

2(c)

13.8o

2(d) 327 cm2

3(a)

$2860.72

3(b) $43 475

3(c)

$43 667.98

3(d) $7.72

4(a)

13 + 23 + 33 + 43 + 53 = 225 = 152 = ( 1 + 2 + 3 + 4 + 5 )2 = 

4(bii)

p=

5(a)

1 6

5 ×6    2 

n( n + 1) 2

4(c) n = 25 5(b)

4(d)

2 3

5(di)

2

4(bi) 3025

500 is not a perfect square number 1 11

(dii) 0

(diii)

2 11

(div)

5 22

6(a)

75 x

6(b)

75 x − 0.4

7(ai)

124o

7(aii) 112o

8(a)

32.0o

9(ai)

62.8π cm2

6(d) 2.037 or −1.637 7(aiii) 34o

8(b) 148.0o 9(aii) 65.6π cm3

10(a) a = 2.5, b = 2.1

7(aiv) 56o 8(c)

9(b) 9.00 cm

10(c) x = 3.8 to 4.1

6(e) 36.7 l

6(f) 21.0 l

7(b) Alternate angles

26.6o 9(c) 6.27 cm

10(d) −2.5 to −3

9(d) $1.21

10(e) k = 0.16 to

0.2 11(ai) 36 marks

11(aii) 44 marks

11(ci) 45, 75, 95, 95, 40 11(di) 42

11(aiii) 20

11(aiv) 21.1 %

11(b) 57

11(cii) mean = 34.5, SD = 14.3

11(dii) 42

11(e) The median for School Q is higher which means that the students from School Q did better in the examination than students from School P. The interquartile range for School Q is higher which means that the marks are more widespread. 11(f) No. Because the median for School P is lower indicating that their results are poorer than School’s Q.

chijsectp.4S/E.prelim.emath2.2009