1

1 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]

[Total: 4]

2
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 cm3.

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°.

[The volume, V, of a cone with radius r and height h is .]

................................................... [6]
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[Total: 6]

3
O

60°
NOT TO
24 cm SCALE

P Q

The diagram shows a sector OPQ of a circle with centre O and radius 24 cm.
The sector angle is 60°.

A cone is made from this sector by joining OP to OQ.

NOT TO
SCALE

P
Q

Calculate the volume of the cone.

[The volume, V, of a cone with radius r and height h is .]

................................................... cm3 [6]

 3

[Total: 6]

4
O

53°
NOT TO
9.5 cm
A B SCALE

X Y

The diagram shows a sector OXY of a circle with centre O and radius 9.5 cm.
The sector angle is 53°.
A lies on OX, B lies on OY and OA = OB.

(a) Show that the area of the sector is 41.7 cm2 , correct to 1 decimal place.

[2]

(b) The area of triangle OAB is of the area of sector OXY.

Calculate OA.

OA = ................................................... cm [4]

[Total: 6]
 4

5 A bag contains 15 000 cm3 of sand.

Some of this sand is used to completely fill a hole in the shape of a cylinder.
The hole is 30 cm deep and has radius 10 cm.

Calculate the percentage of the sand from the bag that is used.

................................................... % [3]

[Total: 3]

The diagram shows a sector of a circle with centre O, radius 8 cm and sector angle 165°.

(a) Calculate the total perimeter of the sector.

................................................... cm [3]
 5

(b) The surface area of a sphere is the same as the area of the sector.

Calculate the radius of the sphere.

[The surface area, A, of a sphere with radius r is .]

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

(c)

A cone is made from the sector by joining OA to OB.

(i) Calculate the radius, r, of the cone.

r = ................................................... cm [2]
 6

(ii) Calculate the volume of the cone.

[The volume, V, of a cone with radius r and height h is .]

................................................... cm3 [4]

[Total: 13]

7 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.

[The volume, V, of a sphere with radius r is .]

h = ................................................... [3]

[Total: 3]
 7

8 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.

[The volume, V, of a sphere with radius r is .]

r = ................................................... [3]

[Total: 3]

9
A solid cylinder has radius x cm and height cm.
The surface area of a sphere with radius R cm is equal to the total surface area of the cylinder.

Find an expression for R in terms of x.

[The surface area, A, of a sphere with radius r is .]

R = ................................................... [3]

[Total: 3]
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10

Water flows at a speed of 20 cm/s along a rectangular channel into a lake.

The width of the channel is 15 cm.
The depth of the water is 2.5 cm.

Calculate the amount of water that flows from the channel into the lake in 1 hour.
Give your answer in litres.

................................................... litres [4]

[Total: 4]
 9

11

A prism with a right-angled triangle as its cross-section has a volume of 1000 cm3.

Calculate the value of x.

x = ................................................... [4]

[Total: 4]
 10

12 A sphere with radius x cm has a volume of 1000 cm3.

Calculate the value of x.

[The volume, V, of a sphere with radius r is .]

x = ................................................... [3]

[Total: 3]

13 A cube with side length x cm has a volume of 1000 cm3.

Calculate the value of x.

x = ................................................... [1]

[Total: 1]

14

The diagram shows the surface of a garden pond, made from a rectangle and two semicircles.
The rectangle measures 3 m by 1.2 m.

(a) Calculate the area of this surface.

................................................... m2 [3]
 11

(b) The pond is a prism and the water in the pond has a depth of 20 cm.

Calculate the number of litres of water in the pond.

................................................... litres [3]

(c) After a rainfall, the number of litres of water in the pond is 1007.

Calculate the increase in the depth of water in the pond.

Give your answer in centimetres.

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

[Total: 9]

15
 12

The diagram shows a field ABCDE.

(a) Calculate the perimeter of the field ABCDE.

................................................... m [4]

(b) Calculate angle ABD.

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]
 13

(d) Calculate the area of the field ABCDE.

Give your answer in hectares.
[1 hectare = 10 000 m2]

................................................... hectares [4]

[Total: 16]

16 The volume of a solid metal sphere is 24 430 cm3.

(a) Calculate the radius of the sphere.

[The volume, V, of a sphere with radius r is .]

................................................... cm [3]
 14

(b) The metal sphere is placed in an empty tank.

The tank is a cylinder with radius 50 cm, standing on its circular base.
Water is poured into the tank to a depth of 60 cm.

Calculate the number of litres of water needed.

................................................... litres [3]

[Total: 6]

17 A tank is a cuboid measuring 1.8 m by 1.5 m by 1.2 m.

Water flows from a pipe into this empty tank at a rate of 200 cm3 per second.

Find the time it takes to fill the tank.

Give your answer in hours and minutes.

.............................. hours .............................. minutes [4]

[Total: 4]
 15

18 A cone with radius x cm and slant height cm has a volume of 1000 cm3.

Calculate the value of x.

[The volume, V, of a cone with radius r and height h is .]

x = ................................................... [4]

[Total: 4]
 16

19 Brad travelled from his home in New York to Chamonix.

• He left his home at 16 30 and travelled by taxi to the airport in New York.
This journey took 55 minutes and had an average speed of 18 km/h.

• He then travelled by plane to Geneva, departing from New York at 22 15.

The flight path can be taken as an arc of a circle of radius 6400 km with a sector angle of 55.5°.
The local time in Geneva is 6 hours ahead of the local time in New York.
Brad arrived in Geneva at 11 25 the next day.

• To complete his journey, Brad travelled by bus from Geneva to Chamonix.

This journey started at 13 00 and took 1 hour 36 minutes.
The average speed was 65 km/h.
The local time in Chamonix is the same as the local time in Geneva.

Find the overall average speed of Brad’s journey from his home in New York to Chamonix.
Show all your working and give your answer in km/h.

................................................... km/h [11]

[Total: 11]
 17

20

The diagram shows a hemispherical bowl of radius 5.6 cm and a cylindrical tin of height 10 cm.

(a) Show that the volume of the bowl is 368 cm3, correct to the nearest cm3.

[The volume, V, of a sphere with radius r is .]

[2]

(b) The tin is completely full of soup.

When all the soup is poured into the empty bowl, 80% of the volume of the bowl is filled.

Calculate the radius of the tin.

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

[Total: 6]

21

The diagram shows a sector of a circle of radius 3.8 cm.

 18

The arc length is 7.7 cm.

(a) Calculate the value of y.

y = ................................................... [2]

(b) Calculate the area of the sector.

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

[Total: 4]

22

The diagram shows a cone with radius 1.75 cm and height 6 cm.
 19

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

[The curved surface area, A, of a cone with radius r and slant height l is .]

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

(b)

The cone contains salt to a depth of 4.5 cm.

The top layer of the salt forms a circle that is parallel to the base of the cone.

(i) Show that the volume of the salt inside the cone is 18.9 cm3, correct to 1 decimal place.
[The volume, V, of a cone with radius r and height h is .]

[4]
 20

(ii) The salt is removed from the cone at a constant rate of 200 mm3 per second.

Calculate the time taken for the cone to be completely emptied.

Give your answer in seconds, correct to the nearest second.

................................................... s [3]

[Total: 12]

23

The diagram shows a company logo made from a rectangle and a major sector of a circle.
The circle has centre O and radius OA.
OA = OD = 0.5 cm and AB = 1.5 cm.
E is a point on OC such that OE = 0.25 cm and angle OED = 90°.
 21

(a) Calculate the perimeter of the logo.

................................................... cm [5]

(b) Calculate the area of the logo.

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

(c) A mathematically similar logo is drawn.

The area of this logo is 77.44 cm2.

(i) Calculate the radius of the major sector in this logo.

................................................... cm [3]
 22

(ii) A gold model is made.

This model is a prism with a cross-section of area 77.44 cm2.

This gold model is 15 mm thick.

One cubic centimetre of gold has a mass of 19 grams.

Calculate the mass of the gold model in kilograms.

................................................... kg [3]

[Total: 14]

24

The diagram shows a prism with length 18 cm and volume 253.8 cm3.
The cross-section of the prism is a right-angled triangle with base 6 cm and height h cm.

(a) (i) Show that the value of h is 4.7 .

[3]
 23

(ii) Calculate the value of x.

x = ................................................... [2]

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

................................................... cm2 [6]

[Total: 11]

25

The diagram shows a circle, centre O.

The straight line ABC is a tangent to the circle at B.
OB = 8 cm, AB = 15 cm and BC = 22.4 cm.
AO crosses the circle at X and OC crosses the circle at Y.
 24

(a) Calculate angle XOY.

Angle XOY = ................................................... [5]

(b) Calculate the length of the arc XBY.

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

(c) Calculate the total area of the two shaded regions.

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

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[Total: 11]

26 The lake behind a dam has an area of 55 hectares.

When the gates in the dam are open, water flows out at a rate of 75 000 litres per second.

(a) Show that 90 million litres of water flows out in 20 minutes.

[1]

(b) Beneath the surface, the lake has vertical sides.

Calculate the drop in the water level of the lake when the gates are open for 20 minutes.
Give your answer in centimetres.
[1 hectare = 104 m2, 1000 litres = 1 m3]

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

[Total: 4]
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27

The sector of a circle has radius 8.5 m and angle 76°.

Calculate the perimeter of the sector.

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

[Total: 3]

28

A solid metal cone has radius 10 cm and height 36 cm.

(a) Calculate the volume of this cone.

[The volume, V, of a cone with radius r and height h is .]

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

 27

(b) The cone is cut, parallel to its base, to give a smaller cone.

The volume of the smaller cone is half the volume of the original cone.
The smaller cone is melted down to make two different spheres.
The ratio of the radii of these two spheres is 1 : 2.

Calculate the radius of the smaller sphere.

[The volume, V, of a sphere with radius r is .]

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

[Total: 6]

29 The diagram shows a right-angled triangle ABC.

The area of this triangle is 30 cm2.

 28

(a) Show that .

[3]

(b) Use factorisation to solve the equation .

x = .............................. or x = .............................. [3]

(c) Calculate BC.

BC = ................................................... cm [3]

[Total: 9]

30

The diagram shows a solid cone.

The radius is 8 cm and the slant height is 17 cm.
 29

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

[The curved surface area, A, of a cone with radius r and slant height l is .]

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

(b) Calculate the volume of the cone.

[The volume, V, of a cone with radius r and height h is .]

................................................... cm3 [4]

(c) The cone is made of wood and 1 cm3 of the wood has a mass of 0.8 g.

Calculate the mass of the cone.

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

(d) The cone is placed in a box.

The total mass of the cone and the box is 1.2 kg.

Calculate the mass of the box.

Give your answer in grams.

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

[Total: 8]
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31

The diagram shows three solids.

The base radius of the cone is 6 cm and the slant height is 12 cm.
The radius of the sphere is x cm and the radius of the hemisphere is y cm.
The total surface area of each solid is the same.

(a) Show that the total surface area of the cone is 108π cm2.

[The curved surface area, A, of a cone with radius r and slant height l is .]

[2]

(b) Find the value of x and the value of y.

[The surface area, A, of a sphere with radius r is .]

x = ...................................................

y = ................................................... [4]

[Total: 6]
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32

Water flows through a cylindrical pipe at a speed of 8 cm/s.

The radius of the circular cross-section is 1.5 cm and the pipe is always completely full of water.

Calculate the amount of water that flows through the pipe in 1 hour.
Give your answer in litres.

................................................... litres [4]

[Total: 4]

33 A solid hemisphere has volume 230 cm3.

(a) Calculate the radius of the hemisphere.

[The volume, V, of a sphere with radius r is .]

................................................... cm [3]
 32

(b) A solid cylinder with radius 1.6 cm is attached to the hemisphere to make a toy.

The total volume of the toy is 300 cm3.

(i) Calculate the height of the cylinder.

................................................... cm [3]
 33

(ii) A mathematically similar toy has volume 19 200 cm3.

Calculate the radius of the cylinder for this toy.

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

[Total: 9]

34

In the pentagon ABCDE, angle ACB = angle AED = 90°.

Triangle ACD is equilateral with side length 12 cm.
DE = BC = 6 cm.
 34

(a) Calculate angle BAE.

Angle BAE = ................................................... [4]

(b) Calculate AB.

AB = ................................................... cm [2]

(c) Calculate AE.

AE = ................................................... cm [3]

(d) Calculate the area of the pentagon.

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

[Total: 13]

35 In this question, all measurements are in metres.

 35

The diagram shows a right-angled triangle.

(a) Show that .

[3]

(b) Solve .
Show all your working and give your answers correct to 2 decimal places.

x = .............................. or x = .............................. [4]

(c) Calculate the perimeter of the triangle.

................................................... m [2]
 36

(d) Calculate the smallest angle of the triangle.

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

[Total: 11]

36

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]
 37

(b) Calculate angle OAC.

Angle OAC = ................................................... [3]

(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]

[Total: 16]
 38

37

P, Q and R are points on the circumference of the circle, centre O.

PO is parallel to QR and angle POQ = 48°.

(a) Find angle OPR.

Angle OPR = ................................................... [2]

(b) The radius of the circle is 5.4 cm.

Calculate the length of the major arc PQ.

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

[Total: 5]
 39

38 A solid metal prism with volume 500 cm3 is melted and made into 6 identical spheres.

Calculate the radius of each sphere.

[The volume, V, of a sphere with radius r is .]

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

[Total: 3]

39

The diagram shows a solid prism with length 15.2 cm.

The cross-section of this prism is a regular hexagon with side 7 cm.

(a) Calculate the volume of the prism.

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

 40

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

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

[Total: 8]

40

The vertices of a square ABCD lie on the circumference of a circle, radius 8 cm.

(a) Calculate the area of the square.

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

(b) (i) Calculate the area of the shaded segment.

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

 41

(ii) Calculate the perimeter of the shaded segment.

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

[Total: 9]