1 (a) Write 120 as a product of its prime factors.
(b) Given that 90 = 2 × 32 × 5, use prime factors to find the Highest Common
Factor (HCF) and the Lowest Common Multiple (LCM) of 120 and 90.
2 Simplify fully:
(a) (2𝑥 3 𝑦 2 )4
20𝑎 7𝑏3
(b) 4𝑎 2𝑏2
3 (a) Evaluate 5−2 .
1
(b) Evaluate 642 .
(c) Find the value of 𝑥 if 3𝑥 = 243.
4 In the diagram below, line AB is parallel to line CD. Line EF is a transversal that
intersects AB at G and CD at H. Angle AGH = 110∘ .
(Diagram not shown, but assume a standard parallel lines cut by transversal
setup)
(a) Find the size of angle GHC. Give a reason for your answer.
(b) Find the size of angle DHF. Give a reason for your answer.
5 (a) Calculate the sum of the interior angles of a regular pentagon.
(b) Calculate the size of each exterior angle of a regular nonagon (9-sided
polygon).
6 A triangular prism has a cross-sectional area (the triangle) with a base of 6 cm
and a perpendicular height of 4 cm. The length of the prism is 10 cm. Calculate
the volume of the prism.
7 A cuboid has length 5 cm, width 3 cm, and height 4 cm. Calculate its total surface
area.
8 (a) Calculate the gradient of the line segment joining the points 𝐴(2,5) and
𝐵(4,1).
(b) Find the equation of the line passing through points A and B. Give your
answer in the form 𝑦 = 𝑚𝑥 + 𝑐.
9 Line L1 has the equation 𝑦 = 2𝑥 + 3. Line L2 is parallel to L1 and passes through
the point (1, −1). Find the equation of Line L2.
10 Share £150 in the ratio 2: 3: 5.
11 The ratio of boys to girls in a school choir is 4: 5. If there are 180 girls in the choir,
how many boys are there?
12 (a) Write 37,600,000 in standard form.
(b) Write 2.05 × 10−4 as an ordinary number.
13 Calculate (4 × 105 ) × (5 × 10−2 ). Give your answer in standard form.
14 In a right-angled triangle, the two shorter sides are 7 cm and 24 cm. Calculate the
length of the hypotenuse.
15 In a right-angled triangle, one angle is 60∘ and the hypotenuse is 12 cm. Calculate
the length of the side opposite the 60∘ angle. Give your answer correct to 1
decimal place.
16 A ladder 5 m long leans against a vertical wall. The foot of the ladder is 2 m from
the base of the wall on horizontal ground. Calculate the angle the ladder makes
with the ground. Give your answer correct to 1 decimal place.
17 Calculate 35% of £240.
18 A phone originally costs £500. Its price is reduced to £420 in a sale. Calculate the
percentage decrease in price.
19 After a 15% price increase, a train ticket costs £55.20. What was the original price
of the ticket?
20 Sarah invests £3000 in a savings account that pays compound interest at a rate of
2.5% per annum. How much will she have in the account after 3 years? Give your
answer to the nearest penny.
21 Expand and simplify (𝑥 + 7)(𝑥 − 3).
22 Factorise 𝑥 2 + 4𝑥 − 21.
23 Solve 𝑥 2 − 9𝑥 + 18 = 0 by factorising.
24 Solve the equation 2𝑥 2 + 5𝑥 − 4 = 0. Give your solutions correct to 2 decimal
−𝑏±√𝑏2 −4𝑎𝑐
places. (You may use the quadratic formula 𝑥 = ).
2𝑎
25 Sketch the graph of 𝑦 = 𝑥 2 − 2𝑥 − 8. Label the coordinates of the points where
the graph crosses the x-axis and y-axis, and the coordinates of the turning point.
26 The length of a rectangle is 4 cm more than its width. The area of the rectangle is
77 cm2 . If the width is 𝑥 cm:
(a) Form a quadratic equation in terms of 𝑥.
(b) Solve the equation to find the dimensions (width and length) of the rectangle.
27 The number of sweets in 10 bags are: 23,25,22,26,23,28,29,22,23,27. Find:
(a) the mode
(b) the median
(c) the mean
(d) the range.
28 The times, in seconds, taken by 11 students to solve a puzzle are:
15,18,22,25,28,30,32,35,38,40,45.
(a) Find the lower quartile (𝑄1 ).
(b) Find the upper quartile (𝑄3 ).
(c) Calculate the interquartile range (IQR).
29 A box plot summarises the heights (in cm) of a group of year 9 students. The
minimum height is 145 cm, lower quartile is 152 cm, median is 158 cm, upper
quartile is 165 cm, and maximum height is 175 cm.
(a) What is the range of the heights?
(b) What is the interquartile range?
(c) What percentage of students are taller than 152 cm?
30 The marks obtained by 15 students in a test are as follows:
78,62,85,71,68,90,75,66,82,79,60,88,75,69,93.
(a) Construct an ordered stem and leaf diagram to represent this data.
(b) Find the median mark.
(c) What is the range of the marks?
