May/June 2025

Calculators are not permitted in this paper.

The four operations

Write $228$ in prime factor form.

Types of number

For a small box, the mass is $x\text{ kg}$. For a large box, the mass is $y\text{ kg}$.

Equations

On the grid, triangle $A$ and triangle $B$ are shown.

Transformations

During a sale, a shop cuts every price by $20\%$.

Percentages

$A$ has coordinates $(-4,5)$, while $B$ has coordinates $(6,1)$. $\vec{BC} = \begin{pmatrix}-3\\-4\end{pmatrix}$.

Vectors in two dimensions

Simplify $\sqrt{175}-\sqrt{28}$.

Surds

On one day, a group of 80 people each note how long the trip from home to work takes. The cumulative frequency diagram displays the results.

Cumulative frequency diagrams

The equation for line $L$ is $5y+3x=10$.

Perpendicular lines

The diagram displays the cyclist’s journey on a speed-time graph.

Graphs in practical situations

Simplify the expression $\dfrac{3x^2-12}{2x^2+11x+14}$.

Algebraic fractions

A bag holds 11 balls. Of these, 5 are blue and 4 are yellow. The remaining balls are green. One ball is selected from the bag at random.

Introduction to probability

Calculate $0.1\dot{7} + \frac{5}{9}$. Give your answer as a fraction in lowest terms.

Fractions, decimals and percentages

$\vec{OA}=\vec{a}$ and $\vec{OB}=\vec{b}$. Point $X$ lies on $AB$ with $AX:XB=3:2$. $OXC$ forms a straight line. $AC$ is parallel to $OB$.

Vector geometry

Express $x^2+4x-12$ in the form $(x+a)^2+b$.

Sketching curves

$OAB$ forms a minor sector of a circle, with centre $O$. $OCD$ forms a major sector of another circle, with centre $O$. $OCA$ and $ODB$ are straight lines. $OC=6\text{ cm}$ and $OA=9\text{ cm}$. The minor arc $AB$ measures $5\pi\text{ cm}$.

Circles, arcs and sectors

The area of a single face of the cube is $9\text{ cm}^2$.

Surface area and volume

AEC and BED lie on straight lines. $ED = EC$.

Angles

Solve for $x$ in $5(4-x)=35$.

Equations

The scale diagram gives the locations of two villages, $A$ and $B$. The scale is $1\text{ cm}$ for $5\text{ km}$.

Scale drawings

Evaluate the value of $\sqrt[3]{125}$.

Powers and roots

Asha notes the distance she walks and the time she spends on each of 10 walks. The table displays her results.

Scatter diagrams

The diagram represents a rectangle measuring $87.1\text{ mm}$ by $23.6\text{ mm}. $

Estimation

Express $0.07$ as a fraction.

Fractions, decimals and percentages

A spinner may stop on red, green, yellow or blue. The table gives the probability attached to each result.

Introduction to probability

A green light flashes at 12-minute intervals. A red light flashes at 45-minute intervals. Both lights flash at the same time at $9\,\text{am}$.

Types of number

The vector $\vec{PQ}$ has the form $\begin{pmatrix}4 \\ -2\end{pmatrix}$.

Vectors in two dimensions

The diagram depicts a trapezium, and the lengths are given in centimetres.

Area and perimeter

Find the numerical value of $3^3$.

Indices I

Calculate $70\%$ of $120$.

Percentages

The cumulative frequency diagram shows the marks obtained by $80$ students in an exam.

Cumulative frequency diagrams

Rewrite the formula so that $x$ is the subject. $ax = \frac{3x + 2}{5}$.

Algebraic manipulation

The diagram gives the speed-time graph for a section of a car’s journey.

Graphs in practical situations

The diagram depicts triangle $A$ alongside triangle $B$.

Transformations

Shown here are the ice cream flavours preferred by 20 children.

Classifying statistical data

The sectors OAB and OCD are parts of circles that each have centre $O$ and an angle of $60^{\circ}$. The radius of sector OAB is $6\,\text{cm}$. The radius of sector OCD is $9\,\text{cm}$.

Circles, arcs and sectors

The equation of line $L$ is $2y = 5x + 3$.

Equations of linear graphs

Simplify the expression $\frac{x^2 - 9}{5x^2 - 11x - 12}$.

Algebraic fractions

Expand and simplify the product $(x - 3)(2x + 5)(x + 2)$.

Algebraic manipulation

The diagram depicts quadrilateral $OACB$. $\vec{OA} = \vec{a}$ and $\vec{OB} = \vec{b}$. $OB$ runs parallel to $AC$, and $AC = 2OB$.

Vector geometry

Simplify $\sqrt{300} - \sqrt{48}$.

Surds

The diagrams are not drawn to scale.

Angles

Find the value of $5x + 3y$ for $x = 4$ and $y = -2$.

Introduction to algebra

State

Types of number

Points $A$, $B$ and $C$ have been drawn on the grid.

Equations of linear graphs

Asif is creating a rectangular lawn that measures $14\,\text{m}$ by $19\,\text{m}$. He uses grass seed to establish the lawn. The grass seed costs $\$0.34$ for each $1\,\text{m}^2$.

Estimation

Solve the equation $4x - 7 = 9$.

Equations

Work out the value of $8 \div 0.1$.

Fractions, decimals and percentages

The diagram shows a section of a pattern that has rotational symmetry of order 4.

Symmetry

Simplify the expression $\frac{7x^2y^5}{x^3y^2}$.

Indices II

Zaya puts $\$4500$ into a savings account. The account gives simple interest at a rate of $3.2\%$ per year. Calculate the value of the investment after $5$ years.

Exponential growth and decay

Factorise the expression $7hx + 6fy - 21fx - 2hy$.

Algebraic manipulation

The bag contains $9$ red counters and $3$ green counters. Lee selects two counters at random from the bag, without replacement.

Probability of combined events

$PQRS$ forms a cyclic quadrilateral. The line $APB$ is tangent to the circle at $P$. Angle $SPB = 37^{\circ}$ and angle $PSQ = 85^{\circ}$.

Circle theorems I

The first four patterns belong to a sequence built from black beads and white beads.

Sequences

Kenya has a population of $5.71 \times 10^7$ people. Its land area is $6 \times 10^5\text{ km}^2$. Population density means the number of people in each $\text{km}^2$.

Standard form

For $f(x) = \frac{10-x}{3}$, determine $f(-8)$.

Functions

300 students are involved in a competition. The table gives the details of their scores.

Averages and measures of spread

Fill in the table of values for $y = x^3 - 5x + 4$.

Sketching curves

Write eighteen thousand and twelve in figures.

Types of number

The quadrilaterals $ABCD$ and $EFGD$ are rectangles.

Equations

The diagram shows a solid made by attaching a cone to a hemisphere. The cone has diameter $14\text{ cm}$ and the hemisphere has diameter $14\text{ cm}$. The solid has a total height of $24\text{ cm}$.

Surface area and volume

The diagram represents a triangular field $ABC$. Here, $AB = 150\text{ m}$, $BC = 187\text{ m}$, and $\angle ABC = 112^{\circ}$.

Non-right-angled triangles

Zara travels a distance of $500\text{ m}$, correct to the nearest $5\text{ m}$. The journey lasts $24.7\text{ s}$, correct to the nearest $0.1\text{ s}$.

Limits of accuracy

Write $\frac{5}{2x+1} - \frac{2}{4x-3}$ as one fraction in simplest form.

Algebraic fractions

Express $175\text{ ml} : 2.5\text{ litres}$ in its simplest form.

Ratio and proportion

A sketch showing three parallel lines crossed by transversals is given (not to scale).

Angles

Convert $6.1\text{ m}^2$ into $\text{cm}^2$.

Units of measure

In a survey, students must select the kind of movie they prefer most. The table presents the outcomes.

Relative and expected frequencies

Chris plans to change $\$350$ into euros (€) at the bank. The bank issues euros only in €5 multiples. The exchange rate is $\$1 = €0.92$.

Ratio and proportion

Calculate the measure of the interior angle of a regular octagon.

Angles

On the Venn diagram, shade the area shown by $(A \cap B)'$.

Sets

State $4.2358$ correct to $2$ decimal places.

Limits of accuracy

Let $f(x) = 3x - 5$ and $g(x) = 2 - 6x$ be given.

Functions

Take $\mathcal{E} = \{2,3,4,5,6,7,8,9,10,11,12\}$, $A = \{x : x \text{ is a prime number}\}$, and $B = \{x : x \text{ is a factor of } 36\}$.

Sets

The first four terms of a sequence are $16, 13, 10, 7$.

Sequences

Express $1.23 \times 10^{-4}$ in ordinary-number form.

Standard form

Factorise the expression $5x^2 + 15xy$.

Algebraic manipulation

Find the three inequalities that describe the unshaded region $R$.

Equations of linear graphs

The table provides data on the ages of the 80 gym members.

Averages and measures of spread

The points $B$, $D$, $E$ and $F$ all lie on one circle. $AC$ is a tangent to the circle at $B$. Angle $BEF = 50^{\circ}$ and $EF = BF$.

Circle theorems I

The diagram represents a cuboid. Its length is $x$ cm. Its height is $3$ times its length. Its width is $4$ cm less than its length.

Equations

The figure contains triangle $ABC$ along with triangle $ABD$. Here $AB = 12\text{ cm}$, $BD = 7\text{ cm}$ and $AD = 6\text{ cm}$. Also, angle $DAC = 32^{\circ}$ and angle $BCA = 48^{\circ}$.

Non-right-angled triangles

A rectangle with side lengths $2.4\text{ cm}$ and $5.6\text{ cm}$ is enlarged using a scale factor of $3.25$.

Transformations

Solve the equation $\frac{3x}{x-1} + \frac{4}{x+2} = 3$.

Algebraic fractions

A rectangle has a width of $5.4\text{ cm}$, rounded to the nearest $0.1\text{ cm}$.

Limits of accuracy

The diagram depicts a pyramid $ABCV$. Its base is the equilateral triangle $ABC$, with each side measuring $34\text{ cm}$. The perpendicular height $VO$ meets the base at right angles. Also, $VA = VB = VC = 82\text{ cm}$.

Pythagoras' theorem and trigonometry in 3D

The box has $13$ pencils in total. Among them, $4$ are red, $7$ are green and $2$ are yellow. Two pencils are then selected at random from the box without replacement.

Probability of combined events

$104$ dollars are split between Ang and Bou in the ratio $7:6$.

Ratio and proportion

Shade one small square so that the diagram has rotational symmetry of order $2$.

Symmetry

Convert $6300\text{ cm}$ into metres.

Units of measure

Simplify the expression $5a + 3b + 2a - 7b$.

Algebraic manipulation

Write down every integer value of $x$ that makes the inequality $-3 \leq x < 1$ true.

Inequalities

A bag has $4$ black tiles and $6$ white tiles. One tile is picked at random from the bag and then put back. A second tile is then picked at random.

Probability of combined events

Ameerah places $480$ dollars into a savings account. Simple interest is paid at a rate of $3.6\%$ per year. Calculate the value of the investment after $5$ years.

Exponential growth and decay