IGCSE CLASSIFIED PAST PAPERS
MR.YASSER ELSAYED
Cambridge International Education CIE
Extended mathematics 0580
PAPER
4
Part 2
Follow me on the following links:
Get some educational videos on youtube
01
02
03
https://www.youtube.com/channel/UCDSmq_Y2zlYDCPXmgXZq3Aw
in ligula.
Download the other parts for the book and
the solutions from our facebook group
https://www.facebook.com/groups/722104471882963
in ligula.
For more info contact me on
0 0 2 01 2 01 3 2 2 2 9 7
STAR WAY
your way to the star
MATHS
IGCSEmaths1@gmail.com
Paper 4 (2)
Contents
1- Solid Geometry ................................................................
(3)
2- Trigonometry and Bearing .............................................. (76)
3- Geometric Constructions ................................................ (155)
4- Vectors and Matrices ....................................................... (165)
5- Transformations ................................................................ (206)
6- Sets and Probability ....................... .................................. (258)
7- Statistics ........................................................................... (317)
Mr.Yasser Elsayed
002 012 013 222 97
2
Solid Geometry
Mr.Yasser Elsayed
002 012 013 222 97
3
1) June 2010 V1
7
(a) Calculate the volume of a cylinder of radius 31 centimetres and length 15 metres.
Give your answer in cubic metres.
Answer(a)
m3 [3]
(b) A tree trunk has a circular cross-section of radius 31 cm and length 15 m.
One cubic metre of the wood has a mass of 800 kg.
Calculate the mass of the tree trunk, giving your answer in tonnes.
Answer(b)
tonnes [2]
(c)
NOT TO
SCALE
plastic
sheet
D
C
E
The diagram shows a pile of 10 tree trunks.
Each tree trunk has a circular cross-section of radius 31 cm and length 15 m.
A plastic sheet is wrapped around the pile.
C is the centre of one of the circles.
CE and CD are perpendicular to the straight edges, as shown.
Mr.Yasser Elsayed
002 012 013 222 97
4
(i) Show that angle ECD = 120°.
Answer(c)(i)
[2]
(ii) Calculate the length of the arc DE, giving your answer in metres.
Answer(c)(ii)
m [2]
(iii) The edge of the plastic sheet forms the perimeter of the cross-section of the pile.
The perimeter consists of three straight lines and three arcs.
Calculate this perimeter, giving your answer in metres.
Answer(c)(iii)
m [3]
(iv) The plastic sheet does not cover the two ends of the pile.
Calculate the area of the plastic sheet.
Answer(c)(iv)
Mr.Yasser Elsayed
002 012 013 222 97
m2 [1]
5
2) June 2010 V2
6
A spherical ball has a radius of 2.4 cm.
(a) Show that the volume of the ball is 57.9 cm3, correct to 3 significant figures.
[The volume V of a sphere of radius r is V =
4 3
πr . ]
3
Answer(a)
[2]
(b)
NOT TO
SCALE
Six spherical balls of radius 2.4 cm fit exactly into a closed box.
The box is a cuboid.
Find
(i) the length, width and height of the box,
Answer(b)(i)
cm,
cm,
cm
[3]
cm3
[1]
Answer(b)(iii)
cm3
[1]
Answer(b)(iv)
cm2
[2]
(ii) the volume of the box,
Answer(b)(ii)
(iii) the volume of the box not occupied by the balls,
(iv) the surface area of the box.
Mr.Yasser Elsayed
002 012 013 222 97
6
(c)
NOT TO
SCALE
The six balls can also fit exactly into a closed cylindrical container, as shown in the diagram.
Find
(i) the volume of the cylindrical container,
Answer(c)(i)
cm3
[3]
cm3
[1]
(ii) the volume of the cylindrical container not occupied by the balls,
Answer(c)(ii)
(iii) the surface area of the cylindrical container.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(c)(iii)
cm2 [3]
7
3) June 2010 V3
8
NOT TO
SCALE
3 cm
6 cm
10 cm
A solid metal cuboid measures 10 cm by 6 cm by 3 cm.
(a) Show that 16 of these solid metal cuboids will fit exactly into a box which has internal
measurements 40 cm by 12 cm by 6 cm.
Answer(a)
[2]
(b) Calculate the volume of one metal cuboid.
cm3
[1]
Answer(c)(i)
g
[2]
Answer(c)(ii)
kg
8
Answer(b)
(c) One cubic centimetre of the metal has a mass of 8 grams.
The box has a mass of 600 grams.
Calculate the total mass of the 16 cuboids and the box in
(i) grams,
(ii) kilograms.
Mr.Yasser
Elsayed
002 012 013 222 97
[1]
(d) (i)
Calculate the surface area of one of the solid metal cuboids.
Answer(d)(i)
cm 2
[2]
(ii) The surface of each cuboid is painted. The cost of the paint is $25 per square metre
.
Calculate the cost of painting all 16 cuboids.
[3]
Answer(d)(ii) $
(e) One of the solid metal cuboids is melted down.
Some of the metal is used to make 200 identical solid spheres of radius 0.5 cm.
Calculate the volume of metal from this cuboid which is not used.
[The volume, V, of a sphere of radius r is V =
4
3
π r 3.]
Answer(e)
cm 3 [3]
3
(f) 50 cm of metal is used to make 20 identical solid spheres of radius r .
Calculate the radius r.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(f) r =
cm
[3]
9
4) November 2010 V1
4
NOT TO
SCALE
4m
1.5 m
2m
l
An open water storage tank is in the shape of a cylinder on top of a cone.
The radius of both the cylinder and the cone is 1.5 m.
The height of the cylinder is 4 m and the height of the cone is 2 m.
(a) Calculate the total surface area of the outside of the tank.
[The curved surface area, A, of a cone with radius r and slant height l is A = πrl. ]
Answer(a)
m2
[6]
(b) The tank is completely full of water.
(i) Calculate the volume of water in the tank and show that it rounds to 33 m3, correct to the
nearest whole number.
1
[The volume, V, of a cone with radius r and height h is V = πr2h.]
3
Answer(b)(i)
Mr.Yasser Elsayed
002 012 013 222 97
[4]
10
(ii)
NOT TO
SCALE
0.5 m
The cross-section of an irrigation channel is a semi-circle of radius 0.5 m.
The 33 m3 of water from the tank completely fills the irrigation channel.
Calculate the length of the channel.
Answer(b)(ii)
m
[3]
litres
[1]
s
[2]
(c) (i) Calculate the number of litres in a full tank of 33 m3.
Answer(c)(i)
(ii) The water drains from the tank at a rate of 1800 litres per minute.
Calculate the time, in minutes and seconds, taken to empty the tank.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(c)(ii)
min
11
5) November 2010 V2
4
(a)
4 cm
NOT TO
SCALE
13 cm
The diagram shows a cone of radius 4 cm and height 13 cm.
It is filled with soil to grow small plants.
Each cubic centimetre of soil has a mass of 2.3g.
(i) Calculate the volume of the soil inside the cone.
[The volume, V, of a cone with radius r and height h is V =
1
3
π r 2 h .]
Answer(a)(i)
cm3
[2]
Answer(a)(ii)
g
[1]
(ii) Calculate the mass of the soil.
(iii) Calculate the greatest number of these cones which can be filled completely using 50 kg
of soil.
Answer(a)(iii)
[2]
(b) A similar cone of height 32.5 cm is used for growing larger plants.
Calculate the volume of soil used to fill this cone.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(b)
cm3
[3]
12
(c)
NOT TO
SCALE
12 cm
Some plants are put into a cylindrical container with height 12 cm and volume 550 cm3 .
Calculate the radius of the cylinder.
Answer(c)
Mr.Yasser Elsayed
002 012 013 222 97
cm
[3]
13
6) November 2010 V3
8
NOT TO
SCALE
3 cm
12 cm
The diagram shows a solid made up of a hemisphere and a cylinder.
The radius of both the cylinder and the hemisphere is 3 cm.
The length of the cylinder is 12 cm.
(a) (i) Calculate the volume of the solid.
4
[ The volume, V, of a sphere with radius r is V = πr 3 .]
3
Answer(a)(i)
cm3 [4]
(ii) The solid is made of steel and 1 cm3 of steel has a mass of 7.9 g.
Calculate the mass of the solid.
Give your answer in kilograms.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(a)(ii)
kg
[2]
14
(iii) The solid fits into a box in the shape of a cuboid, 15 cm by 6 cm by 6 cm.
Calculate the volume of the box not occupied by the solid.
Answer(a)(iii)
cm3 [2]
(b) (i) Calculate the total surface area of the solid.
You must show your working.
[ The surface area, A, of a sphere with radius r is A = 4πr 2 .]
Answer(b)(i)
cm2 [5]
(ii) The surface of the solid is painted.
The cost of the paint is $0.09 per millilitre.
One millilitre of paint covers an area of 8 cm2.
Calculate the cost of painting the solid.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(b)(ii) $
[2]
15
7) June 2011 V1
6
F
NOT TO
SCALE
C
D
E
14 cm
36 cm
A
B
19 cm
In the diagram, ABCDEF is a prism of length 36 cm.
The cross-section ABC is a right-angled triangle.
AB = 19 cm and AC = 14 cm.
Calculate
(a) the length BC,
Answer(a) BC =
cm [2]
Answer(b)
cm2 [4]
Answer(c)
cm3 [2]
Answer(d) CE =
cm [2]
(b) the total surface area of the prism,
(c) the volume of the prism,
(d) the length CE,
(e) the angle between the line CE and the base ABED.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(e)
[3]
16
8) June 2011 V2
7
(a)
V
B
C
A
F
E
V
C
B
NOT TO
SCALE
A
D
D
2.5 cm
9.5 cm
2.5 cm
F
F
E
2.5 cm
E
A solid pyramid has a regular hexagon of side 2.5cm as its base.
Each sloping face is an isosceles triangle with base 2.5 cm and height 9.5 cm.
Calculate the total surface area of the pyramid.
Answer(a)
cm
2
[4]
(b)
O
55°
15 cm
A
NOT TO
SCALE
B
A sector OAB has an angle of 55° and a radius of 15 cm.
2
Calculate the area of the sector and show that it rounds to 108 cm , correct to 3 significant figures.
Answer (b)
[3]
Mr.Yasser Elsayed
002 012 013 222 97
17
(c)
15 cm
NOT TO
SCALE
The sector radii OA and OB in part (b) are joined to form a cone.
(i) Calculate the base radius of the cone.
[The curved surface area, A, of a cone with radius r and slant height l is A = πrl.]
Answer(c)(i)
cm [2]
(ii) Calculate the perpendicular height of the cone.
Answer(c)(ii)
cm [3]
(d)
7.5 cm
NOT TO
SCALE
A solid cone has the same dimensions as the cone in part (c).
A small cone with slant height 7.5 cm is removed by cutting parallel to the base.
Calculate the volume of the remaining solid.
[The volume, V, of a cone with radius r and height h is V =
Mr.Yasser Elsayed
002 012 013 222 97
Answer(d)
1
3
πr2h.]
cm3 [3]
18
9) November 2011 V1
4
r
8 cm
NOT TO
SCALE
s
2.7 cm
20 cm
The diagram shows a plastic cup in the shape of a cone with the end removed.
The vertical height of the cone in the diagram is 20 cm.
The height of the cup is 8 cm.
The base of the cup has radius 2.7 cm.
(a) (i) Show that the radius, r, of the circular top of the cup is 4.5 cm.
Answer(a)(i)
[2]
(ii) Calculate the volume of water in the cup when it is full.
[The volume, V, of a cone with radius r and height h is V =
Mr.Yasser Elsayed
002 012 013 222 97
Answer(a)(ii)
1
3
πr2h.]
cm3 [4]
19
(b) (i) Show that the slant height, s, of the cup is 8.2 cm.
Answer(b)(i)
[3]
(ii) Calculate the curved surface area of the outside of the cup.
[The curved surface area, A, of a cone with radius r and slant height l is A = πrl.]
Answer(b)(ii)
Mr.Yasser Elsayed
002 012 013 222 97
cm2 [5]
20
10) November 2011 V2
4
Boris has a recipe which makes 16 biscuits.
The ingredients are
160 g flour,
160 g sugar,
240 g butter,
200 g oatmeal.
(a) Boris has only 350 grams of oatmeal but plenty of the other ingredients.
(i) How many biscuits can he make?
Answer(a)(i)
[2]
(ii) How many grams of butter does he need to make this number of biscuits?
Answer(a)(ii)
g [2]
(b) The ingredients are mixed together to make dough.
This dough is made into a sphere of volume 1080 cm3.
Calculate the radius of this sphere.
[The volume, V, of a sphere of radius r is V =
Mr.Yasser Elsayed
002 012 013 222 97
4
3
πr3.]
Answer(b)
cm [3]
21
(c)
20 cm
1.8 cm
30 cm
NOT TO
SCALE
The 1080 cm3 of dough is then rolled out to form a cuboid 20 cm × 30 cm × 1.8 cm.
Boris cuts out circular biscuits of diameter 5 cm.
(i) How many whole biscuits can he cut from this cuboid?
Answer(c)(i)
[1]
(ii) Calculate the volume of dough left over.
Answer(c)(ii)
Mr.Yasser Elsayed
002 012 013 222 97
cm3 [3]
22
11) November 2011 V2
6
NOT TO
SCALE
10 cm
h cm
9 cm
A solid cone has diameter 9 cm, slant height 10 cm and vertical height h cm.
(a) (i) Calculate the curved surface area of the cone.
[The curved surface area, A, of a cone, radius r and slant height l is A = πrl.]
cm2 [2]
Answer(a)(i)
(ii) Calculate the value of h, the vertical height of the cone.
Answer(a)(ii) h =
[3]
(b)
NOT TO
SCALE
9 cm
3 cm
Sasha cuts off the top of the cone, making a smaller cone with diameter 3 cm.
This cone is similar to the original cone.
(i) Calculate the vertical height of this small cone.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(b)(i)
cm [2]
23
(ii) Calculate the curved surface area of this small cone.
Answer(b)(ii)
cm2
[2]
(c)
NOT TO
SCALE
12 cm
9 cm
The shaded solid from part (b) is joined to a solid cylinder with diameter 9 cm
and height 12 cm.
Calculate the total surface area of the whole solid.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(c)
cm2 [5]
24
12) November 2011 V3
1
0.8 m
0.5 m
NOT TO
SCALE
1.2 m
1.2 m
d
0.4 m
A rectangular tank measures 1.2 m by 0.8 m by 0.5 m.
(a) Water flows from the full tank into a cylinder at a rate of 0.3 m3/min.
Calculate the time it takes for the full tank to empty.
Give your answer in minutes and seconds.
Answer(a)
Mr.Yasser Elsayed
002 012 013 222 97
min
s [3]
25
(b) The radius of the cylinder is 0.4 m.
Calculate the depth of water, d, when all the water from the rectangular tank is in the cylinder.
Answer(b) d =
m [3]
(c) The cylinder has a height of 1.2 m and is open at the top.
The inside surface is painted at a cost of $2.30 per m2.
Calculate the cost of painting the inside surface.
Answer(c) $
Mr.Yasser Elsayed
002 012 013 222 97
[4]
26
13) November 2011 V3
6
Q
P
NOT TO
SCALE
3 cm
D
C
4 cm
A
B
12 cm
The diagram shows a triangular prism of length 12 cm.
The rectangle ABCD is horizontal and the rectangle DCPQ is vertical.
The cross-section is triangle PBC in which angle BCP = 90°, BC = 4 cm and CP = 3 cm.
(a) (i) Calculate the length of AP.
Answer(a)(i) AP =
cm [3]
(ii) Calculate the angle of elevation of P from A.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(a)(ii)
[2]
27
(b) (i) Calculate angle PBC.
Answer(b)(i) Angle PBC =
[2]
(ii) X is on BP so that angle BXC = 120°.
Calculate the length of XC.
Answer(b)(ii) XC =
Mr.Yasser Elsayed
002 012 013 222 97
cm [3]
28
14) June 2012 V1
10
24 cm
NOT TO
SCALE
9 cm
A solid metal cone has base radius 9 cm and vertical height 24 cm.
(a) Calculate the volume of the cone.
[The volume, V, of a cone with radius r and height h is V =
1
3
πr2h.]
Answer(a)
cm3 [2]
(b)
16 cm
NOT TO
SCALE
9 cm
A cone of height 8 cm is removed by cutting parallel to the base, leaving the solid shown above.
Show that the volume of this solid rounds to 1960 cm3, correct to 3 significant figures.
Answer (b)
[4]
(c) The 1960 cm3 of metal in the solid in part (b) is melted and made into 5 identical cylinders,
each of length 15 cm.
Show that the radius of each cylinder rounds to 2.9 cm, correct to 1 decimal place.
Answer (c)
Mr.Yasser Elsayed
002 012 013 222 97
[4]
29
15) June 2012 V3
5
NOT TO
SCALE
3 cm
3 cm
6 cm
8 cm
The diagram shows two solid spheres of radius 3 cm lying on the base of a cylinder of radius 8 cm.
Liquid is poured into the cylinder until the spheres are just covered.
[The volume, V, of a sphere with radius r is V =
4
3
πr3.]
(a) Calculate the volume of liquid in the cylinder in
(i) cm3,
Answer(a)(i)
cm3 [4]
Answer(a)(ii)
litres [1]
(ii) litres.
Mr.Yasser Elsayed
002 012 013 222 97
30
(b) One cubic centimetre of the liquid has a mass of 1.22 grams.
Calculate the mass of the liquid in the cylinder.
Give your answer in kilograms.
Answer(b)
kg [2]
(c) The spheres are removed from the cylinder.
Calculate the new height of the liquid in the cylinder.
Answer(c)
Mr.Yasser Elsayed
002 012 013 222 97
cm [2]
31
16) November 2012 V1
5
(a)
NOT TO
SCALE
20 cm
24 cm
46 cm
Jose has a fish tank in the shape of a cuboid measuring 46 cm by 24 cm by 20 cm.
Calculate the length of the diagonal shown in the diagram.
Answer(a)
cm [3]
(b) Maria has a fish tank with a volume of 20 000 cm3.
Write the volume of Maria’s fish tank as a percentage of the volume of Jose’s fish tank.
Answer(b)
%
[3]
(c) Lorenzo’s fish tank is mathematically similar to Jose’s and double the volume.
Calculate the dimensions of Lorenzo’s fish tank.
Answer(c)
cm by
cm by
cm [3]
(d) A sphere has a volume of 20 000 cm3. Calculate its radius.
4
[The volume, V, of a sphere with radius r is V = πr3.]
3
Mr.Yasser Elsayed
002 012 013 222 97
Answer(d)
cm [3]
32
17) November 2012 V1
8
A rectangular piece of card has a square of side 2 cm removed from each corner.
2 cm
2 cm
NOT TO
SCALE
(2x + 3) cm
(x + 5) cm
(a) Write expressions, in terms of x, for the dimensions of the rectangular card before the squares
are removed from the corners.
Answer(a)
cm by
(b) The diagram shows a net for an open box.
Show that the volume, V cm3, of the open box is given by the formula
cm [2]
V = 4x2 + 26x + 30 .
Answer(b)
Mr.Yasser Elsayed
002 012 013 222 97
[3]
33
(c) (i) Calculate the values of x when V = 75.
Show all your working and give your answers correct to two decimal places.
or x =
Answer(c)(i) x =
[5]
(ii) Write down the length of the longest edge of the box.
Answer(c)(ii)
Mr.Yasser Elsayed
002 012 013 222 97
cm [1]
34
18) November 2012 V3
3
A metal cuboid has a volume of 1080 cm3 and a mass of 8 kg.
(a) Calculate the mass of one cubic centimetre of the metal.
Give your answer in grams.
Answer(a)
g [1]
Answer(b)
cm [2]
(b) The base of the cuboid measures 12 cm by 10 cm.
Calculate the height of the cuboid.
(c) The cuboid is melted down and made into a sphere with radius r cm.
(i) Calculate the value of r.
[The volume, V, of a sphere with radius r is V =
Mr.Yasser Elsayed
002 012 013 222 97
4
3
πr 3.]
Answer(c)(i) r =
[3]
35
(ii) Calculate the surface area of the sphere.
[The surface area, A, of a sphere with radius r is A = 4πr 2.]
Answer(c)(ii)
cm2 [2]
(d) A larger sphere has a radius R cm.
The surface area of this sphere is double the surface area of the sphere with radius r cm in
part (c).
R
.
Find the value of
r
Answer(d)
Mr.Yasser Elsayed
002 012 013 222 97
[2]
36
19) June 2013 V2
9
(a)
NOT TO
SCALE
12 cm
4 cm
The diagram shows a prism of length 12 cm.
The cross section is a regular hexagon of side 4 cm.
Calculate the total surface area of the prism.
Answer(a) ........................................ cm2 [4]
(b) Water flows through a cylindrical pipe of radius 0.74 cm.
It fills a 12 litre bucket in 4 minutes.
(i) Calculate the speed of the water through the pipe in centimetres per minute.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(b)(i) .................................. cm/min [4]
37
(ii) When the 12 litre bucket is emptied into a circular pool, the water level rises by 5 millimetres .
Calculate the radius of the pool correct to the nearest centimetre.
Answer(b)(ii) ......................................... cm [5]
Mr.Yasser Elsayed
002 012 013 222 97
38
20) June 2013 V3
4
I
NOT TO
SCALE
H
J
F
40 cm
7 cm
E
22 cm
G
EFGHIJ is a solid metal prism of length 40 cm.
The cross section EFG is a right-angled triangle.
EF = 7 cm and EG = 22 cm.
(a) Calculate the volume of the prism.
Answer(a) ........................................ cm3 [2]
(b) Calculate the length FJ.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(b) FJ = ......................................... cm [4]
39
(c) Calculate the angle between FJ and the base EGJH of the prism.
Answer(c) ............................................... [3]
(d) The prism is melted and made into spheres.
Each sphere has a radius 1.5 cm.
Work out the greatest number of spheres that can be made.
4
[The volume, V, of a sphere with radius r is V = πr3.]
3
Answer(d) ............................................... [3]
(e) (i) A right-angled triangle is the cross section of another prism.
This triangle has height 4.5 cm and base 11.0 cm.
Both measurements are correct to 1 decimal place.
Calculate the upper bound for the area of this triangle.
Answer(e)(i) ........................................ cm2 [2]
(ii) Write your answer to part (e)(i) correct to 4 significant figures.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(e)(ii) ........................................ cm2 [1]
40
21) November 2013 V1
3
NOT TO
SCALE
h
13 cm
5 cm
(a) The diagram shows a cone of radius 5 cm and slant height 13 cm.
(i) Calculate the curved surface area of the cone.
[The curved surface area, A, of a cone with radius r and slant height l is A = πrl.]
Answer(a)(i) ........................................ cm2 [2]
(ii) Calculate the perpendicular height, h, of the cone.
Answer(a)(ii) h = ......................................... cm [3]
(iii) Calculate the volume of the cone.
[The volume, V, of a cone with radius r and height h is V =
1
3
πr2h.]
Answer(a)(iii) ........................................ cm3 [2]
(iv) Write your answer to part (a)(iii) in cubic metres.
Give your answer in standard form.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(a)(iv) .......................................... m3 [2]
41
(b)
A
O
NOT TO
SCALE
13 cm
h
O
5 cm
B
The cone is now cut along a slant height and it opens out to make the sector AOB of a circle.
Calculate angle AOB.
Answer(b) Angle AOB = ............................................... [4]
Mr.Yasser Elsayed
002 012 013 222 97
42
22) November 2013 V2
4
8 cm
O
A
42°
NOT TO
SCALE
8 cm
B
h cm
A wedge of cheese in the shape of a prism is cut from a cylinder of cheese of height h cm.
The radius of the cylinder, OA, is 8 cm and the angle AOB = 42°.
(a) (i) The volume of the wedge of cheese is 90 cm3.
Show that the value of h is 3.84 cm correct to 2 decimal places.
Answer(a)(i)
[4]
(ii) Calculate the total surface area of the wedge of cheese.
Answer(a)(ii) ........................................ cm2 [5]
(b) A mathematically similar wedge of cheese has a volume of 22.5 cm3.
Calculate the height of this wedge.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(b) ......................................... cm [3]
43
23) November 2013 V3
3
A rectangular metal sheet measures 9 cm by 7 cm.
A square, of side x cm, is cut from each corner.
The metal is then folded to make an open box of height x cm.
9 cm
x cm
7 cm
NOT TO
SCALE
x cm
x cm
(a) Write down, in terms of x, the length and width of the box.
Answer(a) Length = ...............................................
Width = ............................................... [2]
(b) Show that the volume, V , of the box is
4x3 – 32x 2 + 63x
.
Answer(b)
[2]
Mr.Yasser Elsayed
002 012 013 222 97
44
24) November 2013 V3
6
Sandra has designed this open container.
The height of the container is 35 cm.
NOT TO
SCALE
35 cm
The cross section of the container is designed from three semi-circles with diameters 17.5 cm, 6.5 cm
and 24 cm.
17.5 cm
6.5 cm
NOT TO
SCALE
(a) Calculate the area of the cross section of the container.
Answer(a) ........................................ cm2 [3]
(b) Calculate the external surface area of the container, including the base.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(b) ........................................ cm2 [4]
45
(c) The container has a height of 35 cm.
Calculate the capacity of the container.
Give your answer in litres.
Answer(c) ...................................... litres [3]
(d) Sandra’s container is completely filled with water.
All the water is then poured into another container in the shape of a cone.
The cone has radius 20 cm and height 40 cm.
20 cm
NOT TO
SCALE
r
40 cm
h
(i) The diagram shows the water in the cone.
Show that
r=
h .
2
Answer(d)(i)
[1]
(ii) Find the height, h, of the water in the cone.
1
[The volume, V, of a cone with radius r and height h is V = 3 πr 2h.]
Mr.Yasser Elsayed
002 012 013 222 97
Answer(d)(ii) h = ......................................... cm [3]
46
25) June 2014 V1
3
(a) The running costs for a papermill are $75 246.
This amount is divided in the ratio labour costs : materials = 5 : 1.
Calculate the labour costs.
Answer(a) $ ................................................ [2]
(b) In 2012 the company made a profit of $135 890.
In 2013 the profit was $150 675.
Calculate the percentage increase in the profit from 2012 to 2013.
Answer(b) ............................................ % [3]
(c) The profit of $135 890 in 2012 was an increase of 7% on the profit in 2011.
Calculate the profit in 2011.
Answer(c) $ ................................................ [3]
(d)
2 cm
NOT TO
SCALE
21 cm
30 cm
Paper is sold in cylindrical rolls.
There is a wooden cylinder of radius 2 cm and height 21 cm in the centre of each roll.
The outer radius of a roll of paper is 30 cm.
(i) Calculate the volume of paper in a roll.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(d)(i) ......................................... cm3 [3]
47
(ii) The paper is cut into sheets which measure 21 cm by 29.7 cm.
The thickness of each sheet is 0.125 mm.
(a) Change 0.125 millimetres into centimetres.
Answer(d)(ii)(a) .......................................... cm [1]
(b) Work out how many whole sheets of paper can be cut from a roll.
Answer(d)(ii)(b) ................................................ [4]
Mr.Yasser Elsayed
002 012 013 222 97
48
26) June 2014 V2
5
8 cm
NOT TO
SCALE
12 cm
10 cm
4 cm
The diagram shows a cylinder with radius 8 cm and height 12 cm which is full of water.
A pipe connects the cylinder to a cone.
The cone has radius 4 cm and height 10 cm.
(a) (i) Calculate the volume of water in the cylinder.
Show that it rounds to 2410 cm3 correct to 3 significant figures.
Answer(a)(i)
[2]
(ii) Change 2410 cm3 into litres.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(a)(ii) ....................................... litres [1]
49
(b) Water flows from the cylinder along the pipe into the cone at a rate of 2 cm3 per second.
Calculate the time taken to fill the empty cone.
Give your answer in minutes and seconds correct to the nearest second.
1
[The volume, V, of a cone with radius r and height h is V = 3 πr 2h.]
Answer(b) .................. min .................. s [4]
(c) Find the number of empty cones which can be filled completely from the full cylinder.
Answer(c) ................................................ [3]
Mr.Yasser Elsayed
002 012 013 222 97
50
27) June 2014 V3
10 (a)
8 cm
NOT TO
SCALE
r cm
The three sides of an equilateral triangle are tangents to a circle of radius r cm.
The sides of the triangle are 8 cm long.
Calculate the value of r.
Show that it rounds to 2.3, correct to 1 decimal place.
Answer(a)
[3]
(b)
8 cm
NOT TO
SCALE
12 cm
The diagram shows a box in the shape of a triangular prism of height 12 cm.
The cross section is an equilateral triangle of side 8 cm.
Calculate the volume of the box.
Mr.Yasser Elsayed
002 012 013 222 97
51
Answer(b) ......................................... cm3 [4]
(c) The box contains biscuits.
Each biscuit is a cylinder of radius 2.3 centimetres and height 4 millimetres.
Calculate
(i) the largest number of biscuits that can be placed in the box,
Answer(c)(i) ................................................ [3]
(ii) the volume of one biscuit in cubic centimetres,
Answer(c)(ii) ......................................... cm3 [2]
(iii) the percentage of the volume of the box not filled with biscuits.
Answer(c)(iii) ............................................ % [3]
Mr.Yasser Elsayed
002 012 013 222 97
52
28) November 2014 V2
7
75 cm
NOT TO
SCALE
55 cm
120 cm
The diagram shows a water tank in the shape of a cuboid measuring 120 cm by 55 cm by 75 cm.
The tank is filled completely with water.
(a) Show that the capacity of the water tank is 495 litres.
Answer(a)
[2]
(b) (i) The water from the tank flows into an empty cylinder at a uniform rate of 750 millilitres per second.
Calculate the length of time, in minutes, for the water to be completely emptied from the tank.
Answer(b)(i) ......................................... min [2]
(ii) When the tank is completely empty, the height of the water in the cylinder is 112 cm.
NOT TO
SCALE
112 cm
Calculate the radius of the cylinder.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(b)(ii) .......................................... cm [3]
53
29) June 2015 V2
4
(a) A sector of a circle has radius 12 cm and an angle of 135°.
(i) Calculate the length of the arc of this sector.
Give your answer as a multiple of π.
NOT TO
SCALE
135°
12 cm
Answer(a)(i) .......................................... cm [2]
(ii) The sector is used to make a cone.
(a) Calculate the base radius, r.
12 cm
h
NOT TO
SCALE
r
Answer(a)(ii)(a) r = .......................................... cm [2]
(b) Calculate the height of the cone, h.
Answer(a)(ii)(b) h = .......................................... cm [3]
(b) The diagram shows a plant pot.
It is made by removing a small cone from a larger cone and adding a circular base.
NOT TO
SCALE
Mr.Yasser Elsayed
002 012 013 222 97
54
This is the cross section of the plant pot.
15 cm
(i) Find l.
35 cm
8 cm
l
NOT TO
SCALE
Answer(b)(i) l = .......................................... cm [3]
(ii) Calculate the total surface area of the outside of the plant pot.
[The curved surface area, A, of a cone with radius r and slant height l is A = πrl .]
Answer(b)(ii) ......................................... cm2 [3]
(c) Some cones are mathematically similar.
For these cones, the mass, M grams, is proportional to the cube of the base radius, r cm.
One of the cones has mass 1458 grams and base radius 4.5 cm.
(i) Find an expression for M in terms of r
.
Answer(c)(i) M = ................................................ [2]
(ii) Two of the cones have radii in the ratio 2 : 3.
Write down
the ratio of their masses.
Mr.Yasser
Elsayed
002 012 013 222 97
Answer(c)(ii) ................. : ................. [1]
55
30) June 2015 V3
8
(a) A cylindrical tank contains 180 000 cm3 of water.
The radius of the tank is 45 cm.
45 cm
Calculate the height of water in the tank.
NOT TO
SCALE
Answer(a) ........................................... cm [2]
(b)
D
NOT TO
SCALE
C
70 cm
40 cm
150 cm
A
50 cm
B
The diagram shows an empty tank in the shape of a horizontal prism of length 150 cm.
The cross section of the prism is an isosceles trapezium ABCD.
AB = 50 cm, CD = 70 cm and the vertical height of the trapezium is 40 cm.
(i)
Calculate the volume of the tank.
Answer(b)(i) .......................................... cm3 [3]
(ii)
Write your answer to part (b)(i) in litres.
Answer(b)(ii) ........................................ litres [1]
(c) The 180 000 cm3 of water flows from the tank in part (a) into the tank in part (b) at a rate of 15 cm3/s.
Calculate the time this takes.
Give your answer in hours and minutes.
Mr.Yasser Elsayed
002 012 013 222 97
56
Answer(c) ................ h ................ min [3]
(d)
70 cm
D
40 cm
C
x cm
F
E
NOT TO
SCALE
h cm
A
50 cm
B
The 180 000 cm3 of water reaches the level EF as shown above.
EF = x cm and the height of the water is h cm.
(i)
Using the properties of similar triangles, show that h = 2(x – 50).
Answer(d)(i)
[2]
(ii)
Using h = 2(x – 50), show that the shaded area, in cm2, is x2 – 2500.
Answer(d)(ii)
[1]
(iii)
Find the value of x.
Answer(d)(iii) x = ................................................. [2]
(iv)
Find the value of h.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(d)(iv) h = ................................................. [1]
57
31) November 2015 V3
3
The diagram shows a horizontal water trough in the shape of a prism.
NOT TO
SCALE
35 cm
12 cm
6 cm
120 cm
25 cm
The cross section of this prism is a trapezium.
The trapezium has parallel sides of lengths 35 cm and 25 cm and a perpendicular height of 12 cm.
The length of the prism is 120 cm.
(a) Calculate the volume of the trough.
Answer(a) ......................................... cm3 [3]
(b) The trough contains water to a depth of 6 cm.
(i) Show that the volume of water is 19 800 cm3.
Answer (b)(i)
[2]
(ii) Calculate the percentage of the trough that contains water.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(b)(ii) ............................................ % [1]
58
(c) The water is drained from the trough at a rate of 12 litres per hour.
Calculate the time it takes to empty the trough.
Give your answer in hours and minutes.
Answer(c) ................. h ................. min [4]
(d) The water from the trough just fills a cylinder of radius r cm and height 3r cm.
Calculate the value of r.
Answer(d) r = ................................................ [3]
(e) The cylinder has a mass of 1.2 kg.
1 cm3 of water has a mass of 1 g.
Calculate the total mass of the cylinder and the water.
Give your answer in kilograms.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(e) ........................................... kg [2]
59
32) March 2015 V2
8
(a) The diagram shows a sector of a circle
with centre O and radius 24 cm.
A
NOT TO
SCALE
x°
24 cm
(i) The total perimeter of the sector is 68 cm.
Calculate the value of x.
O
B
Answer(a)(i) x = ................................................ [3]
(ii) The points A and B of the sector are joined together to
make a hollow cone.
The arc AB becomes the circumference of the base of the cone.
O
NOT TO
SCALE
AB
Calculate the volume of the cone.
1
[The volume, V, of a cone with radius r and height h is V = 3 πr2h.]
Mr.Yasser Elsayed
002 012 013 222 97
Answer(a)(ii) ......................................... cm3 [6]
60
Q
(b)
NOT TO
SCALE
M
P
X
O
8 cm
Y
The diagram shows a shape made from a square, a quarter circle and a semi-circle.
OPXY is a square of side 8 cm.
OPQ is a quarter circle, centre O.
The line OMQ is the diameter of the semi-circle.
Calculate the area of the shape.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(b) ......................................... cm2 [5]
61
33) March 2016 V2
10 (a) The ten circles in the diagram each have radius 1 cm.
The centre of each circle is marked with a dot.
P
Calculate the height of triangle PQR.
NOT TO
SCALE
Q
R
8 cm
............................................. cm [3]
(b) Mr Patel uses whiteboard pens that are
cylinders of radius 1 cm.
A
(i) The diagram shows 10 pens stacked in a tray.
The tray is 8 cm wide.
The point A is the highest point in the stack.
NOT TO
SCALE
Find the height of A above the base, BC, of the tray.
B
8 cm
C
............................................ cm [1]
(ii) The diagram shows a box that holds one pen.
The box is a prism of length 12 cm.
The cross section of the prism is an
equilateral triangle.
The pen touches each of the three rectangular
faces of the box.
NOT TO
SCALE
12 cm
Calculate the volume of this box.
Mr.Yasser Elsayed
002 012 013 222 97
........................................... cm3 [5]
62
34) June 2016 V1
4
(a) Calculate the volume of a metal sphere of radius 15 cm and show that it rounds to 14 140 cm3, correct
to 4 significant figures.
[The volume, V, of a sphere with radius r is V = 43 rr 3.]
[2]
(b) (i)
The sphere is placed inside an empty cylindrical tank of radius 25 cm and height 60 cm.
The tank is filled with water.
25 cm
NOT TO
SCALE
60 cm
Calculate the volume of water required to fill the tank.
........................................... cm3 [3]
(ii)
The sphere is removed from the tank.
NOT TO
SCALE
d
Calculate the depth, d, of water in the tank.
Mr.Yasser Elsayed
002 012 013 222 97
63
d = ........................................... cm [2]
(c) The sphere is melted down and the metal is made into a solid cone of height 54 cm.
(i)
Calculate the radius of the cone.
[The volume, V, of a cone with radius r and height h is V = 13 rr 2 h .]
............................................ cm [3]
(ii)
Calculate the total surface area of the cone.
[The curved surface area, A , of a cone with radius r and slant height l is A = rrl .]
Mr.Yasser Elsayed
002 012 013 222 97
........................................... cm2 [4]
64
35) June 2016 V2
6
The diagram shows a cuboid.
F
G
E
H
B
A
30 cm
NOT TO
SCALE
C
35 cm
60 cm
D
AD = 60 cm, CD = 35 cm and CG = 30 cm.
(a) Write down the number of planes of symmetry of this cuboid.
.................................................. [1]
(b) (i)
Work out the surface area of the cuboid.
........................................... cm2 [3]
(ii)
Write your answer to part (b)(i) in square metres.
............................................. m2 [1]
(c) Calculate
(i)
the length AG,
Mr.Yasser Elsayed
002 012 013 222 97
AG = ............................................ cm [4]
65
(ii)
the angle between AG and the base ABCD.
.................................................. [3]
(d) (i)
Show that the volume of the cuboid is 63 000 cm3.
[1]
(ii)
A cylinder of height 40 cm has the same volume as the cuboid.
Calculate the radius of the cylinder.
............................................. cm [3]
Mr.Yasser Elsayed
002 012 013 222 97
66
36) June 2016 V3
9
A
NOT TO
SCALE
12 cm
O
145°
B
The diagram shows a sector, centre O, and radius 12 cm.
(a) Calculate the area of the sector.
........................................... cm2 [3]
(b) The sector is made into a cone by joining OA to OB.
Calculate the volume of the cone.
1
[The volume, V, of a cone with base radius r and height h is V = rr 2 h .]
3
Mr.Yasser Elsayed
002 012 013 222 97
........................................... cm3 [6]
67
37) June 2017 V1
5
(a) The diagram shows a cylindrical container used to serve coffee in a hotel.
18 cm
NOT TO
SCALE
50 cm
The container has a height of 50 cm and a radius of 18 cm.
(i)
Calculate the volume of the cylinder and show that it rounds to 50 900 cm3, correct to 3 significant
figures.
[2]
(ii)
30 litres of coffee are poured into the container.
Work out the height, h, of the empty space in the container.
NOT TO
SCALE
h
h = ......................................... cm [3]
Mr.Yasser Elsayed
002 012 013 222 97
68
(iii)
Cups in the shape of a hemisphere are filled with coffee from the container.
The radius of a cup is 3.5 cm.
NOT TO
SCALE
3.5 cm
Work out the maximum number of these cups that can be completely filled from the 30 litres of
coffee in the container.
4
[The volume, V, of a sphere with radius r is V = rr 3 .]
3
................................................. [4]
(b) The hotel also uses glasses in the shape of a cone.
r
8.4 cm
NOT TO
SCALE
The capacity of each glass is 95 cm3.
(i)
Calculate the radius, r, and show that it rounds to 3.3 cm, correct to 1 decimal place.
1
[The volume, V, of a cone with radius r and height h is V = rr 2 h .]
3
[3]
(ii)
Calculate the curved surface area of the cone.
[The curved surface area, A, of a cone with radius r and slant height l is A = rrl .]
Mr.Yasser Elsayed
002 012 013 222 97
......................................... cm2 [4]
69
38) November 2017 V1
8
l
h
NOT TO
SCALE
5 mm
The diagram shows a solid made from a hemisphere and a cone.
The base diameter of the cone and the diameter of the hemisphere are each 5 mm.
(a) The total surface area of the solid is
115r
mm2.
4
Show that the slant height, l, is 6.5 mm.
[The curved surface area, A, of a cone with radius r and slant height l is A = rrl.]
[The surface area, A, of a sphere with radius r is A = 4rr2.]
[4]
(b) Calculate the height, h, of the cone.
h = ......................................... mm [3]
Mr.Yasser Elsayed
002 012 013 222 97
© UCLES 2017
0580/41/O/N/17
70
(c) Calculate the volume of the solid.
[The volume, V, of a cone with radius r and height h is V =
[The volume, V, of a sphere with radius r is V =
4 3
rr .]
3
1 2
rr h.]
3
.........................................mm3 [4]
(d) The solid is made from gold.
1 cubic centimetre of gold has a mass of 19.3 grams.
The value of 1 gram of gold is $38.62 .
Calculate the value of the gold used to make the solid.
$ ................................................. [3]
Mr.Yasser Elsayed
002 012 013 222 97
© UCLES 2017
0580/41/O/N/17
71
[Turn over
39) June 2018 V1
6
A solid hemisphere has volume 230 cm3.
(a) Calculate the radius of the hemisphere.
4
[The volume, V, of a sphere with radius r is V = rr 3 .]
3
.......................................... cm [3]
(b) A solid cylinder with radius 1.6 cm is attached to the hemisphere to make a toy.
NOT TO
SCALE
The total volume of the toy is 300 cm3.
(i)
Calculate the height of the cylinder.
Mr.Yasser Elsayed
002 012 013 222 97
.......................................... cm [3]
72
(ii)
A mathematically similar toy has volume 19 200 cm3.
Calculate the radius of the cylinder for this toy.
.......................................... cm [3]
Mr.Yasser Elsayed
002 012 013 222 97
73
40) June 2020 V2
8 (a)
C
R
NOT TO
SCALE
A
8 cm
B
P
12 cm
Q
Triangle ABC is mathematically similar to triangle PQR.
The area of triangle ABC is 16 cm2.
(i) Calculate the area of triangle PQR.
.......................................... cm2 [2]
(ii) The triangles are the cross-sections of prisms which are also mathematically similar.
The volume of the smaller prism is 320 cm3.
Calculate the length of the larger prism.
............................................ cm [3]
Mr.Yasser Elsayed
002 012 013 222 97
74
(b) A cylinder with radius 6 cm and height h cm has the same volume as a sphere with radius 4.5 cm.
Find the value of h.
4
[The volume, V, of a sphere with radius r is V = rr 3 .]
3
h = ................................................ [3]
(c) A solid metal cube of side 20 cm is melted down and made into 40 solid spheres, each of radius
r cm.
Find the value of r.
4
[The volume, V, of a sphere with radius r is V = rr 3 .]
3
r = ................................................ [3]
7x
cm.
2
The surface area of a sphere with radius R cm is equal to the total surface area of the cylinder.
(d) A solid cylinder has radius x cm and height
Find an expression for R in terms of x.
[The surface area, A, of a sphere with radius r is A = 4rr 2 .]
Mr.Yasser Elsayed
002 012 013 222 97
R = ................................................ [3]
75
Trigonometry
and Bearing
Mr.Yasser Elsayed
002 012 013 222 97
76
1) June 2010 V1
5
D
30°
C
NOT TO
SCALE
24 cm
40°
40°
A
B
26 cm
ABCD is a quadrilateral and BD is a diagonal.
AB = 26 cm, BD = 24 cm, angle ABD = 40°, angle CBD = 40° and angle CDB = 30°.
(a) Calculate the area of triangle ABD.
Answer(a)
cm2
[2]
Answer(b)
cm
[4]
Answer(c)
cm
[4]
cm
[2]
(b) Calculate the length of AD.
(c) Calculate the length of BC.
(d) Calculate the shortest distance from the point C to the line BD.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(d)
77
2) June 2010 V2
5
North
A
NOT TO
SCALE
180 km
115 km
H
90 km
30°
T
70°
R
The diagram shows some straight line distances between Auckland (A), Hamilton (H), Tauranga (T)
and Rotorua (R).
AT = 180 km, AH = 115 km and HT = 90 km.
(a) Calculate angle HAT.
Show that this rounds to 25.0°, correct to 3 significant figures.
Answer(a)
[4]
(b) The bearing of H from A is 150°.
Find the bearing of
(i) T from A,
(ii) A fromElsayed
T.
Mr.Yasser
002 012 013 222 97
Answer(b)(i)
[1]
Answer(b)(ii)
[1]
78
(c) Calculate how far T is east of A.
Answer(c)
km
[3]
Answer(d)
km
[3]
(d) Angle THR = 30° and angle HRT = 70°.
Calculate the distance TR.
(e) On a map the distance representing HT is 4.5cm.
The scale of the map is 1 : n.
Calculate the value of n.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(e) n =
[2]
79
3) June 2010 V3
C
2
B
8 cm
NOT TO
SCALE
5 cm
3 cm
D
A
11 cm
In the quadrilateral ABCD, AB = 3 cm, AD = 11 cm and DC = 8 cm.
The diagonal AC = 5 cm and angle BAC = 90°.
Calculate
(a) the length of BC,
Answer(a) BC =
cm
[2]
(b) angle ACD,
[4]
Answer(b) Angle ACD =
(c) the area of the quadrilateral ABCD.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(c)
cm2
[3]
80
4) November 2010 V1
6
(a)
A
P
19.5 cm
B
16.5 cm
C
Q
11 cm
NOT TO
SCALE
R
The diagram shows a toy boat.
AC = 16.5 cm, AB = 19.5 cm and PR = 11 cm.
Triangles ABC and PQR are similar.
(i) Calculate PQ.
Answer(a)(i) PQ =
cm [2]
Answer(a)(ii) BC =
cm [3]
(ii) Calculate BC.
(iii) Calculate angle ABC.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(a)(iii) Angle ABC =
[2]
81
(iv) The toy boat is mathematically similar to a real boat.
The length of the real boat is 32 times the length of the toy boat.
The fuel tank in the toy boat holds 0.02 litres of diesel.
Calculate how many litres of diesel the fuel tank of the real boat holds.
litres
Answer(a)(iv)
[2]
(b)
E
F
32°
143°
105 m
NOT TO
SCALE
67 m
70°
D
G
The diagram shows a field DEFG, in the shape of a quadrilateral, with a footpath along the
diagonal DF.
DF = 105 m and FG = 67 m.
Angle EDF = 70U, angle EFD = 32U and angle DFG = 143U.
(i) Calculate DG.
Answer(b)(i) DG =
m
[4]
Answer(b)(ii) EF =
m
[4]
(ii) Calculate EF.
Mr.Yasser Elsayed
002 012 013 222 97
82
5) November 2010 V2
6
L
5480 km
D
NOT TO
SCALE
165°
3300 km
C
The diagram shows the positions of London (L), Dubai (D) and Colombo (C).
(a) (i) Show that LC is 8710 km correct to the nearest kilometre.
Answer(a)(i)
[4]
(ii) Calculate the angle CLD.
Answer(a)(ii) Angle CLD =
Mr.Yasser Elsayed
002 012 013 222 97
[3]
83
(b) A plane flies from London to Dubai and then to Colombo.
It leaves London at 01 50 and the total journey takes 13 hours and 45 minutes.
The local time in Colombo is 7 hours ahead of London.
Find the arrival time in Colombo.
Answer(b)
[2]
(c) Another plane flies the 8710 km directly from London to Colombo at an average speed of
800 km/h.
How much longer did the plane in part (b) take to travel from London to Colombo?
Give your answer in hours and minutes, correct to the nearest minute.
Answer(c)
Mr.Yasser Elsayed
002 012 013 222 97
h
min
[4]
84
6) November 2010 V3
2
R
4 km
Q
NOT TO
SCALE
7 km
4.5 km
S
85°
40°
P
The diagram shows five straight roads.
PQ = 4.5 km, QR = 4 km and PR = 7 km.
Angle RPS = 40° and angle PSR = 85°.
(a) Calculate angle PQR and show that it rounds to 110.7°.
Answer(a)
[4]
(b) Calculate the length of the road RS and show that it rounds to 4.52 km.
Answer(b)
[3]
(c) Calculate the area of the quadrilateral PQRS.
[Use the value of 110.7° for angle PQR and the value of 4.52 km for RS.]
Mr.Yasser Elsayed
002 012 013 222 97
Answer(c)
km2
[5]
85
7) June 2011 V1
1
(b) The route for the sponsored walk in winter is triangular.
North
B 110°
NOT TO
SCALE
C
A
(i) Senior students start at A, walk North to B, then walk on a bearing 110° to C.
They then return to A.
AB = BC.
Calculate the bearing of A from C.
Answer(b)(i)
[3]
(ii)
North
B 110°
NOT TO
SCALE
110°
C
4 km
A
AB = BC = 6 km.
Junior students follow a similar path but they only walk 4 km North from A, then 4 km on a
bearing 110° before returning to A.
Senior students walk a total of 18.9 km.
Calculate the distance walked by junior students.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(b)(ii)
km [3]
86
8) June 2011 V1
4
(a)
12 cm
H
G
NOT TO
SCALE
6 cm
14 cm
F
The diagram shows triangle FGH, with FG = 14 cm, GH = 12 cm and FH = 6 cm.
(i) Calculate the size of angle HFG.
Answer(a)(i) Angle HFG =
[4]
(ii) Calculate the area of triangle FGH.
Answer(a)(ii)
Mr.Yasser Elsayed
002 012 013 222 97
cm2 [2]
87
(b)
Q
18 cm
R
NOT TO
SCALE
12 cm
117°
P
The diagram shows triangle PQR, with RP = 12 cm, RQ = 18 cm and angle RPQ = 117°.
Calculate the size of angle RQP.
Answer(b) Angle RQP =
Mr.Yasser Elsayed
002 012 013 222 97
[3]
88
9) June 2011 V2
2
B
C
A
1.7 m
D
F
NOT TO
SCALE
G
1.5 m
E
2m
H
The diagram shows a box ABCDEFGH in the shape of a cuboid measuring 2 m by 1.5 m by 1.7 m.
(a) Calculate the length of the diagonal EC
.
Answer(a) EC =
m [4]
(b) Calculate the angle between EC and the base EFGH.
Answer(b)
[3]
(c) (i) A rod has length 2.9 m, correct to 1 decimal place.
What is the upper bound for the length of the rod?
Answer(c)(i)
m [1]
(ii) Will the rod fit completely in the box?
Give a reason for your answer.
Answer(c)(ii)
Mr.Yasser
Elsayed
002 012 013 222 97
[1]
89
10) June 2011 V2
3
(a)
North
C
North
A
The scale drawing shows the positions of two towns A and C on a map.
On the map, 1 centimetre represents 20 kilometres.
(i) Find the distance in kilometres from town A to town C.
Answer(a)(i)
km [2]
(ii) Measure and write down the bearing of town C from town A.
Answer(a)(ii)
[1]
(iii) Town B is 140 km from town C on a bearing of 150°.
Mark accurately the position of town B on the scale drawing.
[2]
(iv) Find the bearing of town C from town B.
Answer(a)(iv)
[1]
(v) A lake on the map has an area of 0.15 cm2.
Work out the actual area of the lake.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(a)(v)
km2 [2]
90
(b) A plane leaves town C at 11 57 and flies 1500 km to another town, landing at 14 12.
Calculate the average speed of the plane.
Answer(b)
km/h [3]
(c)
Q
NOT TO
SCALE
1125 km
790 km
P
1450 km
R
The diagram shows the distances between three towns P, Q and R.
Calculate angle PQR.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(c)Angle PQR =
[4]
91
11) November 2011 V1
6
C
26°
B
95 m
79 m
A
NOT TO
SCALE
77°
D
120 m
The quadrilateral ABCD represents an area of land.
There is a straight road from A to C.
AB = 79 m, AD = 120 m and CD = 95 m.
Angle BCA = 26° and angle CDA = 77°.
(a) Show that the length of the road, AC, is 135 m correct to the nearest metre.
Answer(a)
[4]
(b) Calculate the size of the obtuse angle ABC.
Answer(b) Angle ABC =
Mr.Yasser Elsayed
002 012 013 222 97
[4]
92
(c) A straight path is to be built from B to the nearest point on the road AC.
Calculate the length of this path.
Answer(c)
m [3]
(d) Houses are to be built on the land in triangle ACD.
Each house needs at least 180 m2 of land.
Calculate the maximum number of houses which can be built.
Show all of your working.
Answer(d)
Mr.Yasser Elsayed
002 012 013 222 97
[4]
93
12) November 2011 V2
8
D
NOT TO
SCALE
C
5m
45°
B
A
3m
Parvatti has a piece of canvas ABCD in the shape of an irregular quadrilateral.
AB = 3 m, AC = 5 m and angle BAC = 45°.
(a) (i) Calculate the length of BC and show that it rounds to 3.58 m, correct to 2 decimal places.
You must show all your working.
Answer(a)(i)
[4]
(ii) Calculate angle BCA.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(a)(ii) Angle BCA =
[3]
94
(b) AC = CD and angle CDA = 52°.
(i) Find angle DCA.
Answer(b)(i) Angle DCA =
[1]
(ii) Calculate the area of the canvas.
m2 [3]
Answer(b)(ii)
(c) Parvatti uses the canvas to give some shade.
She attaches corners A and D to the top of vertical poles, AP and DQ, each of height 2 m.
Corners B and C are pegged to the horizontal ground.
AB is a straight line and angle BPA = 90°.
D
A
3m
2m
2m
B
NOT TO
SCALE
C
P
Q
Calculate angle PAB.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(c) Angle PAB =
[2]
95
13) June 2012 V1
2
North
K
NOT TO
SCALE
108°
4 km
9 km
M
L
Three buoys K, L and M show the course of a boat race.
MK = 4 km, KL = 9 km and angle MKL = 108°.
(a) Calculate the distance ML.
Answer(a) ML =
km [4]
Answer(b)(i)
km [3]
Answer(b)(ii)
[2]
(b) The bearing of L from K is 125°.
(i) Calculate how far L is south of K.
(ii) Find the three figure bearing of K from M.
Mr.Yasser Elsayed
002 012 013 222 97
96
14) June 2012 V2
11
(c)
A
D
31 cm
50°
B
50°
22 cm
NOT TO
SCALE
100°
C
The frame of a child’s bicycle is made from metal rods.
ABC is an isosceles triangle with base 22 cm and base angles 50°.
Angle ACD = 100° and CD = 31 cm.
Calculate the length AD.
Answer(c) AD =
Mr.Yasser Elsayed
002 012 013 222 97
cm [6]
97
15) June 2012 V3
2
North
D
95°
10 km
40°
A
NOT TO
SCALE
12 km
30°
C
17 km
B
The diagram shows straight roads connecting the towns A, B, C and D.
AB = 17 km, AC = 12 km and CD = 10 km.
Angle BAC = 30° and angle ADC = 95°.
(a) Calculate angle CAD.
Answer(a) Angle CAD =
[3]
(b) Calculate the distance BC.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(b) BC =
km [4]
98
(c) The bearing of D from A is 040°.
Find the bearing of
(i) B from A,
Answer(c)(i)
[1]
Answer(c)(ii)
[1]
(ii) A from B.
(d) Angle ACB is obtuse.
Calculate angle BCD.
Answer(d) Angle BCD =
Mr.Yasser Elsayed
002 012 013 222 97
[4]
99
16) November 2012 V2
2
A
32 m
B
43 m
NOT TO
SCALE
64 m
C
D
The diagram represents a field in the shape of a quadrilateral ABCD.
AB = 32 m, BC = 43 m and AC = 64 m.
(a) (i) Show clearly that angle CAB = 37.0° correct to one decimal place.
Answer(a)(i)
[4]
(ii) Calculate the area of the triangle ABC.
Answer(a)(ii)
m2 [2]
(b) CD = 70 m and angle DAC = 55°.
Calculate the perimeter of the whole field ABCD.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(b)
m [6]
100
17) November 2012 V3
6
A
16 cm
B
NOT TO
SCALE
25 cm
C
The area of triangle ABC is 130 cm2.
AB = 16 cm and BC = 25 cm.
(a) Show clearly that angle ABC = 40.5°, correct to one decimal place.
Answer (a)
[3]
(b) Calculate the length of AC.
Answer(b) AC =
cm [4]
(c) Calculate the shortest distance from A to BC.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(c)
cm [2]
101
18) June 2013 V1
6
B
A
30°
52°
E
15.7 cm
NOT TO
SCALE
16.5 cm
C
23.4 cm
D
In the diagram, BCD is a straight line and ABDE is a quadrilateral.
Angle BAC = 90°, angle ABC = 30° and angle CAE = 52°.
AC = 15.7 cm, CE = 16.5 cm and CD = 23.4 cm.
(a) Calculate BC.
Answer(a) BC = ......................................... cm [3]
(b) Use the sine rule to calculate angle AEC.
Show that it rounds to 48.57°, correct to 2 decimal places.
Answer(b)
[3]
Mr.Yasser Elsayed
002 012 013 222 97
102
(c) (i) Show that angle ECD = 40.6°, correct to 1 decimal place.
Answer(c)(i)
[2]
(ii) Calculate DE.
Answer(c)(ii) DE = ......................................... cm [4]
(d) Calculate the area of the quadrilateral ABDE.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(d) ........................................ cm2 [4]
103
19) June 2013 V1
A
7
(a)
NOT TO
SCALE
(2x + 3) cm
(x + 2) cm
B
C
In triangle ABC, AB = (x + 2) cm and AC = (2x + 3) cm.
sin ACB =
9
16
Find the length of BC.
Answer(a) BC = ......................................... cm [6]
(b) A bag contains 7 white beads and 5 red beads.
(i) The mass of a red bead is 2.5 grams more than the mass of a white bead.
The total mass of all the 12 beads is 114.5 grams.
Find the mass of a white bead and the mass of a red bead.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(b)(i) White ............................................ g
104
Red ............................................ g [5]
(ii) Two beads are taken out of the bag at random, without replacement.
Find the probability that
(a) they are both white,
Answer(b)(ii)(a) ............................................... [2]
(b) one is white and one is red.
Answer(b)(ii)(b) ............................................... [3]
Mr.Yasser Elsayed
002 012 013 222 97
105
20) June 2013 V2
6
(a)
L
15 cm
N
12 cm
NOT TO
SCALE
21 cm
M
The diagram shows triangle LMN with LM = 12 cm, LN = 15 cm and MN = 21 cm.
(i) Calculate angle LMN.
Show that this rounds to 44.4°, correct to 1 decimal place.
Answer(a)(i)
[4]
(ii) Calculate the area of triangle LMN.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(a)(ii) ........................................ cm2 [2]
106
(b)
Q
6.4 cm
P
82°
NOT TO
SCALE
43°
R
The diagram shows triangle PQR with PQ = 6.4 cm, angle PQR = 82° and angle QPR = 43°.
Calculate the length of PR.
Answer(b) PR = ......................................... cm [4]
Mr.Yasser Elsayed
002 012 013 222 97
107
21) June 2013 V2
11 Sidney draws the triangle OP1 P2.
OP 1 = 3 cm and P 1 P2 = 1 cm.
Angle OP1 P2 = 90°.
O
NOT TO
SCALE
3 cm
P1
1 cm
P2
(a) Show that OP2 = 10 cm.
Answer(a)
[1]
(b) Sidney now draws the lines P2 P3 and OP3
Triangle OP2 P3 is mathematically similar
to triangle OP1 P2
O
.
NOT TO
SCALE
.
3 cm
P3
P1
(i) Write down the length of P2 P3 in the form
1 cm
P2
a
where a and b are integers.
b
Answer(b)(i) P2 P3 = ......................................... cm [1]
(ii) Calculate the length of OP3 giving your answer in the form
c
where c and d are integers.
d
Answer(b)(ii) OP3 = ......................................... cm [2]
(c) Sidney continues to add
mathematically similar triangles
to his drawing.
P5
O
P4
Find the length of OP5.
3 cm
NOT TO
SCALE
P3
P1
Mr.Yasser Elsayed
002 012 013 222 97
1 cm
P2
Answer(c) OP = ......................................... cm [2]
108
(d) (i) Show that angle P1OP2 = 18.4°, correct to 1 decimal place.
Answer(d)(i)
[2]
(ii) Write down the size of angle P2OP3.
Answer(d)(ii) Angle P2OP3 = ............................................... [1]
(iii) The last triangle Sidney can draw without covering his first triangle is triangle OP(n–1) Pn.
P5
O
P4
NOT TO
SCALE
P3
P1
P2
P(n–1)
Pn
Calculate the value of n.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(d)(iii) n = ............................................... [3]
109
22) November 2013 V1
4
D
C
32°
70 m
NOT TO
SCALE
40°
A
55 m
B
The diagram shows a school playground ABCD.
ABCD is a trapezium.
AB = 55 m, BD = 70 m, angle ABD = 40° and angle BCD = 32°.
(a) Calculate AD.
Answer(a) AD = ........................................... m [4]
(b) Calculate BC.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(b) BC = ........................................... m [4]
110
(c) (i) Calculate the area of the playground ABCD.
Answer(c)(i) .......................................... m2 [3]
(ii) An accurate plan of the school playground is to be drawn to a scale of 1: 200 .
Calculate the area of the school playground on the plan.
Give your answer in cm2.
Answer(c)(ii) ........................................ cm2 [2]
(d) A fence, BD, divides the playground into two areas.
Calculate the shortest distance from A to BD.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(d) ........................................... m [2]
111
23) November 2013 V2
2
B
2.4 m
C
NOT TO
SCALE
6.46 m
1.8 m
A
D
8.6 m
The diagram shows the cross section, ABCD, of a ramp.
(a) Calculate angle DBC.
Answer(a) Angle DBC = ............................................... [2]
(b) (i) Show that BD is exactly 3 m.
Answer(b)(i)
[2]
(ii) Use the cosine rule to calculate angle ABD.
Answer(b)(ii) Angle ABD = ............................................... [4]
(c) The ramp is a prism of width 4 m.
Calculate the volume of this prism.
Mr.Yasser Elsayed
002 012 013 222 97
112
Answer(c) .......................................... m3 [3]
24) November 2013 V3
2
A field, ABCD, is in the shape of a quadrilateral.
A footpath crosses the field from A to C.
C
26°
B
NOT TO
SCALE
55 m
65°
32°
A
62 m
122°
D
(a) Use the sine rule to calculate the distance AC and show that it rounds to 119.9 m,
correct to 1 decimal place.
Answer(a)
[3]
(b) Calculate the length of BC.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(b) BC = ........................................... m [4]
113
(c) Calculate the area of triangle ACD.
Answer(c) .......................................... m2 [2]
(d) The field is for sale at $4.50 per square metre.
Calculate the cost of the field.
Answer(d) $ ............................................... [3]
Mr.Yasser Elsayed
002 012 013 222 97
114
25) June 2014 V1
5
S
North
Scale: 2 cm to 3 km
P
L
In the scale drawing, P is a port, L is a lighthouse and S is a ship.
The scale is 2 centimetres represents 3 kilometres.
(a) Measure the bearing of S from P.
Answer(a) ................................................ [1]
(b) Find the actual distance of S from L.
Answer(b) .......................................... km [2]
(c) The bearing of L from S is 160°.
Calculate the bearing of S from L.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(c) ................................................ [1]
115
(d) Work out the scale of the map in the form 1 : n.
Answer(d) 1 : ................................................ [2]
(e) A boat B is
●
equidistant from S and L
●
equidistant from the lines PS and SL.
and
On the diagram, using a straight edge and compasses only, construct the position of B.
[5]
(f) The lighthouse stands on an island of area 1.5 cm2 on the scale drawing.
Work out the actual area of the island.
Answer(f) ......................................... km2 [2]
Mr.Yasser Elsayed
002 012 013 222 97
116
26) June 2014 V2
3
C
90 m
D
80 m
95 m
NOT TO
SCALE
49°
A
55°
B
The diagram shows a quadrilateral ABCD.
Angle BAD = 49° and angle ABD = 55°.
BD = 80 m, BC = 95 m and CD = 90 m.
(a) Use the sine rule to calculate the length of AD.
Answer(a) AD = ............................................ m [3]
(b) Use the cosine rule to calculate angle BCD.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(b) Angle BCD = ................................................ [4]
117
(c) Calculate the area of the quadrilateral ABCD.
Answer(c) ........................................... m2 [3]
(d) The quadrilateral represents a field.
Corn seeds are sown across the whole field at a cost of $3250 per hectare.
Calculate the cost of the corn seeds used.
1 hectare = 10 000 m2
Answer(d) $ ................................................ [3]
Mr.Yasser Elsayed
002 012 013 222 97
118
27) June 2014 V3
3
(a)
P
12 cm
X
17 cm
NOT TO
SCALE
Q
R
The diagram shows triangle PQR with PQ = 12 cm and PR = 17 cm.
The area of triangle PQR is 97 cm2 and angle QPR is acute.
(i) Calculate angle QPR.
Answer(a)(i) Angle QPR = ................................................ [3]
(ii) The midpoint of PQ is X.
Use the cosine rule to calculate the length of XR.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(a)(ii) XR = .......................................... cm [4]
119
(b)
9.4 cm
42°
a cm
NOT TO
SCALE
37°
Calculate the value of a.
Answer(b) a = ................................................ [4]
(c)
sin x = cos 40°, 0° Y x Y 180°
Find the two values of x.
Answer(c) x = .................. or x = .................. [2]
Mr.Yasser Elsayed
002 012 013 222 97
120
28) November 2014 V1
7
(a) The diagram shows a circle with two chords, AB and CD, intersecting at X.
B
C
NOT TO
SCALE
X
A
D
(i) Show that triangles ACX and DBX are similar.
Answer(a)(i)
[2]
(ii) AX = 3.2 cm, BX = 12.5 cm, CX = 4 cm and angle AXC = 110°.
(a) Find DX.
Answer(a)(ii)(a) DX = .......................................... cm [2]
(b) Use the cosine rule to find AC.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(a)(ii)(b) AC = .......................................... cm [4]
121
(c) Find the area of triangle BXD.
Answer(a)(ii)(c) ......................................... cm2 [2]
(b)
D
NOT TO
SCALE
C
30 m
37°
A
31°
B
In the diagram, BC represents a building 30 m tall.
A flagpole, DC, stands on top of the building.
From a point, A, the angle of elevation of the top of the building is 31°.
The angle of elevation of the top of the flagpole is 37°.
Calculate the height, DC, of the flagpole.
Answer(b) ............................................ m [5]
Mr.Yasser Elsayed
002 012 013 222 97
122
29) November 2014 V2
7
(c)
x cm
75 cm
NOT TO
SCALE
m
45 c
1
55 cm
120 cm
A rod of length 145 cm is placed inside the water tank.
One end of the rod is in the bottom corner of the tank as shown.
The other end of the rod is x cm below the top corner of the tank as shown.
Calculate the value of x.
Answer(c) x = ................................................ [4]
(d) Calculate the angle that the rod makes with the base of the tank.
Answer(d) ................................................ [3]
Mr.Yasser Elsayed
002 012 013 222 97
123
30) November 2014 V2
North
8
NOT TO
SCALE
P
58 km
L
74 km
North
Q
A ship sails from port P to port Q.
Q is 74 km from P on a bearing of 142°.
A lighthouse, L, is 58 km from P on a bearing of 110°.
(a) Show that the distance LQ is 39.5 km correct to 1 decimal place.
Answer(a)
[5]
(b) Use the sine rule to calculate angle PQL.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(b) Angle PQL = ................................................ [3]
124
(c) Find the bearing of
(i) P from Q,
Answer(c)(i) ................................................ [2]
(ii) L from Q.
Answer(c)(ii) ................................................ [1]
(d) The ship takes 2 hours and 15 minutes to sail the 74 km from P to Q.
Calculate the average speed in knots.
[1 knot = 1.85 km/h]
Answer(d) ....................................... knots [3]
(e) Calculate the shortest distance from the lighthouse to the path of the ship.
Answer(e) .......................................... km [3]
Mr.Yasser Elsayed
002 012 013 222 97
125
31) November 2014 V3
1
(a) ABCD is a trapezium.
11 cm
A
B
NOT TO
SCALE
4.7 cm
D
C
2.6 cm
17 cm
(i) Calculate the length of AD.
Answer(a)(i) AD = .......................................... cm [2]
(ii) Calculate the size of angle BCD.
Answer(a)(ii) Angle BCD = ................................................ [3]
(iii) Calculate the area of the trapezium ABCD.
Answer(a)(iii) ......................................... cm2 [2]
(b) A similar trapezium has perpendicular height 9.4 cm.
Calculate the area of this trapezium.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(b) ......................................... cm2 [3]
126
32) June 2015 V1
5
(a) Andrei stands on level horizontal ground, 294 m from the foot of a vertical tower which is 55 m high.
(i)
Calculate the angle of elevation of the top of the tower.
Answer(a)(i) ................................................. [2]
(ii)
Andrei walks a distance x metres directly towards the tower.
The angle of elevation of the top of the tower is now 24.8°.
Calculate the value of x.
Answer(a)(ii) x = ................................................. [4]
Mr.Yasser Elsayed
002 012 013 222 97
127
(b) The diagram shows a pyramid with a horizontal rectangular base.
NOT TO
SCALE
y
4m
3m
4.8 m
The rectangular base has length 4.8 m and width 3 m and the height of the pyramid is 4 m.
Calculate
(i)
y, the length of a sloping edge of the pyramid,
Answer(b)(i) y = ............................................. m [4]
(ii)
the angle between a sloping edge and the rectangular base of the pyramid.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(b)(ii) ................................................ [2]
128
33) June 2015 V1
7
P
(a)
NOT TO
SCALE
8.4 cm
Q
62°
7.6 cm
R
In the triangle PQR, QR = 7.6 cm and PR = 8.4 cm.
Angle QRP = 62°.
Calculate
(i) PQ,
Answer(a)(i) PQ = ........................................... cm [4]
(ii)
the area of triangle PQR.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(a)(ii) .......................................... cm 2 [2]
129
(b)
North
H
NOT TO
SCALE
North
63 km
G
J
The diagram shows the positions of three small islands G, H and J.
The bearing of H from G is 045°.
The bearing of J from G is 126°.
The bearing of J from H is 164°.
The distance HJ is 63 km.
Calculate the distance GJ.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(b) GJ = .......................................... km [5]
130
34) June 2015 V2
6
The diagram shows the positions of two ships, A and B, and a coastguard station, C.
North
A
B
95.5 km
NOT TO
SCALE
83.1 km
101°
C
(a) Calculate the distance, AB, between the two ships.
Show that it rounds to 138 km, correct to the nearest kilometre.
Answer(a)
[4]
(b) The bearing of the coastguard station C from ship A is 146°.
Calculate the bearing of ship B from ship A.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(b) ................................................ [4]
131
(c)
L
North
46.2 km
NOT TO
SCALE
45°
21°
B
At noon, a lighthouse, L, is 46.2 km from ship B on the bearing 021°.
Ship B sails north west.
Calculate the distance ship B must sail from its position at noon to be at its closest distance to the
lighthouse.
Answer(c) .......................................... km [2]
Mr.Yasser Elsayed
002 012 013 222 97
132
35) November 2015 V1
T
3
(a)
60 m
50 m
NOT TO
SCALE
130°
A
B
70 m
C
A, B and C are points on horizontal ground.
BT is a vertical pole.
AT = 60 m, AB = 50 m, BC = 70 m and angle ABC = 130°.
(i) Calculate the angle of elevation of T from C.
Answer(a)(i) ................................................ [5]
(ii) Calculate the length AC.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(a)(ii) AC = ............................................ m [4]
133
(iii) Calculate the area of triangle ABC.
Answer(a)(iii) ........................................... m2 [2]
(b)
Y
12 cm
22 cm
X
NOT TO
SCALE
45 cm
A cuboid has length 45 cm, width 22 cm and height 12 cm.
Calculate the length of the straight line XY.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(b) XY = .......................................... cm [4]
134
36) November 2015 V1
7
The scale drawing shows the positions of three towns A, B and C on a map.
The scale of the map is 1 centimetre represents 10 kilometres.
C
North
A
North
Scale: 1 cm to 10 km
B
(a) Find the actual distance AB.
Answer(a) .......................................... km [1]
(b) Measure the bearing of A from B.
Answer(b) ................................................ [1]
(c) Write the scale 1 cm to 10 km in the form 1 : n.
Answer(c) 1 : ................................................ [1]
(d) A national park lies inside the triangle ABC.
The four boundaries of the national park are
•
•
•
•
equidistant from C and B
equidistant from AC and CB
15 km from CB
along AB.
On the scale drawing, shade the region which represents the national park.
Leave in your construction arcs.
[7]
(e) On the scale drawing, a lake inside the national park has area 0.4 cm2.
Calculate the actual area of the lake.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(e) ......................................... km2 [2]
135
37) November 2015 V2
A
4
NOT TO
SCALE
B
D
C
The diagram shows a tent ABCD.
The front of the tent is an isosceles triangle ABC, with AB = AC.
The sides of the tent are congruent triangles ABD and ACD.
(a) BC = 1.2 m and angle ABC = 68°.
Find AC.
Answer(a) AC = ............................................ m [3]
(b) CD = 2.3 m and AD = 1.9 m.
Find angle ADC.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(b) Angle ADC = ............................................... [4]
136
(c) The floor of the tent, triangle BCD, is also an isosceles triangle with BD = CD.
Calculate the area of the floor of the tent.
Answer(c) ...........................................m2 [4]
(d) When the tent is on horizontal ground, A is a vertical distance 1.25 m above the ground.
Calculate the angle between AD and the ground.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(d) ............................................... [3]
137
38) November 2015 V3
5
K
40°
North
65°
680 km
D
NOT TO
SCALE
2380 km
M
1560 km
C
The diagram shows some distances between Mumbai (M), Kathmandu (K), Dhaka (D) and Colombo (C).
(a) Angle CKD = 65°.
Use the cosine rule to calculate the distance CD.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(a) CD = .......................................... km [4]
138
(b) Angle MKC = 40°.
Use the sine rule to calculate the acute angle KMC.
Answer(b) Angle KMC = ................................................ [3]
(c) The bearing of K from M is 050°.
Find the bearing of M from C.
Answer(c) ................................................ [2]
(d) A plane from Colombo to Mumbai leaves at 21 15 and the journey takes 2 hours 24 minutes.
(i) Find the time the plane arrives at Mumbai.
Answer(d)(i) ................................................ [1]
(ii) Calculate the average speed of the plane.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(d)(ii) ....................................... km/h [2]
139
39) March 2015 V2
X
5
(a)
NOT TO
SCALE
5.4 cm
Y
62°
Z
16 cm
Show that the area of triangle XYZ is 38.1 cm2, correct to 1 decimal place.
Answer(a)
[2]
(b)
NOT TO
SCALE
48°
6.7 cm
x°
8.4 cm
Calculate the value of x.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(b) x =
................................................
[4]
140
(c)
North
A
NOT TO
SCALE
P
B
Ship A is 180 kilometres from port P on a bearing of 063°.
Ship B is 245 kilometres from P on a bearing of 146°.
Calculate AB, the distance between the two ships.
Answer(c) .......................................... km [5]
Mr.Yasser Elsayed
002 012 013 222 97
141
40) June 2016 V3
5
NOT TO
SCALE
A
North
510 km
B
720 km
40°
C
A plane flies from A to C and then from C to B.
AC = 510 km and CB = 720 km.
The bearing of C from A is 135° and angle ACB = 40°.
(a) Find the bearing of
(i)
B from C,
................................................... [2]
(ii)
C from B.
................................................... [2]
(b) Calculate AB and show that it rounds to 464.7 km, correct to 1 decimal place.
[4]
(c) Calculate angle ABC.
Mr.Yasser Elsayed
002 012 013 222 97
Angle ABC = .................................................. [3]
142
41) June 2017 V1
8
(a)
North
A 110°
38 km
50 km
C
North
NOT TO
SCALE
B
280°
A, B and C are three towns.
The bearing of B from A is 110°.
The bearing of C from B is 280°.
AC = 38 km and AB = 50 km .
(i)
Find the bearing of A from B.
................................................. [2]
(ii)
Calculate angle BAC.
Angle BAC = ................................................ [5]
(iii)
A road is built from A to join the straight road BC.
Calculate the shortest possible length of this new road.
.......................................... km [3]
Mr.Yasser Elsayed
002 012 013 222 97
143
15
(b) Town A has a rectangular park.
The length of the park is x m.
The width of the park is 25 m shorter than the length.
The area of the park is 2200 m2.
(i)
Show that x 2 - 25x - 2200 = 0 .
[1]
(ii)
Solve x 2 - 25x - 2200 = 0 .
Show all your working and give your answers correct to 2 decimal places.
x = ..................... or x = ..................... [4]
Mr.Yasser Elsayed
002 012 013 222 97
144
42) November 2017 V1
10
B
8.5 cm
A
60°
46°
12.5 cm
x cm
76°
C
NOT TO
SCALE
58°
D
The diagram shows a quadrilateral ABCD.
(a) The length of AC is x cm.
Use the cosine rule in triangle ABC to show that 2x2 – 17x – 168 = 0.
[4]
(b) Solve the equation 2x2 – 17x – 168 = 0.
Show all your working and give your answers correct to 2 decimal places.
Mr.Yasser Elsayed
002 012 013 222 97
x = .......................... or x = .......................... [4]
145
(c) Use the sine rule to calculate the length of CD.
CD = .......................................... cm [3]
(d) Calculate the area of the quadrilateral ABCD.
..........................................cm2 [3]
Mr.Yasser Elsayed
002 012 013 222 97
146
43) June 2018 V2
5
O
A
NOT TO
SCALE
8 cm
7 cm
78°
C
B
The diagram shows a design made from a triangle AOC joined to a sector OCB.
AC = 8 cm, OB = OC = 7 cm and angle ACO = 78°.
(a) Use the cosine rule to show that OA = 9.47 cm, correct to 2 decimal places.
[4]
(b) Calculate angle OAC.
Angle OAC = ................................................ [3]
Mr.Yasser Elsayed
002 012 013 222 97
147
9
(c) The perimeter of the design is 29.5 cm.
Show that angle COB = 41.2°, correct to 1 decimal place.
[5]
(d) Calculate the total area of the design.
......................................... cm2 [4]
Mr.Yasser Elsayed
002 012 013 222 97
148
44) June 2019 V1
3
North
C
D
170 m
120 m
150 m
NOT TO
SCALE
E
50 m
A
100 m
B
The diagram shows a field ABCDE.
(a) Calculate the perimeter of the field ABCDE.
................................................ m [4]
(b) Calculate angle ABD.
Mr.Yasser Elsayed
002 012 013 222 97
149
Angle ABD = .......................................................... [4]
(c)
(i)
Calculate angle CBD.
Angle CBD = .................................................... [2]
(ii)
The point C is due north of the point B.
Find the bearing of D from B.
.................................................... [2]
(d) Calculate the area of the field ABCDE.
Give your answer in hectares.
[1 hectare = 10 000 m2]
...................................... hectares [4]
Mr.Yasser Elsayed
002 012 013 222 97
150
45) June 2020 V2
5
North
D
NOT TO
SCALE
A 140°
450 m
400 m
B
350 m
C
The diagram shows a field ABCD.
The bearing of B from A is 140°.
C is due east of B and D is due north of C.
AB = 400 m, BC = 350 m and CD = 450 m.
(a) Find the bearing of D from B.
................................................. [2]
Mr.Yasser Elsayed
002 012 013 222 97
151
(b) Calculate the distance from D to A.
............................................. m [6]
(c) Jono runs around the field from A to B, B to C, C to D and D to A.
He runs at a speed of 3 m/s.
Calculate the total time Jono takes to run around the field.
Give your answer in minutes and seconds, correct to the nearest second.
.................. min .................. s [4]
Mr.Yasser Elsayed
002 012 013 222 97
152
46) November 2020 V1
6
D
287.9 m
North
C
38°
168 m
NOT TO
SCALE
205.8 m
192 m
B
A
The diagram shows a field, ABCD, on horizontal ground.
BC = 192 m, CD = 287.9 m, BD = 168 m and AD = 205.8 m.
(a) (i) Calculate angle CBD and show that it rounds to 106.0°, correct to 1 decimal place.
[4]
(ii) The bearing of D from B is 038°.
Find the bearing of C from B.
................................................. [1]
(iii) A is due east of B.
Calculate the bearing of D from A.
Mr.Yasser Elsayed
002 012 013 222 97
................................................. [5]
153
(b) (i) Calculate the area of triangle BCD.
............................................ m2 [2]
(ii) Tomas buys the triangular part of the field, BCD.
The cost is $35 750 per hectare.
Calculate the amount he pays.
Give your answer correct to the nearest $100.
[1 hectare = 10 000 m2]
$ ................................................ [2]
Mr.Yasser Elsayed
002 012 013 222 97
154
Geometric
Constructions
Mr.Yasser Elsayed
002 012 013 222 97
155
1) November 2010 V3
5
C
D
B
A
The diagram shows an area of land ABCD used for a shop, a car park and gardens.
(a) Using a straight edge and compasses only, construct
(i) the locus of points equidistant from C and from D,
[2]
(ii) the locus of points equidistant from AD and from AB.
[2]
(b) The shop is on the land nearer to D than to C and nearer to AD than to AB.
Write the word SHOP in this region on the diagram.
[1]
(c) (i) The scale of the diagram is 1 centimetre to 20 metres.
The gardens are the part of the land less than 100 m from B.
Draw the boundary for the gardens.
[1]
(ii) The car park is the part of the land not used for the shop and not used for the gardens.
Shade the car park region on the diagram.
Mr.Yasser Elsayed
002 012 013 222 97
[1]
156
2) June 2011 V3
8
D
C
A
B
(a) Draw accurately the locus of points, inside the quadrilateral ABCD, which are 6 cm from the
point D.
[1]
(b) Using a straight edge and compasses only, construct
(i) the perpendicular bisector of AB,
[2]
(ii) the locus of points, inside the quadrilateral, which are equidistant from AB and from BC. [2]
(c) The point Q is equidistant from A and from B and equidistant from AB and from BC.
(i) Label the point Q on the diagram.
[1]
(ii) Measure the distance of Q from the line AB.
Answer(c)(ii)
cm [1]
(d) On the diagram, shade the region inside the quadrilateral which is
•
•
less than 6 cm from D
and
nearer to A than to B
and
nearer to AB than to BC.
Mr.Yasser
Elsayed
•
002 012 013 222 97
[1]
157
3) June 2012 V2
9
F
E
Scale 1 : 10 000
H
G
The diagram is a scale drawing of a park EFGH. The scale is 1 : 10 000.
A statue is to be placed in the park so that it is
•
nearer to G than to H
•
nearer to HG than to FG
•
more than 550 metres from F.
Construct accurately the boundaries of the region R in which the statue can be placed.
Leave in all your construction arcs and shade the region R.
Mr.Yasser Elsayed
002 012 013 222 97
[7]
158
4) June 2013 V3
2
(a) In this question show all your construction arcs and use only a ruler and compasses to draw
the boundaries of your region.
This scale drawing shows the positions of four towns, P, Q, R and S, on a map where 1 cm represents
10 km.
North
P
Q
Scale: 1 cm to 10 km
S
R
A nature reserve lies in the quadrilateral PQRS.
The boundaries of the nature reserve are:
●
●
●
●
equidistant from Q and from R
equidistant from PS and from PQ
60 km from R
along QR
.
Mr.Yasser
Elsayed
(ii) Measure the bearing of S from P.
002 012 013 222 97
(i) Shade the region which represents the nature reserve.
[7]
159
Answer(a)(ii) ............................................... [1]
(b) A circular lake in the nature reserve has a radius of 45 m.
(i) Calculate the area of the lake.
Answer(b)(i) .......................................... m2 [2]
(ii)
NOT TO
SCALE
A fence is placed along part of the circumference of the lake.
This arc subtends an angle of 210° at the centre of the circle.
Calculate the length of the fence.
Answer(b)(ii) ........................................... m [2]
Mr.Yasser Elsayed
002 012 013 222 97
160
5) June 2015 V1
10
The diagram is a scale drawing of three straight roads, AB, BC and CD.
The scale is 1 : 5000.
C
D
A
B
Scale 1 : 5000
(a) Find the actual length of the road BC .
Give your answer in metres.
Answer(a) ............................................. m [2]
(b) Another straight road starts at M , the midpoint of AB.
This road is perpendicular to AB and it meets the road CD at X.
Using a straight edge and compasses only, construct MX.
Mr.Yasser Elsayed
002 012 013 222 97
[2]
161
(c) There is a park in the area enclosed by the four roads.
The park is
and
•
less than 290 m from B
•
nearer to CD than to CB.
Using a ruler and compasses only, construct the boundaries of the park.
Leave in all your construction arcs and label the park P
Mr.Yasser Elsayed
002 012 013 222 97
[5]
.
162
6) March 2016 V2
2
In this question use a ruler and compasses only.
Show all your construction arcs.
The diagram shows a triangular field ABC.
The scale is 1 centimetre represents 50 metres.
C
A
B
Scale : 1 cm to 50 m
(a) Construct the locus of points that are equidistant from A and B.
[2]
(b) Construct the locus of points that are equidistant from the lines AB and AC.
[2]
(c) The two loci intersect at the point E.
Construct the locus of points that are 250 m from E.
[2]
(d) Shade any region inside the field ABC that is
and
•
more than 250 m from E
closer to AC than to AB.
Mr.Yasser• Elsayed
002 012 013 222 97
[2]
163
7) March 2016 V2
2
The scale drawing shows two boundaries, AB and BC, of a field ABCD.
The scale of the drawing is 1 cm represents 8 m.
C
B
A
Scale: 1 cm to 8 m
(a) The boundaries CD and AD of the field are each 72 m long.
(i)
Work out the length of CD and AD on the scale drawing.
.......................................... cm [1]
(ii)
Using a ruler and compasses only, complete accurately the scale drawing of the field.
[2]
(b) A tree in the field is
and
•
equidistant from A and B
•
equidistant from AB and BC.
On the scale drawing, construct two lines to find the position of the tree.
Use a straight edge and compasses only and leave in your construction arcs.
Mr.Yasser Elsayed
002 012 013 222 97
[4]
164
Vectors and Matrices
Mr.Yasser Elsayed
002 012 013 222 97
165
1) June 2010 V2
2
3
6
(a) p =   and q =   .
 3
 2
(i)
Find, as a single column vector, p + 2q.
Answer(a)(i)










[2]
(ii) Calculate the value of | p + 2q |.
Answer(a)(ii)
(b)
[2]
C
NOT TO
SCALE
M
O
L
V
In the diagram, CM = MV and OL = 2LV.
O is the origin.
= c and
=v
.
Find, in terms of c and v, in their simplest forms
(i)
,
Answer(b)(i)
[2]
Answer(b)(ii)
[2]
Answer(b)(iii)
[2]
(ii) the position vector of M ,
(iii)
.
Mr.Yasser Elsayed
002 012 013 222 97
166
2) November 2010 V1
7
(b)
y
B (4,4)
NOT TO
SCALE
A (2,1)
x
O
(i) Write down
as a column vector.
Answer(b)(i)
(ii)
=










[1]
0
=   .
7
 
Work out
as a column vector.
Answer(b)(ii)
Mr.Yasser Elsayed
002 012 013 222 97
=










[2]
167
(c)
R
NOT TO
SCALE
r
P
O
T
Q
t
= r and
= t.
P is on RT such that RP : PT = 2 : 1.
2
Q is on OT such that OQ = OT.
3
Write the following in terms of r and/or t.
Simplify your answers where possible.
(i)
Answer(c)(i)
=
[1]
Answer(c)(ii)
=
[2]
Answer(c)(iii)
=
[2]
(ii)
(iii)
(iv) Write down two conclusions you can make about the line segment QP.
Answer(c)(iv)
[2]
Mr.Yasser Elsayed
002 012 013 222 97
168
3) November 2010 V3
4
(a)
 2 3

 4 5
A= 
 2
B=  
7
C = (1 2 )
Find the following matrices.
(i) AB
Answer(a)(i)
[2]
Answer(a)(ii)
[2]
Answer(a)(iii)
[2]
(ii) CB
(iii) A-1, the inverse of A
1
0
.
_
0 1 
(b) Describe fully the single transformation represented by the matrix 
Answer(b)
[2]
(c) Find the 2 by 2 matrix that represents an anticlockwise rotation of 90° about the origin.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(c)














[2]
169
4) November 2010 V3
9
(a)
y
3
A
2
1
–4
–3
–2
–1 0
–1
C
1
2
3
4
5
x
–2
–3
–4
B
The points A (5, 3), B (1, –4) and C (–4, –2) are shown in the diagram.
as a column vector.
(i) Write
Answer(a)(i)
(ii) Find
–
=














[1]














[2]
as a single column vector.
Answer(a)(ii)
(iii) Complete the following statement.
–
(iv) Calculate
=
[1]
Answer(a)(iv)
[2]
170
.
Mr.Yasser Elsayed
002 012 013 222 97
(b)
D
u
C
NOT TO
SCALE
t
M
A
B
ABCD is a trapezium with DC parallel to AB and DC =
1
AB.
2
M is the midpoint of BC.
= t and
= u.
Find the following vectors in terms of t and / or u.
Give each answer in its simplest form.
(i)
Answer(b)(i)
=
[1]
(ii)
Answer(b)(ii)
=
[2]
Answer(b)(iii)
=
[2]
(iii)
Mr.Yasser Elsayed
002 012 013 222 97
171
5) June 2011 V3
10 (a)
C
L
D
NOT TO
SCALE
N
M
q
A
B
p
ABCD is a parallelogram.
L is the midpoint of DC, M is the midpoint of BC and N is the midpoint of LM.
= p and
= q.
(i) Find the following in terms of p and q, in their simplest form.
(a)
Answer(a)(i)(a)
=
[1]
Answer(a)(i)(b)
=
[2]
Answer(a)(i)(c)
=
[2]
(b)
(c)
(ii) Explain why your answer for
Answer(a)(ii)
Mr.Yasser Elsayed
002 012 013 222 97
shows that the point N lies on the line AC.
[1]
172
(b)
F
G
2x°
(x + 15)°
H
J
NOT TO
SCALE
75°
E
EFG is a triangle.
HJ is parallel to FG.
Angle FEG = 75°.
Angle EFG = 2x° and angle FGE = (x + 15)°.
(i) Find the value of x.
Answer(b)(i) x =
[2]
Answer(b)(ii) Angle HJG =
[1]
(ii) Find angle HJG.
Mr.Yasser Elsayed
002 012 013 222 97
173
6) November 2011 V3
11 (a)
y
5
4
3
Q
2
P
1
x
–3
–2
–1
0
1
2
3
4
5
The points P and Q have co-ordinates (–3, 1) and (5, 2).
(i) Write
as a column vector.
Answer(a)(i)
(ii)
=










[1]
− 1 
= 2 
 1
[1]
Mark the point R on the grid.
(iii) Write down the position vector of the point P.
Answer(a)(iii)
Mr.Yasser Elsayed
002 012 013 222 97










[1]
174
(b)
U
L
NOT TO
SCALE
u
M
O
= u and
2
=
K is on UV so that
3
M is the midpoint of KL.
In the diagram,
K
V
v
= v.
and L is on OU so that
=
3
.
4
Find the following in terms of u and v, giving your answers in their simplest form.
(i)
Answer(b)(i)
=
[4]
Answer(b)(ii)
=
[2]
(ii)
Mr.Yasser Elsayed
002 012 013 222 97
175
7) June 2012 V2
7
(a) P is the point (2, 5) and
 
=  3 .
 − 2
Write down the co-ordinates of Q.
Answer(a) (
,
) [1]
(b)
D
C
B
E
c
O
NOT TO
SCALE
M
A
3a
O is the origin and OABC is a parallelogram.
M is the midpoint of AB.
= 3a and CE =
= c,
1
CB.
3
OED is a straight line with OE : ED = 2 : 1 .
Find in terms of a and c, in their simplest forms
(i)
,
Answer(b)(i)
=
[1]
Answer(b)(ii)
[2]
(ii) the position vector of M,
(iii)
(iv)
,
Answer(b)(iii)
=
[1]
Answer(b)(iv)
=
[2]
.
(c) Write down two facts about the lines CD and OB.
Mr.Yasser Elsayed
002 012 013 222 97
Answer (c)
[2]
176
8) November 2012 V1
6
(a)
 − 2

 3
a= 
 2

− 7
b= 
 − 10 

 21
c= 
(i) Find 2a + b.
Answer(a)(i)










[1]
(ii) Find ö=b ö.
Answer(a)(ii)
[2]
(iii) ma + nb = c
Find the values of m and n.
Show all your working.
Answer(a)(iii) m =
Mr.Yasser Elsayed
002 012 013 222 97
n=
[6]
177
(b)
P
X
NOT TO
SCALE
O
Y
Q
In the diagram, OX : XP = 3 : 2 and OY : YQ = 3 : 2 .
= p and
= q.
(i) Write
(ii) Write
in terms of p and q.
Answer(b)(i)
=
[1]
Answer(b)(ii)
=
[1]
in terms of p and q.
(iii) Complete the following sentences.
The lines XY and PQ are
The triangles OXY and OPQ are
The ratio of the area of triangle OXY to the area of triangle OPQ is
Mr.Yasser Elsayed
002 012 013 222 97
:
[3]
178
9) November 2012 V2
6
 3
.
 − 5
(a) Calculate the magnitude of the vector 
Answer(a)
[2]
(b)
y
16
14
12
10
8
R
P
6
4
2
0
x
2
4
6
8
10
12
14
16
18
(i) The points P and R are marked on the grid above.
 3
 . Draw the vector
 − 5
=
(ii) Draw the image of vector
(c)
= 2a + b and
Find
on the grid above.
[1]
after rotation by 90° anticlockwise about R.
[2]
= 3b O a.
in terms of a and b. Write your answer in its simplest form.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(c)
=
[2]
179
(d)
 − 2
 and
 5
 5
.
 − 1
=
Write
=
as a column vector.
Answer(d)
=










[2]
(e)
A
NOT TO
SCALE
M
B
X
C
= b and
(i) Find
= c.
in terms of b and c.
Answer(e)(i)
=
[1]
(ii) X divides CB in the ratio 1 : 3 .
M is the midpoint of AB.
in terms of b and c.
Find
Show all your working and write your answer in its simplest form.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(e)(ii)
=
[4]
180
10) June 2013 V2
7
5
A=e o
7
B = (6
– 4)
2 4
o
C=e
1 3
2 9
o
D=e
-1 -3
(a) Calculate the result of each of the following, if possible.
If a calculation is not possible, write “not possible” in the answer space.
(i) 3A
Answer(a)(i)
[1]
Answer(a)(ii)
[1]
Answer(a)(iii)
[2]
Answer(a)(iv)
[1]
Answer(a)(v)
[2]
Answer(b)
[2]
181
(ii) AC
(iii) BA
(iv) C + D
(v) D2
(b) Calculate C–1, the inverse of C.
Mr.Yasser Elsayed
002 012 013 222 97
11) November 2013 V1
5
(b)
P
Q
NOT TO
SCALE
p
R
O
s
S
In the pentagon OPQRS, OP is parallel to RQ and OS is parallel to PQ.
PQ = 2OS and OP = 2RQ.
= p and
= s.
O is the origin,
Find, in terms of p and s, in their simplest form,
(i) the position vector of Q,
Answer(b)(i) ............................................... [2]
(ii)
.
Answer(b)(ii)
= ............................................... [2]
(c) Explain what your answers in part (b) tell you about the lines OQ and SR.
Answer(c) .................................................................................................................................. [1]
Mr.Yasser Elsayed
002 012 013 222 97
182
12) November 2013 V3
7
(a) The co-ordinates of P are (–4, –4) and the co-ordinates of Q are (8, 14).
(i) Find the gradient of the line PQ.
Answer(a)(i) ............................................... [2]
(ii) Find the equation of the line PQ.
Answer(a)(ii) ............................................... [2]
(iii) Write
as a column vector.
Answer(a)(iii)
(iv) Find the magnitude of
=
f
p
[1]
.
Answer(a)(iv) ............................................... [2]
Mr.Yasser Elsayed
002 012 013 222 97
183
(b)
T
A
NOT TO
SCALE
R
4a
O
In the diagram,
B
3b
= 4a and
= 3b.
1
R lies on AB such that
= 5 (12a + 6b).
T is the point such that
= 2
3
.
(i) Find the following in terms of a and b, giving each answer in its simplest form.
(a)
Answer(b)(i)(a)
= ............................................... [1]
Answer(b)(i)(b)
= ............................................... [2]
Answer(b)(i)(c)
= ............................................... [1]
(b)
(c)
(ii) Complete the following statement.
The points O, R and T are in a straight line because ................................................................
........................................................................................................................................... [1]
(iii) Triangle OAR and triangle TBR are similar.
Find the value of
area of triangle TBR
.
area of triangle OAR
Mr.Yasser Elsayed
002 012 013 222 97
184
Answer(b)(iii) ............................................... [2]
13) June 2014 V1
1
A= f
3
1
-
2
p
1
C= e
2
o
5
-
B = (–2
5)
2 0
p
D= f
0 2
(a) Work out, when possible, each of the following.
If it is not possible, write ‘not possible’ in the answer space.
(i) 2A
Answer(a)(i)
[1]
Answer(a)(ii)
[1]
Answer(a)(iii)
[2]
Answer(a)(iv)
[2]
(ii) B + C
(iii) AD
(iv) A–1, the inverse of A
.
(b) Explain why it is not possible to work out CD.
Answer(b) ........................................................................................................................................... [1]
(c) Describe fully the single transformation represented by the matrix D.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(c) ............................................................................................................................................
.............................................................................................................................
................................ [3]
185
14) June 2014 V1
11 (a)
-3
=e o
4
(i) P is the point (–2, 3).
Work out the co-ordinates of Q .
Answer(a)(i) (............. , .............) [1]
(ii) Work out 
, the magnitude of
.
Answer(a)(ii) ................................................ [2]
Mr.Yasser Elsayed
002 012 013 222 97
186
(b)
C
Y
NOT TO
SCALE
A
N
a
B
b
O
OACB is a parallelogram.
= a and
= b.
2
AN : NB = 2 : 3 and AY = 5 AC.
(i) Write each of the following in terms of a and/or b.
Give your answers in their simplest form.
(a)
Answer(b)(i)(a)
= ................................................ [2]
Answer(b)(i)(b)
= ................................................ [2]
(b)
(ii) Write down two conclusions you can make about the line segments NY and BC.
Answer(b)(ii) ...............................................................................................................................
..................................................................................................................................................... [2]
Mr.Yasser Elsayed
002 012 013 222 97
187
15) June 2014 V3
5
(a)
y
5
A
4
3
2
B
1
x
0
1
2
3
4
5
6
8
7
(i) Write down the position vector of A.
f
Answer(a)(i)
(ii) Find ì
ì , the magnitude of
.
p
[1]
Answer(a)(ii) ................................................ [2]
(b)
S
NOT TO
SCALE
Q
q
O
p
R
P
O is the origin,
= p and
= q.
OP is extended to R so that OP = PR.
OQ is extended to S so that OQ = QS.
(i) Write down
in terms of p and q.
Answer(b)(i)
= ................................................ [1]
(ii) PS and RQ intersect at M and RM = 2MQ.
Use vectors to find the ratio PM : PS, showing all your working.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(b)(ii) PM : PS = ....................... : ....................... [4]
188
16) November 2014 V1
8
B
P
A
NOT TO
SCALE
Q
9b
6a
C
3c
O
= 9b and
= 6a,
In the diagram, O is the origin and
The point P lies on AB such that
= 3b – 2a.
The point Q lies on BC such that
= 2c – 6b .
= 3c.
(a) Find, in terms of b and c, the position vector of Q
Give your answer in its simplest form.
.
Answer(a) ................................................ [2]
Mr.Yasser Elsayed
002 012 013 222 97
189
(b) Find, in terms of a and c, in its simplest form
(i)
(ii)
,
Answer(b)(i)
= ................................................ [1]
Answer(b)(ii)
= ................................................ [2]
.
(c) Explain what your answers in part (b) tell you about PQ and AC.
Answer(c) ............................................................................................................................................
............................................................................................................................. ................................ [2]
Mr.Yasser Elsayed
002 012 013 222 97
190
17) November 2014 V3
P=f
5
0 -1
p
1 0
1 -2
p
Q=f
0 1
R=e
-3
o
5
(a) Work out
(i) 4P,
Answer(a)(i)
[1]
Answer(a)(ii)
[1]
Answer(a)(iii)
[2]
Answer(a)(iv)
[2]
(ii) P – Q,
(iii) P2,
(iv) QR.
1
(b) Find the matrix S, so that QS = f
0
0
p.
1
Mr.Yasser Elsayed
002 012 013 222 97
Answer(b)
191
[3]
18) June 2015 V2
10 (a)
=c
5
m
-8
(i) Find the value of 
.
Answer(a)(i) 
 = ................................................ [2]
(ii) Q is the point (2, –3).
Find the co-ordinates of the point P.
Answer(a)(ii) (...................... , ......................) [1]
(b)
A
NOT TO
SCALE
M
a
L
O
N
B
b
In the diagram, M is the midpoint of AB and L is the midpoint of OM.
The lines OM and AN intersect at L and ON = 13 OB.
= a and
= b.
(i) Find, in terms of a and b, in its simplest form,
(a)
(b)
(c)
,
Answer(b)(i)(a)
= ................................................ [2]
Answer(b)(i)(b)
= ................................................ [1]
Answer(b)(i)(c)
= ................................................ [2]
,
.
Mr.Yasser Elsayed
002 012 013 222 97
192
(ii) Find the ratio AL : AN in its simplest form.
Answer(b)(ii) ................ : ................ [3]
(c)
y
4
3
A
2
1
–7
–6
–5
–4
–3
–2
–1 0
–1
1
2
3
4
5
x
–2
–3
–4
B
–5
(i) On the grid, draw the image of triangle A after the transformation represented by the
matrix f
- 1.5 0
p.
0 -1.5
[3]
(ii) Find the 2 × 2 matrix which represents the transformation that maps triangle A onto triangle B.
Answer(c)(ii)
Mr.Yasser Elsayed
002 012 013 222 97
f
p
193
[2]
19) June 2015 V3
9
P=c
2
1
3
m
4
Q=c
1
0
2
m
3
R=c
0
1
u
m
v
S=c
w
8
3
m
2
(a) Work out PQ.
Answer(a)
f
p
[2]
Answer(b)
f
p
[2]
(b) Find Q –1.
(c) PR = RP
Find the value of u and the value of v.
Answer(c) u = .................................................
v = ................................................. [3]
(d) The determinant of S is 0.
Find the value of w.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(d) w = ................................................. [2]
194
20) November 2015 V1
10
C
NOT TO
SCALE
b
M
a
A
X
B
BC = a and AC = b.
(a) Find AB in terms of a and b.
Answer(a) AB = ................................................ [1]
(b) M is the midpoint of BC.
X divides AB in the ratio 1 : 4.
Find XM in terms of a and b.
Show all your working and write your answer in its simplest form.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(b) XM = ................................................ [4]
195
21) March 2016 V2
S
9
R
T
G
Q
NOT TO
SCALE
y
O
x
P
O is the origin and OPQRST is a regular hexagon.
OP = x and OT = y.
(a) Write down, in terms of x and/or y, in its simplest form,
(i)
QR ,
(ii)
PQ,
QR = ................................................. [1]
PQ = ................................................. [1]
(iii) the position vector of S.
.................................................. [2]
(b) The line SR is extended to G so that SR : RG = 2 : 1.
Find GQ, in terms of x and y, in its simplest form.
GQ = ................................................. [2]
(c) M is the midpoint of OP.
(i) Find MG , in terms of x and y, in its simplest form.
MG = ................................................. [2]
(ii) H is a point on TQ such that TH : HQ = 3 : 1.
Use vectors to show that H lies on MG.
Mr.Yasser Elsayed
002 012 013 222 97
196
[2]
22) June 2016 V1
7
R
M
Q
T
NOT TO
SCALE
r
O
p
P
OPQR is a rectangle and O is the origin.
M is the midpoint of RQ and PT : TQ = 2 : 1.
OP = p and OR = r.
(a) Find, in terms of p and/or r, in its simplest form
(i)
MQ,
MQ = .................................................. [1]
(ii)
MT ,
MT = .................................................. [1]
(iii)
OT .
OT = .................................................. [1]
(b) RQ and OT are extended to meet at U.
Find the position vector of U in terms of p and r.
Give your answer in its simplest form.
Mr.Yasser Elsayed
002 012 013 222 97
................................................... [2]
197
(c)
2k
MT = c m and MT = 180 .
-k
Find the positive value of k.
k = .................................................. [3]
Mr.Yasser Elsayed
002 012 013 222 97
198
23) June 2016 V3
2 0
A = f- 1 5p
3 -4
8
1 3
m
B=c
-1 5
7
C=c m
-4
D = ^2 5h
(a) Work out each of the following if the answer is possible.
If a calculation is not possible, write “not possible” in the answer space.
(i)
BA
[1]
(ii)
2A
[1]
(iii)
CD
[2]
(iv)
DC
[2]
(v)
B2
[2]
(b) Find B–1, the inverse of B.
Mr.Yasser Elsayed
002 012 013 222 97
f
p
199
[2]
24) November 2017 V1
11
2 -3
m
A =c
1 4
(a)
Find
(i)
(ii)
A2,
f
p
[2]
f
p
[2]
A–1, the inverse of A.
-1 0
m.
(b) Describe fully the single transformation represented by the matrix c
0 1
..............................................................................................................................................................
.............................................................................................................................................................. [2]
(c) Find the matrix that represents a clockwise rotation of 90º about the origin.
f
Mr.Yasser Elsayed
002 012 013 222 97
p
200
[2]
19
(d)
C
A
O
NOT TO
SCALE
P
a
b
B
In the diagram, O is the origin and P lies on AB such that AP : PB = 3 : 4.
OA = a and OB = b .
(i)
Find OP , in terms of a and b, in its simplest form.
OP = ................................................ [3]
(ii)
The line OP is extended to C such that OC = m OP and BC = ka.
Find the value of m and the value of k.
m = ................................................
k = ................................................ [2]
Mr.Yasser Elsayed
002 012 013 222 97
201
25) June 2018 V1
11
8
AB = c m
-7
4
OA = c m
3
(a)
-3
AC = c m
6
Find
(i)
OB ,
OB = ............................................... [3]
(ii)
BC .
BC =
(b)
S
R
b
P
f
p
[2]
NOT TO
SCALE
X
a
Q
PQRS is a parallelogram with diagonals PR and SQ intersecting at X.
PQ = a and PS = b .
Find QX in terms of a and b.
Give your answer in its simplest form.
QX = ............................................... [2]
Mr.Yasser Elsayed
002 012 013 222 97
202
17
M=c
(c)
2 5
m
1 8
Calculate
(i)
(ii)
M2 ,
M2 =
f
p
[2]
M -1 =
f
p
[2]
M -1 .
Mr.Yasser Elsayed
002 012 013 222 97
203
26) June 2020 V2
2
4
p =e o
5
(a)
-2
q =e o
7
(i) Find 2p + q .
f
p
[2]
(ii) Find p .
................................................. [2]
-3
(b) A is the point (4, 1) and AB = e o.
1
Find the coordinates of B.
( ...................... , ...................... ) [1]
(c) The line y = 3x - 2 crosses the y-axis at G.
Write down the coordinates of G.
( ...................... , ...................... ) [1]
Mr.Yasser Elsayed
002 012 013 222 97
© UCLES 2020
0580/42/M/J/20
204
5
(d)
D
NOT TO
SCALE
T
M
O
C
In the diagram, O is the origin, OT = 2TD and M is the midpoint of TC.
OC = c and OD = d .
Find the position vector of M.
Give your answer in terms of c and d in its simplest form.
................................................. [3]
Mr.Yasser Elsayed
002 012 013 222 97
© UCLES 2020
0580/42/M/J/20
205
[Turn over
Transformations
Mr.Yasser Elsayed
002 012 013 222 97
206
1) June 2010 V1
3
y
8
7
T
6
5
4
3
2
1
–8
–7
–6
–5
–4
–3
–2
–1
0
1
2
3
4
5
6
7
8
–1
–2
P
–3
–4
–5
–6
Q
–7
–8
(a) On the grid, draw the enlargement of the triangle T, centre (0, 0), scale factor
Mr.Yasser Elsayed
002 012 013 222 97
1
2
.
[2]
207
 −1 0
 represents a transformation.
 0 1
(b) The matrix 
 − 1 0  8 8 2 

 .
 0 1  4 8 8 
(i) Calculate the matrix product 
Answer(b)(i)
[2]
(ii) On the grid, draw the image of the triangle T under this transformation.
[2]
(iii) Describe fully this single transformation.
Answer(b)(iii)
[2]
(c) Describe fully the single transformation which maps
(i) triangle T onto triangle P,
Answer(c)(i)
[2]
(ii) triangle T onto triangle Q.
Answer(c)(ii)
[3]
(d) Find the 2 by 2 matrix which represents the transformation in part (c)(ii).
Answer(d)
Mr.Yasser Elsayed
002 012 013 222 97








[2]
208
2) June 2010 V2
y
4
9
8
7
6
5
V
4
3
2
T
1
4
–
–3
–2
–1
x
0
1
2
3
4
5
6
7
8
9
–1
–2
–3
–4
–5
U
–6
(a) On the grid, draw
the translation of triangle T by the vector  7  ,
−
(i)
3 
(ii) the rotation of triangle T about (0, 0), through 90° clockwise.
[2]
[2]
(b) Describe fully the single transformation that maps
(i) triangle T onto triangle U,
Answer(b)(i)
(ii) triangle T onto triangle V
[2]
Mr.Yasser
Elsayed
Answer(b)(ii)
002 012 013 222 97
.
[3]
209
(c) Find the 2 by 2 matrix which represents the transformation that maps
(i) triangle T onto triangle U,
Answer(c)(i)












[2]
Answer(c)(ii)












[2]
Answer(c)(iii)












[1]
(ii) triangle T onto triangle V,
(iii) triangle V onto triangle T.
Mr.Yasser Elsayed
002 012 013 222 97
210
3) June 2010 V3
4
y
12
11
10
9
8
7
6
5
P
4
3
T
Q
2
1
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
1
–
2
–
3
–
4
–
(a) Draw the reflection of triangle T in the line y = 6.
Label the image A.
[2]
4
(b) Draw the translation of triangle T by the vector   .
6
 
Label the image B.
−
Mr.Yasser Elsayed
002 012 013 222 97
[2]
211
(c) Describe fully the single transformation which maps triangle B onto triangle T.
Answer(c)
[2]
(d) (i) Describe fully the single transformation which maps triangle T onto triangle P.
Answer(d)(i)
[3]
(ii) Complete the following statement.
Area of triangle P =
× Area of triangle T
[1]
(e) (i) Describe fully the single transformation which maps triangle T onto triangle Q.
Answer(e)(i)
[3]
(ii) Find the 2 by 2 matrix which represents the transformation mapping triangle T onto
triangle Q.
Answer(e)(ii)
Mr.Yasser Elsayed
002 012 013 222 97










[2]
212
4) November 2010 V1
2
(a)
y
5
4
3
2
A
1
–5
–4
–3
–2
–1
0
1
2
3
4
5
x
–1
–2
–3
–4
–5
(i) Draw the image when triangle A is reflected in the line y = 0.
Label the image B.
[2]
(ii) Draw the image when triangle A is rotated through 90U anticlockwise about the origin.
Label the image C.
[2]
(iii) Describe fully the single transformation which maps triangle B onto triangle C.
Answer(a)(iii)
[2]
0 −1 
(b) Rotation through 90U anticlockwise about the origin is represented by the matrix M = 
.
1 0
(i) Find M–1, the inverse of matrix M.
–1
Answer(b)(i) M =










[2]
(ii) Describe fully the single transformation represented by the matrix M–1.
Mr.Yasser
Elsayed
Answer(b)(ii)
002 012 013 222 97
[2]
213
5) November 2010 V2
8
(a)
y
8
6
4
A
A
2
–8
–6
–4
–2
0
2
4
6
8
–2
–4
–6
–8
Draw the images of the following transformations on the grid above.
(i) Translation of triangle A by the vector 
3
 . Label the image B.
 −7 
[2]
(ii) Reflection of triangle A in the line x = 3. Label the image C.
[2]
(iii) Rotation of triangle A through 90° anticlockwise around the point (0, 0).
Label the image D.
[2]
(iv) Enlargement of triangle A by scale factor –4, with centre (0, 1).
Label the image E.
[2]
Mr.Yasser Elsayed
002 012 013 222 97
214
(b) The area of triangle E is k × area of triangle A.
Write down the value of k.
Answer(b) k =
[1]
(c)
y
5
4
3
2
1
F
x
–5
–4
–3
–2
–1
0
1
2
3
4
5
–1
–2
–3
–4
–5
(i) Draw the image of triangle F under the transformation represented by the
 1 3
 .
 0 1
matrix M = 
[3]
(ii) Describe fully this single transformation.
Answer(c)(ii)
[3]
(iii) Find M–1, the inverse of the matrix M.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(c)(iii)










[2]
215
6) June 2011 V1
y
4
5
A
B
3
2
1
5
–
–4
–3
–2
–1 0
–1
x
2
1
3
4
5
6
7
C
–2
–3
–4
–5
–6
(a) On the grid above, draw the image of
 3
,
 2
(i) shape A after translation by the vector 
−
[2]
−
(ii) shape A after reflection in the line x = 1 .
[2]
−
(b) Describe fully the single transformation which maps
(i) shape A onto shape B,
Answer(b)(i)
[3]
(ii) shape A onto shape C.
Answer(b)(ii)
[3]
(c) Find the matrix representing the transformation which maps shape A onto shape B.
Answer(c)






 1
 0
(d) Describe fully the single transformation represented by the matrix 
Answer(d)
Mr.Yasser
Elsayed
002 012 013 222 97
−
−
0 
1 
[2]
.
[3]
216
7) June 2011 V2
8
(a)
A
P
Draw the enlargement of triangle P with centre A and scale factor 2.
[2]
(b)
y
Q
R
x
0
(i) Describe fully the single transformation which maps shape Q onto shape R.
Answer(b)(i)
[3]
(ii) Find the matrix which represents this transformation.
Answer(b)(ii)






[2]
(c)
y
S
T
0
x
Describe fully the single transformation which maps shape S onto shape T.
Answer(c)
Mr.Yasser
Elsayed
002 012 013 222 97
[3]
217
8) June 2011 V3
2
y
6
X
5
4
3
2
1
–6
–5
–4
–3
–2
–1
0
1
2
3
4
5
x
6
–1
–2
–3
–4
–5
–6
(a) (i) Draw the reflection of shape X in the x-axis. Label the image Y.
[2]
(ii) Draw the rotation of shape Y, 90° clockwise about (0, 0). Label the image Z.
[2]
(iii) Describe fully the single transformation that maps shape Z onto shape X.
Answer(a)(iii)
[2]
(b) (i) Draw the enlargement of shape X, centre (0, 0), scale factor
1
2
.
[2]
(ii) Find the matrix which represents an enlargement, centre (0, 0), scale factor
Answer(b)(ii)



1
2
.



[2]
(c) (i) Draw the shear of shape X with the x-axis invariant and shear factor –1.
[2]
(ii) Find the matrix which represents a shear with the x-axis invariant and shear factor –1.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(c)(ii)






[2]
218
9) November 2011 V1
7
y
8
6
4
B
2
–8
–6
–4
–2
0
2
4
6
8
x
C
–2
–4
D
–6
A
–8
(a) Describe fully the single transformation which maps
(i) triangle A onto triangle B,
Answer(a)(i)
[2]
(ii) triangle A onto triangle C,
Answer(a)(ii)
Mr.Yasser Elsayed
Answer(a)(iii)
002 012 013 222 97
[3]
(iii) triangle A onto triangle D.
[3]
219
(b) Draw the image of
− 5 
 ,
 2
(i) triangle B after a translation of 
[2]
1 0
 .
0
2


(ii) triangle B after a transformation by the matrix 
[3]
1 0
 .
0 2
(c) Describe fully the single transformation represented by the matrix 
Answer(c)
[3]
Mr.Yasser Elsayed
002 012 013 222 97
220
10) November 2011 V2
3
y
9
8
7
6
5
4
3
2
A
T
1
x
9
–
–8
–7
–6
–5
–4
–3
–2
–1 0
–1
1
2
3
4
5
6
7
8
9
–2
–3
–4
–5
–6
–7
–8
–9
Triangles T and A are drawn on the grid above.
(a) Describe fully the single transformation that maps triangle T onto triangle A.
Answer(a)
[2]
(b) (i) Draw the image of triangle T after a rotation of 90° anticlockwise about the point (0,0).
Label the image B.
[2]
(ii) Draw the image of triangle T after a reflection in the line x + y = 0.
Label the image C.
[2]
(iii) Draw the image of triangle T after an enlargement with centre (4, 5) and scale factor 1.5.
Mr.Yasser
Elsayed
Label the image D.
002 012 013 222 97
[2]
221
(c) (i) Triangle T has its vertices at co-ordinates (2, 1), (6, 1) and (6, 3).
1 0 
.
1 1 
Transform triangle T by the matrix 
Draw this image on the grid and label it E.
[3]
1 0 
.
1 1 
(ii) Describe fully the single transformation represented by the matrix 
Answer(c)(ii)
[3]
(d) Write down the matrix that transforms triangle B onto triangle T.
Answer(d)
Mr.Yasser Elsayed
002 012 013 222 97










[2]
222
11) November 2011 V3
4
y
6
4
P
2
W
x
–6
–4
–2
0
2
4
6
–2
–4
–6
(a) Draw the reflection of shape P in the line y = x.
[2]
− 2 
 .
 1
(b) Draw the translation of shape P by the vector 
[2]
(c) (i) Describe fully the single transformation that maps shape P onto shape W.
Answer(c)(i)
[3]
(ii) Find the 2 by 2 matrix which represents this transformation.
Answer(c)(ii)










[2]
1 0
 .
0 2
(d) Describe fully the single transformation represented by the matrix 
Mr.Yasser
Elsayed
Answer(d)
002 012 013 222 97
[3]
223
12) June 2012 V1
y
10
7
9
8
7
6
5
4
P
3
2
1
5
–
–4
–3
–2
–1
0
1
2
3
4
5
6
7
8
9
x
–1
R
–2
–3
Q
–4
–5
–6
(a) Describe fully
(i) the single transformation which maps triangle P onto triangle Q ,
Answer(a)(i)
[3]
(ii) the single transformation which maps triangle Q onto triangle R,
Answer(a)(ii)
[3]
(iii) the single transformation which maps triangle R onto triangle P.
Answer(a)(iii)
Mr.Yasser
Elsayed
002 012 013 222 97
[3]
224
(b) On the grid, draw the image of
 − 4
,
 − 5
(i) triangle P after translation by 
[2]
(ii) triangle P after reflection in the line x = −1 .
[2]
(c) (i) On the grid, draw the image of triangle P after a stretch, scale factor 2 and the y-axis as the
invariant line.
[2]
(ii) Find the matrix which represents this stretch.
Answer(c)(ii)
Mr.Yasser Elsayed
002 012 013 222 97










[2]
225
13) June 2012 V3
3
y
11
10
9
8
7
6
5
4
Q
3
2
P
1
x
–3
–2
–1
0
1
2
3
4
5
6
7
8
9
10
11
12
 5
(a) Draw the translation of triangle P by   .
 3
[2]
(b) Draw the reflection of triangle P in the line x = 6 .
[2]
(c) (i) Describe fully the single transformation that maps triangle P onto triangle Q.
Answer(c)(i)
[3]
(ii) Find the 2 by 2 matrix which represents the transformation in part(c)(i).
Answer(c)(ii)










(d) (i) Draw the stretch of triangle P with scale factor 3 and the x-axis as the invariant line.
[2]
[2]
(ii) Find the 2 by 2 matrix which represents a stretch, scale factor 3 and x-axis invariant.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(d)(ii)










[2]
226
14) November 2012 V3
2
(a)
y
8
7
X
6
5
4
3
2
1
–8
–7
–6
–5
–4
–3
–1 0
–2
x
1
2
3
4
5
6
7
8
–1
–2
Y
–3
–4
–5
–6
–7
–8
–9
 − 11
.
 − 1
(i) Draw the translation of triangle X by the vector 
(ii) Draw the enlargement of triangle Y with centre (–6, – 4) and scale factor
Mr.Yasser Elsayed
002 012 013 222 97
[2]
1
2
.
[2]
227
(b)
y
8
7
W
6
X
5
4
3
2
1
–8 –7 –6 –5 –4 –3 –2 –1 0
–1
Y
1
2
3
4
5
6
7
8
x
–2
–3
–4
Z
–5
–6
–7
–8
–9
Describe fully the single transformation that maps
(i) triangle X onto triangle Z,
Answer(b)(i)
[2]
(ii) triangle X onto triangle Y,
Answer(b)(ii)
[3]
(iii) triangle X onto triangle W.
Answer(b)(iii)
[3]
(c) Find the matrix that represents the transformation in part (b)(iii).
Mr.Yasser Elsayed
002 012 013 222 97
Answer(c)










[2]
228
15) June 2013 V1
y
4
9
8
7
6
5
4
3
Q
2
1
8 –7 –6 –5 –4 –3 –2 –1 0
–1
–
1
2
3
4
5
6
7
8
x
–2
R
–3
–4
–5
–6
(a) Describe fully the single transformation that maps shape Q onto shape R .
Answer(a) ................................................................................................................................. [3]
5
(b) (i) Draw the image when shape Q is translated by the vector e o .
4
[2]
(ii) Draw the image when shape Q is reflected in the line x = 2.
[2]
(iii) Draw the image when shape Q is stretched, factor 3, x-axis invariant.
[2]
(iv) Find the 2 × 2 matrix that represents a stretch of factor 3, x-axis invariant.
Answer(b)(iv)
e
o
[2]
0 1
o.
(c) Describe fully the single transformation represented by the matrix e
1 0
Mr.Yasser Elsayed
002 012 013 222 97
Answer(c) .................................................................................................................................. [2]
229
16) June 2013 V2
2
(a)
y
6
5
4
3
Q
2
1
x
–7
–6
–5
–4
–3
–2
–1 0
1
2
3
4
5
6
7
8
–1
–2
–3
–4
–5
P
–6
–7
–8
(i) Describe fully the single transformation which maps shape P onto shape Q.
Answer(a)(i) ...................................................................................................................... [2]
(ii) On the grid above, draw the image of shape P after reflection in the line y = –1.
[2]
(iii) On the grid above, draw the image of shape P under the transformation represented by the
matrix e0 -1 o .
[3]
1 0
Mr.Yasser Elsayed
002 012 013 222 97
230
(b)
y
10
9
8
7
6
5
4
3
M
L
2
1
–4
–3
–2
–1 0
1
2
3
4
5
6
7
8
9
10
11
12
x
–1
–2
–3
–4
–5
(i) Describe fully the single transformation which maps shape M onto shape L.
Answer(b)(i) ...................................................................................................................... [3]
(ii) On the grid above, draw the image of shape M after enlargement by scale factor 2,
centre (5, 0).
[2]
Mr.Yasser Elsayed
002 012 013 222 97
231
17) June 2013 V3
7
y
10
9
8
7
6
5
4
3
2
A
B
1
0
x
1
2
3
4
5
6
7
8
(a) (i) Draw the image of shape A after a stretch, factor 3, x-axis invariant.
[2]
(ii) Write down the matrix representing a stretch, factor 3, x-axis invariant.
Answer(a)(ii)
e
o
[2]
(b) (i) Describe fully the single transformation which maps shape A onto shape B.
Answer(b)(i) ...................................................................................................................... [3]
(ii) Write down the matrix representing the transformation which maps shape A onto shape B.
Answer(b)(ii)
Mr.Yasser Elsayed
002 012 013 222 97
e
o
[2]
232
18) November 2013 V1
5
(a)
y
10
9
8
7
6
5
4
3
2
T
U
1
0
x
1
2
3
4
5
6
7
8
9
10
(i) Draw the reflection of triangle T in the line y = 5.
[2]
(ii) Draw the rotation of triangle T about the point (4, 2) through 180°.
[2]
(iii) Describe fully the single transformation that maps triangle T onto triangle U.
Answer(a)(iii) .................................................................................................................... [3]
(iv) Find the 2 × 2 matrix which represents the transformation in part (a)(iii).
Answer(a)(iv)
Mr.Yasser Elsayed
002 012 013 222 97
f
p
[2]
233
19) November 2013 V1
9
y
8
7
6
5
4
D
C
3
2
B
–8
–7
A
1
–6
–5
–4
–3
–2
–1 0
x
1
2
3
4
5
6
7
8
–1
–2
–3
–4
–5
–6
–7
–8
(a) Describe fully the single transformation that maps triangle A onto
(i) triangle B,
Answer(a)(i) ...................................................................................................................... [2]
(ii) triangle C,
Answer(a)(ii) ..................................................................................................................... [2]
(iii) triangle D.
.................................................................................................................... [3]
Mr.YasserAnswer(a)(iii)
Elsayed
002 012 013 222 97
234
(b) On the grid, draw
(i) the rotation of triangle A about (6, 0) through 90° clockwise,
[2]
(ii) the enlargement of triangle A by scale factor –2 with centre (0, –1),
[2]
(iii) the shear of triangle A by shear factor –2 with the y-axis invariant.
[2]
(c) Find the matrix that represents the transformation in part (b)(iii).
Answer(c)
Mr.Yasser Elsayed
002 012 013 222 97
f
p
[2]
235
20) June 2014 V1
7
y
4
3
A
2
1
–6
–5
–4
–3
–2
–1 0
x
1
2
3
4
5
6
–1
–2
–3
–4
–5
(a) On the grid,
-5
(i) draw the image of shape A after a translation by the vector e o ,
-4
[2]
(ii) draw the image of shape A after a rotation through 90° clockwise about the origin.
[2]
2 0
p.
(b) (i) On the grid, draw the image of shape A after the transformation represented by the matrix f
0 1
[3]
2 0
p.
(ii) Describe fully the single transformation represented by the matrix f
0 1
Answer(b)(ii) ...............................................................................................................................
..................................................................................................................................................... [3]
Mr.Yasser Elsayed
002 012 013 222 97
236
21) June 2014 V2
4
y
8
7
6
5
4
3
Q
2
1
–8
–7
–6
–5
–4
–3
–2
–1 0
x
1
2
3
4
5
6
7
8
–1
–2
–3
–4
–5
–6
–7
–8
(a) Draw the reflection of shape Q in the line x = –1 .
[2]
(b) (i) Draw the enlargement of shape Q, centre (0, 0), scale factor –2 .
[2]
(ii) Find the 2 × 2 matrix that represents an enlargement, centre (0, 0), scale factor –2 .
Mr.Yasser Elsayed
002 012 013 222 97
Answer(b)(ii)
f
p
237
[2]
(c) (i) Draw the stretch of shape Q, factor 2, x-axis invariant.
[2]
(ii) Find the 2 × 2 matrix that represents a stretch, factor 2, x-axis invariant.
Answer(c)(ii)
f
p
[2]
Answer(c)(iii)
f
p
[2]
(iii) Find the inverse of the matrix in part (c)(ii).
(iv) Describe fully the single transformation represented by the matrix in part (c)(iii).
Answer(c)(iv) ..............................................................................................................................
..................................................................................................................................................... [3]
Mr.Yasser Elsayed
002 012 013 222 97
238
22) November 2014 V1
3
y
8
7
6
5
4
3
2
A
1
–8
–7
–6
–5
–4
–3
–2
–1 0
–1
1
2
3
4
5
6
7
8
x
–2
–3
B
–4
–5
–6
–7
–8
(a) Draw the image when triangle A is reflected in the line x = 0.
[1]
(b) Draw the image when triangle A is rotated through 90° anticlockwise about (–4, 0).
[2]
(c) (i) Describe fully the single transformation that maps triangle A onto triangle B.
Answer(c)(i) ................................................................................................................................
..................................................................................................................................................... [3]
(ii) Complete the following statement.
Area of triangle A : Area of triangle B = .................... : ....................
Mr.Yasser Elsayed
002 012 013 222 97
[2]
239
(d) Write down the matrix that represents a stretch, factor 4 with the y-axis invariant.
Answer(d)
f
p
[2]
(e) (i) On the grid, draw the image of triangle A after the transformation represented by the
1 0
o.
matrix e
2 1
[3]
(ii) Describe fully this single transformation.
Answer(e)(ii) ...............................................................................................................................
..................................................................................................................................................... [3]
1 0
o.
(iii) Find the inverse of the matrix e
2 1
Answer(e)(iii)
Mr.Yasser Elsayed
002 012 013 222 97
f
p
240
[2]
23) November 2014 V2
4
y
8
7
6
B
5
A
4
3
2
1
–8
–7
–6
–5
–4
–3
–2
–1
0
1
2
3
4
5
6
7
8
x
–1
–2
–3
–4
–5
–6
–7
–8
(a) Describe fully the single transformation that maps triangle A onto triangle B.
Answer(a) ...........................................................................................................................................
............................................................................................................................................................. [3]
Mr.Yasser Elsayed
002 012 013 222 97
241
(b) On the grid, draw the image of
(i) triangle A after a reflection in the line x = –3,
[2]
(ii) triangle A after a rotation about the origin through 270° anticlockwise,
[2]
-1
(iii) triangle A after a translation by the vector e o .
-5
[2]
(c) M is the matrix that represents the transformation in part (b)(ii).
(i) Find M.
Answer(c)(i) M =
f
p
[2]
(ii) Describe fully the single transformation represented by M–1, the inverse of M.
Answer(c)(ii) ...............................................................................................................................
..................................................................................................................................................... [2]
Mr.Yasser Elsayed
002 012 013 222 97
242
24) June 2015 V1
y
3
7
6
5
4
C
3
2
A
1
–12 –11 –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0
–1
1
2
3
4
5
6
x
–2
–3
B
–4
–5
–6
–7
(a) Draw the image of
(i)
shape A after a translation by f
-1
p,
3
(ii)
shape A after a rotation through 180° about the point (0, 0),
(iii)
1
shape A after the transformation represented by the matrix f
0
[2]
[2]
0
p.
-1
[3]
(b) Describe fully the single transformation that maps shape A onto shape B.
Answer(b) .................................................................................................................................................
.............................................................................................................................................................. [3]
(c) Find the matrix which represents the transformation that maps shape A onto shape C.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(c)
f
p
243
[2]
25) June 2015 V3
1
y
6
U
5
4
3
T
2
V
1
–5
–4
–3
–2
–1 0
1
2
3
4
5
6
7
x
–1
–2
–3
–4
–5
–6
(a) On the grid, draw the image of
(i)
triangle T after a reflection in the line x = –1,
[2]
(ii)
triangle T after a rotation through 180° about (0, 0).
[2]
(b) Describe fully the single transformation that maps
(i)
triangle T onto triangle U,
Answer(b)(i) ......................................................................................................................................
...................................................................................................................................................... [2]
(ii)
triangle T onto triangle V.
Answer(b)(ii) .....................................................................................................................................
......................................................................................................................................................
[3]
Mr.Yasser
Elsayed
002 012 013 222 97
244
26) November 2015 V3
2
y
8
7
6
5
4
3
2
T
1
–8
–7
–6
–5
–4
–3
–2
–1
0
1
2
3
4
5
6
x
–1
U
–2
–3
–4
W
–5
–6
(a) On the grid, draw the image of
-4
(i) triangle T after a translation by the vector e o ,
4
[2]
(ii) triangle T after a reflection in the line y = – 1.
[2]
Mr.Yasser Elsayed
002 012 013 222 97
245
(b) Describe fully the single transformation that maps triangle T onto triangle U.
Answer(b) ...........................................................................................................................................
............................................................................................................................................................. [3]
(c) (i) Describe fully the single transformation that maps triangle T onto triangle W.
Answer(c)(i) ................................................................................................................................
..................................................................................................................................................... [2]
(ii) Find the 2 × 2 matrix that represents the transformation in part (c)(i).
Answer(c)(ii)
Mr.Yasser Elsayed
002 012 013 222 97
f
p
246
[2]
27) March 2015 V2
y
7
7
6
A
5
4
3
2
1
–7 –6 –5 –4 –3 –2 –1 0
–1
1
2
3
4
5
6
7
8
x
–2
–3
B
–4
–5
–6
–7
C
–8
(a) Describe fully the single transformation that maps
(i) flag A onto flag B,
Answer(a)(i) ................................................................................................................................
..................................................................................................................................................... [3]
(ii) flag A onto flag C.
Answer(a)(ii) ...............................................................................................................................
..................................................................................................................................................... [3]
2
(b) Draw the image of flag A after a translation by the vector e o .
1
[2]
(c) Draw the image of flag A after a reflection in the line x = 1.
[2]
(d) Describe fully the single transformation represented by the matrix e
1 0
o.
0 -1
Mr.Yasser
Elsayed
Answer(d) ...........................................................................................................................................
............................................................................................................................................................. [2]
002 012
013 222 97
247
28) March 2016 V2
6
y
8
7
6
Z
5
4
X
3
Y
2
1
–5
–4
–3
–2
–1
0
1
2
3
4
5
6
7
x
–1
–2
(a) Describe fully the single transformation that maps
(i) triangle X onto triangle Y,
.......................................................................................................................................................
....................................................................................................................................................... [3]
(ii) triangle X onto triangle Z.
.......................................................................................................................................................
....................................................................................................................................................... [3]
(b) (i) Draw the image of triangle X after a translation by the vector c
-5
m.
3
Label this triangle P.
[2]
(ii) Draw the reflection of triangle P in the line y = 3.
Mr.Yasser Elsayed
002 012 013 222 97
(c) Draw the image of triangle X after the transformation represented by the matrix c
[2]
0 –1
m.
1 0
[3]
248
29) June 2016 V1
2
(a)
y
6
5
4
3
Q
2
1
–7
–6
–5
–4
–2
–3
–1
0
1
2
3
4
5
6
7
8
9
10
–1
T
–2
–3
–4
–5
–6
(i)
5
Draw the image of triangle T after a translation by the vector c m .
-2
[2]
(ii)
Draw the image of triangle T after a reflection in the line y = 1.
[2]
(iii)
Describe fully the single transformation that maps triangle T onto triangle Q.
......................................................................................................................................................
...................................................................................................................................................... [3]
Mr.Yasser Elsayed
002 012 013 222 97
249
x
1 2
m
(b) M = c
3 4
(i)
(ii)
(iii)
4 3
m
N=c
1 k
1 3
m
P=c
0 6
Work out M + P.
f
p
[1]
f
p
[2]
Work out PM.
M = N
Find the value of k.
k = .................................................. [3]
(c)
(i)
0 -1
m.
Describe fully the single transformation represented by the matrix c
1 0
......................................................................................................................................................
...................................................................................................................................................... [3]
(ii)
Find the matrix which represents a reflection in the line y = x.
Mr.Yasser Elsayed
002 012 013 222 97
f
p
250
[2]
30) June 2016 V2
3
(a)
y
5
4
3
A
2
1
–4
–3
–2
0
–1
1
2
3
4
5
6
7
x
–1
–2
–3
–4
–5
–6
On the grid, draw the image of
(i)
shape A after a reflection in the line x = 1,
[2]
(ii)
shape A after an enlargement with scale factor –2, centre (0, 1),
[2]
(iii)
shape A after the transformation represented by the matrix c
0 -1
m.
1 0
(b) Describe fully the single transformation represented by the matrix c
[3]
3 0
m.
0 3
..............................................................................................................................................................
.............................................................................................................................................................. [3]
Mr.Yasser Elsayed
002 012 013 222 97
251
31) June 2016 V3
6
y
8
U
6
4
V
2
–8
–6
–4
–2
0
2
4
6
8
x
–2
T
–4
–6
(a) (i)
Draw the image of triangle T after a reflection in the line x = 0.
[2]
(ii)
Draw the image of triangle T after a rotation through 90° clockwise about (–2, –1).
[2]
(iii)
Describe fully the single transformation that maps triangle T onto triangle U.
......................................................................................................................................................
...................................................................................................................................................... [2]
(iv)
Describe fully the single transformation that maps triangle T onto triangle V.
......................................................................................................................................................
...................................................................................................................................................... [3]
(b) (i)
Find the matrix that represents the transformation in part (a)(i).
f
(ii)
p
[2]
Describe fully the single transformation represented by the inverse of the matrix in part (b)(i).
......................................................................................................................................................
......................................................................................................................................................
[2]
Mr.Yasser
Elsayed
002 012 013 222 97
252
32) June 2017 V1
3
y
8
7
B
6
5
4
3
2
A
1
–6
–5
–4
–3
–2
–1
0
1
2
3
4
5
6
7
8
x
–1
–2
–3
–4
–5
–6
(a) (i)
Draw the image of triangle A after reflection in the line x = 4 .
[2]
(ii)
Draw the image of triangle A after rotation of 90° anticlockwise about (0, 0).
[2]
(iii)
1
Draw the image of triangle A after translation by the vector c m .
-5
[2]
(b) Describe fully the single transformation that maps triangle A onto triangle B.
..............................................................................................................................................................
.............................................................................................................................................................. [3]
(c) Find the matrix that represents the transformation in part (a)(ii).
Mr.Yasser Elsayed
002 012 013 222 97
f
p
253
[2]
(d) Point P has co-ordinates (4, 1).
-1 0
1 0
m and G = c
m represent transformations.
F =c
0 1
0 2
(i)
Find G(P), the image of P after the transformation represented by G.
(....................... , .......................) [2]
(ii)
Find GF(P).
(....................... , .......................) [3]
(iii)
Find the matrix Q such that
GQ (P) = P .
f
Mr.Yasser Elsayed
002 012 013 222 97
p
254
[3]
33) June 2018 V1
4
y
8
7
6
5
B
4
3
A
2
1
–6
–5
–4
–3
–2
C
–1 0
–1
1
2
3
4
5
6
7
8
x
–2
–3
–4
–5
D
–6
(a) Describe fully the single transformation that maps
(i)
triangle A onto triangle B,
.....................................................................................................................................................
..................................................................................................................................................... [2]
(ii)
triangle A onto triangle C,
.....................................................................................................................................................
..................................................................................................................................................... [3]
(iii)
triangle A onto triangle D.
.....................................................................................................................................................
..................................................................................................................................................... [3]
(b) On the grid, draw the image of triangle A after an enlargement by scale factor 2, centre ^7, 3h .
Mr.Yasser Elsayed
002 012 013 222 97
255
[2]
34) June 2018 V2
3
y
6
5
4
3
B
2
1
–7
–6
–5
–4
–3
–2
–1
0
1
2
3
4
5
6
7
8
x
–1
–2
–3
A
–4
–5
–6
(a) (i)
Draw the image of triangle A after a reflection in the line x = 2.
[2]
(ii)
Draw the image of triangle A after a translation by the vector c
[2]
(iii)
1
Draw the image of triangle A after an enlargement by scale factor - , centre (3, 1).
2
-2
m.
4
[3]
(b) Describe fully the single transformation that maps triangle A onto triangle B.
..............................................................................................................................................................
.............................................................................................................................................................. [3]
0 -1
m.
(c) Describe fully the single transformation represented by the matrix c
-1 0
..............................................................................................................................................................
.............................................................................................................................................................. [2]
Mr.Yasser Elsayed
002 012 013 222 97
256
35) November 2020 V1
1
y
7
6
C
5
4
3
A
2
1
–7 –6 –5 –4 –3 –2 –1 0
–1
B
1
2
3
4
5
6
x
–2
–3
–4
–5
–6
–7
–8
8
(a) Draw the image of shape A after a translation by the vector e o.
-6
[2]
(b) Draw the image of shape A after a reflection in the line y =- 1.
[2]
(c) Describe fully the single transformation that maps shape A onto shape B.
.....................................................................................................................................................
..................................................................................................................................................... [3]
(d) Describe fully the single transformation that maps shape A onto shape C.
.....................................................................................................................................................
..................................................................................................................................................... [3]
Mr.Yasser Elsayed
002 012 013 222 97
257
Sets and Probability
Mr.Yasser Elsayed
002 012 013 222 97
258
1) June 2010 V1
4
A
B
Box A contains 3 black balls and 1 white ball.
Box B contains 3 black balls and 2 white balls.
(a) A ball can be chosen at random from either box.
Complete the following statement.
There is a greater probability of choosing a white ball from Box
.
Explain your answer.
Answer(a)
[1]
(b) Abdul chooses a box and then chooses a ball from this box at random.
The probability that he chooses box A is
2
3
.
(i) Complete the tree diagram by writing the four probabilities in the empty spaces.
BOX
COLOUR
1
4
2
3
white
A
black
white
B
black
Mr.Yasser Elsayed
002 012 013 222 97
[4]
259
(ii) Find the probability that Abdul chooses box A and a black ball.
Answer(b)(ii)
[2]
(iii) Find the probability that Abdul chooses a black ball.
Answer(b)(iii)
[2]
(c) Tatiana chooses a box and then chooses two balls from this box at
random (without replacement).
The probability that she chooses box A is
2
3
.
Find the probability that Tatiana chooses two white balls.
Answer(c)
Mr.Yasser Elsayed
002 012 013 222 97
[2]
260
2) June 2010 V2
3
2
2
1
1
10
1
The diagram shows a spinner with six numbered sections.
Some of the sections are shaded.
Each time the spinner is spun it stops on one of the six sections.
It is equally likely that it stops on any one of the sections.
(a) The spinner is spun once.
Find the probability that it stops on
(i) a shaded section,
Answer(a)(i)
[1]
Answer(a)(ii)
[1]
Answer(a)(iii)
[1]
(ii) a section numbered 1,
(iii) a shaded section numbered 1,
(iv) a shaded section or a section numbered 1.
Answer(a)(iv)
Mr.Yasser Elsayed
002 012 013 222 97
[1]
261
(b) The spinner is now spun twice.
Find the probability that the total of the two numbers is
(i) 20,
Answer(b)(i)
[2]
Answer(b)(ii)
[2]
(ii) 11.
(c) (i) The spinner stops on a shaded section.
Find the probability that this section is numbered 2.
Answer(c)(i)
[1]
(ii) The spinner stops on a section numbered 2.
Find the probability that this section is shaded.
Answer(c)(ii)
[1]
(d) The spinner is now spun until it stops on a section numbered 2.
The probability that this happens on the nth spin is
16
.
243
Find the value of n.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(d) n =
[2]
262
3) June 2010 V3
3
1
2
4
4
1
1
3
2
4
1
The diagram shows a circular board, divided into 10 numbered sectors.
When the arrow is spun it is equally likely to stop in any sector.
(a) Complete the table below which shows the probability of the arrow stopping at each number.
Number
1
Probability
2
3
0.2
4
0.3
[1]
(b) The arrow is spun once.
Find
(i) the most likely number,
Answer(b)(i)
[1]
Answer(b)(ii)
[1]
(ii) the probability of a number less than 4.
Mr.Yasser Elsayed
002 012 013 222 97
263
(c) The arrow is spun twice.
Find the probability that
(i) both numbers are 2,
Answer(c)(i)
[1]
Answer(c)(ii)
[2]
Answer(c)(iii)
[3]
(ii) the first number is 3 and the second number is 4,
(iii) the two numbers add up to 4.
(d) The arrow is spun several times until it stops at a number 4.
Find the probability that this happens on the third spin.
Answer(d)
Mr.Yasser Elsayed
002 012 013 222 97
[2]
264
4) November 2010 V2
9
A bag contains 7 red sweets and 4 green sweets.
Aimee takes out a sweet at random and eats it.
She then takes out a second sweet at random and eats it.
(a) Complete the tree diagram.
First sweet
Second sweet
6
10
7
11
red
..........
..........
..........
red
green
red
green
..........
green
[3]
(b) Calculate the probability that Aimee has taken
(i) two red sweets,
Answer(b)(i)
[2]
Answer(b)(ii)
[3]
(ii) one sweet of each colour.
Mr.Yasser Elsayed
002 012 013 222 97
265
(c) Aimee takes a third sweet at random.
Calculate the probability that she has taken
(i) three red sweets,
Answer(c)(i)
[2]
Answer(c)(ii)
[3]
(ii) at least one red sweet.
Mr.Yasser Elsayed
002 012 013 222 97
266
5) November 2010 V3
6
Sacha either walks or cycles to school.
On any day, the probability that he walks to school is
3
.
5
(a) (i) A school term has 55 days.
Work out the expected number of days Sacha walks to school.
Answer(a)(i)
[1]
(ii) Calculate the probability that Sacha walks to school on the first 5 days of the term.
Answer(a)(ii)
[2]
(b) When Sacha walks to school, the probability that he is late is
When he cycles to school, the probability that he is late is
1
.
8
1
.
4
(i) Complete the tree diagram by writing the probabilities in the four spaces provided.
1
4
3
5
walks
..........
..........
..........
late
not late
late
cycles
Mr.Yasser Elsayed
002 012 013 222 97
..........
not late
[3]
267
(ii) Calculate the probability that Sacha cycles to school and is late.
Answer(b)(ii)
[2]
(iii) Calculate the probability that Sacha is late to school.
Answer(b)(iii)
Mr.Yasser Elsayed
002 012 013 222 97
[2]
268
6) June 2011 V1
2
In this question give all your answers as fractions.
The probability that it rains on Monday is
3
5
.
If it rains on Monday, the probability that it rains on Tuesday is
4
7
.
If it does not rain on Monday, the probability that it rains on Tuesday is
5
7
.
(a) Complete the tree diagram.
Monday
Tuesday
Rain
Rain
No rain
Rain
No rain
No rain
[3]
(b) Find the probability that it rains
(i) on both days,
Answer(b)(i)
[2]
Answer(b)(ii)
[2]
Answer(b)(iii)
[2]
(ii) on Monday but not on Tuesday,
(iii) on only one of the two days.
(c) If it does not rain on Monday and it does not rain on Tuesday, the probability that it does not
1
.
rain on Wednesday is
4
Calculate the probability that it rains on at least one of the three days.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(c)
[3]
269
7) June 2011 V3
7
Katrina puts some plants in her garden.
The probability that a plant will produce a flower is
7
.
10
If there is a flower, it can only be red, yellow or orange.
2
1
When there is a flower, the probability it is red is
and the probability it is yellow is .
3
4
(a) Draw a tree diagram to show all this information.
Label the diagram and write the probabilities on each branch.
Answer(a)
[5]
(b) A plant is chosen at random.
Find the probability that it will not produce a yellow flower.
Answer(b)
[3]
(c) If Katrina puts 120 plants in her garden, how many orange flowers would she expect?
Mr.Yasser Elsayed
002 012 013 222 97
Answer(c)
[2]
270
8) November 2011 V2
9
(a) Emile lost 2 blue buttons from his shirt.
A bag of spare buttons contains 6 white buttons and 2 blue buttons.
Emile takes 3 buttons out of the bag at random without replacement.
Calculate the probability that
(i) all 3 buttons are white,
Answer(a)(i)
[3]
Answer(a)(ii)
[3]
(ii) exactly one of the 3 buttons is blue.
Mr.Yasser Elsayed
002 012 013 222 97
271
(b) There are 25 buttons in another bag.
This bag contains x blue buttons.
Two buttons are taken at random without replacement.
7
.
The probability that they are both blue is
100
(i) Show that x2 O x O 42 = 0.
Answer (b)(i)
[4]
(ii) Factorise x2 O x O 42.
Answer(b)(ii)
[2]
(iii) Solve the equation x2 O x O 42 = 0.
or x =
Answer(b)(iii) x =
[1]
(iv) Write down the number of buttons in the bag which are not blue.
Answer(b)(iv)
Mr.Yasser Elsayed
002 012 013 222 97
[1]
272
9) November 2011 V3
9
Set A
S U M S
Set B
M I
N U S
The diagram shows two sets of cards.
(a) One card is chosen at random from Set A and replaced.
(i) Write down the probability that the card chosen shows the letter M.
Answer(a)(i)
[1]
(ii) If this is carried out 100 times, write down the expected number of times the card chosen
shows the letter M.
Answer(a)(ii)
[1]
(b) Two cards are chosen at random, without replacement, from Set A.
Find the probability that both cards show the letter S.
Answer(b)
[2]
(c) One card is chosen at random from Set A and one card is chosen at random from Set B.
Find the probability that exactly one of the two cards shows the letter U.
Answer(c)
[3]
(d) A card is chosen at random, without replacement, from Set B until the letter shown is either
I or U.
Find the probability that this does not happen until the 4th card is chosen.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(d)
[2]
273
10) June 2012 V1
= {1, 2, 3, 4, 5, 6, 7, 8, 9}
8
E = { x : x is an even number}
F = {2, 5, 7}
2
G = {x : x
O
13x + 36 = 0}
(a) List the elements of set E.
Answer(a) E = {
} [1]
Answer(b) n(F ) =
[1]
Answer(c)(i)
[2]
(b) Write down n( F ).
2
(c) (i) Factorise x
O
13x + 36.
(ii) Using your answer to part (c)(i), solve x 2
O
13x + 36 = 0 to find the two elements of G.
Answer(c)(ii) x =
(d) Write all the elements of
[1]
or x =
in their correct place in the Venn diagram.
E
F
G
[2]
(e) Use set notation to complete the following statements.
(i) F ∩ G =
[1]
(ii) 7
[1]
E
E
F) = 6
(iii) n(Elsayed
Mr.Yasser
002 012 013 222 97
[1]
274
11) June 2012 V2
8
In all parts of this question give your answer as a fraction in its lowest terms.
(a) (i) The probability that it will rain today is
1
.
3
What is the probability that it will not rain today?
Answer(a)(i)
(ii) If it rains today, the probability that it will rain tomorrow is
[1]
2
5
.
If it does not rain today, the probability that it will rain tomorrow is
1
6
.
Complete the tree diagram.
Today
Tomorrow
Rain
Rain
No rain
Rain
No rain
No rain
[2]
(b) Find the probability that it will rain on at least one of these two days.
Answer(b)
[3]
(c) Find the probability that it will rain on only one of these two days.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(c)
[3]
275
12) June 2012 V3
6
H
C
30
150
20
40
= {240 passengers who arrive on a flight in Cyprus}
H = {passengers who are on holiday}
C = {passengers who hire a car}
(a) Write down the number of passengers who
(i) are on holiday,
Answer(a)(i)
[1]
Answer(a)(ii)
[1]
Answer(b)
[1]
Answer(c)(i)
[1]
Answer(c)(ii)
[1]
(ii) hire a car but are not on holiday.
(b) Find the value of n(H ∪ CV ).
(c) One of the 240 passengers is chosen at random.
Write down the probability that this passenger
(i) hires a car,
(ii) is on holiday and hires a car.
Mr.Yasser Elsayed
002 012 013 222 97
276
(d) Give your answers to this part correct to 4 decimal places.
Two of the 240 passengers are chosen at random.
Find the probability that
(i) they are both on holiday,
Answer(d)(i)
[2]
(ii) exactly one of the two passengers is on holiday.
Answer(d)(ii)
[3]
(e) Give your answer to this part correct to 4 decimal places.
Two passengers are chosen at random from those on holiday.
Find the probability that they both hire a car.
Answer(e)
Mr.Yasser Elsayed
002 012 013 222 97
[3]
277
13) November 2012 V1
3
90 students are asked which school clubs they attend.
D = {students who attend drama club}
M = {students who attend music club}
S = { students who attend sports club}
39 students attend music club.
26 students attend exactly two clubs.
35 students attend drama club.
D
M
10
........
13
5
........
........
........
23
S
(a) Write the four missing values in the Venn diagram.
[4]
(b) How many students attend
(i) all three clubs,
Answer(b)(i)
[1]
Answer(b)(ii)
[1]
Answer(c)(i)
[1]
Answer(c)(ii)
[1]
(ii) one club only?
(c) Find
(i) n(D ∩ M ),
(ii) n((D ∩ M ) ∩ S' ).
Mr.Yasser Elsayed
002 012 013 222 97
278
(d) One of the 90 students is chosen at random.
Find the probability that the student
(i) only attends music club,
Answer(d)(i)
[1]
Answer(d)(ii)
[1]
(ii) attends both music and drama clubs.
(e) Two of the 90 students are chosen at random without replacement.
Find the probability that
(i) they both attend all three clubs,
Answer(e)(i)
[2]
(ii) one of them attends sports club only and the other attends music club only.
Answer(e)(ii)
Mr.Yasser Elsayed
002 012 013 222 97
[3]
279
14) November 2012 V2
9
(a)
= {25 students in a class}
F = {students who study French}
S = {students who study Spanish}
16 students study French and 18 students study Spanish.
2 students study neither of these.
(i) Complete the Venn diagram to show this information.
F
.....
S
.....
.....
.....
[2]
(ii) Find n(F ').
Answer(a)(ii)
[1]
Answer(a)(iii)
[1]
(iii) Find n(F ∩ S)'.
(iv) One student is chosen at random.
Find the probability that this student studies both French and Spanish.
Answer(a)(iv)
[1]
(v) Two students are chosen at random without replacement.
Find the probability that they both study only Spanish.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(a)(v)
[2]
280
(b) In another class the students all study at least one language from French, German and Spanish.
No student studies all three languages.
The set of students who study German is a proper subset of the set of students who study
French.
4 students study both French and German.
12 students study Spanish but not French.
9 students study French but not Spanish.
A total of 16 students study French.
(i) Draw a Venn diagram to represent this information.
[4]
(ii) Find the total number of students in this class.
Answer(b)(ii)
Mr.Yasser Elsayed
002 012 013 222 97
[1]
281
15) November 2012 V3
7
(a)
1
2
2
3
4
Two discs are chosen at random without replacement from the five discs shown in the diagram.
(i) Find the probability that both discs are numbered 2 .
Answer(a)(i)
[2]
(ii) Find the probability that the numbers on the two discs have a total of 5 .
Answer(a)(ii)
[3]
(iii) Find the probability that the numbers on the two discs do not have a total of 5.
Answer(a)(iii)
[1]
(b) A group of international students take part in a survey on the nationality of their parents.
E = {students with an English parent}
F = {students with a French parent}
E
F
n( ) = 50, n(E) = 15, n(F ) = 9 and n(E ∪ F )' = 33 .
(i) Find n(E ∩ F ).
Answer(b)(i)
[1]
Answer(b)(ii)
[1]
(ii) Find n(E' ∪ F ).
(iii) A student is chosen at random.
Find the probability that this student has an English parent and a French parent.
Answer(b)(iii)
[1]
(iv) A student who has a French parent is chosen at random.
Find the probability that this student also has an English parent.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(b)(iv)
[1]
282
16) June 2013 V2
8
(a)
In this question, give all your answers as fractions.
5
When Ivan goes to school in winter, the probability that he wears a hat is .
8
2
If he wears a hat, the probability that he wears a scarf is .
3
1
If he does not wear a hat, the probability that he wears a scarf is .
6
(a) Complete the tree diagram.
........
........
Scarf
Hat
No scarf
........
........
........
Scarf
No hat
........
No scarf
[3]
(b) Find the probability that Ivan
(i) does not wear a hat and does not wear a scarf,
Answer(b)(i) ............................................... [2]
(ii) wears a hat but does not wear a scarf,
Answer(b)(ii) ............................................... [2]
(iii) wears a hat or a scarf but not both.
Answer(b)(iii) ............................................... [2]
7
.
10
Calculate the probability that Ivan does not wear all three of hat, scarf and gloves.
(c) If Ivan wears a hat and a scarf, the probability that he wears gloves is
Mr.Yasser Elsayed
002 012 013 222 97
Answer(c) ............................................... [3]
283
17) June 2013 V3
6
In a box there are 7 red cards and 3 blue cards.
A card is drawn at random from the box and is not replaced.
A second card is then drawn at random from the box.
(a) Complete this tree diagram.
First card
Second card
........
7
10
Red
Red
Blue
........
........
........
Red
Blue
........
Blue
[3]
(b) Work out the probability that the two cards are of different colours.
Give your answer as a fraction.
Answer(b) ............................................... [3]
Mr.Yasser Elsayed
002 012 013 222 97
284
18) November 2013 V1
6
E
N
L
A
R
G
E
M
E
N
T
Prettie picks a card at random from the 11 cards above and does not replace it.
She then picks a second card at random and does not replace it.
(a) Find the probability that she picks
(i) the letter L and then the letter G,
Answer(a)(i) ............................................... [2]
(ii) the letter E twice,
Answer(a)(ii) ............................................... [2]
(iii) two letters that are the same.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(a)(iii) ............................................... [2]
285
(b) Prettie now picks a third card at random.
Find the probability that the three letters
(i) are all the same,
Answer(b)(i) ............................................... [2]
(ii) do not include a letter E,
Answer(b)(ii) ............................................... [2]
(iii) include exactly two letters that are the same.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(b)(iii) ............................................... [5]
286
19) November 2013 V3
5
3
(b) The probability that Chaminda uses the internet on any day is 5 .
3
The probability that Niluka uses the internet on any day is 4 .
(i) Complete the tree diagram.
Chaminda
Niluka
3
4
3
5
Uses the
internet
........
........
........
Does not
use the
internet
........
Uses the
internet
Does not
use the
internet
Uses the
internet
Does not
use the
internet
[2]
(ii) Calculate the probability, that on any day, at least one of the two students uses the internet.
Answer(b)(ii) ............................................... [3]
(iii) Calculate the probability that Chaminda uses the internet on three consecutive days.
Answer(b)(iii) ............................................... [2]
Mr.Yasser Elsayed
002 012 013 222 97
287
20) June 2014 V1
4
T
B
11
9
x
6–x
4
P
In the Venn diagram,
= {children in a nursery}
B = {children who received a book for their birthday}
T = {children who received a toy for their birthday}
P = {children who received a puzzle for their birthday}
x children received a book and a toy and a puzzle.
6 children received a toy and a puzzle.
(a) 4 children received a book and a toy.
5 children received a book and a puzzle.
7 children received a puzzle but not a book and not a toy.
Complete the Venn diagram above.
[3]
(b) There are 40 children in the nursery.
Using the Venn diagram, write down and solve an equation in x.
Answer(b)
Mr.Yasser Elsayed
002 012 013 222 97
[3]
288
(c) Work out
(i) the probability that a child, chosen at random, received a book but not a toy and not a puzzle,
Answer(c)(i) ................................................ [1]
(ii) the number of children who received a book and a puzzle but not a toy,
Answer(c)(ii) ................................................ [1]
(iii) n(B),
Answer(c)(iii) ................................................ [1]
(iv) n(B ∪ P),
Answer(c)(iv) ................................................ [1]
(v) n(B ∪ T ∪ P)'.
Answer(c)(v) ................................................ [1]
(d)
T
B
P
Shade the region B ∩ (T ∪ P)'.
Mr.Yasser Elsayed
002 012 013 222 97
[1]
289
21) June 2014 V1
6
(a) A square spinner is biased.
The probabilities of obtaining the scores 1, 2, 3 and 4 when it is spun are given in the table.
Score
Probability
1
2
3
4
0.1
0.2
0.4
0.3
(i) Work out the probability that on one spin the score is 2 or 3.
Answer(a)(i) ................................................ [2]
(ii) In 5000 spins, how many times would you expect to score 4 with this spinner?
Answer(a)(ii) ................................................ [1]
(iii) Work out the probability of scoring 1 on the first spin and 4 on the second spin.
Answer(a)(iii) ................................................ [2]
(b) In a bag there are 7 red discs and 5 blue discs.
From the bag a disc is chosen at random and not replaced.
A second disc is then chosen at random.
Work out the probability that at least one of the discs is red.
Give your answer as a fraction.
Mr.Yasser Elsayed
002 012 013 222 97
290
Answer(b) ................................................ [3]
22) June 2014 V2
9
1
If the weather is fine the probability that Carlos is late arriving at school is 10 .
1
If the weather is not fine the probability that he is late arriving at school is 3 .
3
The probability that the weather is fine on any day is 4 .
(a) Complete the tree diagram to show this information.
Weather
Arriving at school
1
10
3
4
Late
Fine
Not late
........
........
........
Late
Not fine
........
Not late
[3]
(b) In a school term of 60 days, find the number of days the weather is expected to be fine.
Answer(b) ................................................ [1]
(c) Find the probability that the weather is fine and Carlos is late arriving at school.
Answer(c) ................................................ [2]
(d) Find the probability that Carlos is not late arriving at school.
Answer(d) ................................................ [3]
(e) Find the probability that the weather is not fine on at least one day in a school week of 5 days.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(e) ................................................ [2]
291
23) June 2014 V3
6
In this question, give all your answers as fractions.
N
A
T
I
O
N
The letters of the word NATION are printed on 6 cards.
(a) A card is chosen at random.
Write down the probability that
(i) it has the letter T printed on it,
Answer(a)(i) ................................................ [1]
(ii) it does not have the letter N printed on it,
Answer(a)(ii) ................................................ [1]
(iii) the letter printed on it has no lines of symmetry.
Answer(a)(iii) ................................................ [1]
(b) Lara chooses a card at random, replaces it, then chooses a card again.
Calculate the probability that only one of the cards she chooses has the letter N printed on it.
Answer(b) ................................................ [3]
(c) Jacob chooses a card at random and does not replace it.
He continues until he chooses a card with the letter N printed on it.
Find the probability that this happens when he chooses the 4th card.
Mr.Yasser Elsayed
002 012 013 222 97
292
Answer(c) ................................................ [3]
24) November 2014 V2
10 Kenwyn plays a board game.
Two cubes (dice) each have faces numbered 1, 2, 3, 4, 5 and 6.
In the game, a throw is rolling the two fair 6-sided dice and then adding the numbers on their top faces.
This total is the number of spaces to move on the board.
For example, if the numbers are 4 and 3, he moves 7 spaces.
(a) Giving each of your answers as a fraction in its simplest form, find the probability that he moves
(i) two spaces with his next throw,
Answer(a)(i) ................................................ [2]
(ii) ten spaces with his next throw.
Answer(a)(ii) ................................................ [3]
(b) What is the most likely number of spaces that Kenwyn will move with his next throw?
Explain your answer.
Answer(b) .................... because .........................................................................................................
............................................................................................................................................................. [2]
Mr.Yasser Elsayed
002 012 013 222 97
293
(c)
95
96
97
98
99
100
Go back
3 spaces
WIN
To win the game he must move exactly to the 100th space.
Kenwyn is on the 97th space.
If his next throw takes him to 99, he has to move back to 96.
If his next throw takes him over 100, he stays on 97.
Find the probability that he reaches 100 in either of his next two throws.
Answer(c) ................................................ [5]
Mr.Yasser Elsayed
002 012 013 222 97
294
25) November 2014 V3
4
Yeung and Ariven compete in a triathlon race.
3
The probability that Yeung finishes this race is 5 .
2
The probability that Ariven finishes this race is 3 .
(a) (i) Which of them is more likely to finish this race?
Give a reason for your answer.
Answer(a)(i) ...................................................... because ..........................................................
..................................................................................................................................................... [1]
(ii) Find the probability that they both finish this race.
Answer(a)(ii) ................................................ [2]
(iii) Find the probability that only one of them finishes this race.
Answer(a)(iii) ................................................ [3]
Mr.Yasser Elsayed
002 012 013 222 97
295
(b) After the first race, Yeung competes in two further triathlon races.
(i) Complete the tree diagram.
First race
Second race
6
7
Third race
7
10
Finishes
........
Does not
finish
7
10
Finishes
........
Does not
finish
7
10
Finishes
........
Does not
finish
7
10
Finishes
........
Does not
finish
Finishes
Finishes
3
5
........
6
7
........
Does not
finish
Finishes
Does not
finish
........
Does not
finish
[3]
(ii) Calculate the probability that Yeung finishes all three of his races.
Answer(b)(ii) ................................................ [2]
(iii) Calculate the probability that Yeung finishes at least one of his races.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(b)(iii) ................................................ [3]
296
26) June 2015 V1
4
30 students were asked if they had a bicycle (B), a mobile phone (M ) and a computer (C).
The results are shown in the Venn diagram.
B
M
2
x
4
1
7
6
3
2
C
(a) Work out the value of x .
Answer(a) x = ................................................. [1]
(b) Use set notation to describe the shaded region in the Venn diagram.
Answer(b) ................................................. [1]
(c) Find n(C
(M
B)).
Answer(c) ................................................. [1]
(d) A student is chosen at random.
(i) Write down the probability that the student is a member of the set M  .
Answer(d)(i) ................................................. [1]
(ii) Write down the probability that the student has a bicycle.
Answer(d)(ii) ................................................. [1]
(e) Two students are chosen at random from the students who have computers.
Find the probability that each of these students has a mobile phone but no bicycle.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(e) ................................................ [3]
297
27) June 2015 V2
11 Gareth has 8 sweets in a bag.
4 sweets are orange flavoured, 3 are lemon flavoured and 1 is strawberry flavoured.
(a) He chooses two of the sweets at random.
Find the probability that the two sweets have different flavours.
Answer(a) ................................................ [4]
(b) Gareth now chooses a third sweet.
Find the probability that none of the three sweets is lemon flavoured.
Answer(b) ................................................ [2]
Mr.Yasser Elsayed
002 012 013 222 97
298
28) June 2015 V3
5
A
A
A
A
B
B
C
(a) One of these 7 cards is chosen at random.
Write down the probability that the card
(i)
shows the letter A,
(ii)
shows the letter A or B,
(iii)
does not show the letter B.
Answer(a)(i) ................................................. [1]
Answer(a)(ii) ................................................. [1]
Answer(a)(iii) ................................................. [1]
(b) Two of the cards are chosen at random, without replacement.
Find the probability that
(i)
both show the letter A,
Answer(b)(i) ................................................. [2]
(ii) the two letters are different.
Answer(b)(ii) ................................................. [3]
(c) Three of the cards are chosen at random, without replacement.
Find the probability that the cards do not show the letter C.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(c) ................................................. [2]
299
29) March 2015 V2
2
(a) x is an integer.
= {x: 1  x  10}
A
A = {x: x is a factor of 12}
B
B = {x: x is an odd number}
C = {x: x is a prime number}
(i) Complete the Venn diagram to show this information.
C
A
B
C
[3]
(ii) Use set notation to complete each statement.
6 ...................... A
B
A
A
C = ......................
A' = ......................
(iii) Find n(B).
Mr.Yasser Elsayed
002 012 013 222 97
[3]
Answer(a)(iii) ................................................ [1]
300
(b)
X
Y
q
p
r
s
t
u
w
v
Z
(i) Use set notation to complete the statement.
{u, v} ...................... Z
(ii) Shade X
(Z
Y )'.
Mr.Yasser Elsayed
002 012 013 222 97
[1]
[1]
301
30) March 2015 V2
6
In this question write any probability as a fraction.
Navpreet has 15 cards with a shape drawn on each card.
5 cards have a square, 6 cards have a triangle and 4 cards have a circle drawn on them.
(a) Navpreet selects a card at random.
Write down the probability that the card has a circle drawn on it.
Answer(a) ................................................ [1]
(b) Navpreet selects a card at random and replaces it.
She does this 300 times.
Calculate the number of times she expects to select a card with a circle drawn on it.
Answer(b) ................................................ [1]
(c) Navpreet selects a card at random, replaces it and then selects another card.
Calculate the probability that
(i) one card has a square drawn on it and the other has a circle drawn on it,
Answer(c)(i) ................................................ [3]
(ii) neither card has a circle drawn on it.
Answer(c)(ii) ................................................ [3]
(d) Navpreet selects two cards at random, without replacement.
Calculate the probability that
(i) only one card has a triangle drawn on it,
Answer(d)(i) ................................................ [3]
(ii) the two cards have different shapes drawn on them.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(d)(ii) ................................................ [4]
302
31) March 2016 V2
3
(a) Davinder asked some people if they ate mangoes, pineapples or bananas last week.
M = { people who ate mangoes }
P = { people who ate pineapples }
B = { people who ate bananas }
The Venn diagram shows some of the information.
M
P
5
7
......
4
......
1
......
12
B
19 people said they ate mangoes.
6 people said they ate only pineapples.
18 people said they ate exactly two of the three types of fruit.
(i) Write the three missing values in the Venn diagram.
[3]
(ii) Find the total number of people Davinder asked.
.................................................. [1]
(iii) Find n(M
P).
.................................................. [1]
(iv) One person is chosen at random from the people who ate mangoes.
Write down the probability that this person also ate bananas.
Mr.Yasser Elsayed
002 012 013 222 97
.................................................. [2]
303
32) June 2016 V2
5
Kiah plays a game.
The game involves throwing a coin onto a circular board.
Points are scored for where the coin lands on the board.
5
10
20
If the coin lands on part of a line or misses the board then 0 points are scored.
The table shows the probabilities of Kiah scoring points on the board with one throw.
Points scored
20
10
5
0
Probability
x
0.2
0.3
0.45
(a) Find the value of x.
x = ................................................. [2]
(b) Kiah throws a coin fifty times.
Work out the expected number of times she scores 5 points.
.................................................. [1]
(c) Kiah throws a coin two times.
Calculate the probability that
(i)
she scores either 5 or 0 with her first throw,
.................................................. [2]
(ii)
she scores 0 with her first throw and 5 with her second throw,
Mr.Yasser Elsayed
002 012 013 222 97
.................................................. [2]
304
(iii)
she scores a total of 15 points with her two throws.
.................................................. [3]
(d) Kiah throws a coin three times.
Calculate the probability that she scores a total of 10 points with her three throws.
.................................................. [5]
Mr.Yasser Elsayed
002 012 013 222 97
305
33) November 2017 V1
9
(a) A bag contains red beads and green beads.
There are 80 beads altogether.
The probability that a bead chosen at random is green is 0.35 .
(i)
Find the number of red beads in the bag.
................................................. [2]
(ii)
Marcos chooses a bead at random and replaces it in the bag.
He does this 240 times.
Find the number of times he would expect to choose a green bead.
................................................. [1]
(b) A different bag contains 2 blue marbles, 3 yellow marbles and 4 white marbles.
Huma chooses a marble at random, notes the colour, then replaces it in the bag.
She does this three times.
Find the probability that
(i)
all three marbles are yellow,
................................................. [2]
(ii)
all three marbles are different colours.
................................................. [3]
Mr.Yasser Elsayed
002 012 013 222 97
306
(c) Another bag contains 2 green counters and 3 pink counters.
Teresa chooses three counters at random without replacement.
Find the probability that she chooses more pink counters than green counters.
................................................. [4]
Mr.Yasser Elsayed
002 012 013 222 97
307
34) June 2018 V1
9
The probability that it will rain tomorrow is
5
.
8
If it rains, the probability that Rafael walks to school is
1
.
6
If it does not rain, the probability that Rafael walks to school is
7
.
10
(a) Complete the tree diagram.
Walks
........
Rains
........
........
Does not walk
Walks
........
........
Does not rain
........
Does not walk
[3]
(b) Calculate the probability that it will rain tomorrow and Rafael walks to school.
................................................ [2]
(c) Calculate the probability that Rafael does not walk to school.
................................................ [3]
Mr.Yasser Elsayed
002 012 013 222 97
308
10
(a) In 2017, the membership fee for a sports club was $79.50 .
This was an increase of 6% on the fee in 2016.
Calculate the fee in 2016.
$ ............................................... [3]
(b) On one day, the number of members using the exercise machines was 40, correct to the nearest 10.
Each member used a machine for 30 minutes, correct to the nearest 5 minutes.
Calculate the lower bound for the number of minutes the exercise machines were used on this day.
......................................... min [2]
(c) On another day, the number of members using the exercise machines (E), the swimming pool (S) and
the tennis courts (T ) is shown on the Venn diagram.
E
20
7
5
4
33
8
16
(i)
S
T
Find the number of members using only the tennis courts.
................................................ [1]
(ii)
Find the number of members using the swimming pool.
................................................ [1]
(iii)
A member using the swimming pool is chosen at random.
Find the probability that this member also uses the tennis courts and the exercise machines.
................................................ [2]
(iv)
Find n ^T + ^E , Shh .
Mr.Yasser Elsayed
002 012 013 222 97
................................................ [1]
309
35) June 2019 V2
3
The probability that Andrei cycles to school is r.
(a) Write down, in terms of r, the probability that Andrei does not cycle to school.
............................................... [1]
(b) The probability that Benoit does not cycle to school is 1.3 - r.
The probability that both Andrei and Benoit do not cycle to school is 0.4 .
(i)
Complete the equation in terms of r.
(.........................) # (.........................) = 0.4
(ii)
[1]
Show that this equation simplifies to 10r 2 - 23r + 9 = 0 .
[3]
(iii)
Solve by factorisation 10r 2 - 23r + 9 = 0 .
r = ................... or r = ................... [3]
(iv)
Find the probability that Benoit does not cycle to school.
Mr.Yasser Elsayed
002 012 013 222 97
............................................... [1]
310
36) June 2019 V3
8
(a) Angelo has a bag containing 3 white counters and x black counters.
He takes two counters at random from the bag, without replacement.
(i)
Complete the following statement.
The probability that Angelo takes two black counters is
x #
x+3
(ii)
.
[2]
The probability that Angelo takes two black counters is 7 .
15
(a) Show that 4x 2
-
25x
-
21 = 0.
[4]
(b) Solve by factorisation.
4x 2
-
25x
-
21 = 0
x = .................... or x = ................. [3]
(c) Write down the number of black counters in the bag.
............................................... [1]
Mr.Yasser Elsayed
002 012 013 222 97
311
(b) Esme has a bag with 5 green counters and 4 red counters.
She takes three counters at random from the bag without replacement.
Work out the probability that the three counters are all the same colour.
............................................... [4]
Mr.Yasser Elsayed
002 012 013 222 97
312
37) June 2020 V2
7 Tanya plants some seeds.
The probability that a seed will produce flowers is 0.8 .
When a seed produces flowers, the probability that the flowers are red is 0.6 and the probability that the
flowers are yellow is 0.3 .
(a) Tanya has a seed that produces flowers.
Find the probability that the flowers are not red and not yellow.
................................................. [1]
(b) (i) Complete the tree diagram.
Produces
flowers
Colour
Red
...............
0.8
Yes
...............
...............
...............
Yellow
Other
colours
No
[2]
(ii) Find the probability that a seed chosen at random produces red flowers.
................................................. [2]
Mr.Yasser Elsayed
002 012 013 222 97
313
15
(iii) Tanya chooses a seed at random.
Find the probability that this seed does not produce red flowers and does not produce yellow
flowers.
................................................. [3]
(c) Two of the seeds are chosen at random.
Find the probability that one produces flowers and one does not produce flowers.
................................................. [3]
Mr.Yasser Elsayed
002 012 013 222 97
314
38) November 2020 V1
9 (a) There are 32 students in a class.
5 do not study any languages.
15 study German (G).
18 study Spanish (S).
G
S
(i) Complete the Venn diagram to show this information.
[2]
(ii) A student is chosen at random.
Find the probability that the student studies Spanish but not German.
................................................. [1]
(iii) A student who studies German is chosen at random.
Find the probability that this student also studies Spanish.
................................................. [1]
Mr.Yasser Elsayed
002 012 013 222 97
315
(b) A bag contains 54 red marbles and some blue marbles.
36% of the marbles in the bag are red.
Find the number of blue marbles in the bag.
................................................. [2]
(c) Another bag contains 15 red beads and 10 yellow beads.
Ariana picks a bead at random, records its colour and replaces it in the bag.
She then picks another bead at random.
(i) Find the probability that she picks two red beads.
................................................. [2]
(ii) Find the probability that she does not pick two red beads.
................................................. [1]
(d) A box contains 15 red pencils, 8 yellow pencils and 2 green pencils.
Two pencils are picked at random without replacement.
Find the probability that at least one pencil is red.
................................................. [3]
Mr.Yasser Elsayed
002 012 013 222 97
316
Statistics
Mr.Yasser Elsayed
002 012 013 222 97
317
1) June 2010 V1
2
40 students are asked about the number of people in their families.
The table shows the results.
Number of people in family
2
3
4
5
6
7
Frequency
1
1
17
12
6
3
(a) Find
(i) the mode,
Answer(a)(i)
[1]
Answer(a)(ii)
[1]
Answer(a)(iii)
[3]
(ii) the median,
(iii) the mean.
(b) Another n students are asked about the number of people in their families.
The mean for these n students is 3.
Find, in terms of n, an expression for the mean number for all (40 + n) students.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(b)
[2]
318
2) June 2010 V1
6
The masses of 60 potatoes are measured.
The table shows the results.
Mass (m grams)
10 I m Y 20
20 I m Y 40
40 I m Y 50
Frequency
10
30
20
(a) Calculate an estimate of the mean.
g
Answer(a)
[4]
(b) On the grid, draw an accurate histogram to show the information in the table.
2
Frequency
density
1
0
m
10
Mr.Yasser Elsayed
002 012 013 222 97
20
30
40
50
Mass (grams)
[3]
319
3) June 2010 V2
7
200 students were asked how many hours they exercise each week.
The table shows the results.
Time (t hours)
0ItY5
Number of
students
12
5ItY10 10ItY15 15ItY20 20ItY25 25ItY30 30ItY35 35ItY40
15
23
30
40
35
25
20
(a) Calculate an estimate of the mean.
h
Answer(a)
[4]
(b) Use the information in the table above to complete the cumulative frequency table.
Time (t hours)
Cumulative frequency
t Y=5 t Y=10 t Y=15 t Y=20 t Y=25 t Y=30 t Y=35 t Y=40
12
Mr.Yasser Elsayed
002 012 013 222 97
27
50
80
120
200
[1]
320
200
180
160
Cumulative frequency
140
120
100
80
60
40
20
0
5
10
15
20
Time (t hours)
25
30
35
40
(c) On the grid, draw a cumulative frequency diagram to show the information in the table in
part (b).
[4]
(d) On your cumulative frequency diagram show how to find the lower quartile.
[1]
(e) Use your cumulative frequency diagram to find
(i) the median,
Answer(e)(i)
[1]
Answer(e)(ii)
[1]
Answer(e)(iii)
[1]
(ii) the inter-quartile range,
(iii) the 64th percentile,
(iv) the number of students who exercise for more than 17 hours.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(e)(iv)
[2]
321
4) June 2010 V3
7
(a) The table shows how many books were borrowed by the 126 members of a library group in a
month.
Number of books
11
12
13
14
15
16
Number of members
(frequency)
35
28
22
18
14
9
Find the mode, the median and the mean for the number of books borrowed.
Answer(a) mode =
median =
[6]
mean =
(b) The 126 members record the number of hours they read in one week.
The histogram shows the results.
Frequency
density
15
10
5
0
Mr.Yasser
Elsayed
5
8
002 012 013 222 97
h
10
12
16
20
322
(i) Use the information from the histogram to complete the frequency table.
Number of
hours (h)
0IhY5
5IhY8
Frequency
8 I h Y 10
10 I h Y 12
12 I h Y 16
16 I h Y 20
20
24
10
[3]
(ii) Use the information in this table to calculate an estimate of the mean number of hours.
Show your working.
Answer(b)(ii)
Mr.Yasser Elsayed
002 012 013 222 97
hours
[4]
323
5) November 2010 V1
5
The cumulative frequency table shows the distribution of heights, h centimetres, of 200 students.
Y130 Y140 Y150 Y160 Y165 Y170 Y180 Y190
Height (h cm)
Cumulative frequency
0
10
50
95
115
145
180
200
(a) Draw a cumulative frequency diagram to show the information in the table.
200
160
120
Cumulative
frequency
80
40
0
130
140
150
160
170
180
190
Height (h cm)
[4]
(b) Use your diagram to find
(i) the median,
Answer(b)(i)
cm
[1]
Answer(b)(ii)
cm
[1]
Answer(b)(iii)
cm
[1]
(ii) the upper quartile,
(iii) the interquartile range.
(c) (i) One of the 200 students is chosen at random.
Mr.Yasser Elsayed
002 012 013 222 97
Use the table to find the probability that the height of this student is greater than 170 cm.
Give your answer as a fraction.
Answer(c)(i)
[1]
324
(ii) One of the 200 students is chosen at random and then a second student is chosen at random
from the remaining students.
Calculate the probability that one has a height greater than 170 cm and the other has a
height of 140 cm or less.
Give your answer as a fraction.
Answer(c)(ii)
[3]
(d) (i) Complete this frequency table which shows the distribution of the heights of the 200
students.
Height (h cm)
130IhY140 140IhY150 150IhY160 160IhY165 165IhY170 170IhY180 180Ih
Frequency
10
40
45
20
[2]
(ii) Complete this histogram to show the distribution of the heights of the 200 students.
6
5
4
Frequency
3
density
2
1
0
140
150
Mr.Yasser130 Elsayed
002 012 013 222 97
160
170
180
190
Height (h cm)
325
[3]
6) November 2010 V2
3
80 boys each had their mass, m kilograms, recorded.
The cumulative frequency diagram shows the results.
80
60
Cumulative
40
frequency
20
m
0
30
40
50
60
70
80
90
Mass (kg)
(a) Find
(i) the median,
Answer(a)(i)
kg
[1]
Answer(a)(ii)
kg
[1]
Answer(a)(iii)
kg
[1]
(ii) the lower quartile,
(iii) the interquartile range.
(b) How many boys had a mass greater than 60kg?
Mr.Yasser Elsayed
002 012 013 222 97
Answer(b)
[2]
326
(c) (i) Use the cumulative frequency graph to complete this frequency table.
Mass, m
Frequency
30 I m Y 40
8
40 I m Y 50
50 I m Y 60
14
60 I m Y 70
22
70 I m Y 80
80 I m Y 90
10
[2]
(ii) Calculate an estimate of the mean mass.
Answer(c)(ii)
Mr.Yasser Elsayed
002 012 013 222 97
kg
[4]
327
7) November 2010 V3
10 (a) For a set of six integers, the mode is 8, the median is 9 and the mean is 10.
The smallest integer is greater than 6 and the largest integer is 16.
Find the two possible sets of six integers.
Answer(a) First set
Second set
,
,
,
,
,
,
,
,
,
,
[5]
(b) One day Ahmed sells 160 oranges.
He records the mass of each orange.
The results are shown in the table.
Mass (m grams) 50 < m Y 80
Frequency
30
80 < m Y 90
90 < m Y 100
100 < m Y 120
120 < m Y 150
35
40
40
15
(i) Calculate an estimate of the mean mass of the 160 oranges.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(b)(i)
g [4]
328
(ii) On the grid, complete the histogram to show the information in the table.
5
4
3
Frequency
density
2
1
m
0
50
60
70
80
90
100
110
120
130
140
150
Mass (grams)
[4]
Mr.Yasser Elsayed
002 012 013 222 97
329
8) June 2011 V1
8
The table below shows the marks scored by a group of students in a test.
Mark
11
12
13
14
15
16
17
18
Frequency
10
8
16
11
7
8
6
9
(a) Find the mean, median and mode.
Answer(a) mean =
median =
mode =
[6]
(b) The table below shows the time (t minutes) taken by the students to complete the test.
0 I=t Y=10
Time (t)
Frequency
2
10 I=t Y=20 20 I=t Y=30 30 I=t Y=40 40 I=t Y=50 50 I=t Y=60
19
16
14
15
9
(i) Cara rearranges this information into a new table.
Complete her table.
Time (t)
0 I=t Y=20
20 I=t Y=40
40 I=t Y=50
50 I=t Y=60
Frequency
9
[2]
(ii) Cara wants to draw a histogram to show the information in part (b)(i).
Complete the table below to show the interval widths and the frequency densities.
0 I=t Y=20
Interval
width
Frequency
density
Mr.Yasser Elsayed
002 012 013 222 97
20 I=t Y=40
40 I=t Y=50
50 I=t Y=60
10
0.9
[3]
330
(c) Some of the students were asked how much time they spent revising for the test.
10 students revised for 2.5 hours, 12 students revised for 3 hours and n students revised for
4 hours.
The mean time that these students spent revising was 3.1 hours.
Find n.
Show all your working .
Answer(c) n =
Mr.Yasser Elsayed
002 012 013 222 97
[4]
331
9) June 2011 V2
6
Time
(t mins)
0 I t Y 20
20 I t Y 35
35 I t Y 45
45 I t Y 55
55 I t Y 70
70 I t Y 80
Frequency
6
15
19
37
53
20
The table shows the times taken, in minutes, by 150 students to complete their homework on one day.
(a) (i) In which interval is the median time?
Answer(a)(i)
[1]
(ii) Using the mid-interval values 10, 27.5, ……..calculate an estimate of the mean time.
min [3]
Answer(a)(ii)
(b) (i) Complete the table of cumulative frequencies.
Time
(t mins)
t Y 20
t Y 35
Cumulative
frequency
6
21
t Y 45
t Y 55
t Y 70
t Y 80
[2]
(ii) On the grid, label the horizontal axis from 0 to 80, using the scale 1 cm represents 5 minutes
and the vertical axis from 0 to 150, using the scale 1 cm represents 10 students.
Draw a cumulative frequency diagram to show this information.
Mr.Yasser Elsayed
002 012 013 222 97
[5]
332
(c) Use your graph to estimate
(i) the median time,
Answer(c)(i)
min
[1]
Answer(c)(ii)
min
[2]
(ii) the inter-quartile range,
(iii) the number of students whose time was in the range 50 I t Y 60,
Answer(c)(iii)
[1]
(iv) the probability, as a fraction, that a student, chosen at random, took longer than 50 minutes,
Answer(c)(iv)
[2]
(v) the probability, as a fraction, that two students, chosen at random, both took longer than 50
minutes.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(c)(v)
[2]
333
10) June 2011 V3
6
200
180
160
140
120
Cumulative
100
frequency
80
60
40
20
0
m
1
2
3
4
5
6
7
8
9
10
Mass (kilograms)
The masses of 200 parcels are recorded.
The results are shown in the cumulative frequency diagram above.
(a) Find
(i) the median,
Answer(a)(i)
kg [1]
Answer(a)(ii)
kg [1]
Answer(a)(iii)
kg [1]
(ii) the lower quartile,
(iii) the inter-quartile range,
(iv) the number of parcels with a mass greater than 3.5 kg.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(a)(iv)
[2]
334
(b) (i) Use the information from the cumulative frequency diagram to complete the grouped
frequency table.
Mass (m ) kg
0 ImY4
Frequency
36
4 ImY 6
6I m Y7
7 I m Y 10
50
[2]
(ii) Use the grouped frequency table to calculate an estimate of the mean.
Answer(b)(ii)
kg [4]
(iii) Complete the frequency density table and use it to complete the histogram.
Mass (m ) kg
0 ImY4
Frequency
density
9
4 ImY 6
6I m Y7
7 I m Y 10
16.7
40
35
30
25
Frequency
20
density
15
10
5
0
m
1
2
3
Mr.Yasser Elsayed
002 012 013 222 97
4
5
6
7
8
9
10
Mass (kilograms)
[4]
335
11) November 2011 V1
3
The table shows information about the heights of 120 girls in a swimming club.
Height (h metres)
Frequency
1.3 I h Y 1.4
4
1.4 I h Y 1.5
13
1.5 I h Y 1.6
33
1.6 I h Y 1.7
45
1.7 I h Y 1.8
19
1.8 I h Y 1.9
6
(a) (i) Write down the modal class.
Answer(a)(i)
m [1]
(ii) Calculate an estimate of the mean height. Show all of your working.
Answer(a)(ii)
m [4]
(b) Girls from this swimming club are chosen at random to swim in a race.
Calculate the probability that
(i) the height of the first girl chosen is more than 1.8 metres,
Answer(b)(i)
[1]
(ii) the heights of both the first and second girl chosen are 1.8 metres or less.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(b)(ii)
[3]
336
(c) (i) Complete the cumulative frequency table for the heights.
Height (h metres)
Cumulative frequency
h Y 1.3
0
h Y 1.4
4
h Y 1.5
17
h Y 1.6
50
h Y 1.7
h Y 1.8
114
h Y 1.9
[1]
(ii) Draw the cumulative frequency graph on the grid.
120
110
100
90
80
70
Cumulative
frequency
60
50
40
30
20
10
h
0
1.3
1.4
1.5
1.6
1.7
Height (m)
1.8
1.9
[3]
(d) Use your graph to find
(i) the median height,
Mr.Yasser
Elsayed
(ii) the 30th
percentile.
002 012 013 222 97
Answer(d)(i)
m [1]
Answer(d)(ii)
m [1]
337
12) November 2011 V2
5
(a) The times, t seconds, for 200 people to solve a problem are shown in the table.
Time (t seconds)
0 I t Y 20
20 I t Y 40
40 I t Y 50
50 I t Y 60
60 I t Y 70
70 I t Y 80
80 I t Y 90
90 I t Y 100
Frequency
6
12
20
37
42
50
28
5
Calculate an estimate of the mean time.
Answer(a)
s [4]
(b) (i) Complete the cumulative frequency table for this data.
Time
(t seconds)
Cumulative
Frequency
t Y 20
t Y 40
t Y 50
6
18
38
t Y 60
t Y 70
t Y 80
t Y 90
t Y 100
167
[2]
(ii) Draw the cumulative frequency graph on the grid opposite to show this data.
[4]
(c) Use your cumulative frequency graph to find
(i) the median time,
Answer(c)(i)
s [1]
Answer(c)(ii)
s [1]
Answer(c)(iii)
s [1]
(ii) the lower quartile,
(iii) the inter-quartile range,
(iv) how many people took between 65 and 75 seconds to solve the problem,
Answer(c)(iv)
(v) how many
people took longer than 45 seconds to solve the problem.
Mr.Yasser
Elsayed
Answer(c)(v)
002 012 013 222 97
[1]
[2]
338
200
180
160
140
120
Cumulative
frequency 100
80
60
40
20
0
t
20
Mr.Yasser Elsayed
002 012 013 222 97
40
60
80
100
Time (seconds)
339
13) November 2011 V3
7
The times, t minutes, taken for 200 students to cycle one kilometre are shown in the table.
Time (t minutes)
0 I=t=Y=2
2 I=t=Y=3
3 I=t=Y=4
4 I=t=Y=8
Frequency
24
68
72
36
(a) Write down the class interval that contains the median.
Answer(a)
[1]
(b) Calculate an estimate of the mean.
Show all your working.
Answer(b)
Mr.Yasser Elsayed
002 012 013 222 97
min [4]
340
(c) (i) Use the information in the table opposite to complete the cumulative frequency table.
Time (t minutes)
tY2
Cumulative frequency
24
tY3
tY4
tY8
200
[1]
(ii) On the grid, draw a cumulative frequency diagram.
200
180
Cumulative frequency
160
140
120
100
80
60
40
20
0
t
1
2
3
4
Time (minutes)
5
6
7
8
[3]
(iii) Use your diagram to find the median, the lower quartile and the inter-quartile range.
Answer(c)(iii) Median =
min
Lower quartile =
min
Inter-quartile range =
Mr.Yasser Elsayed
002 012 013 222 97
min [3]
341
14) June 2012 V1
Felix asked 80 motorists how many hours their journey took that day.
He used the results to draw a cumulative frequency diagram.
5
Cumulative
frequency
80
70
60
50
40
30
20
10
t
0
1
2
3
4
5
6
7
8
Time (hours)
(a) Find
(i) the median,
Answer(a)(i)
h [1]
Answer(a)(ii)
h [1]
Answer(a)(iii)
h [1]
342
(ii) the upper quartile,
(iii) the inter-quartile range.
Mr.Yasser Elsayed
002 012 013 222 97
(b) Find the number of motorists whose journey took more than 5 hours but no more than 7 hours.
Answer(b)
[1]
(c) The frequency table shows some of the information about the 80 journeys.
Time in hours (t)
0ItY2
2ItY3
3ItY4
Frequency
20
25
18
4ItY5
5ItY6
(i) Use the cumulative frequency diagram to complete the table above.
6ItY8
[2]
(ii) Calculate an estimate of the mean number of hours the 80 journeys took.
Answer(c)(ii)
h [4]
(d) On the grid, draw a histogram to represent the information in your table in part (c).
Mr.Yasser Elsayed
002 012 013 222 97
[5]
343
15) June 2012 V2
1
Mathematics mark
30
50
35
25
5
39
48
40
10
15
English mark
26
39
35
28
9
37
45
33
16
12
The table shows the test marks in Mathematics and English for 10 students.
(a) (i) On the grid, complete the scatter diagram to show the Mathematics and English marks for
the 10 students. The first four points have been plotted for you.
50
40
English
mark
30
20
10
0
5
10
15
20
25
30
35
40
45
50
Mathematics mark
[2]
(ii) What type of correlation does your scatter diagram show?
Answer(a)(ii)
(iii) Draw a line of best fit on the grid.
[1]
[1]
(iv) Ann missed the English test but scored 22 marks in the Mathematics test.
Use your line of best fit to estimate a possible English mark for Ann.
Answer(a)(iv)
[1]
(b) Show that the mean English mark for the 10 students is 28.
Answer(b)
[2]
(c) Two new students do the English test. They both score the same mark.
The mean English mark for the 12 students is 31.
Calculate the English mark for the new students.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(c)
[3]
344
16) June 2012 V3
4
(a) In a football league a team is given 3 points for a win, 1 point for a draw and 0 points for a loss.
The table shows the 20 results for Athletico Cambridge.
Points
3
1
0
Frequency
10
3
7
(i) Find the median and the mode.
Answer(a)(i) Median =
[3]
Mode =
(ii) Thomas wants to draw a pie chart using the information in the table.
Calculate the angle of the sector which shows the number of times Athletico Cambridge
were given 1 point.
Answer(a)(ii)
[2]
(b) Athletico Cambridge has 20 players.
The table shows information about the heights (h centimetres) of the players.
Height (h cm)
Frequency
170 I h Y 180
180 I h Y 190
190 I h Y 200
5
12
3
Calculate an estimate of the mean height of the players.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(b)
cm [4]
345
17) November 2012 V1
1
B, C or D
A or A*
A or A*
(x + 18)°
x°
B, C or D
NOT TO
SCALE
72°
60° E, F or G
E, F or G
Girls
Boys
The pie charts show information on the grades achieved in mathematics by the girls and boys at a
school.
(a) For the Girls’ pie chart, calculate
(i) x,
Answer(a)(i) x =
[2]
Answer(a)(ii)
[1]
(ii) the angle for grades B, C or D.
(b) Calculate the percentage of the Boys who achieved grades E, F or G.
Answer(b)
% [2]
(c) There were 140 girls and 180 boys.
(i) Calculate the percentage of students (girls and boys) who achieved grades A or A*.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(c)(i)
% [3]
346
(ii) How many more boys than girls achieved grades B, C or D?
Answer(c)(ii)
[2]
(d) The table shows information about the times, t minutes, taken by 80 of the girls to complete
their mathematics examination.
Time taken ( t minutes)
Frequency
40 I t Y 60
60 I t Y 80
80 I t Y 120
120 I t Y 150
5
14
29
32
(i) Calculate an estimate of the mean time taken by these 80 girls to complete the examination.
Answer(d)(i)
min [4]
(ii) On a histogram, the height of the column for the interval 60 I t Y 80 is 2.8 cm.
Calculate the heights of the other three columns.
Do not draw the histogram.
Answer(d) (ii) 40 I t Y 60 column height =
cm
80 I t Y 120 column height =
cm
Mr.Yasser Elsayed120 I t Y 150 column height =
002 012 013 222 97
cm [4]
347
18) November 2012 V2
5
(a) A farmer takes a sample of 158 potatoes from his cr op. He records the mass of each potato and
the results are shown in the table.
Mass (m grams)
Frequency
0 I m Y 40
6
40 I m Y 80
10
80 I m Y 120
28
120 I m Y 160
76
160 I m Y 200
22
200 I m Y 240
16
Calculate an estimate of the mean mass.
Show all your working.
Answer(a)
g [4]
(b) A new frequency table is made from the results shown in the table in part (a).
Mass (m grams)
Frequency
0 I m Y 80
80 I m Y 200
200 I m Y 240
(i) Complete the table above.
16
Mr.Yasser
Elsayed
(ii) On
the grid opposite, complete the histogram to show the information in this new table.
002 012 013 222 97
348
[2]
1.2
1.0
0.8
Frequency
0.6
density
0.4
0.2
0
m
40
80
120
160
200
Mass (grams)
240
[3]
(c) A bag contains 15 potatoes which have a mean mass of 136 g.
The farmer puts 3 potatoes which have a mean mass of 130 g into the bag.
Calculate the mean mass of all the potatoes in the bag.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(c)
g [3]
349
19) November 2012 V3
9
200 students take a Mathematics examination.
The cumulative frequency diagram shows information about the times taken, t minutes, to complete
the examination.
200
190
180
170
160
150
140
130
120
110
Cumulative
frequency
100
90
80
70
60
50
40
30
20
10
0
Mr.Yasser Elsayed
002 012 013 222 97
30
40
t
50
60
70
80
90
Time (minutes)
350
(a) Find
(i) the median,
Answer(a)(i)
min [1]
Answer(a)(ii)
min [1]
Answer(a)(iii)
min [1]
(ii) the lower quartile,
(iii) the inter-quartile range,
(iv) the number of students who took more than 1 hour.
Answer(a)(iv)
[2]
(b) (i) Use the cumulative frequency diagram to complete the grouped frequency table.
Time,
t minutes
30 I t Y 40
Frequency
9
40 I t Y 50
50 I t Y 60
60 I t Y 70
70 I t Y 80
80 I t Y 90
16
28
108
28
[1]
(ii) Calculate an estimate of the mean time taken by the 200 students to complete the
examination.
Show all your working.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(b)(ii)
min [4]
351
20) June 2013 V1
3
200 students estimate the mass (m grams) of a coin.
The cumulative frequency diagram shows the results.
200
180
160
140
120
Cumulative
frequency
100
80
60
40
20
0
m
1
2
Mr.Yasser Elsayed
002 012 013 222 97
3
4
5
6
7
8
9
10
Mass (grams)
352
(a) Find
(i) the median,
Answer(a)(i) ............................................ g [1]
(ii) the upper quartile,
Answer(a)(ii) ............................................ g [1]
(iii) the 80th percentile,
Answer(a)(iii) ............................................ g [1]
(iv) the number of students whose estimate is 7 g or less.
Answer(a)(iv) ............................................... [1]
(b) (i) Use the cumulative frequency diagram to complete the frequency table.
Mass (m grams)
Frequency
0<mĞ2
2<mĞ4
40
4<mĞ6
6<mĞ8
8 < m Ğ 10
2
[2]
(ii) A student is chosen at random.
The probability that the student estimates that the mass is greater than M grams is 0.3.
Find the value of M.
Answer(b)(ii) M = ............................................... [2]
Mr.Yasser Elsayed
002 012 013 222 97
353
21) June 2013 V1
5
Height (h cm)
150 < h Ğ 160
160 < h Ğ 165
165 < h Ğ 180
180 < h Ğ 190
5
9
18
10
Frequency
The table shows information about the heights of a group of 42 students.
(a) Using mid-interval values, calculate an estimate of the mean height of the students.
Show your working.
Answer(a) ......................................... cm [3]
(b) Write down the interval which contains the lower quartile.
Answer(b) ............................................... [1]
(c) Complete the histogram to show the information in the table.
One column has already been drawn for you.
2
Frequency
density
1
0
Mr.Yasser Elsayed
002 012 013 222 97
150
155
160
165
170
175
180
185
190
Height (cm)
[4]
354
22) June 2013 V3
9
Sam asked 80 people how many minutes their journey to work took on one day.
The cumulative frequency diagram shows the times taken (m minutes).
80
70
60
50
Cumulative
40
frequency
30
20
10
0
m
10
20
30
40
50
Time (minutes)
(a) Find
(i) the median,
Answer(a)(i) ........................................ min [1]
(ii) the lower quartile,
Mr.Yasser
Elsayed
(iii) the inter-quartile
range.
002 012 013 222 97
Answer(a)(ii) ........................................ min [1]
Answer(a)(iii) ........................................ min [1]
355
(b) One of the 80 people is chosen at random.
Find the probability that their journey to work took more than 35 minutes.
Give your answer as a fraction.
Answer(b) ............................................... [2]
(c) Use the cumulative frequency diagram to complete this frequency table.
Time (m minutes)
Frequency
0 < m Y 10
10 < m Y 15 15 < m Y 30 30 < m Y 40 40 < m Y 50
30
12
18
[2]
(d) Using mid-interval values, calculate an estimate of the mean journey time for the 80 people.
Answer(d) ........................................ min [3]
(e) Use the table in part (c) to complete the histogram to show the times taken by the 80 people.
One column has already been completed for you.
4
3
Frequency
density
2
1
0
m
10
Mr.Yasser Elsayed
002 012 013 222 97
20
30
Time (minutes)
40
50
[5]
356
23) November 2013 V1
7
120 students are asked to answer a question.
The time, t seconds, taken by each student to answer the question is measured.
The frequency table shows the results.
0 < t Y 10
Time
Frequency
10 < t Y 20 20 < t Y 30 30 < t Y 40 40 < t Y 50 50 < t Y 60
6
44
40
14
10
6
(a) Calculate an estimate of the mean time.
Answer(a) ............................................ s [4]
(b) (i) Complete the cumulative frequency table.
t Y 10
Time
Cumulative frequency
t Y 20
t Y 30
t Y 40
6
t Y 50
t Y 60
104
120
[2]
(ii) On the grid below, draw a cumulative frequency diagram to show this information.
120
100
80
Cumulative
60
frequency
40
20
0
t
Mr.Yasser Elsayed
002 012 013 222 97
10
20
30
40
50
60
Time (seconds)
[3]
357
(iii) Use your cumulative frequency diagram to find the median, the lower quartile and
the 60th percentile.
Answer(b)(iii)
Median ............................................ s
Lower quartile ............................................ s
60th percentile ............................................ s [4]
(c) The intervals for the times taken are changed.
(i) Use the information in the frequency table on the opposite page to complete this new table.
0 < t Y 20
Time
Frequency
20 < t Y 30
30 < t Y 60
40
[2]
(ii) On the grid below, complete the histogram to show the information in the new table.
One column has already been drawn for you.
4
3.5
3
2.5
Frequency
density
2
1.5
1
0.5
0
t
10
20
30
40
50
60
Time (seconds)
[3]
Mr.Yasser Elsayed
002 012 013 222 97
358
24) November 2013 V3
5
(a) 80 students were asked how much time they spent on the internet in one day.
This table shows the results.
Time (t hours)
Number of students
0<tY1
1<tY2
2<tY3
3<tY5
5<tY7
7 < t Y 10
15
11
10
19
13
12
(i) Calculate an estimate of the mean time spent on the internet by the 80 students.
Answer(a)(i) ..................................... hours [4]
(ii) On the grid, complete the histogram to show this information.
16
14
12
10
Frequency
density 8
6
4
2
0
t
1
2
3
4
5
6
7
8
9
10
Time (hours)
[4]
Mr.Yasser Elsayed
002 012 013 222 97
359
25) June 2014 V1
9
80
70
60
50
Cumulative
frequency
40
30
20
10
0
t
10
20
30
40
50
Time (minutes)
The times (t minutes) taken by 80 people to complete a charity swim were recorded.
The results are shown in the cumulative frequency diagram above.
(a) Find
(i) the median,
Answer(a)(i) ......................................... min [1]
Mr.Yasser Elsayed
002 012 013 222 97
(ii) the inter-quartile range,
360
Answer(a)(ii) ......................................... min [2]
(iii) the 70th percentile.
Answer(a)(iii) ......................................... min [2]
(b) The times taken by the 80 people are shown in this grouped frequency table.
Time (t minutes)
Frequency
0 < t Ğ 20
20 < t Ğ 30
30 < t Ğ 45
45 < t Ğ 50
12
21
33
14
(i) Calculate an estimate of the mean time.
Answer(b)(i) ......................................... min [4]
(ii) Draw a histogram to represent the grouped frequency table.
4
3
Frequency
density
2
1
0
10
Mr.Yasser Elsayed
002 012 013 222 97
t
20
30
40
50
Time (minutes)
361
[4]
26) June 2014 V2
7
(a)
1.0
0.8
Frequency 0.6
density
0.4
0.2
0
m
10
20
30
40
50
60
70
80
90
100
Mass (grams)
The histogram shows some information about the masses (m grams) of 39 apples.
(i) Show that there are 12 apples in the interval 70 < m Y 100 .
Answer(a)(i)
[1]
(ii) Calculate an estimate of the mean mass of the 39 apples.
Answer(a)(ii) ............................................. g [5]
(b) The mean mass of 20 oranges is 70 g.
One orange is eaten.
The mean mass of the remaining oranges is 70.5 g.
Find the mass of the orange that was eaten.
Mr.Yasser Elsayed
002 012 013 222 97
362
Answer(b) ............................................. g [3]
27) June 2014 V3
2
4
3
Frequency
density
2
1
0
10
20
30
40
50
60
Amount ($x)
A survey asked 90 people how much money they gave to charity in one month.
The histogram shows the results of the survey.
(a) Complete the frequency table for the six columns in the histogram.
Amount ($x)
0 < x Y 10
Frequency
4
[5]
(b) Use your frequency table to calculate an estimate of the mean amount these 90 people gave to charity.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(b) $ ................................................ [4]
363
28) November 2014 V1
6
A company tested 200 light bulbs to find the lifetime, T hours, of each bulb.
The results are shown in the table.
Lifetime
(T hours)
Number
of bulbs
0 < T Y 1000
10
1000 < T Y 1500
30
1500 < T Y 2000
55
2000 < T Y 2500
72
2500 < T Y 3500
33
(a) Calculate an estimate of the mean lifetime for the 200 light bulbs.
Answer(a) ...................................... hours [4]
(b) (i) Complete the cumulative frequency table.
Lifetime (T hours)
T Y 1000
T Y 1500
T Y 2000
T Y 2500
T Y 3500
Number of bulbs
[2]
Mr.Yasser Elsayed
002 012 013 222 97
364
(ii) On the grid, draw a cumulative frequency diagram to show this information.
200
150
Cumulative
frequency
100
50
0
T
500
1000
1500
2000
2500
3000
3500
Lifetime (hours)
[3]
(iii) The company says that the average lifetime of a bulb is 2200 hours.
Estimate the number of bulbs that lasted longer than 2200 hours.
Answer(b)(iii) ................................................ [2]
(c) Robert buys one energy saving bulb and one halogen bulb.
9
The probability that the energy saving bulb lasts longer than 3500 hours is 10 .
3
The probability that the halogen bulb lasts longer than 3500 hours is 5 .
Work out the probability that exactly one of the bulbs will last longer than 3500 hours.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(c) ................................................ [4]
365
29) November 2014 V2
3
The time, t seconds, taken for each of 50 chefs to cook an omelette is recorded.
Time
(t seconds)
20 < t Y 25
25 < t Y 30
30 < t Y 35
35 < t Y 40
40 < t Y 45
45 < t Y 50
Frequency
2
6
7
19
9
7
(a) Write down the modal time interval.
Answer(a) .............................................. s [1]
(b) Calculate an estimate of the mean time.
Show all your working.
Answer(b) .............................................. s [4]
Mr.Yasser Elsayed
002 012 013 222 97
366
(c) A new frequency table is made from the results shown in the table opposite.
Time
(t seconds)
20 < t Y 35
35 < t Y 40
40 < t Y 50
Frequency
(i) Complete the table.
[1]
(ii) On the grid, draw a histogram to show the information in this new table.
4
3
Frequency
2
density
1
t
0
20
25
30
35
40
45
50
Time (seconds)
[3]
Mr.Yasser Elsayed
002 012 013 222 97
367
30) November 2014 V3
9
(a) Ricardo asks some motorists how many litres of fuel they use in one day.
The numbers of litres, correct to the nearest litre, are shown in the table.
Number of litres
16
17
18
19
20
Number of motorists
11
10
p
4
8
(i) For this table, the mean number of litres is 17.7 .
Calculate the value of p.
Answer(a)(i) p = ................................................ [4]
(ii) Find the median number of litres.
Answer(a)(ii) ....................................... litres [1]
(b) Manuel completed a journey of 320 km in his car.
The fuel for the journey cost $1.28 for every 6.4 km travelled.
(i) Calculate the cost of fuel for this journey.
Answer(b)(i) $ ................................................. [2]
(ii) When Manuel travelled 480 km in his car it used 60 litres of fuel.
Manuel’s car used fuel at the same rate for the journey of 320 km.
Calculate the number of litres of fuel the car used for the journey of 320 km.
Answer(b)(ii) ....................................... litres [2]
(iii) Calculate the cost per litre of fuel used for the journey of 320 km.
Mr.Yasser Elsayed
002 012 013 222 97
368
Answer(b)(iii) $ ................................................. [2]
(c) Ellie drives a car at a constant speed of 30 m/s correct to the nearest 5 m/s.
She maintains this speed for 5 minutes correct to the nearest 10 seconds.
Calculate the upper bound of the distance in kilometres that Ellie could have travelled.
Answer(c) .......................................... km [5]
Mr.Yasser Elsayed
002 012 013 222 97
369
31) June 2015 V1
6
The table shows the time, t minutes, that 400 people take to complete a test.
Time taken
(t mins)
0  t  10
Frequency
10
(a) (i)
10  t  24 24  t  30 30  t  40 40  t  60 60  t  70
90
135
85
70
10
Write down the modal time interval.
Answer(a)(i) .......................................... min [1]
(ii)
Calculate an estimate of the mean time taken to complete the test.
Answer(a)(ii) .......................................... min [4]
(b) (i)
Complete the table of cumulative frequencies.
Time taken
(t mins)
t  10
t  24
Cumulative
frequency
10
100
t  30
t  40
t  60
t  70
400
[2]
(ii)
On the grid opposite, draw a cumulative frequency diagram to show this information.
Mr.Yasser Elsayed
002 012 013 222 97
370
400
350
300
250
Cumulative
frequency
200
150
100
50
0
10
(c) Use your graph to estimate
(i)
the median time,
(ii)
the inter-quartile range,
20
30
40
Time taken (minutes)
50
60
70
t
[3]
Answer(c)(i) .......................................... min [1]
Answer(c)(ii) .......................................... min [2]
(iii)
the 15th percentile,
Answer(c)(iii) .......................................... min [2]
(iv) the number
of people who took more than 50 minutes.
Mr.Yasser
Elsayed
Answer(c)(iv) ................................................. [2]
002 012 013 222 97
371
32) June 2015 V2
7
(a) A group of 50 students estimated the mass, M grams, of sweets in a jar.
The results are shown in the table.
Mass (M grams)
Number of students
0 < M  200
5
200 < M  300
9
300 < M  350
18
350 < M  400
12
400 < M  500
6
(i) Calculate an estimate of the mean.
Answer(a)(i) ..................................... grams [4]
(ii) Complete this histogram to show the information in the table.
0.4
0.3
Frequency
density
0.2
0.1
0
100
Mr.Yasser Elsayed
002 012 013 222 97
200
300
Mass (grams)
400
500
M
372
[3]
(b) A group of 50 adults also estimated the mass, M grams, of the sweets in the jar.
The histogram below shows information about their estimates.
Use the histograms to make two comparisons between the distributions of the estimates of the students
and the adults.
0.4
0.3
Frequency
density
0.2
0.1
0
100
200
300
Mass (grams)
400
500
M
Answer(b)
1 ..........................................................................................................................................................
.............................................................................................................................................................
2 ..........................................................................................................................................................
............................................................................................................................................................. [2]
Mr.Yasser Elsayed
002 012 013 222 97
373
33) June 2015 V3
4
The table shows the times, t minutes, taken by 200 students to complete an IGCSE paper.
Time (t minutes)
Frequency
40  t  60
60  t  70
70  t  75
75  t  90
10
50
80
60
(a) By using mid-interval values, calculate an estimate of the mean time.
Answer(a) .......................................... min [3]
(b) On the grid, draw a histogram to show the information in the table.
20
18
16
14
12
Frequency
10
density
8
6
4
2
0
40
50
Mr.Yasser Elsayed
002 012 013 222 97
60
Time (minutes)
70
80
90
t
[4]
374
34) November 2015 V1
6
120 students take a mathematics examination.
(a) The time taken, m minutes, for each student to answer question 1 is shown in this table.
Time (m minutes)
Frequency
0<mG1
72
1<mG2
21
2<mG3
9
3<mG4
11
4<mG5
5
5<mG6
2
Calculate an estimate of the mean time taken.
Answer(a) .......................................... min [4]
(b) (i) Using the table in part (a), complete this cumulative frequency table.
Time (m minutes)
Cumulative frequency
mG1
72
mG2
Mr.Yasser Elsayed
002 012 013 222 97
mG3
mG4
mG5
mG6
120
[2]
[4]
375
(ii) Draw a cumulative frequency diagram to show the time taken.
120
110
100
90
80
Cumulative 70
frequency
60
50
40
30
20
10
0
1
2
Mr.Yasser Elsayed
002 012 013 222 97
3
Time (minutes)
4
5
6
m
[3]
[4]
376
(iii) Use your cumulative frequency diagram to find
(a) the median,
Answer(b)(iii)(a) .......................................... min [1]
(b) the inter-quartile range,
Answer(b)(iii)(b) .......................................... min [2]
(c) the 35th percentile.
Answer(b)(iii)(c) .......................................... min [2]
(c) A new frequency table is made from the table shown in part (a).
Time (m minutes)
0<mG1
Frequency
72
1<mG3
3<mG6
(i) Complete the table above.
[2]
(ii) A histogram was drawn and the height of the first block representing the time 0 < m G 1 was 3.6 cm.
Calculate the heights of the other two blocks.
Answer(c)(ii) ................. cm and ................. cm [3]
Mr.Yasser Elsayed
002 012 013 222 97
[4]
377
35) November 2015 V2
3
Leo measured the rainfall each day, in millimetres, for 120 days.
The cumulative frequency table shows the results.
Rainfall (r mm)
r  20
r  25
r  35
r  40
r  60
r  70
5
13
72
90
117
120
Cumulative
frequency
(a) On the grid below, draw a cumulative frequency diagram to show these results.
120
100
80
Cumulative
frequency 60
40
20
0
10
20
30
40
50
60
70
r
Rainfall (mm)
[3]
(b) (i)
Find the median.
Answer(b)(i) .........................................mm [1]
(ii)
Use your diagram to find the number of days when the rainfall was more than 50 mm.
Mr.Yasser Elsayed
002 012 013 222 97
Answer(b)(ii) ............................................... [2]
378
(c) Use the information in the cumulative frequency table to complete the frequency table below.
Rainfall (r mm)
Frequency
0  r  20
20  r  25
5
25  r  35
35  r  40
59
40  r  60
60  r  70
3
[2]
(d) Use your frequency table to calculate an estimate of the mean.
You must show all your working.
Answer(d) ........................................mm [4]
(e) In a histogram drawn to show the information in the table in part (c), the frequency density for the
interval 25  r  35 is 5.9 .
Calculate the frequency density for the intervals 20  r  25 , 40  r  60 and 60  r  70 .
Answer(e) 20  r  25 ...............................................
Mr.Yasser Elsayed
002 012 013 222 97
40  r  60 ...............................................
60  r  70 ............................................... [4]
379
36) November 2015 V3
6
The table shows information about the masses, m grams, of 160 apples.
Mass (m grams)
30 < m  80
80 < m  100
100 < m  120
120 < m  200
Frequency
50
30
40
40
(a) Calculate an estimate of the mean.
Answer(a) ............................................. g [4]
(b) On the grid, complete the histogram to show the information in the frequency table.
2.5
2
1.5
Frequency
density
1
0.5
0
Mr.Yasser Elsayed
002 012 013 222 97
40
80
120
Mass (grams)
160
200
m
380 [3]
(c) An apple is chosen at random from the 160 apples.
Find the probability that its mass is more than 120 g.
Answer(c) ................................................ [1]
(d) Two apples are chosen at random from the 160 apples, without replacement.
Find the probability that
(i) they both have a mass of more than 120 g,
Answer(d)(i) ................................................ [2]
(ii) one has a mass of more than 120 g and one has a mass of 80 g or less.
Answer(d)(ii) ................................................ [3]
Mr.Yasser Elsayed
002 012 013 222 97
381
37) March 2015 V2
9
The table shows the height, h cm, of 40 children in a class.
Height (h cm)
120 < h  130
130 < h  140
140 < h  144
144 < h  150
150 < h  170
3
14
4
6
13
Frequency
(a) Write down the class interval containing the median.
Answer(a) ................................. < h  ................................. [1]
(b) Calculate an estimate of the mean height.
Answer(b) .......................................... cm [4]
(c) Complete the histogram.
2
1.5
Frequency
density
1
0.5
0
120
130
Mr.Yasser Elsayed
002 012 013 222 97
140
150
160
170
h
Height (cm)
[4]
382
38) March 2016 V2
4
The cumulative frequency diagram shows information about the time taken, t minutes, by 60 students to
complete a test.
60
50
40
Cumulative
frequency 30
20
10
0
10
20
30
40
50
60
70
80
90
100
t
Time taken (minutes)
(a) Find
(i) the median,
........................................... min [1]
(ii) the inter-quartile range,
........................................... min [2]
(iii) the 40th percentile,
........................................... min [2]
(iv) the number of students who took more than 80 minutes to complete the test.
Mr.Yasser Elsayed
002 012 013 222 97
.................................................. [2]
383
(b) Use the cumulative frequency diagram to complete the frequency table below.
Time taken
(t minutes)
0  t  40
Frequency
8
40  t  60
60  t  70
70  t  80
80  t  90
90  t  100
4
[3]
(c) On the grid below, complete the histogram to show the information in the table in part (b).
3
2
Frequency
density
1
0
10
20
30
Mr.Yasser Elsayed
002 012 013 222 97
40
50
60
70
Time taken (minutes)
80
90
100
t
[4]
384
39) June 2016 V1
3
(a) 200 students estimate the volume, V m 3, of a classroom.
The cumulative frequency diagram shows their results.
200
180
160
140
Cumulative
frequency
120
100
80
60
40
20
0
50
100
150
200
250
300
350
400
450
500
V
Volume (m3)
Find
(i)
the median,
............................................. m3 [1]
(ii)
the lower quartile,
............................................. m3 [1]
(iii)
the inter-quartile range,
............................................. m3 [1]
Mr.Yasser
Elsayed
(iv) the number
of students who estimate that the volume is greater than 300 m .
...................................................
002 012 013 222 97
385 [2]
3
40) June 2016 V3
4
Coins are put into a machine to pay for parking cars.
The probability that the machine rejects a coin is 0.05 .
(a) Adhira puts 2 coins into the machine.
(i) Calculate the probability that the machine rejects both coins.
................................................... [2]
(ii)
Calculate the probability that the machine accepts at least one coin.
................................................... [1]
(b) Raj puts 4 coins into the machine.
Calculate the probability that the machine rejects exactly one coin.
................................................... [3]
(c) The table shows the amount of money, $a, received for parking each day for 200 days.
Amount ($a)
Frequency
200 1 a G 250 250 1 a G 300 300 1 a G 350 350 1 a G 400 400 1 a G 450 450 1 a G 500
13
19
27
56
62
23
Calculate an estimate of the mean amount of money received each day.
m2.
Mr.Yasser Elsayed
002 012 013 222 97
$ ................................................... [4]
386
2
(d) The histogram shows the length of time that 200 cars were parked.
1.2
1.1
1
0.9
0.8
0.7
Frequency
density 0.6
0.5
0.4
0.3
0.2
0.1
0
50
100
150
200
250
300
350
400
Time in minutes
(i)
Calculate the number of cars that were parked for 100 minutes or less.
................................................... [1]
(ii)
Calculate the percentage of cars that were parked for more than 250 minutes.
m2.
...............................................% [2]
Mr.Yasser Elsayed
002 012 013 222 97
387
(b) The 200 students also estimate the total area, A m2, of the windows in the classroom.
The results are shown in the table.
Area (A m2)
20 1 A G 60
60 1 A G 100
100 1 A G 150
150 1 A G 250
Frequency
32
64
80
24
(i)
Calculate an estimate of the mean.
Show all your working.
............................................. m2 [4]
(ii)
Complete the histogram to show the information in the table.
2
Frequency
1
density
0
50
100
150
200
250
A
Area (m2)
[4]
(iii)
Two of the 200 students are chosen at random.
Find the probability that they both estimate that the area is greater than 100 m2.
Mr.Yasser Elsayed
002 012 013 222 97
................................................... [2]
388
41) June 2017 V1
2
The time taken for each of 90 cars to complete one lap of a race track is shown in the table.
Time (t seconds)
Frequency
70 1 t G 71
71 1 t G 72
72 1 t G 73
73 1 t G 74
74 1 t G 75
17
24
21
18
10
(a) Write down the modal time interval.
............... 1 t G ............. [1]
(b) Calculate an estimate of the mean time.
.............................................. s [4]
(c)
(i)
Complete the cumulative frequency table.
Time (t seconds)
Cumulative frequency
t G 71
t G 72
t G 73
t G 74
t G 75
17
[2]
Mr.Yasser Elsayed
002 012 013 222 97
389
(ii)
On the grid, draw a cumulative frequency diagram to show this information.
90
80
70
60
Cumulative
frequency
50
40
30
20
10
0
70
71
72
73
74
75
t
Time (seconds)
[3]
(iii)
Find the median time.
.............................................. s [1]
(iv)
Find the inter-quartile range.
.............................................. s [2]
(d) One lap of the race track measures 3720 metres, correct to the nearest 10 metres.
A car completed the lap in 75 seconds, correct to the nearest second.
Calculate the upper bound for the average speed of this car.
Give your answer in kilometres per hour.
Mr.Yasser Elsayed
002 012 013 222 97
....................................... km/h [4]
390
42) November 2017 V1
5
The histogram shows the distribution of the masses, m grams, of 360 apples.
Key: the shaded
square represents
10 apples
Frequency
density
0
140
160
180
200
220
240
m
Mass (grams)
(a) Use the histogram to complete the frequency table.
Mass (m grams)
Number of apples
140 < m G 170
170 < m G 180
180 < m G 190
190 < m G 210
92
210 < m G 240
42
[3]
Mr.Yasser Elsayed
002 012 013 222 97
391
(b) Calculate an estimate of the mean mass of the 360 apples.
.............................................. g [4]
Mr.Yasser Elsayed
002 012 013 222 97
© UCLES 2017
0580/41/O/N/17
[Turn over
392
42) June 2018 V2
2
The time taken for each of 120 students to complete a cooking challenge is shown in the table.
Time (t minutes)
20 1 t G 25
25 1 t G 30
30 1 t G 35
35 1 t G 40
40 1 t G 45
44
32
28
12
4
Frequency
(a) (i)
Write down the modal time interval.
................... 1 t G ................... [1]
(ii)
Write down the interval containing the median time.
................... 1 t G ................... [1]
(iii)
Calculate an estimate of the mean time.
......................................... min [4]
(iv)
A student is chosen at random.
Find the probability that this student takes more than 40 minutes.
................................................. [1]
(b) (i)
Complete the cumulative frequency table.
Time (t minutes)
Cumulative
frequency
t G 20
t G 25
0
44
t G 30
t G 35
t G 40
t G 45
[2]
Mr.Yasser Elsayed
002 012 013 222 97
393
(ii)
On the grid, draw a cumulative frequency diagram to show this information.
120
110
100
90
80
70
Cumulative
frequency 60
50
40
30
20
10
0
20
25
30
35
Time (minutes)
40
45
t
[3]
(iii)
Find the median time.
......................................... min [1]
(iv)
Find the interquartile range.
......................................... min [2]
(v)
Find the number of students who took more than 37 minutes to complete the cooking challenge.
Mr.Yasser Elsayed
002 012 013 222 97
................................................. [2]
394
43) June 2019 V1
4
(a) The test scores of 14 students are shown below.
21
21
23
26
25
21
22
20
21
23
23
27
24
21
(i) Find the range, mode, median and mean of the test scores.
Range = ....................................................
Mode
= ....................................................
Median = ....................................................
Mean = .................................................... [6]
(ii) A student is chosen at random.
Find the probability that this student has a test score of more than 24.
.................................................... [1]
(b) Petra records the score in each test she takes.
The mean of the first n scores is x.
The mean of the first ( n – 1) scores is (x + 1).
Find the nth score in terms of n and x.
Give your answer in its simplest form.
Mr.Yasser Elsayed
002 012 013 222 97
.................................................... [3]
395
(c) During one year the midday temperatures, t°C, in Zedford were recorded.
The table shows the results.
Temperature (t°C)
Number of days
(i)
0 1 t G 10
10 1 t G 15
15 1 t G 20
20 1 t G 25
25 1 t G 35
50
85
100
120
10
Calculate an estimate of the mean.
............................................... °C [4]
(ii)
Complete the histogram to show the information in the table.
25
20
Frequency
density
15
10
5
0
0
5
10
15
20
Temperature (°C)
25
30
35
t
[4]
Mr.Yasser Elsayed
002 012 013 222 97
396
44) June 2020 V2
3 The speed, v km/h, of each of 200 cars passing a building is measured.
The table shows the results.
Speed (v km/h)
Frequency
0 1 v G 20 20 1 v G 40 40 1 v G 45 45 1 v G 50 50 1 v G 60 60 1 v G 80
16
34
62
58
26
4
(a) Calculate an estimate of the mean.
........................................ km/h [4]
(b) (i) Use the frequency table to complete the cumulative frequency table.
Speed (v km/h)
v G 20
v G 40
16
50
Cumulative frequency
v G 45
v G 50
v G 60
v G 80
196
200
[1]
(ii) On the grid, draw a cumulative frequency diagram.
200
180
160
140
120
Cumulative
frequency 100
80
60
40
20
0
0
10
20
Mr.Yasser Elsayed
002 012 013 222 97
© UCLES 2020
30 40 50
Speed (km/h)
0580/42/M/J/20
60
70
80 v
397
[3]
7
(iii) Use your diagram to find an estimate of
(a) the upper quartile,
........................................ km/h [1]
(b) the number of cars with a speed greater than 35 km/h.
................................................. [2]
(c) Two of the 200 cars are chosen at random.
Find the probability that they both have a speed greater than 50 km/h.
................................................. [2]
(d) A new frequency table is made by combining intervals.
Speed (v km/h)
Frequency
0 1 v G 40 40 1 v G 50 50 1 v G 80
50
120
30
On the grid, draw a histogram to show the information in this table.
15
Frequency
density
10
5
0
0
10
20
30 40 50
Speed (km/h)
60
70
80 v
[3]
Mr.Yasser Elsayed
002 012 013 222 97
© UCLES 2020
0580/42/M/J/20
[Turn over
398
45) November 2020 V1
3 (a)
Women
Men
0
60
120
180
240
300
360
420
Time (minutes)
The box-and-whisker plots show the times spent exercising in one week by a group of women and
a group of men.
Below are two statements comparing these times.
For each one, write down whether you agree or disagree, giving a reason for your answer.
Agree or
disagree
Statement
Reason
On average, the women
spent less time exercising
than the men.
The times for the women
show less variation than
the times for the men.
[2]
(b) The frequency table shows the times, t minutes, each of 100 children spent exercising in one week.
Time (t minutes)
Frequency
0 1 t G 60
41
60 1 t G 100 100 1 t G 160 160 1 t G 220 220 1 t G 320
24
23
8
4
(i) Calculate an estimate of the mean time.
Mr.Yasser Elsayed
002 012 013 222 97
.......................................... min [4]
399
(ii) The information in the frequency table is shown in this cumulative frequency diagram.
100
80
60
Cumulative
frequency
40
20
0
0
60
120
180
240
300
360 t
Time (minutes)
Use the cumulative frequency diagram to find an estimate of
(a) the 60th percentile,
.......................................... min [1]
(b) the number of children who spent more than 3 hours exercising.
................................................. [2]
(iii) A histogram is drawn to show the information in the frequency table.
The height of the bar for the interval 60 1 t G 100 is 10.8 cm.
Calculate the height of the bar for the interval 160 1 t G 220 .
Mr.Yasser Elsayed
002 012 013 222 97
............................................ cm [2]
400