Diagram NOT
accurately drawn
10 cm
11 cm
12 cm
3.5 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.
Bemrose Community School
1
..............................cm
(Total 4 marks)
2.
B
Diagram NOT
accurately drawn
C
A
10.4 cm
10.4 cm
120°
O
The diagram shows a sector OABC of a circle with centre O.
OA = OC = 10.4 cm.
Angle AOC = 120°.
Calculate the area of the shaded segment ABC.
Give your answer correct to 3 significant figures.
…………………….cm2
(Total 4 marks)
Bemrose Community School
2
3.
O
Diagram NOT
accurately drawn
40°
9 cm
The diagram shows a sector of a circle, centre O.
The radius of the circle is 9 cm.
The angle at the centre of the circle is 40°.
Find the perimeter of the sector.
Leave your answer in terms of π.
...........................cm
(Total 4 marks)
Bemrose Community School
3
4.
The diagram below shows a 6-sided shape.
All the corners are right angles.
All measurements are given in centimetres.
Diagram NOT accurately drawn
The area of the shape is 25 cm2.
Show that
6x2 + 17x – 39 = 0
(Total 3 marks)
Bemrose Community School
4
5.
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)
Bemrose Community School
5
6.
The radius of the base of a cone is x cm and its height is h cm.
The radius of a sphere is 2x cm.
The volume of the cone and the volume of the sphere are equal.
Express h in terms of x.
Give your answer in its simplest form.
h = ………………..
(Total 3 marks)
Bemrose Community School
6
7.
Diagram NOT accurately drawn
B
C
A
10.4 cm
10.4 cm
120°
O
The diagram shows a sector OABC of a circle with centre O.
OA = OC = 10.4 cm.
Angle AOC = 120°.
(a)
Calculate the length of the arc ABC of the sector.
Give your answer correct to 3 significant figures.
.....................................cm
(3)
Bemrose Community School
7
(b)
Calculate the area of the shaded segment ABC.
Give your answer correct to 3 significant figures.
.....................................cm2
(4)
(Total 7 marks)
8.
The diagram shows a sector of a circle with a radius of x cm and centre O.
PQ is an arc of the circle.
Angle POQ = 120.
O
x cm
P
120°
Diagram NOT
accurately drawn
Q
(a)
Write down an expression in terms of  and x for
(i)
the area of this sector,
................................
(ii)
the arc length of this sector.
................................
(2)
Bemrose Community School
8
The sector is the net of the curved surface of this cone.
Arc PQ forms the circumference of the circle that makes the base of the cone.
h cm
x cm
The curved surface area of the cone is A cm2.
The volume of the cone is V cm3.
The height of the cone is h cm.
Given that V = 3A,
(b)
find the value of h.
.............................
(3)
(Total 5 marks)
Bemrose Community School
9
9.
y
Diagram NOT
accurately drawn
2
y = a sin x°
1
0
90°
180°
270°
360°
x
–1
–2
y = cos x° + b
The diagram shows part of two graphs.
The equation of one graph is
The equation of the other graph is
(a)
y = a sin x
y = cos x + b
Use the graphs to find the value of a and the value of b.
a = .............................
b = .............................
(2)
(b)
Use the graphs to find the values of x in the range 0°  x  720 when
a sin x = cos x + b.
x = ..........................................
(2)
Bemrose Community School
10
(c)
Use the graphs to find the value of a sin x – (cos x + b) when x = 450.
.............................
(2)
(Total 6 marks)
Bemrose Community School
11
10.
F
E
Diagram NOT
accurately drawn
C
30°
D
60 cm
m
60 c
B
A
The diagram represents a prism.
AEFD is a rectangle.
ABCD is a square.
EB and FC are perpendicular to plane ABCD.
AB = 60 cm.
AD = 60 cm.
Angle ABE = 90°.
Angle BAE = 30°.
Calculate the size of the angle that the line DE makes with the plane ABCD.
Give your answer correct to 1 decimal place.
..........................°
Bemrose Community School
12
(Total 4 marks)
Bemrose Community School
13
11.
Diagram NOT accurately drawn
D
A
54°
28°
25 m
Bemrose Community School
B
C
14
The diagram shows a vertical tower DC on horizontal ground ABC.
ABC is a straight line.
The angle of elevation of D from A is 28°.
The angle of elevation of D from B is 54°.
AB = 25 m.
Calculate the height of the tower.
Give your answer correct to 3 significant figures.
..................................... m
(Total 5 marks)
Bemrose Community School
15
12.
x+2
Diagram NOT
accurately drawn
x–5
x+6
The diagram shows a trapezium.
The lengths of three of the sides of the trapezium are x – 5, x + 2 and x + 6.
All measurements are given in centimetres.
The area of the trapezium is 36 cm2.
(a)
Show that x2 – x – 56 = 0
(4)
Bemrose Community School
16
(b)
(i)
Solve the equation x2 – x – 56 = 0
…………………………
(ii)
Hence find the length of the shortest side of the trapezium.
…………………… cm
(4)
(Total 8 marks)
Bemrose Community School
17
13.
y
100
y = a – bcos(kt)
80
60
40
20
t
O
30
60
90
120
The graph of y = a – bcos(kt), for values of t between 0° and 120°, is drawn on the grid.
Use the graph to find an estimate for the value of
(i)
a,
.....................................
(ii)
b,
....................................
(iii)
k.
....................................
(Total 3 marks)
Bemrose Community School
18
14.
H
G
E
F
3 cm
Diagram NOT
accurately drawn
D
A
5 cm
C
7 cm
B
The diagram represents a cuboid ABCDEFGH.
AB = 5 cm.
BC = 7 cm.
AE = 3 cm.
(a)
Calculate the length of AG.
Give your answer correct to 3 significant figures.
...................................... cm
(2)
Bemrose Community School
19
(b)
Calculate the size of the angle between AG and the face ABCD.
Give your answer correct to 1 decimal place.
........................................
(2)
(Total 4 marks)
Bemrose Community School
20
15.
Diagram NOT
accurately drawn
T
x
x+5
O
x+8
A
AT is a tangent at T to a circle, centre O.
OT = x cm, AT = (x + 5) cm, OA = (x + 8) cm.
(a)
Show that
x2 – 6x – 39 = 0
(4)
(b)
Solve the equation x2 – 6x – 39 = 0 to find the radius of the circle.
Give your answer correct to 3 significant figures.
..................... cm
(3)
Bemrose Community School
21
(Total 7 marks)
Bemrose Community School
22
16.
Diagram NOT
accurately drawn
3 cm
3 cm
The radius of a sphere is 3 cm.
The radius of the base of a cone is also 3 cm.
The volume of the sphere is 3 times the volume of the cone.
Work out the curved surface area of the cone.
Give your answer as a multiple of .
Bemrose Community School
23
…………………………… cm2
(Total 7 marks)
Bemrose Community School
24
17.
6 cm
4 cm
Diagram NOT
accurately drawn
A
B
Cylinder A and cylinder B are mathematically similar.
The length of cylinder A is 4 cm and the length of cylinder B is 6 cm.
The volume of cylinder A is 80 cm3.
Calculate the volume of cylinder B.
………………………… cm3
(Total 3 marks)
Bemrose Community School
25
18.
Lm
x°
Elliot did an experiment to find the value of g m/s2, the acceleration due to gravity.
He measured the time, T seconds, that a block took to slide L m down a smooth slope
of angle x°.
He then used the formula
g=
2L
T sin x
2
to calculate an estimate for g.
T = 1.3 correct to 1 decimal place.
L = 4.50 correct to 2 decimal places.
x = 30 correct to the nearest integer.
(a)
Calculate the lower bound and the upper bound for the value of g.
Give your answers correct to 3 decimal places.
Lower bound .........................................
Upper bound ..........................................
(4)
Bemrose Community School
26
(b)
Use your answers to part (a) to write down the value of g to a suitable degree of accuracy.
Explain your reasoning.
..............................................................................................................................
..............................................................................................................................
....................................
(1)
(Total 5 marks)
19.
This is a sketch of the curve with equation y =f(x).
It passes through the origin O.
y
y = f(x)
x
O
×
A (2, –4)
The only vertex of the curve is at A (2, –4)
(a)
Write down the coordinates of the vertex of the curve with equation
(i)
y = f(x – 3),
(...... , ......)
(ii)
y = f(x) – 5,
(...... , ......)
(iii)
y = –f(x),
(...... , ......)
(iv)
y = f(2x).
(...... , ......)
(4)
Bemrose Community School
27
The curve with equation y = x2 has been translated to give the curve y = f(x).
(b)
Find f(x) in terms of x.
f(x) = ..........................................
(4)
(Total 8 marks)
20.
x
x
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)
Bemrose Community School
28
The equation
x3 + x2 = 230
has a solution between x = 5 and x = 6.
(b)
Use a trial and improvement method to find this solution.
Give your answer correct to 1 decimal place.
You must show all your working.
x = ...........................
(4)
(Total 6 marks)
21.
A straight line has equation
y=
1
x +1
2
The point P lies on the straight line.
P has a y-coordinate of 5.
(a)
Find the x-coordinate of P.
............................
(2)
Bemrose Community School
29
(b)
Write down the equation of a different straight line that is parallel to y =
1
x + 1.
2
.........................................
(1)
(c)
Rearrange y =
1
x + 1 to make x the subject.
2
.................................
(2)
(Total 5 marks)
22.
A
X
Diagram NOT
accurately drawn
8 cm
C
70°
15 cm
B
In triangle ABC,
AC = 8 cm,
CB = 15 cm,
Angle ACB = 70°.
Bemrose Community School
30
(a)
Calculate the area of triangle ABC.
Give your answer correct to 3 significant figures.
............................... cm2
(2)
X is the point on AB such that angle CXB = 90°.
(b)
Calculate the length of CX.
Give your answer correct to 3 significant figures.
................................ cm
(4)
(Total 6 marks)
Bemrose Community School
31