12 questions · 100 marks · ≈ 2 hours
Charles asks each of the students in his class which flavour of ice cream they prefer.
There are 30 students in his class.
He uses the information he collects to draw this accurate pie chart.
mint
fudge
vanilla
strawberry
chocolate
Find how many of the 30 students prefer chocolate ice cream.
The trapezium PQRS is a scale drawing of Abdul’s garden.
Construct the locus of all points inside the trapezium that are 6 metres from P
Q
P
Scale: 1 cm represents 2 metres
S
R
The trapezium PQRS is a scale drawing of Abdul’s garden.
Construct the locus of all points inside the trapezium that are 12 metres from QR
Q
P
Scale: 1 cm represents 2 metres
S
R
The trapezium PQRS is a scale drawing of Abdul’s garden.
Using ruler and compasses only and showing all your construction lines, construct
the locus of all points inside the trapezium that are equidistant from PS and RS
Abdul is going to put a fountain in his garden.
The fountain must be
greater than 6 metres from P
greater than 12 metres from QR
closer to PS than to RS
Q
P
Scale: 1 cm represents 2 metres
S
R
The trapezium PQRS is a scale drawing of Abdul’s garden.
Show, by shading, the region in which Abdul can place the fountain.
Label the region R
Write 34 700 000 in standard form.
as an ordinary number.
Write 9 8 10 3
.
5 4 10121
x
.
2 10120
y
Calculate xy
Give your answer in standard form.
Calculate x
y
Give your answer in standard form.
Question 3 continued
B
C
Diagram NOT
5 m
accurately drawn
50°
D
A
5 m
Figure 1
Figure 1 shows the section of a park, ABCD, where ADB is a sector of a circle centre D
= 5m
50
AD
BD
ADB
BCD
CDA
=
90
Calculate the area, in m² to 3 significant figures, of the sector ADB
Grass is to be grown in this section ABCD of the park.
2
35 g of grass seed is needed for each 1
m
B
C
Diagram NOT
5 m
accurately drawn
50°
D
A
5 m
Figure 1
Figure 1 shows the section of a park, ABCD, where ADB is a sector of a circle centre D
= 5m
50
AD
BD
ADB
BCD
CDA
=
90
Calculate, in g to the nearest g, the weight of grass seed needed to cover
this section of the park.
Show all your working.
Question 4 continued
A company buys blankets, cushions and towels in the ratios
number of blankets: number of cushions: number of towels = 2 3
: :6
The company buys a total of 3850 of these items.
Calculate the number of blankets that the company buys.
The company sells each cushion for £15.95
The company makes a profit of 16% on each cushion.
A company buys blankets, cushions and towels in the ratios
number of blankets: number of cushions: number of towels = 2 3
: :6
The company buys a total of 3850 of these items.
Calculate, in £, how much the company pays for each cushion.
The company buys 1500 small towels for £7 each and 600 large towels for £11 each.
In January, it sells 3
of the small towels for £15 each and 95% of the large towels
5
for £18 each. The remaining towels were given away for free.
A company buys blankets, cushions and towels in the ratios
number of blankets: number of cushions: number of towels = 2 3
: :6
The company buys a total of 3850 of these items.
Calculate the percentage profit that the company makes in January.
Give your answer to one decimal place.
The company buys some items from a new supplier.
The company spends $9460
Using the exchange rate of £1 = $1.23
A company buys blankets, cushions and towels in the ratios
number of blankets: number of cushions: number of towels = 2 3
: :6
The company buys a total of 3850 of these items.
calculate the amount, in £ to the nearest £, that the company spends with this
new supplier.
Question 5 continued
Question 5 continued
Question 5 continued
The Venn diagram shows information about the number of students in a school who are
members of the Athletics Club (A), the Brass Band (B) and the Chess Club (C)
15 students are not members of any of these clubs.
E
C
A
B
x
25
10
21
37
15
There are twice as many members of the Chess Club than there are members of the
Brass Band.
Find the value of x
The Venn diagram shows information about the number of students in a school who are
members of the Athletics Club (A), the Brass Band (B) and the Chess Club (C)
15 students are not members of any of these clubs.
E
C
A
B
x
25
10
21
37
15
There are twice as many members of the Chess Club than there are members of the
Brass Band.
Find the number of these students who are not members of the Chess Club.
The Venn diagram shows information about the number of students in a school who are
members of the Athletics Club (A), the Brass Band (B) and the Chess Club (C)
15 students are not members of any of these clubs.
E
C
A
B
x
25
10
21
37
15
There are twice as many members of the Chess Club than there are members of the
Brass Band.
Find the number of these students who are members of all three clubs.
'
The Venn diagram shows information about the number of students in a school who are
members of the Athletics Club (A), the Brass Band (B) and the Chess Club (C)
15 students are not members of any of these clubs.
E
C
A
B
x
25
10
21
37
15
There are twice as many members of the Chess Club than there are members of the
Brass Band.
Find n B
C
At the start of 2022, the Drama Club had 80 members.
From the start of 2022 until the start of 2023, the number of members of the
Drama Club increased by 12.5%
From the start of 2023 until the start of 2024, the number of members of the
Drama Club increased by 30%
The Venn diagram shows information about the number of students in a school who are
members of the Athletics Club (A), the Brass Band (B) and the Chess Club (C)
15 students are not members of any of these clubs.
E
C
A
B
x
25
10
21
37
15
There are twice as many members of the Chess Club than there are members of the
Brass Band.
Calculate the number of students who were members of the Drama Club in 2024
Question 6 continued
Bag A contains 10 red counters and 5 yellow counters.
Two counters are taken at random from bag A.
Complete the tree diagram below.
First counter
Second counter
red
red
10
15
yellow
red
yellow
yellow
Bag A contains 10 red counters and 5 yellow counters.
Two counters are taken at random from bag A.
Calculate the probability that the two counters taken are the same colour.
Bag B contains 4 red counters and 3 yellow counters.
The two counters that have been taken from bag A are both placed into bag B so that
there are a total of 9 counters in bag B.
A counter is then taken at random from bag B.
Given that each of the two counters taken from bag A and the one counter taken from
bag B are all the same colour,
Bag A contains 10 red counters and 5 yellow counters.
Two counters are taken at random from bag A.
calculate the probability that the three counters are all red.
Question 7 continued
Question 7 continued
Question 7 continued
y cm
2x cm
Figure 2
Figure 2 shows a solid cuboid with length 2x cm, width x cm and height y cm
The volume of the cuboid is a cm³, where a is a constant.
The total surface area of the cuboid is S cm² 
a
3
Show that S
2
x
4
x
Given that the minimum value of S is 400
y cm
2x cm
Figure 2
Figure 2 shows a solid cuboid with length 2x cm, width x cm and height y cm
The volume of the cuboid is a cm³, where a is a constant.
The total surface area of the cuboid is S cm² 
a
3
Find, to 3 significant figures, the value of a
Question 8 continued
Question 8 continued
Question 8 continued
X
M
E
F
A
D
O
B
C
Figure 3
Figure 3 shows a solid right pyramid ABCDEFX
The base ABCDEF of the pyramid is horizontal and is a regular hexagon of side 3 cm
O is the midpoint of the base.
The vertex X of the pyramid is vertically above point O such that OX = 5 cm
M is the midpoint of XD
Find the area, in cm² to one decimal place, of the hexagonal base.
X
M
Diagram NOT
accurately drawn
E
F
A
D
O
B
C
Figure 3
Figure 3 shows a solid right pyramid ABCDEFX
The base ABCDEF of the pyramid is horizontal and is a regular hexagon of side 3 cm
O is the midpoint of the base.
The vertex X of the pyramid is vertically above point O such that OX = 5 cm
M is the midpoint of XD
Calculate the size, in degrees to 3 significant figures, of ∠AMX
Show all your working.
1
Area of triangle
ab
C
=
sin
2
Cosine rule :
2
2
2
bc
A
a
b
c
2
cos
Question 9 continued
Question 9 continued
Question 9 continued
f( )
3
x
x
x
3
2
2 is a factor of f( )
Use the factor theorem to show that that x
x
x completely.
f( )
3
x
x
x
3
2
2 is a factor of f( )
Hence, factorise f( )
3
f( )
3
x
x
x
3
2
2 is a factor of f( )
Complete the table of values for y
x
x
2
3
Give your values of y to one decimal place where necessary.
f( )
3
x
x
x
3
2
2 is a factor of f( )
On the grid opposite, plot the points from your completed table and join them to
form a smooth curve.
f( )
3
x
x
x
3
2
2 is a factor of f( )
By drawing a suitable straight line on the grid, find an estimate, to one decimal
place, for the solution of the equation
≤
≤
3x
in the interval −2
x
x
2
7
6
0
2 5
.
Question 10 continued
y
8
3 x
Turn over for a spare grid if you need to redraw your curve.
Question 10 continued
Question 10 continued
Only use this grid if you need to redraw your curve.
y
8
3 x
The functions f, g and h are defined as
1
2
f : x
x
x
−
<
<
↦6
0
3
2
x
1
g : x
↦
x 4
2
x
x
x
6
8
2
x
5
x > 0
h : x
↦
x
2
3
Write g(x) as a single fraction in its simplest form.
Show clear algebraic working.
The functions f, g and h are defined as
1
2
f : x
x
x
−
<
<
↦6
0
3
2
x
1
g : x
↦
x 4
2
x
x
x
6
8
2
x
5
x > 0
h : x
↦
x
2
3
Find hf
The functions f, g and h are defined as
1
2
f : x
x
x
−
<
<
↦6
0
3
2
x
1
g : x
↦
x 4
2
x
x
x
6
8
2
x
5
x > 0
h : x
↦
x
2
3
Find the inverse function h–1 in the form h −1 :
x ↦
...
Give your answer in its simplest form.
The functions f, g and h are defined as
1
2
f : x
x
x
−
<
<
↦6
0
3
2
x
1
g : x
↦
x 4
2
x
x
x
6
8
2
x
5
x > 0
h : x
↦
x
2
3
Solve the equation hh( )
x
x
=
Show your working clearly.
2
b
b
ac
4
2
ax
bx
c
x
Solutions of
0
are
a
2
Question 11 continued
Question 11 continued
Question 11 continued
Triangle A and triangle E are drawn on the grid opposite.
Describe fully the single transformation that maps triangle A onto triangle E
Triangle A is transformed to triangle B under a reflection in the y-axis.
Triangle A and triangle E are drawn on the grid opposite.
On the grid, draw and label triangle B
4
Triangle A is transformed to triangle C under a translation with the vector
3
Triangle A and triangle E are drawn on the grid opposite.
On the grid, draw and label triangle C
Triangle A and triangle E are drawn on the grid opposite.
Triangle A is transformed to triangle D under the transformation with matrix M where
M
2
1
1
3
Triangle A and triangle E are drawn on the grid opposite.
On the grid, draw and label triangle D
Triangle F is transformed to triangle A under the transformation with matrix N where
k
5
N
1
Triangle F is transformed to triangle D under the transformation with matrix P
Given that the determinant of P is 5
calculate the value of k
a
b
Determinant of matrix
ad
bc
=
c
d
Question 12 continued
y
x
Turn over for a spare grid if you need to redraw your triangles.
Only use this grid if you need to redraw your triangles.