Answer (a) - WordPress.com · Holy Innocents’ High School, Prelim Exam 2009 Secondary 4Express Elementary Mathematics Paper 1 3 Answer all the questions. E 1 The estimated number

Holy Innocents’ High School, Prelim Exam 2009

Secondary 4Express Elementary Mathematics Paper 1

3

Answer all the questions.

1 The estimated number of spectators, rounded off to 3 significant figures, for a soccer

match is 46 000. Write down

(a) the smallest possible number of spectators,

(b) the largest possible number of spectators.

Answer (a) ……………………………….. [1]

(b) …………….…………………. [1]

2 Simplify 121 842 mmm.

Answer …………………………………… [2]

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4

3 (a) Evaluate

2

12

11

7

)345.0( 2

.

(b) Given xyyx 9876 , calculate the numerical value of y

x.

Answer (a) ……………………………….. [1]

(b) …………….…………………. [1]

4 The pie chart represents the masses of the different ingredients in a chocolate cake.

The ratio of the mass of flour to cocoa used is 2 : 1.

(a) Calculate the value of x.

(b) Given that the combined mass of sugar and flour used is 462 g, calculate the total

mass of the cake.

Give your answer to the nearest gram.

Answer (a) x = ………………………….. [1]

(b) …………….………………. g [1]

Butter

Cocoa

Sugar

3xº

Flour

3xº

xº

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5

5 (a) An aircraft touched down at Changi Airport at 05 35 h on Monday after flying for

4

38 hours.

Find the time at which the aircraft started its journey.

(b) An aircraft is traveling at constant speed of 760 km/h.

Calculate the distance it will travel in 3 hours 10 minutes.

Answer (a) ………………………………. [1]

(b) …………….…………… km [1]

6 In the diagram, which is not drawn to scale, the line QR has the equation 1054 xy .

(a) Find the coordinates of the point Q.

(b) Determine whether the point P

7 ,

5

23 lies on the line. Show your working

clearly.

Answer (a) Q (…………, …………) [1]

Answer (b) …………………………………..…………….………………………………….

……………………………………………………………………………………

……………………………………………………………………………….. [1]

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x

Q

R (2, 0)

y

O



6

7 In the diagram, ABC represents a semicircular card with centre O, diameter 8r cm.

Two equal semicircles are cut out of it to produce the shaded region.

It is given that the area of the shaded region is 37 cm2 and mr , where m is a real

number.

Calculate the value of m.

Answer m = ……………………………… [2]

8 The visible blue light has a wavelength of 475 nanometres.

(a) 475 nanometres can be written as nA 10 metres, where 101 A and n is an

integer.

Find the value of n.

(b) The wavelength of the visible blue light is 0.00475 times the wavelength of the

infrared radiation. Find the wavelength of the infrared radiation.

Give your answer in standard form.

Answer (a) n = …………………………... [1]

(b) …………….………… metres [1]

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8r cm



7

9 Alex has 6x sweets.

Betty has 4 less sweets than Alex.

Clarissa has half as many sweets as Betty.

Write down an expression, in terms of x, for

(a) the number of sweets that Clarissa has,

(b) the number of sweets left, if 3

1 of the total number of sweets were given away.

Answer (a) ………………………………. [1]

(b) …………….………………… [2]

10 (a) On the Venn diagram shown in the answer space, draw a set B such that BA and

BBA .

[1]

(b) 50139 andinteger an is : xxx

P = number prime a is : xx

Q = 4045: xx

(i) List the elements of P.

(ii) Find the value of QPn .

Answer (b) (i) …………………………… [1]

(ii) …..…….………………… [1]

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8

11 Kendy and Leslie are given pocket money everyday. The table below shows the number

of coins they have saved after 3 weeks.

$1 50 cents 20 cents

Kendy 32 25 48

Leslie 20 35 50

(a) Given A =

503520

482532 and B =

1

1

1

,

(i) find the product AB,

(ii) explain what the elements of the product represent.

(b) Write the matrix C such that the elements of the matrix product AC will give the

money each one has saved.

Answer (a) (i) AB = ………………………. [1]

Answer (a)(ii) ……………………………………………………………………………….....

………………………………………………….………………………………

………………………………………………………………………………….

………………………………………………………………………………[1]

(b) C = …….…………………….. [1]

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9

12 It is given that the time taken, t minutes, to download a file from a computer is inversely

proportional to the Internet connection speed, v kB/s.

(a) Which one of the graph below could represent the relation between the time and

the Internet connection speed?

When the Internet connection speed is x kB/s, the time taken to download a particular

file is 5 minutes.

When the connection speed is increased by 200%, find

(b) an expression in terms of x, for the Internet connection speed,

(c) the time taken to download the file.

Answer (a) ……………………………….. [1]

(b) …………….…………… kB/s [1]

(c) …………….…………… mins [1]

v

t

O v

t

O v

t

O

v

t

O v

t

O

Graph I Graph II Graph III

Graph IV Graph V

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13 The diagram shows a pentagon ABCDE. The point F is on AB produced and AE is

parallel to BC.

(a) Calculate the value of y.

(b) Given also that 72FBC , calculate the size of the smallest exterior angle.

Answer (a) y = ……………………......... [2]

(b) …………….………………. ° [1]

14 The heights of 40 pupils in group A are given by the frequency table below.

The mean height of this group of pupils is 152.5 cm.

Height 130120 x 140130 x 150140 x 160150 x 170160 x

Frequency 1 2 8 24 5

(a) Use the above frequency table to calculate an estimate of the standard deviation.

(b) The heights of another group of 40 pupils in group B are summarised below.

Mean height = 152.7 cm

Standard deviation = 11.2 cm

If you are to select players for a school basketball team, which group will you

choose? Give a reason for your decision.

Answer (a) ……………………………….. [2]

Answer (b) …………….……………………………………………………………………….

…………………………………………………………………………………….

…………………………………………………………………………………. [1]

114°

2y°

4y°

A B

C

D

E

F

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15 In the diagram, BAC = 90º and AC is produced to D. Given that tan ABC = 5

12,

(a) calculate AC if AB = 12 cm,

(b) write down the value of BCDcos .

Answer (a) AC = ……………………..cm [2]

(b) BCDcos = ….……………. [2]

16 (a) Factorise completely 32220 qpqqp .

(b) Expand and simplify

yxyxyx 3332

1 2 .

Answer (a) …………………………......... [2]

(b) …………….…………………. [2]

A

B

C D

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17 (a) Given that 44 x and 53 y find

(i) the greatest possible value of 2xy ,

(ii) the least integer value of y

x.

(b) Solve the simultaneous equations.

103 xy

1632 yx

Answer (a) (i) ...………………………….. [1]

(ii) ...………………………….. [1]

(b) x = ………..………………….

y = ……..……………………. [3]

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13

18 (a) The number 126 written as a product of its prime factors is

732126 2 .

(i) Express 105 as a product of its prime factors.

(ii) What is the smallest positive integer value of n for which 126n is a multiple

of 105?

(b) The dimensions of a rectangular box are 168 cm by 132 cm by 84 cm. The box is

to be filled with identical cubes so that there will be no empty space.

(i) Find the longest possible length of each side of a cube.

(ii) Hence, find the number of cubes that the box can contain.

Answer (a) (i) 105 = …………………….. [1]

(ii) n = ……………………….. [1]

(b) (i) ...……………………… cm [2]

(ii) ...………………………….. [1]

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19 Two cones, P and Q are geometrically similar.

The height of cone P is 4

11 times the height of cone Q.

(a) If the height of cone P is 20 cm, find the height of cone Q.

(b) The surface area of the top of cone Q is 56 cm2.

Find the surface area of the top of cone P.

(c) Cone P can hold 1.75 kg of sand.

Find the mass of sand that cone Q can hold.

.

Answer (a) …………………………... cm [1]

(b) …………….…………….. cm2 [2]

(c) …………….…………….. kg [2]

P Q

20 cm

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15

20 The graph shows the relation between the duration of call (t) and the total cost (C)

charged by a service provider, Teleshop.

Answer (a) p = ……………… p represents…………………………………………..

…………….…………………………………………………………………………… [2]

(b) q = ……………… q represents ………………………………………….

…………….…………………………………………………………………………… [2]

40

60

20

80

2 4 6 0 8

Cost (C cents)

Time (t minutes)

Teleshop

Given that the relation between the duration of call and the total cost charged by

Teleshop is represented by qptC ,

(a) state the value of q and explain its significance,

(b) find the value of p and explain its significance.

Another service provider, Phoneshop, charges 30 cents for calls of 3 minutes or less and

then at a constant rate of 15 cents per minute.

(c) On the same axes, draw graph to show the relationship between the duration of call

and the total cost charged by Phoneshop. [1]

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16

21 In the diagram, 2OP p and OQ = 3q.

X is the point on PQ such that PX = PQ3

1.

The line OX when produced, meets PY at Z.

(a) Express as simply as possible, in terms of p and q,

(i) QX ,

(ii) OX .

(b) Y is the point such that PY = OQ2

3.

(i) Show that QY = 2p + 2

3q.

(ii) Hence, explain why OX is parallel to QY .

(c) Calculate

(i) OQX

PXZ

of area

of area,

(ii) the area of OQX , given that the area of OPX is 6 square units.

Answer (a) (i) QX = …………………….. [1]

(ii)OX = …………………….. [1]

Answer (b)(i) …………………………………………………………………………………….

……………………………………………………………………………….. [1]

(ii)………………………………………………………………………………... [1]

(c) (i) ...………………………….. [1]

(ii) ...…………… square units [1]

O

Q

P

X

Y

3q

2p

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17

22 (a) The diagram shows a straight line, l and a quadratic curve )2( pxxy

intersecting at two points A and B

0 ,

2

15 .

The line l cuts the y-axis at the point (0, 11).

(i) State the value of p.

(ii) Find the equation of the line l and hence calculate the x-coordinate of A.

Answer (a) (i) p = ……………………….. [1]

(ii) ...…………………………..

x = ………………………. [3]

(b) Sketch the graph of 232 xy indicating the coordinates of the turning point

clearly.

[2]

A

B O

2

15

11

l

y

x

y

x O

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18

23 The scale drawing in the answer space below shows the position of two towns P and Q

using a scale of 1 cm to represent 1 km.

(a) Given that town R is such that the bearing of R from P is 160° and the bearing of R

from Q is 240°, construct town R on the diagram.

(b) A tree, T is to be planted equidistant from the towns P, Q and R.

(i) Showing your construction clearly, find and label the position of the tree.

(ii) Find the actual distance of the tree from the corners of the three towns,

giving your answer in km.

(iii) Complete the sentence in the answer space below.

Answer (a) and (b) (i)

[4]

Answer (b) (ii)………………………. km [1]

Answer (b)(iii) Since PT = QT = RT, we can draw a ……………… to pass through the

points ….…..… , …………. and ………… [1]

End of Paper

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N

Q



1

Answer Key

1(a) 45 950 13(a) 41y

1(b) 46 049 13(b) 16°

2 232 m 14(a) 8.29

3(a) 45.3 (3 sf) 14(b) Group A because the heights are more

consistent as shown by the smaller standard

deviation.

3(b)

15

1

y

x

15(a)

5

428 cm

4(a)

48x 15(b)

13

12

4(b) 866 g 16(a) qpqpq 45

5(a) 20 50, Sunday

16(b)

2

2

273

2

2

1yxyx 5(b)

3

22406 km

6(a) Q (0,

2

12 )

17(a)(i)

3

25

6(b) P does not lie on the line QR 17(a)(ii) – 4

7 m = 9.25 17(b) 2x , 4y

8(a) n = –7 18(a)(i) 753105

8(b) 4101 metres 18(a)(ii) n = 5

9(a) )23( x sweets 18(b)(i) 12 cm

9(b) 10x – 4 18(b)(ii) 1078 cubes

!0(a)

19(a) 16 cm

19(b) 87.5 cm2

19(c) 0.896 kg

10(b)(i) P {5, 7, 11, 13} 20(a) q = 20, it represents the connection fee

charged for every call regardless of the

duration [basic/initial charged incurred once

a call is made] 10(b)(ii) QPn = 2

11(a)

105

105AB

20(b) p = 8, it represents the additional cost

charged per minute [cost of the call per

minute]

11(b) Elements of AB represent the

total number of coins saved by

Kendy and Leslie respectively.

20(c)

11(c)

20.0

50.0

1

C or

20

50

100

C

12(a) Graph IV 21(a)(i)

3

2(2p – 3q)

12(b) 3x kB/s 21(a)(ii)

3

1(4p + 3q) 12(c)

3

21 mins

A B

30

3

60

5 Time (t mins)

Cost (C cents)



2

21(b)(ii) Since OX =

3

2QY , then OX is parallel to QY .

21(c)(i)

4

1

21(c)(ii) 12 square units

22(a)(i) 11

22(a)(ii) 112 xy

x-coordinate of A is –1

22(b)

23(a)

(b)(i)

22(b)(ii) 6.6 km

22(b)(iii) circle, P, Q, R

y

x

(3, 2)

(0, –7)

T

P

Q

R

Student Name Class Index Number Marks

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

MATHEMATICS 4016 Paper 2 13 August 2009

DURATION: 2 hr 30 min

Additional Materials: 6 Writing Papers 1 Graph paper

INSTRUCTIONS TO STUDENTS Write your index number and name 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. 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 , use either your calculator value or 3.142, unless the question requires the answer in

terms of .

The number of marks is given in brackets [ ] at the end of each question or part question. The total marks for this paper is 100.

_________________________________________________________________________

This paper consists of 10 printed pages and 0 blank page.

HOLY INNOCENTS’ HIGH SCHOOL

2

Mathematical Formulae Compound interest

Total amount

nr

P

1001

Geometry and Measurement

Curved surface area of a cone rl

Surface area of a sphere 24 r

Volume of a cone hr 2

3

1

Volume of a sphere 3

3

4r

Area of triangle ABC Cabsin2

1

Arc length r , where is in radians

Sector area 2

2

1r , where is in radians

Trigonometry

C

c

B

b

A

a

sinsinsin

Abccba cos2222

Statistics

Mean

f

fx

Standard deviation

22

f

fx

f

fx

3

Answer all the questions.

1 (a) Solve the equation 4x2 – 5 = x (x – 14). [3]

(b) Given that s

st

1

1, express s in terms of t. [3]

(c) Given that a2 – b2 = 117 and a + b = 13, find the value of (a – b) 2. [2]

2 Mr Tay paid $50 for his water bills in August when the rate was $x per 3cm . In

September, the price of water was increased by $3 per 3cm . By reducing his usage of

water, Mr Tay managed to pay $50 for his water bill in September.

(a) Write down an expression, in terms of x, for the amount of water used by Mr Tay

in August. [1]

(b) Write down an expression, in terms of x, for the amount of water used by Mr Tay

in September. [1]

(c) If the amount of water used in September was 1 3cm less than the amount used in

August. Form an equation in x and show that it reduces to 015032 xx . [3]

(d) Solve the equation and find the price of water per 3cm in the month of August. [3]

(e) Calculate the amount of water used in the month of September. [1]

3 Mr Tan decides to buy a car costing $105 000. He has two options to pay for the car.

(a) If he pays cash, he will get a discount and needs to pay only $92 400 for the

car. What is the percentage discount? [2]

(b) If he trades in his old car, he will get $37 000 for it. He will then have to pay the

balance of the cost of the new car. His friend is willing to arrange a loan to help

him pay the balance of the cost in equal monthly instalments over 3 years at 5 %

simple interest per annum. Calculate

(i) the total interest of the loan, [2]

(ii) the amount of each monthly instalment to the nearest dollar. [2]

(c) The salesman makes a 25 % profit if he sells the car. Find the cost price of the car.

[2]

(d) A bank is willing to lend Mr Tan $105 000, charging a compound interest of

4.5 % per annum. Calculate the total interest that Mr Tan needs to pay if he

decides to borrow from this bank and can only repay after 3 years. [3]

4

4 The cumulative frequency curve below illustrates the marks obtained, out of 100, by 60

students in a Science Test.

(a) Using the graph, find

(i) the median mark, [1]

(ii) the interquartile range, [2]

(iii) the pass mark if 3

2 of the students passed the test. [1]

(b) The same 60 students took a Mathematics Test. The box and whisker diagram

below illustrates the marks obtained. The maximum mark was again 100.

(i) Compare the marks obtained for Science and Mathematics test in two

different ways. [2]

(ii) Simon said that the Mathematics test was easier than the Science test. Do

you agree? Give a reason for your answer. [1]

10 20 30 40 50 60 70 80 90 100

10

20

30

40

50

60

Mark

Cumulative

Frequency

5

5 The table below refers to a certain series.

N

Row

Number

Series S

Sum of

Series

N+1 N+2 M

1 12 2 2 3 123 = 6

2 12 + 23 8 3 4 234 = 24

3 12 + 23 + 34 20 4 5 345 = 60

6 12 + 23 + … + 67

a 7 8 b

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

(a) Study the table and write down the value of a and the value of b. [2]

(b) Express the relationship between the numbers in column S and those in column

M as a formula involving S and M. [1]

(c) Use your answer to part (b) to find

(i) the value of S when N = 10, [2]

(ii) the value of S for the series

12 + 23 + 34 + 45 + … … + 930. [2]

(d) Suggest a formula, in terms of n, for the sum of the series

12 + 23 + 34 + 45 + … … + n (n+1). [1]

(e) Give a reason why the number 7989 could not appear in the column M. [1]

6

6 (a)

In the diagram, O is the centre of the circle through A, B, C, D and E and TB is

the tangent at B. The diameter AC and chord DB intersect at X. Given that angle

OBC = 38 and angle DCA = 28, calculate

(i) angle ABO, [1]

(ii) angle ABT , [1]

(iii) angle AED, [1]

(iv) angle CXD. [2]

(b)

In the diagram, AODC is a straight line. A, B and D are points on a circle with

centre O. O, B and C are points on a circle with centre D.

(i) Prove that triangle OBD is an equilateral triangle. [2]

(ii) Prove that triangles ABD and CBO are congruent. [3]

O D C

B

A

A

X

E

28°

38°

T

B

D

O

C

7

7 [Take to be 3.142] In a factory, Liquid P is mixed in a trough as shown in the diagram. It is then packed into

cylindrical drums as shown in the diagram.

Trough Cylindrical drum

(a) Find the volume of the trough in cm3. [2]

(b) How many cylindrical drums can a trough full of Liquid P fill? [2]

(c) 7 drums are then packed into a regular hexagonal box, as shown in Diagram I.

WXYZ is one of the vertical sides of the hexagonal box. The height of the

hexagonal box is 40 cm.

Diagram II shows the view of the box of drums from above.

A, B and C are the centres of the circular tops of three adjacent drums which touch

each other. The midpoint of AB and XY are N and M respectively.

(i) Write down the length of AC. [1]

(ii) Calculate the length of CN and CM. [2]

(iii) By first finding XY, calculate the amount of paper needed to cover all 6

vertical sides of the hexagonal box, giving your answer to the nearest

square centimetres. [3]

7.7 m

3.5 m

1.5 m

2.5 m

40 cm

10.5 cm

Diagram I

X Y

Diagram II

40 cm A B

C

N

W

Z

X

Y

M

8

8 (a) There are six red cards and three blue cards in a box.

(i) Joe draws a card from the box at random. Write down the probability that

he will get a blue card. [1]

(ii) David draws 2 cards at random, one after the other, without replacement.

Find the probability that

(a) the first card is blue and the second is red, [1]

(b) the two cards are of the same colour. [2]

(iii) Ali draws one card at a time without replacement, until he gets a blue

card .Find the probability that he will be successful exactly on his third

draw. [2]

(b)

The diagram shows a circle, centre O, of radius 10 cm. The line AC is

perpendicular to the radius OA, and the line OC intersects the circle at B.

Given that 5.0OCA radians, calculate

(i) the length of AC, [2]

(ii) the perimeter of the shaded region. [4]

C A

O

B 10 cm 0.5 radians

9

9

The figure shows the positions A, B, C, and D of four oil rigs. C, A and D lie in a

straight line. Given that AC = 45 km, BC = 85 km, BD = 70 km and ACB = 46 ,

(a) Calculate

(i) the distance AB, [3]

(ii) the size ofADB, [2]

(iii) the area of triangle CBD. [3]

(b) A supply ship S sets sail from C to D in a straight line.

(i) Find the distance of the ship S from C when it is closest to B. [2]

(ii) A coast guard, stationed in a helicopter hovering 2.6 km directly above A,

watches the supply ship S through binoculars. What is the angle of

depression of his view of S, when it is sailing closest to B ? [2]

D

A

45

C 85

70

B

46

10

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

A pot of boiling water is allowed to cool from 100C to room temperature. The

equation relating the temperature yC and the time x hours later is given by the

equation x

y3

8119 . The table below gives some values of x and the corresponding

values of y.

x (hours) 0 2

1 1

2

11 2

2

12 3 4

y (C) 100.0 65.8 46.0 p 28.0 q 22.0 20.0

(a) Find the value of p and of q to 1 decimal place. [1]

(b) Using a scale of 4 cm to represent 1 hour on the x-axis and 2 cm to represent 10C

on the y-axis, draw the graph of x

y3

8119 , for 0 x 4. [3]

(c) Use your graph to solve the equation 643

8119

x. [1]

(d) (i) On the same axes , draw the graph of the straight line y = 10x + 10. [1]

(ii) Use your graphs to find the coordinates of a point on the curve x

y3

8119

at which the gradient of the tangent is equal to 10. [2]

(iii) State briefly what this gradient represents. [1]

(e) (i) On the same axes, draw the graph of 2y = 180 45x. [1]

(ii) Write down the x-coordinates of the points where the graph of

2y = 180 45x meets the graph of x

y3

8119 . [1]

(iii) Write down, but do not simplify, an equation in x which has these values as

its solutions. [1]

End of Paper

11

Answers

1 (a) x = 3

1 (b)

2

2

1

1

t

ts

(c) (a – b)2 = 81

2 (a) x

50 (b)

3

50

x (c)

x

50 –

3

50

x = 1

Simplify to get x2 + 3x – 150 = 0

(d) x = 10.84 or x = – 13.84 (rej) (e) 3.61 cm3

Price of water = $ 10.84

3 (a) 12% (b)(i) $10200 (ii) $2172 (c) $8400 (d) $14822.44

4 (a) (i) median = 64 (ii) Upper Quartile = 73, Lower Quartile = 54

Inter quartile range = 19

(iii) 58

(b) (i) The math marks have a wider spread (higher interquartile range and higher

range).The median mark for Maths is 52 which is lower than the median

mark of the Science test

(ii) No, the median mark of the Math Test is lower than the median mark of the

Science Test.

5 (a) a = 112, b = 336 (b) M = 3s or Ms3

1 c(i) 440, (ii) 9920

(d) )2)(1(3

1 nnns (e) 7989 is not an even no.

6 a(i) 52ABO (ii) 38ABT

(iii) 152AED (iv) 100CXD

(b) (i) OB =OD (radii of circle, centre O) DO=DB (radii

of circle, centre D)

Since OB = OD = DB, triangle OBD is an equilateral triangle.

(ii) ABD = 90 = CBO (angle in a semi-circle)

AD = 2OD = CO (O and D are centres)

BD =BO (sides of equilateral triangle from (i) ) Hence

triangles ABD and CBO are congruent (RHS)

7 (a) 34650000 cm3 (b) 2500

(c) (i) 21 cm (ii) CN = 18.2 cm, CM =28.7 cm

(iii) XY = 33.12 cm, 7949 cm2

8 (a) (i) 3

1 (ii) (a)

4

1 (b)

2

1 (iii)

28

5

12

(b) (i) AC = 18.3 cm (ii) Perimeter = 39.9 cm

9 (a) (i) 62.7 km (ii) 60.9° (iii) 2850 km2

(b) (i) 59.0 km (ii) 10.5°

10 (a) p=34.6, q=24.2 (c) x = 0.55,

(d)(i) (2, 28) (d) (ii) Rate of decrease of temperature when t = 2 hrs. (e)(i) x =

0.25, 3.00 (e) (ii) x

x3

811990

2

45

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