Difficulty level: Easy

1 (a) A heavy ball hangs from a point P, P

P
11 m above horizontal ground, by means
of a thin wire.
11
11
The point D is on the ground vertically below P.
The point B is on the ground 4 m from D.

D
D B
B
44
(i) Calculate the angle of elevation of P from B. [2]

P
P

(ii) The ball swings, with the wire straight, 28°

28°
in the vertical plane PDB. 11
11 X
X
When the ball is at X, directly above B,
DP̂X = 28°.
D
D B
B
Calculate
44
(a) PX, [2]

(b) XB. [3]

4
(b) [The volume of a sphere is 3 πr 3.]

The ball is a sphere of volume 96 cm3.

Calculate its radius. [2]

© UCLES 2009 4024/O2/M/J/09 [Turn over

 12
2

1
12 [Volume of a cone = 3 π r 2h]
[Curved surface area of a cone = π rl]
B
Diagram I shows a solid cone with C
as the centre of its base.
B is the vertex of the cone and A is a point
on the circumference of its base.
AC = 9 cm and BC = 12 cm. Diagram I 12

A C
9

(a) Calculate

(i) AB, [2]

(ii) the total surface area of the cone, [2]

(iii) the volume of the cone. [2]

(b) The cone in Diagram I is cut, parallel B

to the base, to obtain a small cone shown
in Diagram II and a frustum shown in Diagram II
Diagram III.
X 3 Y
Y is the centre of the base of the small cone.
X is the point on the circumference of this X 3 Y
base and on the line AB such that Diagram III
XY = 3 cm.

A C

Calculate

(i) BY, [1]

(ii) AX, [1]

(iii) the circumference of the base of the small cone, [2]

(iv) the volume of the frustum. [2]

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University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local
Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.

© UCLES 2010 4024/22/M/J/10

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3
Section B [48 marks] Do not
write in this
Answer four questions in this section. margin

Each question in this section carries 12 marks.

7 (a) Tuna chunks are sold in cylindrical tins.

The 130 g tin costs $1.00 and the 185 g tin costs $1.50.

Which one is the better value for money?

Show all your working.

Answer ..................................... [2]

(b) A closed cylindrical tin is 11 cm high and the base has a diameter of 7 cm.

11

(i) Calculate the volume of this tin.

Answer ............................. cm3 [2]

© UCLES 2012 4024/21/O/N/12

 15

(ii) Calculate the total external surface area of this tin. Do not
write in this
margin

Answer ..............................cm2 [3]

(iii) In addition to the surface area, a closed tin requires an extra 30 cm2 of metal to allow
the top, bottom and side to be joined together.

Calculate the area of metal required for 30 000 closed tins.

Give your answer in square metres.

Answer .............................. m2 [2]

(c) Two geometrically similar jugs have

volumes of 1000 cm3 and 512 cm3.
They have circular bases.
The diameter of the base of the larger jug
is 9 cm.

Calculate the diameter of the base of the

smaller jug.

Answer .............................. cm [3]

© UCLES 2012 4024/21/O/N/12 [Turn over

4 22

12 (a) For
Examiner’s
r Use

46

A cylindrical tank of height 46 cm and radius r cm has a capacity of 70 litres.

Find the radius correct to the nearest centimetre.

Answer .......................................... cm [3]

(b)

x
4
125° 20
11

A triangular prism has length 20 cm.

The sides of the shaded cross-section are 4 cm, 11 cm and x cm.
The angle between the sides of length 4 cm and 11 cm is 125°.

(i) Calculate the area of the shaded cross-section.

Answer .........................................cm2 [2]

© UCLES 2013 4024/22/M/J/13

5 14

1 2
9 [Volume of a cone = rr h ]
3
[Curved surface area of a cone = πrl]

15

The diagram shows a solid cone of height 15 cm and base radius 6 cm.

(a) Show that the slant height of the cone is 16.2 cm, correct to one decimal place.

[1]

(b) Calculate the total surface area of the cone.

Answer ............................................ cm2 [3]

(c) Calculate the volume of the cone.

Answer ............................................ cm3 [2]

© UCLES 2014 4024/22/M/J/14

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9

1 2
(b) [Volume of a cone = rr h]
3
[Curved surface area of a cone = rrl]

9.5

A cone has height 9.5 cm and volume 115 cm3.

(i) Show that the radius of the base of the cone is 3.4 cm, correct to 1 decimal place.

[2]

(ii) Calculate the curved surface area of the cone.

Answer ....................................cm2 [3]

© UCLES 2018 4024/22/M/J/18 [Turn over

7
5

12 A cone with height 14.8 cm has volume 275 cm3.

Calculate the radius of the cone.

1
[The volume, V, of a cone with radius r and height h is V = rr 2 h .]
3

............................................... cm [3]

13 Factorise.

(a) 7k 2 - 15k

.................................................... [1]

(b) 12 (m + p) + 8 (m + p) 2

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

14 Eric invests an amount in a bank that pays compound interest at a rate of 2.16% per year.
At the end of 5 years, the value of his investment is $6 999.31 .

Calculate the amount Eric invests.

$ .................................................. [3]

© UCLES 2019 0580/22/F/M/19 [Turn over

8 3

6 Expand and simplify (x + 3) (x + 5) .

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

7 Find the gradient of the line that is perpendicular to the line 2y = 3 + 5x.

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

8 When sin x° = 0.36, find

(a) the acute angle x°,

.................................................... [1]

(b) the obtuse angle x°.

.................................................... [1]

9 A cuboid measures 5 cm by 7 cm by 9.5 cm.

NOT TO
SCALE
7 cm

5 cm
9.5 cm

Work out the surface area of this cuboid.

............................................. cm2 [3]

© UCLES 2019 0580/23/O/N/19 [Turn over
9
13

1 2
9 [Volume of a cone = rr h]
3
[Curved surface area of a cone = rrl ]

A cone has radius 6 cm and slant height l cm.

The total surface area of the cone is 84r cm2.

(a) Show that l = 8.

[2]

(b) Calculate the volume of the cone.

.......................................... cm3 [3]

(c) A similar cone has a total surface area of 47.25r cm2.

Find the radius of this cone.

............................................ cm [2]

© UCLES 2021 4024/22/M/J/21 [Turn over

Answers
 Difficulty level: Intermediate

20
30

The diagram shows an open rectangular tank with base 20 cm by 30 cm.

The tank contains 9600 cm3 of water.

(a) (i) State the number of litres of water in the tank. [1]

(ii) Calculate the depth of the water. [2]

(iii) Calculate the total surface area of the tank that is in contact with the water. [2]

(iv) The water had entered the tank through a circular pipe of radius 0.8 cm.
It flowed through the pipe at 25 centimetres per second.

How long did the 9600 cm3 of water take to enter the tank?
Give your answer correct to the nearest second. [3]
4 3
(b) [Volume of a sphere = πr ]
3
250 identical spheres are placed in the bottom of the tank.
Each sphere has a volume of 2.6 cm3.

(i) Calculate by how much the water level in the tank will rise.
Give your answer in millimetres. [2]

(ii) Calculate the radius of one of these spheres. [2]

© UCLES 2010 4024/21/O/N/10

2
9
9
B

C A 5
B 5
5
C D
D
Diagram II

Diagram I

Diagram I shows a cube with a triangular pyramid removed from one vertex.
This triangular pyramid ABCD is shown in Diagram II.
AB = AC = AD = 5 cm.

(a) State the height of this pyramid when the base is triangle ABD. [1]

(b) [The volume of a pyramid = 1 × area of base × height]

3
Calculate

(i) the volume of the pyramid, [2]

(ii) the area of triangle BCD, [3]

(iii) the height of the pyramid when the base is triangle BCD. [3]

(c) An identical triangular pyramid is removed from each of the other 7 vertices of the cube to form
the new solid shown in Diagram III.

Diagram III

The original cube had 6 faces, 8 vertices and 12 edges.

For the new solid, write down the number of

(i) faces, [1]

(ii) vertices, [1]

(iii) edges. [1]

© UCLES 2010 4024/22/O/N/10 [Turn over

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18

11 (a)

4
50° 20

Thediagramshowsasolidtriangularprism.
Alllengthsaregivenincentimetres.

(i) Calculatetheareaofthecross-sectionoftheprism.

Answer .....................................cm2[2]

(ii) Calculatethevolumeoftheprism.

Answer .....................................cm3[1]

(iii) Calculatethetotalsurfaceareaoftheprism.

Answer .....................................cm2[5]

© UCLES 2015 4024/21/O/N/15

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(b) Acylinderhasaheightof70cmandavolumeof0.1m3.

Calculatetheradiusofthecylinder,givingyouranswerincentimetres.

        

Answer ...................................... cm[4]

© UCLES 2015 4024/21/O/N/15

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19

(ii) Solid II is a cone with volume of 3000 cm3.

The perpendicular height of the cone is twice its radius.

Which solid is the taller and by how much?

Solid II

Answer Solid ............ is the taller by .............................. cm [4]

(b) The diagram shows a triangular prism of length 24 cm. 24

Its cross-section is an equilateral triangle with sides 8 cm.

Calculate the total surface area of the prism.

Answer ................................... cm2 [4]

© UCLES 2016 4024/22/M/J/16
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16

4
9 (a) [Volume of a sphere = rr 3 ]
3
[Surface area of a sphere = 4rr 2 ]

24

The diagram shows lamp A.

It is made in the shape of a cylinder with a hemisphere on top.
The radius of the hemisphere and the radius of the cylinder are both 3 cm.
The total height of the lamp is 24 cm.

(i) Show that the volume of lamp A is 650 cm3, correct to 3 significant figures.

[4]

(ii) Calculate the curved surface area of lamp A.

Answer ................................... cm2 [3]

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6
8

1
4 [Volume of cone = rr 2 h ]
3
[Curved surface area of a cone = rrl ]

15

95

The diagram shows a gate post.

It is made in the shape of a cylinder with a cone on top.
The cylinder and the cone each have diameter 8 cm.
The height of the cylinder is 95 cm and the height of the cone is 15 cm.

(a) Calculate the volume of the gate post.

............................................. cm3 [3]

(b) Show that the total curved surface area of the gate post is 2580 cm2, correct to 3 significant figures.

[5]

© UCLES 2019 4024/21/O/N/19

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10

5
H

D
G

2.25 E
C

1.85 F
A
2.10
1.55
B

The diagram shows a garden shed positioned on horizontal ground.

It is in the shape of a prism with trapezium ABCD as its cross-section.
The base of the shed, ABFE, is a rectangle.
AB = 1.55 m, AD = 2.25 m, BC = 1.85 m and BF = 2.10 m.

(a) Calculate the volume of the shed.

............................................ m 3 [3]

© UCLES 2020 4024/21/O/N/20

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15

(b)
Solid A

6
40°

The cross-section of solid A is the sector of a circle of radius 6 cm and angle 40°.
The height of solid A is 5 cm.

(i) Calculate the total surface area of solid A.

.......................................... cm 2 [4]

© UCLES 2020 4024/21/O/N/20 [Turn over

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9

18

7 cm
NOT TO
SCALE

12 cm

The diagram shows a solid made from a cylinder and a hemisphere, both of radius 7 cm.
The cylinder has length 12 cm.

Work out the total surface area of the solid.

[The surface area, A, of a sphere with radius r is A = 4rr 2 .]

........................................... cm2 [4]

© UCLES 2020 0580/21/O/N/20 [Turn over

Answers
 Difficulty level: Hard

1 20

10 A cylindrical candle has a height of 5 cm.

12 mm
Do not
A is the centre of the top of the candle and B is the write in this
A margin
centre of the base of the candle.
The wick runs from B through A and extends
12 mm above A.
5 cm

(a) How many of these candles can be made using a 2 m length of wick?

Answer ........................................ [2]

(b) The wick is in the form of a solid cylinder.

The volume of the wick inside the candle from A to B is 0.2 cm3.

(i) Calculate the radius of the wick.

Give your answer in millimetres.

Answer ................................. mm [3]

© UCLES 2011 4024/21/M/J/11

 21

(ii) One candle was made by pouring candle wax into a cylindrical mould so that it Do not
surrounded the wick. write in this
This mould has an internal radius of 1.9 cm. margin

(a) Calculate the volume of candle wax required to make this candle.

Answer ................................. cm3 [3]

(b) How many of these candles can be made using 3 litres of candle wax?

Answer ........................................ [2]

(c)

length

One of these candles is placed on a rectangular piece of wrapping paper.

The paper is wrapped around the candle so that it covers the outside and there is an
extra 1 cm for an overlap.

What is the length, in centimetres, of paper required to wrap one candle?

Answer .................................. cm [2]

© UCLES 2011 4024/21/M/J/11 [Turn over

2
18

10 Afueltankerdeliversfuelinacylindricalcontaineroflength9.5mandradius0.8m. For
Examiner’s
(a) Afterseveraldeliveries,thefuelremaininginthecontainerisshowninthediagram. Use

9.5

O
0.8
A
B

t = 90c.
AB ishorizontal,Oisthecentreofthecircularcross-sectionand AOB

(i) Calculatethecurvedsurfaceareaofthecontainerthatisincontactwiththefuel.

Answer ......................................... m2[2]

(ii) Calculatethevolumeoffuelremaininginthecontainer.

Answer ......................................... m3[4]

(iii) Calculatethisvolumeremainingasapercentageofthevolumeofthewholecontainer.

Answer  ...........................................%[2]

©UCLES2013 4024/21/O/N/13
 19

(b) Thefuelispumpedthroughacylindricalpipeofradius4.5cmatarateof300cm/s. For

Examiner’s
(i) Calculatethevolumepumpedin1second. Use

Answer  ....................................... cm3[1]

(ii) Calculatethetimetaken,inminutes,topump25000litresoffuel.
Giveyouranswercorrecttothenearestminute.

Answer  ................................. minutes[3]

©UCLES2013 4024/21/O/N/13 [Turn over

3
8

4 3
4 [The volume of a sphere is rr ]
3
(a)

A spoon used for measuring in cookery consists of a hemispherical bowl and a handle.
The internal volume of the hemispherical bowl is 20 cm3.
The handle is of length 5 cm.

(i) Find the internal radius of the hemispherical bowl.

Answer .................................... cm [2]

(ii) The hemispherical bowl of a geometrically similar spoon has an internal volume of 50 cm3.

Find the length of its handle.

Answer .................................... cm [2]

© UCLES 2015 4024/22/O/N/15

 9

(b) [The surface area of a sphere is 4πr2]

An open hemisphere of radius 5.5 cm is used to make a metal kitchen strainer.

50 holes are cut out of the curved surface.
Assume that the piece of metal removed to make each hole is a circle of radius 1.5 mm.

Calculate the external surface area that remains.

Answer .................................. cm2 [3]

© UCLES 2015 4024/22/O/N/15 [Turn over

4
14

Section B[48marks]

Answerfourquestionsinthissection.

Eachquestioninthissectioncarries12marks.

7 (a) ACisadiameterofthecircle,centreO,radius5cm.
t =64°.
ACB B

  Calculatethelengthoftheminorarc BC.
64°
A C
5 O

Answer ..................................... cm[4]

(b)

16.5
rim

15.5

Abakingtrayisanopencylinderofradius15.5cmwitharim.
Theouteredgeoftherimisacircleofradius16.5cm.

© UCLES 2016 4024/21/M/J/16

 15

d mm
15.5 cm

Tomakeapizza,thebakingtrayiscompletelyfilledwithdoughtoadepthofdmm.
Theopencylinderholds500cm3ofdough.

Calculatethedepthofthedough,dmm,givingyouranswercorrecttothenearestmillimetre.

Answer .................................... mm[3]

© UCLES 2016 4024/21/M/J/16 [Turn over

5 18

1 2
11 [ Volume of a cone = πr h ]
3
(a)

3.5 r

20

Solid I

Solid I is a cylinder with a small cylinder removed from its centre, as shown in the diagram.
The height of each cylinder is 20 cm and the radius of the small cylinder is r cm.
The radius of the large cylinder is 3.5 cm greater than the radius of the small cylinder.
The volume of Solid I is 3000 cm3.

(i) Calculate r.

Answer r = .................................... [4]

© UCLES 2016 4024/22/M/J/16

 (ii) Solid II is a cone with volume of 3000 cm3.
The perpendicular height of the cone is twice its radius.

Which solid is the taller and by how much?

Solid II

Answer Solid ............ is the taller by .............................. cm [4]

(b) The diagram shows a triangular prism of length 24 cm.

Its cross-section is an equilateral triangle with sides 8 cm.

Calculate the total surface area of the prism.

Answer ................................... cm2 [4]

© UCLES 2016 4024/22/M/J/16
6 8

4 3
4 [The volume of a sphere is rr ]
3
[The surface area of a sphere is 4rr 2 ]

0.8

1.5

3.8

A hemispherical bowl is made of material that is 0.8 cm thick.

The outside rim of the bowl has radius 9 cm.
The bowl is attached to a base which is a solid cylinder, of radius 3.8 cm and height 1.5 cm.

(a) Calculate the surface area of the inside of the hemispherical bowl.

Answer ................................... cm2 [2]

© UCLES 2016 4024/21/O/N/16

 9

(b) Calculate the total volume of material used to make the bowl and the base.

Answer ................................... cm3 [5]

© UCLES 2016 4024/21/O/N/16 [Turn over

7
12

1 2
8 [Volume of a cone = rr h]
3
[Curved surface area of a cone = rrl ]
4 3
[Volume of a sphere = rr ]
3
[Surface area of a sphere = 4rr2]

18

The diagram shows solid A which is made from a hemisphere joined to a cone of equal radius.
The hemisphere and the cone each have radius 6 cm.
The total height of the solid is 18 cm.

(a) Show that the slant height, x cm, of the cone is 13.4 cm, correct to 1 decimal place.

[2]

(b) Calculate the total surface area of solid A.

Answer ................................... cm2 [3]

© UCLES 2017 4024/21/O/N/17

 13

(c) Calculate the volume of solid A.

Answer ................................... cm3 [3]

© UCLES 2017 4024/21/O/N/17 [Turn over

8 14

1
8 [Volume of cone = rr 2 h ]
3
[Curved surface area of a cone = rrl ]

16

15

12

c 45

The diagram shows a bowl with a circular base.

The curved surface of the bowl is formed by removing a cone with radius 12 cm and height 45 cm from a
larger cone as shown in the diagram.
The radius of the top of the bowl is 16 cm and its height is 15 cm.

(a) Calculate the volume of the bowl.

............................................. cm3 [3]

(b) The slant height of the cone that has been removed is c cm.

Show that c = 46.6, correct to 3 significant figures.

[2]

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(c) The bowl is completely filled with water.

Calculate the total surface area of the bowl that is in contact with the water.

............................................. cm2 [4]

© UCLES 2019 4024/22/O/N/19 [Turn over

9
8

4
4 (a) [Volume of a sphere = rr 3 ]
3
[Surface area of a sphere = 4rr 2 ]

16

The diagram shows a solid formed by joining a cylinder to a hemisphere.

The diameter of the cylinder is 9 cm and its height is 16 cm.

(i) The volume of the hemisphere is equal to the volume of the cylinder.

Show that the radius of the hemisphere is 7.86 cm, correct to 2 decimal places.

[4]

(ii) Calculate the total surface area of the solid.

.......................................... cm 2 [3]

© UCLES 2020 4024/22/O/N/20

Answers