IGCSE Mathematics Model Paper - 2

IGCSE Model Question Paper (Extended) -2

1. The number of listeners to a radio station in 1995 was 350000.

a) In 1996, there were 12% more listeners than 1995.

Calculate the number of listeners in 1996. [2]

b) Each year, the radio station invites its listeners to vote for their favorite

pieces of music.

i) In 1995, 63000 listeners voted.

Calculate the percentage of listeners who voted in 1995. [2]

ii) In 1997, 69000 listeners voted. This was of the number of listeners.

Calculate the number of listeners in 1997. [2]

c) Express, in its simplest form, the ratio

the number of listeners in 1995: 1996 : 1997. [2]

[Give the answer to all of this question as fractions]

2. An examination is set once in every six months. Rojar takes the examination

each time until he passes. Each time, the probability that he passes the

examination is

a) Find the probability that Rojar

i) fails the first examination and passes the second. [1]

ii) passes the examination in either the first attempt or in the second. [2]

After the second attempt, the probability that he passes the examination

for any future attempt is .

b) Find the probability that Rojar fails the first three examinations. [2]

c) Draw a tree diagram for Rojar to pass or fail in his first four attempts. [3]

d) Use your tree diagram to calculate the probability that, Rojar

i) passes the examination within first three attempts. [2]

ii) passes the examination in either the third or in the fourth attempt. [2]

e) Show that the probability of Rojar fails the first ‘n’ attempt is . [2]

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3. [ Answer the whole of this question on a sheet of graph paper]

The tables below give values of f (x) and g (x).

f (x) = x (x + 3)

x – 5 – 4 – 3 – 2 –1.5 –1 0 1 2

f (x) 10 p 0 –2 -2.25 –2 0 4 10

g (x) = 2x

x – 4 – 3 – 2 – 1 0 0.5 1 2 3

g (x) 0.06 0.13 0.25 0.5 1 q 2 4 8

a) Calculate the values of p and q [1]

b) Using a scale of 2 cm to represent 1 unit, draw an x- axis for – 5< x < 3

and using a scale of 1 cm to represent 1 unit, draw the y- axis for

– 3< y < 10.

Draw the graphs of y = f (x) and y = g (x) for – 5 < x < 3

on the same grid. [6]

c) Draw the tangent on the curve y = g (x) at x = 0. Use this to estimate the

gradient of the curve , correct to 2 significant figures. [3]

d) Use your graph to find

(i) fg (0) [1]

(ii) g -1 ( 2.8 ) [1]

e) (i) Write down the two solutions to the equation 2x = x2 + 3x from the graph

and hence write down the range of x for which 2x > x2 + 3x. [3]

(ii) One of these two curves has line symmetry.

Draw the line of symmetry. [1]

f) The point ( -3.8, k) and (h, k) lie on the curve.

Use your graph to find the values of k and h from the curve y = f (x). [2]


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4. Diagram I shows a kite, OABC, AB = 8.8 cm, angle at A= 130o and at B = 40o.

a) i) Show that the length of the minor diagonal AC = 6 cm, to the nearest cm. [2]

ii) Explain why angle AOC = 60o. [1]

iii) The area of kite OABC, to 2 decimal placces. [3]

b) Six of the shapes shown in Diagram I are arranged to form the figure

shown in Diagram II.

For the diagram II,

i) Write down the number of line symmetry. [1]

ii) Write down the order of rotational symmetry. [1]

iii) Calculate the perimeter of the whole shape. [1]

c) Diagram II is a net of a regular hexagonal based pyramid.

Calculate, to 2 decimal places, for this pyramid

i) the surface area. [2]

ii) the base area. [2]

iii) the vertical height. [3]

iv) the volume. [2]


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Diagram IA

O

C

B1300

400

8.8 cm

A

Diagram II

O

BC



5.

The graph shows the relation between the number of units(n) of electricty used and

the total cost (C) of the bill.

a) Use the graph to find

i) the cost of the bill if 185 units are used. [1]

ii) the number of units used when the bill is $ 32.50. [1]

b) Given that the relation is C = pn + q.

i) State the value of q and explain its significance. [2]

ii) find the value of p and explain its significance. [2]

iii) find the total cost of the bill if 1240 units are used. [2]

iv) find the total number of units used if

the total cost of the bill is $ 78.00 [2]

6. A two digit number say an example 57 can be written as .www.praveenglitters.tk/wp/

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Cost in $

0Number of units

50

C

150 200

15

100 250

5

25

35

10

20

30

300

n

40



a) i) A two digit number has the first number is x and the second number is y.

Express the two digit number in terms of x and y. [1]

When the digits are interchanged, the value of new number is 18 less than the actual

number.

ii) Express the new number interms of x and y. [1]

iii) Show that the equation x – y = 2 for the above information. [2]

iv) Given that the sum of the two digits is 14.

Write down another equation interms of x and y. [1]

v) By solving the equations in part (iii) and (iv), write down the actual two digit

number. [3]

b) Solve the equation 3x2 – 10x – 4 = 0. Show all your working and give your

answers correct to 2 decimal places. [3]

7. ABC is a triangular field with

side BC = 48m, angle BAC = 47o

and angle ACB = 87o.

a) Show that the side AC = 47.2 m to 3 significant figures. [3]

Given that P is a point on AC where AP = 27.2 m.

b) Calculate the distance between P and B. [3]

A vertical tree is growing at B and the angle of elevation of the top of the tree

from P is 21.5o.

c) Calculate the height of the tree, to the nearest meter. [2]

Given also that angle BPC = 70o.

d) Calculate the bearing of P from B. [2]

e) Complete the statement. The line BP is the locus of a point which is eqidistant

from …………………………………… [1]

8. [Answer the whole of this question on a sheet of graph paper]www.praveenglitters.tk/wp/

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47oA

C

B

87o*

P

48m



The marks obtained by 120 students in a test. The cumulative frequencies are given

in the table below.

Marks x < 30 x < 50 x < 60 x < 75 x < 90

Cumulative frequency 0 36 81 105 120

a) Using a scale of 2 cm to represent 10 marks, draw a horizontal axis for

30 < x < 90 and vertical scale of 2cm to represent 20 students draw a cumulative

frequency curve to illustrate this information. [4]

b) Use your graph, showing your method clearly, estimate

i) the median mark, the inter quartile range. [3]

ii) A student who scores > 70% is considered as ‘A’ grade.

Estimate the number of ‘A’ grades. [1]

iii) 80% of the students are to be promoted.

Estimate the minimum pass mark. [2]

c) Copy and complete the above table and estimate the mean marks of all 120

students. [3]

d) Calculate the sector angle which is required to draw in a pie chart for

the class 50< x < 60. [1]

9.


Marks 30< x < 50 50< x < 60 60<x < 75 75< x < 90

Frequency 36 45

6

R

r

h



The volume of a frustum can be obtained by a specified formula

a) Calculate in terms of , the volume of a frustum when R = 8 cm, r = 3 cm

and h = 10 cm. [2]

b) Another frustum is made by a wood whose dimensions

R = 7 cm, h = 8 cm, and the volume V = 248 cm3.

i) Show that r2 + 7r – 44 = 0. [3]

ii) Factorize r2 + 7r – 44 [2]

iii) Solve the equation r2 + 7r – 44 = 0. [1]

iv) Calculate the volume of largest cone which can be made from the above

frustum whose volume is 248 cm3. [3]

v) Calculate the percentage of the wood wasted. [2]

\

10.


7

y

x102

10

4 14 -1

12

12 6

2

4

6

8

8

B

O

A

DE

C

M



a) B to A is an enlargement about the center (p, 3). Calculate

i) the scale factor of the enlargement. [1]

ii) the value of p. [1]

b) Describe the single transformation which maps the B on to C

and hence write down the transformation matrix. [3]

c) Calculate the ratio of area of A: B: C in the form 1: m: n,

where m and n are integers. [2]

d) D is a reflection of A.

Find the equation of reflection line. [3]

e) D is the image of E under an anticlockwise rotation.

Write down the angle and the coordinates of center of rotation. [2]

f) B is to be sheared by a scale factor 1, about an invariant line y =1.

Write down the coordinates of image of the point M. [1]


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IGCSE Mathematics Model Paper - 2

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