ALPERTON COMMUNITY SCHOOL
MATHS FACULTY
ACHIEVING GRADE A/A*
EXAM PRACTICE BY TOPIC
WEEK2
Calculator paper
Each set of questions is followed by solutions so you can check & mark your own work
CONTENTS
TOPIC
PAGE
NUMBER
GEOMETRY– Compound
measures ( speed , density )
Page 141
GEOMETRY– Surface area of
a 3-D shape
Page 202
GEOMETRY – Volume of
prisms
Page 200
GEOEMTRY : Volume of
non- prisms
Page 422
GCSE
GRADE
Date
Marks
completed
Density, Mass and Volume
1.
The mass of 5 m3 of copper is 44 800 kg.
(a)
Work out the density of copper. …………………………… kg/m3 (2)
The density of zinc is 7130 kg/m3.
(b)
Work out the mass of 5 m3 of zinc. ………………………… kg (2)
2.
Diagram NOT
accurately drawn
3.8 cm
2.5 cm
An ice hockey puck is in the shape of a cylinder with a radius of 3.8 cm, and a thickness of 2.5 cm.
It is made out of rubber with a density of 1.5 grams per cm3.
Work out the mass of the ice hockey puck. Give your answer correct to 3 significant figures.......... Grams
3.
3 cm
5 cm
10 cm
Diagram NOT accurately drawn
The diagram shows a solid cuboid. The cuboid has length 10 cm, width 8 cm and height 5 cm.
The cuboid is made of wood.
The wood has a density of 0.6 grams per cm3. Work out the mass of the cuboid.
.......................... Grams (Total 4 marks)
4.
60 cm2
24 cm
Diagram NOT accurately drawn
The diagram shows a solid hexagonal prism. The area of the cross-section of the prism is 60 cm2.
The length of the prism is 24 cm.
(a)
Work out the volume of the prism.
.................................... cm3
The prism is made from wood. The prism has a mass of 648 g.
(b)
(2)
Work out the density of the wood.
...................................... g/cm3
(2)
5.
3 cm
3 cm
5 cm
5 cm
5 cm
Diagram NOT accurately drawn
The solid shape, shown in the diagram, is made by cutting a hole all the way through a wooden cube.
The cube has edges of length 5 cm. The hole has a square cross section of side 3 cm.
(a)
Work out the volume of wood in the solid shape. ................................. cm3
(2)
The mass of the solid shape is 64 grams.
(b)
Work out the density of the wood.
................................. grams per cm3 (2) (Total 4 marks)
6.
The density of juice is 4 grams per cm3.
The density of water is 1 gram per cm3. 315 cm3 of drink is made by mixing 15 cm3 of juice with 300
cm3 of water. Work out the density of the drink.
...................................... grams per cm3
SOLUTIONS
1.
(a)
8960
2
44800 ÷ 5
(b)
35650
7130 × 5
2
2.
170
Vol = π × 3.82 × 2.5 = π × 14.44 × 2.5
= 45.36… × 2.5 = 113.411
Mass = “113” × 1.5 = 170.1165
4
3.
10 × 5 × 8 (= 400)
“400” × 0.6 = 240
4
4.
(a)
60 × 24 (= 1440)
= 1440
2
(b)
648 ÷ “1440”
= 0.45
2
(a)
53 – 5 × 3 × 3
125 – 45
5.
(b)
(5 × 5 – 3 × 3) × 5
(25 – 9) × 5
16 × 5 = 80
2
64 ÷ 80
0.8
2
6.
Mass of water = 300 × 1= 300 g
Mass of juice = 15 × 4 = 60g
Total mass = 360
Total volume = 315
Density = 360 ÷ 315
3
Speed, Time & Distance
1.
Kelly runs a distance of 100 metres in a time of 10.52 seconds.
The distance of 100 metres was measured to the nearest metre.
The time of 10.52 seconds was measured to the nearest hundredth of a second.
2.
3.
(a)
Write down the upper bound for the distance of 100 metres. ................................. metres (1)
(b)
Write down the lower bound for the time of 10.52 seconds.................................. seconds (1)
(c)
Calculate the upper bound for Kelly’s average speed. Write down all the figures on your calculator
display.................................. metres per second (2)
(d)
Calculate the lower bound for Kelly’s average speed. Write down all the figures on your calculator
display. ................................. metres per second (2) (Total 6 marks)
A tank contains 480 litres of water. A tap is opened, and water flows out of the tank at the rate of 0.2
litres per second. How long will it take to empty the tank?
40 minutes
96 minutes
960 minutes
2400 minutes
4800 minutes
A
B
C
D
E
A tank contained 48 000 cm3 of salt. The salt was removed from the tank at a constant rate.
It took 2 hours and 40 minutes to empty the tank completely.
At what rate, in cm3 per second, was the salt removed from the tank?
4.
5
6
13
36
300
A
B
C
D
E
There are 960 litres of water in a tank. A workman empties the tank.
The water flows out of the tank at a constant rate of 0.4 litres per second.
How long, in minutes, does it take the workman to empty the tank completely?
5.
40
96
384
960
2400
A
B
C
D
E
Water flows from a container at a constant rate of 0.1 litres per second.
How long does it take to fill a can with 9 litres of water?
6.
9 seconds
90 seconds
9 minutes
10 seconds
90 minutes
A
B
C
D
E
A plane is flying at a speed of 1440 kilometres per hour.
How long, in seconds, will the plane take to fly a distance of 1 kilometre?
0.4 seconds
2.4 seconds
2.5 seconds
4 seconds
24 seconds
A
B
C
D
E
(Total 1 mark
SOLUTIONS:
1.
(a)
100.5
1
(b)
10.515
1
(c)
100.5
= 9.5577746
10.515
2
(d)
99.5
10.525
9.45368..
2
[6]
VOLUME
1.
Diagram NOT
accurately drawn
4 cm
10 cm
The diagram shows a cylinder with a height of 10 cm and a radius of 4 cm.
(a)
Calculate the volume of the cylinder. Give your answer correct to 3 significant figures.
...........................cm3 (2)
The length of a pencil is 13 cm. The pencil cannot be broken.
(b)
Show that this pencil cannot fit inside the cylinder.
(3) (Total 5 marks)
2.
A cuboid has a square base of side x cm. The height of the cuboid is 1 cm more than the length x cm.
The volume of the cuboid is 230 cm3.
(a)
Show that
x3 + x2 = 230
2)
3.
Diagram NOT
accurately drawn
A
A
10 cm
5 cm
B
15 cm
10 cm
5 cm
5 cm
20 cm
B
20 cm
15 cm
The diagram represents a large cone of height 30 cm and base diameter 15 cm.
The large cone is made by placing a small cone A of height 10 cm and base diameter 5 cm on top of a
frustum B.
(a)
Calculate the volume of the frustum B Give your answer correct to 3 significant figures.
.........................cm3 (3)
d cm
Diagram NOT
accurately drawn
h cm
3d cm
The diagram shows a frustum. The diameter of the base is 3d cm and the diameter of the top is d cm.
The height of the frustum is h cm. The formula for the curved surface area, S cm2, of the frustum is
2
2
S = 2πd h  d
(b)
Rearrange the formula to make h the subject.
h = ...................................(3)
Two mathematically similar frustums have heights of 20 cm and 30 cm.
smaller frustum is 450 cm2.
(c)
The surface area of the
Calculate the surface area of the larger frustum..........................cm2 (2)
(Total 8 marks)4
4.
6 cm
Diagram NOT
accurately drawn
10 cm
5 cm
5 cm
The diagram shows a model. The model is a cuboid with a pyramid on top.
The base of the model is a square with sides of length 5 cm. The height of the cuboid in the model is 10
cm. The height of the pyramid in the model is 6 cm
(a)
Calculate the volume of the model.
....................................... cm3 (3)
5.
Diagram NOT accurately drawn
The diagram shows a prism of length 90 cm. The cross section, PQRST, of the prism is a semi-circle
above a rectangle. PQRT is a rectangle. RST is a semi-circle with diameter RT.
PQ = RT = 60 cm. PT = QR = 45 cm. Calculate the volume of the prism.
Give your answer correct to 3 significant figures
....................................... cm3 (Total 4 marks)
6.
The diagram shows a cylinder and a sphere.
The radius of the base of the cylinder is 2x cm and the height of the cylinder is h cm.
The radius of the sphere is 3x cm. The volume of the cylinder is equal to the volume of the sphere.
Express h in terms of x. Give your answer in its simplest form.
h = ........................................ (Total 3 marks)
7.
Diagram
accurately
NOT
drawn
A cone has a base radius of 5 cm and a vertical height of 8 cm.
(a) Calculate the volume of the cone.
Give your answer correct to 3 significant figures. ................................. cm3 (2)
8.
A cylinder has base radius x cm and height 2x cm. A cone has base radius x cm and height
h cm The volume of the cylinder and the volume of the cone are equal. Find h in terms of
x. Give your answer in its simplest form.
h cm
2x cm
h = ............................. (Total 3 marks)
x cm
x cm
9.
The diagram shows a storage tank.
The storage tank consists of a hemisphere on top of a cylinder.
3m
The height of the cylinder is 30 metres. The radius of the cylinder is 3
metres. The radius of the hemisphere is 3 metres.
(a)
3m
Calculate the total volume of the storage tank.
your answer correct to 3 significant figures.
30 m Give
3m
...................................... m3
A sphere has a volume of 500 m3.
(b)
Calculate the radius of the sphere Give your answer correct to 3 significant figures.
....................................... m (3) (Total 6 marks)
10.
4 cm
24 cm
Diagram NOT accurately drawn
A cylinder has a height of 24 cm and a radius of 4 cm. Work out the volume of the cylinder.
Give your answer correct to 3 significant figures.
…………………………… cm3 (Total 2 marks)
11.
A clay bowl is in the shape of a hollow hemisphere.
8 .2 c
m
7.7 cm
Diagram NOT accurately drawn
The external radius of the bowl is 8.2 cm. The internal radius of the bowl is 7.7 cm.
Both measurements are correct to the nearest 0.1 cm. The upper bound for the volume of clay is k cm3.
Find the exact value of k.
k = ………………………..(Total 4 marks)
12.
Diagram NOT accurately drawn
The diagram represents a large cone of height 6 cm and base diameter 18 cm.
The large cone is made by placing a small cone A of height 2 cm and base diameter 6 cm on top of a
frustum B.
Calculate the volume of the frustum B. Give your answer in terms of π.
(Total 4 marks)
.................................
13.
350 cm
1.2 cm
Diagram NOT accurately drawn
The diagram shows a piece of wood. The piece of wood is a prism of length 350 cm.
The cross-section of the prism is a semi-circle with diameter 1.2 cm. Calculate the volume of the piece of
wood. Give your answer correct to 3 significant figures.
………………………… cm3 (Total 4 marks)
14.
Diagram NOT
accurately drawn
10 cm
11 cm
3.5 cm
12 cm
A rectangular container is 12 cm long, 11 cm wide and 10 cm high. The container is filled with water to a
depth of 8 cm.
A metal sphere of radius 3.5 cm is placed in the water. It sinks to the bottom. Calculate the rise in the
water level. Give your answer correct to 3 significant figures...............................cm
(Total 4 marks)
15.
Diagram NOT accurately drawn
r cm
10 cm
The diagram shows a cylinder. The radius of the cylinder is r cm. The length of the cylinder is 10 cm.
The volume of the cylinder is 140 cm3. Work out the value of r. Give your answer correct to 3
significant figures.
r =.....................................(Total 3 marks)
Solutions:
1.
(a)
502  503 cm3
2
V =   42  10
(b)
164  13
3
P2 = 102 + 82
P = 164
2.
(a)
AG
2
x2(x + 1) = 230
3.
(a)
1700
3
  30 
(b)
h
2
2
7.5
2.5
   10 
 1767  65
3
3
S 2  4 2 d 4
4 2 d 2
3
S
 h2  d 2
2d
2
 S 
2
2

  h d
2

d


(c)
1012.5
2
 30 
 20 
   450 or 450   
 30 
 20 
4.
(a)
(b)
3
Area scale factor is 302
290 × 302 = 261000 cm2
261000 ÷ 10000
26.1
3
(
6.
60 × 40 × 2
4800
“4800”=  × 42 × h
"4800"
"50.265..."
= 95.5
6.
2
5 × 5 × 10 = 250
5 × 5 × 6 ÷ 3 = 50
300
1
×  × 302 + 60 × 45) × 90
2
(1/2 × 2827.43 + 2700) × 90
(1413.7.. + 2700) × 90
4113.7.. × 90 = 370234.5...
= 370 000
5.
2
4
5
4
3
 (2 x) 2 h   (3x) 3
4
 (3x) 3
3
h
= 9x
 (2 x) 2
3
7.
8.
πx2(2x) =
(a)
(b)
9.
1
π(x)2h
3
6x
3
Vc    32  30
2
Vh     3 3
3
Vc  Vh  288
905
3
4
   R 3  500
3
500 3
R3 
4
4.92
3
1210 cm3
2
  42  24
10.
156
4
1
1
×  × 92 × 6 – ×  × 3 2 × 2
3
3
162 – 6
11.
198
4
 × 0.62 (1.13…)
"1.13..."
2
“0.565…” × 350
12.
Vol sphere =
4
× π × 3.53
3
= 179.59...
"179.59"
Height =
12  11
1.36
13.
140 =  × r2 × 10
r2 = 140/10
(= 4.4563……)
2.11
4
3