001
9 (a)
O
O
NOT TO
SCALE
x°
A B
2.4 cm AB
The volume of a paper cone of radius 2.4 cm is 95.4 cm 3 .
The paper is cut along the slant height from O to AB.
The cone is opened to form a sector OAB of a circle with centre O.
Calculate the sector angle x°.
1
[The volume, V, of a cone with radius r and height h is V = rr 2 h .]
3
................................................. [6]
(b) An empty fuel tank is filled using a cylindrical pipe with diameter 8 cm.
Fuel flows along this pipe at a rate of 2 metres per second.
It takes 24 minutes to fill the tank.
Calculate the capacity of the tank.
Give your answer in litres.
........................................ litres [4]
002
5 (a) ABCDEFGH is a regular octagon with sides of length 6 cm.
The diagram shows part of the octagon.
O is the centre of the octagon and M is the midpoint of AB.
A M B
NOT TO
SCALE
(i) (a) Show that angle OAM is 67.5°.
[2]
(b) Calculate the area of the octagon.
.......................................... cm 2 [4]
(ii) Find the area of the circle that passes through the vertices of the octagon.
.......................................... cm 2 [3]
003
(2)
NOT TO
SCALE
4m
0.45 m
The diagram shows a horizontal container for water with a uniform cross-section.
The cross-section is a semicircle.
The radius of the semicircle is 0.45 m and the length of the container is 4 m.
(i) Calculate the volume of the container.
........................................... m 3 [2]
(ii)
NOT TO
SCALE
0.3 m
The greatest depth of the water in the container is 0.3 m.
The diagram shows the cross-section.
Calculate the number of litres of water in the container.
Give your answer correct to the nearest integer.
........................................ litres [6]
004
11 (a)
A 28 cm D AD
NOT TO
SCALE
20 cm
N BC
B C
A rectangular sheet of paper ABCD is made into an open cylinder with the edge AB meeting the
edge DC.
AD = 28 cm and AB = 20 cm.
(i) Show that the radius of the cylinder is 4.46 cm, correct to 3 significant figures.
[2]
(ii) Calculate the volume of the cylinder.
......................................... cm 3 [2]
(iii) N is a point on the base of the cylinder, such that BN is a diameter.
Calculate the angle between AN and the base of the cylinder.
................................................. [3]
005
(b) The volume of a solid cone is 310 cm 3 .
The height of the cone is twice the radius of its base.
Calculate the slant height of the cone.
1
[The volume, V, of a cone with radius r and height h is V = rr 2 h .]
3
............................................ cm [5]
006
1 (a) Calculate the volume of
(i) a solid cylinder with radius 6 cm and height 14 cm,
����������������������������������������� cm 3 [2]
(ii) a solid hemisphere with radius 6 cm�
4
[The volume, V, of a sphere with radius r is V = rr 3 �]
3
����������������������������������������� cm 3 [2]
(b)
NOT TO
SCALE
14 cm
6 cm
The cylinder and hemisphere in part (a) are joined to form the solid in the diagram�
The solid is made of steel and 1 cm 3 of steel has a mass of 7�85 g�
(i) Show that 1 cm 3 of steel has a mass of 0�007 85 kg�
[1]
(ii) Calculate the total mass of the solid�
�������������������������������������������� kg [2]
007
(c) 2000 cm 3 of iron is melted down and some of it is used to make 50 spheres with radius 2 cm�
(i) Calculate the percentage of iron that is left over�
4
[The volume, V, of a sphere with radius r is V = rr 3 �]
3
��������������������������������������������� % [3]
(ii) The iron left over is then made into a cube�
Calculate the length of an edge of the cube�
�������������������������������������������� cm [1]
(d) A solid cone has radius 3R cm and slant height 9R cm�
A solid cylinder has radius x cm and height 7x cm�
The total surface area of the cone is equal to the total surface area of the cylinder�
Given that R = kx , find the value of k�
[The curved surface area, A, of a cone with radius r and slant height l is A = rrl �]
k = ������������������������������������������������ [4]
008
10 (a) The lengths of the sides of a triangle are 11.4 cm, 14.8 cm and 15.7 cm, all correct to 1 decimal
place.
Calculate the upper bound of the perimeter of the triangle.
............................................ cm [2]
(b)
15.6 cm NOT TO
150° SCALE
The diagram shows a circle, radius 15.6 cm.
The angle of the minor sector is 150°.
Calculate the area of the minor sector.
.......................................... cm 2 [2]
009
(c)
r cm
NOT TO
x° SCALE
The diagram shows a circle, radius r cm and minor sector angle x°.
The perimeter of the major sector is three times the perimeter of the minor sector.
90 (r - 2)
Show that x = .
r
[4]
010
NOT TO
50 cm SCALE
40 cm
1.2 m
36 cm
The diagram shows a water trough in the shape of a prism.
The prism has a cross-section in the shape of an isosceles trapezium.
The trough is completely filled with water.
(a) Show that the volume of water in the trough is 206.4 litres.
[3]
(b) The water from the trough is emptied at a rate of 600 ml per second.
Calculate the time taken, in minutes and seconds, for the trough to be emptied.
.................... minutes .................... seconds [3]
(c) All the water from the trough is emptied into a vertical cylindrical tank.
The depth of the water in the tank is 84 cm.
(i) Calculate the radius of the tank.
............................................ cm [3]
011
(ii) The tank is 60% full.
Calculate the height of the tank.
............................................ cm [2]
(d)
M
NOT TO
50 cm SCALE
40 cm
1.2 m
A 36 cm
A steel rod AM is placed inside the empty water trough as shown in the diagram.
A is a vertex at the base of the isosceles trapezium and M is the midpoint of the top edge on the
opposite face.
Calculate the length of the steel rod, AM.
AM = ............................................ cm [4]
012
3
12 cm
NOT TO
SCALE
3 cm
The diagram shows a cylinder containing water.
There is a solid metal sphere touching the base of the cylinder.
Half of the sphere is in the water.
The radius of the cylinder is 12 cm and the radius of the sphere is 3 cm.
(a) The sphere is removed from the cylinder and the level of the water decreases by h cm.
Show that h = 0.125 .
4
[The volume, V, of a sphere with radius r is V = rr 3 .]
3
[3]
013
(b) The water in the cylinder is poured into another cylinder of radius R cm.
The depth of the water in this cylinder is 18 cm.
Calculate the value of R.
R = ................................................ [3]
(c) The sphere is melted down and some of the metal is used to make 30 cubes with
edge length 1.5 cm.
Calculate the percentage of metal not used.
4
[The volume, V, of a sphere with radius r is V = rr 3 .]
3
............................................. % [3]
014
2
F
NOT TO
SCALE
D
E C
A B
The diagram shows a solid triangular prism ABCDEF of length 15 cm.
AB = 6.4 cm, EB = 5.7 cm and the volume of the prism is 145 cm 3 .
(a) Show that angle EBA = 32° , correct to the nearest degree.
[3]
(b) Find the length of EA.
.......................................... cm [3]
015
(c) Calculate the shortest distance from E to AB.
............................................ cm [3]
(d) Calculate the angle BF makes with the base, ABCD, of the prism.
................................................. [4]
(e) The prism is made of plastic with density 938 kg/m 3 .
Calculate the mass of the prism in grams.
[Density = mass ' volume ]
.............................................. g [3]
016
4 (a)
P NOT TO
SCALE
8 cm
Q R
24 cm
(i) Calculate the area of triangle PQR.
......................................... cm 2 [2]
(ii) Calculate angle PRQ.
Angle PRQ = ................................................ [2]
(b)
NOT TO
SCALE
11 cm
6 cm
The diagram shows a half-cylinder of radius 6 cm and length 11 cm.
Calculate the volume of the half-cylinder.
......................................... cm 3 [2]
017
(c)
D T C T C
4 cm
S S
O
15 cm NOT TO
X X SCALE
A 20 cm B A B
(i) ABCD is a rectangle with AB = 20 cm and BC = 15 cm.
S, X and T are points on a circle centre O, such that DSA and DTC are tangents to the circle.
The radius of the circle is 4 cm and TX is a diameter of the circle.
The shape DSXT is removed from the corner of the rectangle, leaving the shaded shape shown
in the second diagram.
Calculate the area of the shaded shape.
......................................... cm 2 [5]
(ii) Calculate the perimeter of the shaded shape.
............................................ cm [3]
018
5 (a)
NOT TO
15 cm
SCALE
8 cm
A cone has base diameter 8 cm and perpendicular height 15 cm.
(i) Calculate the volume of the cone.
1
[The volume, V, of a cone with radius r and height h is V = rr 2 h .]
3
......................................... cm 3 [2]
(ii) A label completely covers the curved surface area of the cone.
Calculate the area of the label as a percentage of 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 .]
............................................. % [5]
019
(b)
NOT TO
SCALE
0.45 m
An empty cylindrical container has radius 0.45 m.
300 litres of water is poured into the container at a rate of 375 ml per second.
(i) Find the time taken, in minutes and seconds, for all the water to be poured into the container.
............................. min ............................. s [3]
(ii) Calculate the height of the water in the container.
............................................. m [3]
020
8 (a) A shop sells shirts for $x and jackets for $(x + 27).
The shop sells 4 shirts and 3 jackets for a total of $194.75 .
Write down and solve an equation to find the cost of one shirt.
$ ................................................ [3]
(b) Solve the simultaneous equations.
You must show all your working.
x 2 + 4y = 37
5x + y =- 8
x = ..................... , y = .....................
x = ..................... , y = ..................... [5]
021
(c) A solid cylinder has radius x and height 6x.
A sphere of radius r has the same surface area as the total surface area of the cylinder.
7
Show that r 2 = x 2 .
2
[The surface area, A, of a sphere with radius r is A = 4rr 2 .]
[4]
022
9 (a)
M A B
NOT TO
SCALE
D C N
The diagram shows a shape made from a square ABCD and two equal sectors of a circle.
The square has side 11 cm.
MAB and DCN are straight lines.
(i) Calculate the area of the shape.
......................................... cm 2 [3]
(ii) Calculate the perimeter of the shape.
............................................ cm [3]
023
(b)
H G
E F
NOT TO
SCALE
D
C
A B
The diagram shows a cube ABCDEFGH of edge 7 cm.
Calculate the angle between AG and the base of the cube.
................................................. [4]
