Bowen 2009 Prelim Em P2 + Answers

Class

Full Name

Index Number

BOWEN SECONDARY SCHOOL MATHEMATICS Paper 2 Secondary 4 Exp / 5NA 14th Sep 2009

2 hours 30 minutes

INSTRUCTIONS TO CANDIDATES 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 writing paper. Answer all the questions. Write your answers and working on the writing paper provided. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. 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.

INFORMATION FOR CANDIDATES 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 100. The use of an electronic calculator is expected, where appropriate. You are reminded of the need for clear presentation in your answers. DO NOT OPEN THIS PAPER UNTIL YOU ARE TOLD TO DO SO For Examiner’s Use

This document consists of 10 printed pages (including this cover page). 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 1 2 r h 3 4 Volume of a sphere = r 3 3

Volume of a cone =

Area of triangle ABC =

1 ab sin C 2

Arc length = rθ, where θ is in radians

Sector area =

1 2 r  , 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 =

2

 fx f

 fx 2   fx     f f 

2

Answer all the questions.

1

(a)

(b)

Simplify

Consider the formula

2 3 4   r p q

where r  0,

p  0 and

q0

(i)

Calculate the value of r when p = 5 and q = –6.

[2]

(ii)

Make q the subject of the formula.

[2]

Express 2x2 – 9x –12 in the form of a( x – b )2 + c

[2]

Hence solve the equation 2x2 – 9x –12 = 0 , giving your answers correct to 2 decimal places.

[3]

(c) (i)

(ii)

2

[3]

 8x 4  2 x3 9x

Justin ordered a bed room set for which he paid a down payment of 14% of the selling price.

(a) Calculate the selling price of the bed room set if the balance of the payment is $5 160.

[2]

(b) Justin paid the balance in monthly installments of $266.60 over 2 years. How much extra did he pay through the installment scheme instead of cash?

[2]

(c) Justin could have taken a bank loan to pay for the entire cost of the bed room set. The bank charges 6.5 % simple interest per annum over a period of two years. Would you recommend that he take up the bank loan or the installment scheme? State your reason clearly.

[2]

3

3

In the diagram, A, E, C, G and F lie on a circle. AC and EG are diameters of the circle, centre O. BCD and BEH are tangents to the circle. AEF  38 , EGF  63 and GCD  25 . H A E

38º

F

O

63º G B

C

25º

D

(a) Find (i)

 ECG

[1]

(ii)

 OCG

[1]

(iii)

 CAE

[2]

(iv)

 FEC

[2]

(v)

 CBE

[2]

(b) A point, P, is to be marked on the diagram, on the same side of EF as O, such that EPF  90 . Does the point P lie on the circumference of the circle, inside the circle or outside the circle ? Give a reason for your answer.

[2]

(c) Given that the coordinates of B and E are ( –1 , 2 ) and ( 6 , 4 ) respectively.

(i)

Find the length of BE.

[2]

(ii)

Find the equation of the line parallel to BE and passes through the point ( –5 , 7 ).

[2]

4

4

In the diagram, PR  3PY , X is the midpoint of PZ and Z is the midpoint of QR. QZ  a and PY  b . P

b Y

X

Q

a

Z

R

(a) Express, as simply as possible, in terms of a and / or b, (i)

YR

[1]

(ii)

PZ

[1]

(iii)

XZ

[1]

(iv)

QX

[2]

(v)

QY

[1]

(b) Find the value of (i)

QX QY

[1]

(ii)

the area of PQX the area of PYX

[1]

(iii)

the area of PQX the area of PQR

[1]

5

5

Isaac and Jamie are two salepersons for the fitness programmes of a gymnasium. The new subscriptions that they obtained in April and May are shown in the following tables.

April Package A Package B Package C

Isaac 17 34 19

May Jamie 14 30 21

Package A Package B Package C

Isaac 22 27 19

Jamie 18 33 16

(a) The information for the month of April’s subscriptions can be presented by the matrix,  17 14    A =  34 30   19 21    The information for the month of May is presented by a matrix M . (i)

Write down the matrix M.

[1]

(ii)

Evaluate M  A .

[1]

(iii) Describe what is represented by the elements of M  A.

[1]

(b) The sales commissions for packages A, B and C are $25, $40 and $45 respectively. Write down a matrix T such that the product of TA will show the total amount of commissions for each salesperson in April. [1] (c)

1 It is given that E =   1 (i)

Evaluate ME.

[1]

(ii)

What do the elements in ME represent?

[1]

(d) The prices of packages A, B and C are $400, $750 and $800 respectively. (i)

Write down a matrix P such that PME will give the total amount of sales from the new subscriptions obtained by Isaac and Jamie in May.

[1]

(ii)

Hence, find the total amount of sales obtained by Isaac and Jamie in May.

[2]

6

6

The rectangular water tank whose base measures 45 cm by 12 cm, is height of 14 cm.

7 filled with water to a 9

( Take π to be 3.14 )

14cm

12cm 45cm

(a) Find the capacity of the tank in cm3.

[2]

(b) Ah Heng has some identical lead balls, each of radius 5 cm. He gently puts the balls one at a time into the tank until the water overflows. (i)

Find the volume of each lead ball.

[1]

(ii)

How many lead balls has he put into the tank?

[2]

(iii) What is the volume of water that overflows from the tank?

7

[2]

Grace planned to spend $1 500 on buying some plants, costing $y each. Due to the large number of plants ordered, she was given a discount of $2 for each plant. She eventually spent a total of $1 472, buying more plants than planned. (a) Write down an expression in terms of y for

(i)

the number of plants originally ordered,

[1]

(ii)

the number of plants that was finally purchased

[1]

(b) If the number of plants finally purchased is 4 more than what was originally ordered, write down an equation in y and show that it reduces to y 2  5 y  750  0

[3]

(c) Solve the equation and find the number of plants finally purchased.

[3]

7

8

In the diagram below, A, B, C and D are four points on a level ground, not drawn to scale. A is due north of C. The bearing of D from C is 048o and  CBD = 110 , BD = 336 m, CD = 425 m and AC = 568 m. A

North

D 336

568 B

110

425

48 C

(a) Find

9

(i)

the length of AD,

[2]

(ii)

the length of BC,

[2]

(iii) the bearing of B from D.

[2]

(b) A vertical tower BT, has its base at B. Given that the angle of elevation of T from D is 17o, calculate the height of the tower, in metres.

[2]

(c) Hence, find the angle of depression from T to C.

[2]

The training distances (correct to the nearest kilometres) covered by the footballers of two teams in the Euro 2008 tournament are shown in the table below. Germany Length (km) Frequency

5-7 2

8 - 10 7

11 - 13 4

14 - 16 8

17 - 19 3

Spain Mean = 11.5 km Standard deviation = 4.85 km

(a) Calculate the mean and standard deviation of the training distances covered by the Germany footballers.

[3]

(b) Compare the results of the two teams and explain which team could be fitter.

[2]

8

10 The cumulative frequency curve below gives the distribution of the heights, in cm, of 120 students in Happy Secondary school.

Use the graph to find (a) (i) (ii)

the median height,

[1]

the interquartile range,

[2]

(iii) the number of students whose height is at least 165 cm,

(b) A

[1]

Another 120 students from Lucky Secondary School took part in the survey and the box and whisker diagram below illustrates the distribution of their heights.

(i)

Compare the heights of the students from the 2 secondary schools in two different ways.

[2]

(ii)

Darren said that the students from Lucky Secondary School are generally shorter than the students from Happy Secondary School. Do you agree? Give a reason for your answer.

[2]

9

11 Answer the whole of this question on a sheet of graph paper.

The table below shows the values for the graph of y 

50  2x  20 . x

x

2

2.5

3

4

5

6

7

8

y

p

5

2.7

0.5

q

0.3

1.1

2.3

(a) Find the values of p and q.

[2]

(b) Using a scale of 2 cm to represent 1 unit on the x-axis, 2 cm to represent 1 unit on the y-axis, draw the graph of

[3]

y

50  2x  20 x

for 0 ≤ x ≤ 8.

(c) The point  7.4 , t  lies on the curve. Use your graph to find the value of t.

[1]

(d) By drawing a tangent, find the gradient of the curve at the point where x = 4.

[2]

(e) Use your graph to find the range of values of x for which y is less than 2.

[1]

(f)

[3]

By drawing a suitable line on the graph, solve the equation x 2  21x  50  0 .

End of paper 2 10

Answer key 1a, bi,ii

4 x3

,

r = 30 , q 

1ci, ii

4rp 2 p  3r

2 (x 

4 2 177 , )  9 8

5.58 , 1.08

2a.b

$6 000 , $ 1238.40

2c

$780 , take bank loan

3ai 3aiii 3av 3ci

90o 65o 50o 7.28 units

3aii 3aiv 3b 3cii

4ai 4aiii

2b

4aii 4aiv

65o 52o Inside circle, EF is not a diameter. 2 y  x 5 7 3b – a

4av

2(a–b)

4bi

4bii

3

4biii

5ai

 22 18     27 33   19 16   

5aii

4   5    7 3    0  5  

5aiii

Difference of the new subscriptions between May and April.

5b

25

5ci

 40     60   35   

5cii

Total number of packages A, B and C sold respectively in May.

5di

400

5dii

89000 1 523 cm 3 3 2 456 cm 3 3

1 ( 3b – a ) 2

750

800

3

3 (a–b) 2 3 4 1 4

40 45

6a

9720 cm

6bi

6bii

5

6biii

7ai, ii

1500 1472 , y y2 424 m 250o 30.0o Mean = 12.375 km , SD = 3.60 km

7c

64

8aii 8b

169 m 97.8 m

9b

Germany as their mean is larger and they are more consistent in their training distance. 162 – 152 = 10 cm

8ai 8aiii 8c 9a 10ai 10aiii 10b

157 cm 10aii 20 For Lucky Sec Sch, Median = 160.8 cm and Interquartile range = 166.8 – 152.2 = 14.6 cm Median and interquartile range are higher for Lucky Sec Sch.

10c

No. the median, upper quartile and maximum height are all higher in Lucky Sec Sch.

11a 11d 11f

p= 9 , q =0 – 1.1 Draw y = x + 1

11c 11e Ans. x = 2.7

11

t = 1.5 3.2  x  7.8