1 Find the prime factorisation of 180.
Hence, find the Highest Common Factor (HCF) and Lowest Common Multiple
(LCM) of 180 and 72.
2 A rectangle has a length of 12.5 cm (measured to 1 decimal place) and a width of
8.4 cm (measured to 1 decimal place).
Calculate the lower bound for the area of the rectangle.
3 (a) In a diagram, two parallel lines AB and CD are intersected by a transversal
line XY. The point E is the intersection of AB and XY, and the point F is the
intersection of CD and XY. Angle AEX = 70∘ . Find the size of angle CFE and angle
EFD. Give reasons for your answers.
(b) What is the sum of the interior angles of a regular octagon?
4 Points A, B, C, and D are on the circumference of a circle with centre O. Angle
AOC = 130∘ .
(a) Find the size of angle ABC.
(b) Find the size of angle ADC.
State the circle theorems used for each.
5 A cylinder has a radius of 5 cm and a height of 12 cm. Calculate its volume, giving
your answer in terms of 𝜋.
6 A cone has a radius of 6 cm and a slant height of 10 cm. Calculate its total surface
area, giving your answer in terms of 𝜋. (You may use the formula for the curved
surface area of a cone: 𝐴 = 𝜋𝑟𝑙, where 𝑟 is the radius and 𝑙 is the slant height).
7 Solve the simultaneous equations:
3𝑥 + 2𝑦 = 18
𝑥−𝑦=1
8 (a) Expand and simplify (2𝑥 − 3)(𝑥 + 5).
(b) Factorise completely 3𝑥 2 − 12𝑥.
(c) Solve 𝑥 2 − 7𝑥 + 10 = 0.
9 (a) Express 𝑥 2 + 6𝑥 − 5 in the form (𝑥 + 𝑝)2 + 𝑞, where 𝑝 and 𝑞 are integers.
(b) Sketch the graph of 𝑦 = 𝑥 2 + 6𝑥 − 5, showing the coordinates of the turning
point and the y-intercept.
𝑥 2−9
10 Simplify fully: 2𝑥 2+5𝑥−3
11 (a) In a right-angled triangle, the two shorter sides have lengths 𝑥 cm and 12 cm.
The hypotenuse has length 13 cm. Find the value of 𝑥.
(b) In a different right-angled triangle, one of the angles is 35∘ . The side opposite
this angle is 7 cm long. Find the length of the hypotenuse, correct to 2 decimal
places.
12 In triangle ABC, side AB = 8 cm, side AC = 10 cm, and angle BAC = 40∘ .
(a) Calculate the area of triangle ABC, correct to 2 decimal places.
(b) Calculate the length of side BC, correct to 2 decimal places.
13 (a) Share £72 in the ratio 3: 5.
(b) Convert 5 m2 to cm2 .
(c) Convert 72 km/h to m/s.
14 (a) A car travels 150 km in 2 hours and 30 minutes. Calculate its average speed in
km/h.
(b) An object has a mass of 240 g and a volume of 60 cm3 . Calculate its density in
3
g/cm .
15 (a) Write 0.000573 in standard form.
(b) Write 458000 in standard form.
(c) Calculate (3 × 105 ) × (4 × 10−2 ), giving your answer in standard form.
(d) Calculate (8 × 107 ) ÷ (2 × 103 ), giving your answer in standard form.
16 (a) A price of £120 is increased by 15%. What is the new price?
(b) After a 20% discount, a television costs £320. What was the original price of
the television?
17 (a) Simplify √72.
(b) Expand and simplify (3 + √2)(4 − √2).
6
(c) Rationalise the denominator of .
√3
18 A trapezium has parallel sides of length 7 cm and 11 cm. The perpendicular
height between the parallel sides is 5 cm. Calculate the area of the trapezium.
19 A sector of a circle has a radius of 9 cm and an angle at the centre of 60∘ .
(a) Calculate the arc length of the sector. Give your answer in terms of 𝜋.
(b) Calculate the area of the sector. Give your answer in terms of 𝜋.
20 Let 𝜉 = {integers from 1 to 10 inclusive}, 𝐴 = {prime numbers less than 10}, and
𝐵 = {even numbers less than 10}.
(a) List the members of 𝐴 ∩ 𝐵.
(b) List the members of 𝐴 ∪ 𝐵.
(c) List the members of 𝐵′ .
21 In a class of 30 students, 18 students play football (F), 15 students play tennis (T),
and 6 students play neither football nor tennis.
(a) Draw a Venn diagram to represent this information.
(b) How many students play both football and tennis?
(c) Find 𝑛(𝐹 ∪ 𝑇 ′ ).
22 Consider the sequence: 5,8,11,14, . ..
(a) Find the next two terms of the sequence.
(b) Find an expression for the 𝑛𝑡ℎ term of the sequence.
23 Find the 𝑛𝑡ℎ term of the quadratic sequence: 2,7,16,29,46, . ..
24 Triangle A has vertices at coordinates (1,1), (3,1), and (1,4). Triangle B has
vertices at coordinates (-1,-1), (-3,-1), and (-1,-4). Describe fully the single
transformation that maps triangle A onto triangle B.
25 Shape P has vertices at coordinates (2,2), (4,2), (4,4), and (2,4).
(a) Reflect shape P in the line 𝑦 = 𝑥 to obtain shape Q. Write down the
coordinates of the vertices of shape Q.
−3
(b) Translate shape Q by the vector ( ) to obtain shape R. Write down the
1
coordinates of the vertices of shape R.
26 The ages of 7 people are: 12,15,12,18,25,12,22.
Find:
(a) The mode.
(b) The median.
(c) The mean, correct to 1 decimal place.
27 The number of goals scored by a football team in 20 matches is shown in the
table below:
Number of Goals 0 1 2 3 4
Frequency 5 7 4 3 1
Calculate:
(a) The modal number of goals.
(b) The median number of goals.
(c) The mean number of goals scored per match.
28 The heights of 50 plants were measured. The table below shows a grouped
frequency distribution of these heights:
Height (ℎ cm) Frequency
0 < ℎ ≤ 10 8
10 < ℎ ≤ 20 15
20 < ℎ ≤ 30 18
30 < ℎ ≤ 40 7
40 < ℎ ≤ 50 2
Estimate the mean height of the plants.
29 The test scores (out of 100) of 11 students are listed below:
45,52,58,61,65,70,73,77,82,85,90.
(a) Find the lower quartile (𝑄1 ).
(b) Find the upper quartile (𝑄3 ).
(c) Calculate the interquartile range (IQR).
30 The cumulative frequency graph below shows the times taken by 100 students to
complete a puzzle. (Assume a standard S-shaped cumulative frequency curve is
provided, with time on the x-axis from 0 to 60 minutes, and cumulative
frequency on the y-axis from 0 to 100. Key points on the curve are approximately:
(10,5), (20,25), (30,60), (40,85), (50,95), (60,100)).
Use the graph to estimate:
(a) The median time taken.
(b) The interquartile range of the times.
(c) The number of students who took longer than 45 minutes to complete the
puzzle.
