Oct/Nov 2025

State the reciprocal of $\frac{7}{2}$.

Types of number

The circle’s diameter is 12 cm.

Circles, arcs and sectors

A bag has 10 plums. The average mass of 6 of the plums is 50 g. The average mass of the other 4 plums is 40 g.

Averages and measures of spread

The illustration shows a container that is empty. It is a cuboid with width 10 cm, length 8 cm and height 9 cm. Water enters the container by dripping at a rate of 6 millilitres per second.

Surface area and volume

Find $2\frac{2}{3}+3\frac{3}{4}$. Give your answer as a mixed number in its simplest form.

The four operations

Lia covers 800 m in 105 seconds by cycling. The speed-time graph for this trip is shown in the diagram.

Rates

The cumulative frequency graph gives data about journey times.

Cumulative frequency diagrams

Q, R, S and T lie on a circle with centre O. The line UV is tangent to the circle at T. S, O, Q and V lie on one straight line. Angle QST = 32^{\circ}.

Circle theorems I

The diagram depicts a prism whose cross-section is an isosceles triangle with side lengths 13 cm, 13 cm and base 10 cm. The prism has length 3 cm.

Similarity

Write $0.1\dot{4}$ in fraction form.

Fractions, decimals and percentages

$g(x)=\frac{x}{2}+4$ while $h(x)=3x-1$.

Functions

Some children note how many hours each of them spends watching television on one day. The results are: 1, 2, 2, 2, 6, 7, 7, 8, 10.

Introduction to probability

Express $\frac{5}{2x}-\frac{4}{7x}$ as a single fraction in simplest form.

Algebraic fractions

Evaluate the value of $27^{-\frac{2}{3}}$.

Indices I

This table lists some values of $y=x^3-3x+2$.

Graphs of functions

Work out $\sqrt{8}\times\sqrt{6}+\sqrt{3}$.

Surds

A rectangle and a triangle are shown in the diagram. The rectangle’s length is x cm, and its area is 30 cm^{2}. The triangle’s sides measure (x-1) cm, (x+2) cm and (2x-3) cm.

Equations

Factorise the expression $6x^3+5x^2-4x$.

Algebraic manipulation

The figure depicts triangle OAB. $\overrightarrow{OA}=a$ and $\overrightarrow{OB}=4b$. T lies on AB so that $AT:TB=3:2$.

Vectors in two dimensions

A shirt is priced at $24. During a sale, the price is cut by 15%.

Percentages

Five straight lines are drawn in the diagram: three are horizontal and parallel, while the other two are sloping and parallel. The angles shown are $112^\circ$, $p^\circ$ and $q^\circ$.

Angles

The table presents the ways in which some people get to work.

Statistical charts and diagrams

Sam purchases some chocolate. He eats $\frac{2}{7}$ of it. The mass of the chocolate left is 350 g.

Ratio and proportion

Car hire costs $40 each day, together with $0.30 for every kilometre. Amy hires a car for 10 days and the total amount she pays is $670.

Ratio and proportion

The diagram displays triangle A and triangle B on coordinate axes.

Transformations

By expressing each number correct to 1 significant figure, calculate an estimate for the value of $\frac{\sqrt{102.5}\times 8.7}{27}$.

Estimation

Ruth notes the colour of each of 24 cars. Red, Blue, Silver, Blue, Silver, Silver, White, Silver, Red, Silver, Silver, Blue, Grey, Grey, Silver, Red, White, Red, Blue, Grey, Blue, Silver, Red, Red

Statistical charts and diagrams

Find the value.

The four operations

Shape A and shape B are plotted on the grid.

Transformations

Luca owns these two fair spinners. Spinner A shows 2, 4, 6, 8. Spinner B shows 1, 1, 3, 5.

Probability of combined events

Thomas is employed in a shop. His wage is $12 for every hour he works. He also receives a bonus equal to 5% of the value of the items he sells. On one day, Thomas works 5 hours. His total pay, including the bonus, is $82.

Percentages

A, B, C and D lie on a circle with centre O. EF touches the circle at A, and angle BAF = $65\degree$ while angle AOD = $80\degree$.

Circle theorems I

Given $y=x^3-4x^2+12$.

Graphs of functions

Expand and simplify $(3+\sqrt{2})(1+4\sqrt{2})$.

Algebraic manipulation

Triangles ABC and PQR are similar. In triangle ABC, AB = 6 cm and AC = 7 cm, while in triangle PQR, PQ = 9 cm.

Similarity

The diagram depicts a solid cylinder and a solid hemisphere. The cylinder's radius is the same as the hemisphere's radius. Their volumes are equal. The cylinder has a height of 6 cm.

Surface area and volume

Evaluate the expression $16^{-\frac{3}{4}}$.

Indices I

Carry out the calculation.

The four operations

The functions are $f(x)=ax+3$ and $g(x)=(x+b)^2$.

Functions

OPQR forms a parallelogram. $\overrightarrow{OP}=a$ and $\overrightarrow{OR}=2b$. M lies halfway along QR. N is a point on PQ, with PN:NQ=1:3.

Vectors in two dimensions

Ade and Maha each make cards. On one day, each of them spends 4 hours making cards. Ade needs x minutes for one card.

Equations

The diagram depicts a square-based pyramid. X marks the centre of the square base. Vertex E lies directly above X, and EX = 7 cm. The perpendicular height of triangle BEC is 11 cm.

Right-angled triangles

$ABCD$ forms a parallelogram. $E$ lies on $AB$ and $EC=BC$. Angle $ECB=40\degree$ and angle $DEC=60\degree$.

Similarity

Petra is thinking of a number. It is a multiple of 7. Petra states: If I write my number to the nearest ten, the result is 30.

Limits of accuracy

Arrange these numbers in ascending order, beginning with the smallest: $0.8$, $\frac{7}{9}$, $\frac{17}{20}$, $84.5\%$.

Ordering

Solve this.

Equations

State the inequality shown on the number line.

Inequalities

Points A and B have been placed on the grid.

Gradient of linear graphs

Sara has a bag with 60 red counters, 72 blue counters and 36 green counters. Find the ratio of red counters : blue counters : green counters in the bag. Write the ratio in its simplest form.

Ratio and proportion

Choose from the list: 52, $\sqrt{169}$, 27, $\sqrt{8}$, $\frac{12}{23}$, 49

Types of number

These are the first four dot patterns in the sequence.

Sequences

A has coordinates $(-4,3)$, while B has coordinates $(2,-7)$.

Length and midpoint

Write the number 0.0142 in standard form.

Standard form

Solve the simultaneous equations. You must show all your working. $2a+5b=1.5$ and $3a-10b=42.5$.

Equations

A country's population grows exponentially at a rate of 3.6% per year. On 1st January 2024, the country’s population stands at 5.4 million.

Exponential growth and decay

Let $x=2^2\times3^{n+1}\times7$ and $y=2^n\times3^7\times7^4$. Here, $n$ is a positive integer larger than 2.

Types of number

Point A is $(3,-2)$, while point B is $(5,4)$.

Equations of linear graphs

The table gives the times needed by 80 people to finish a race.

Averages and measures of spread

Express the result as one fraction in lowest terms.

Algebraic fractions

A car covers 320 km, rounded to the nearest 10 kilometres. The trip lasts 5.2 hours, rounded to the nearest 0.1 hour.

Limits of accuracy

Add a single pair of brackets so that the calculation becomes correct: $4.2-3\times5+1.2=7.2$.

The four operations

The diagram presents triangle ABC as a triangular field. AB = 65 m, AC = 75 m and angle CAB = $125^\circ$.

Non-right-angled triangles

Rearrange the formula so that a is the subject. $w=\frac{2a+5}{3a-1}$.

Equations

Sketch the graph for $y=2-3x$.

Sketching curves

Solve the equation $\frac{x}{x+3}-\frac{4}{x-5}=1$.

Equations

Expand, then simplify $(x-1)(x+3)(3x-4)$.

Algebraic manipulation

The cuboid is shown in the diagram. $EH=12$ cm, $HG=10$ cm and $GC=7$ cm.

Pythagoras' theorem and trigonometry in 3D

For this question, every length is measured in centimetres. The diagram shows the net of a cuboid with dimensions 3, 5.2 and 4.7.

Surface area and volume

The table gives the mathematics and physics examination marks for each of the 9 students.

Scatter diagrams

The diagram depicts an isosceles triangle whose apex angle is $56^\circ$.

Angles

Calculate $\frac{12-\sqrt[3]{6}}{0.3^2}$. Give your answer correct to 2 decimal places.

Powers and roots

Calculate the size of the interior angle of a regular decagon.

Geometrical terms

John puts $240 into an account that earns simple interest at a rate of 2.35% per year.

Percentages

For euros (€) and pounds (£), the rate is €1 = £0.88. For dollars ($) and pounds (£), the rate is $1 = £x. Ling converts €500 into dollars and is given $536.58.

Ratio and proportion

Work out $\frac{75.66-40.23}{2.19}$. Round your answer to 2 decimal places.

The four operations

Given $\mathbf{a}=\begin{pmatrix}-3\\4\end{pmatrix}$ and $\mathbf{b}=\begin{pmatrix}8\\-2\end{pmatrix}$.

Vectors in two dimensions

A battery factory produces batteries. A sample of 2000 batteries is checked, and 28 turn out to be faulty. The factory produces 125000 batteries.

Ratio and proportion

Express 372000000 in standard form.

Standard form

The first five terms of a sequence are: 3, 8, 13, 18, 23. Determine the next term in the sequence.

Sequences

$N=2\times3^x\times5^y$. The least common multiple (LCM) of N and 360 equals 16200.

Types of number

The mass of a bag of potatoes is 2.5 kg, stated to the nearest 0.1 kg.

Limits of accuracy

A town’s population falls exponentially by x% each year. It was 120000 on 1 January 2020, and by 1 January 2023 it had reduced to 102885.

Exponential growth and decay

Sketch the graph of each function. Indicate the value where each graph crosses the y-axis.

Graphs of functions

The equation for line L is $y=\frac{x}{5}+3$. The diagram shows lines L and M, the origin O, point A on the x-axis, and point B on the y-axis.

Equations of linear graphs

In a sample of 22 students, they are asked whether they take Economics (E) or History (H). The Venn diagram gives these counts: E only = 7, E∩H = 5, H only = 4, neither = 6.

Probability of combined events

The quiz scores for 12 students are: 16, 27, 20, 15, 25, 21, 10, 24, 35, 16, 32, 22.

Averages and measures of spread

y is inversely proportional to $(x-3)^2$. When x = 7, y = 4.

Proportion

Solve the equation $\frac{2x+1}{x+4}+5=x$. Show every step in your working and give your answers correct to 2 decimal places.

Equations

ABCD is a field, and a straight path runs across it from B to D. Diagram details: DB = 495 m, DC = 620 m, BC = 580 m, angle ABD = 42^{\circ}, angle DAB = 90^{\circ}.

Right-angled triangles

The sketch represents a shaded region inside a large circle with centre O. The large circle has radius 10 cm. OAB is a sector of a smaller circle, also with centre O. The smaller circle has radius 7 cm and the sector angle is 130^{\circ}. Point C lies on the circumference of the large circle, and AC = BC.

Circles, arcs and sectors

Owen measures the mass of 250 onions individually. The histogram displays the outcomes. Histogram classes and frequency densities: 160-200 density 0.7, 200-220 density 3.9, 220-240 density 4.5, 240-300 density 0.9.

Histograms

Idris has 5 kg of rice available. He is cooking rice for 12 people. He needs 75 g of rice for each person.

Ratio and proportion

Triangle ABC has AB = 8.5 cm, AC = 6 cm and BC = 7 cm.

Geometrical constructions

Sofia puts $400 into a savings account. This account pays simple interest at 2.8% per year. Calculate the total amount of interest Sofia receives after 3 years.

Rates

A map is drawn to the scale 1 : 5000, and the path measures 8.3 cm on the map.

Ratio and proportion

Factorise $20x^2-5xy$ into a factorised form.

Algebraic manipulation

Expand and simplify the expression $4(3x+2)+5(x+1)$.

Algebraic manipulation

The diagram displays a regular pentagon and a regular hexagon attached along one side.

Angles