May/June 2023

Calculate how many months are in 5 years.

Time

For $y$ hours of bicycle hire, the cost, $C$, is determined by the formula $C = 12 + 3.5y$. Maria pays $36.50 in order to hire this bicycle.

Equations

$a = \begin{pmatrix}3\\7\end{pmatrix}$ while $b = \begin{pmatrix}-2\\5\end{pmatrix}$.

Coordinates

State the geometric reason that explains why $x$ has the value 52.

Angles

Calculate the volume of a sphere with a diameter of 4.8 cm. [For a sphere of radius $r$, the volume $V$ is given by $V = \frac{4}{3}\pi r^3$.]

Surface area and volume

By expressing each number in the calculation to 1 significant figure.

Limits of accuracy

Eric has four paint colours. The table gives the probability that he uses each colour.

Introduction to probability

Fully factorise $8x^2 - 20x$.

Algebraic manipulation

Write down the first three terms in this sequence.

Sequences

The length, $l$ metres, of a piece of wood is given as 3.6 metres, rounded to the nearest 10 centimetres.

Limits of accuracy

Calculate $1 \div (6.4 \times 10^{-5})$. State your answer in standard form.

Standard form

Write 3752 rounded to the nearest 10.

Limits of accuracy

Without a calculator, find $2\frac{1}{7} \div \frac{5}{9}$. Show every step of your working and write your answer as a mixed number in simplest form.

Fractions, decimals and percentages

A right-angled triangle is shown, with angles marked $(3x - 1)^{\circ}$ and $4x^{\circ}$, together with a right angle. The diagram is labelled NOT TO SCALE.

Angles

The diagram depicts a right-angled triangle with sides 7.5 cm and 12.8 cm, and an angle marked $y^{\circ}$. It is labelled NOT TO SCALE.

Right-angled triangles

Triangles $ABC$ and $DEF$ are similar. For triangle $ABC$, $BC = 7.2$ cm and $AC = 5.6$ cm. For triangle $DEF$, $EF = 8.1$ cm and $DF = h$ cm. The diagrams are marked NOT TO SCALE.

Similarity

$\xi = \{a, b, c, d, e, f, g, h, i, j, k\}$ $F = \{a, c, e, f\}$ $B = \{a, b, c, k\}$

Sets

In a cinema, the price of an adult ticket is $a$, while the price of a child ticket is $c$.

Equations

Each magazine costs $3.40, and Rosina has $15 available to buy as many magazines as she can.

Money

Identify the mathematical name of a 4-sided shape with rotational symmetry of order $2$ and no lines of symmetry.

Symmetry

The values listed are: 21, 8, 15, 32, 3, 29, 19, 45, 8.

Averages and measures of spread

The train departs at 21 43, and the trip lasts 8 hours and 32 minutes.

Time

The numbers given are $\frac{15}{213}$, 0.071, 0.7, 7%.

Fractions, decimals and percentages

Write $\frac{24}{84}$ in simplest form.

Fractions, decimals and percentages

Simplify $3a - 5b - a - 6b$.

Algebraic manipulation

Write down every factor of 18.

Types of number

A right-angled triangular prism is shown in a diagram. The triangular end has sides marked 3 cm, 4 cm and 5 cm, and the prism length is 6 cm. The diagram is NOT TO SCALE. On the 1 cm$^2$ grid, one face has already been drawn for you.

Surface area and volume

On the map, the separation between town A and town B is 3.5 cm. The map scale is $1 : 250\,000$.

Scale drawings

A spinner is spun once. The outcomes may be A, B, C or D. The table lists the probabilities for A, C or D. The values are: A = 0.2, B = blank, C = 0.05, D = 0.35.

Introduction to probability

$\mathbb{Z} = \{x : 1 \leq x < 20\}$, $E = \{\text{even numbers}\}$, $M = \{\text{multiples of }5\}$.

Sets

Without using a calculator, determine $\frac{4}{7} \div 1\frac{5}{21}$. Show every step of your working and present your answer as a fraction in lowest terms.

The four operations

$F$ has coordinates $(1, -4)$, $\vec{FG} = \begin{pmatrix} 8 \\ -3 \end{pmatrix}$ and $\vec{GH} = \begin{pmatrix} -12 \\ 35 \end{pmatrix}$.

Coordinates

$x$ is an integer satisfying $x \geq -3$ and $x < 3$.

Inequalities

Work out the size of one interior angle in a regular 15-sided polygon.

Angles

Express $45\,000$ in standard form.

Standard form

Simplify the expression $18x^{12} \div 3x^{3}$.

Indices I

The diagram depicts a line segment with endpoints marked A and B.

Geometrical constructions

At the station, buses travel either to the port or to the town. The port service departs every 28 minutes, while the town service departs every 48 minutes. At 10:18, one bus for the port and one for the town leave together from the station.

Time

The figure shows two triangles. For triangle ABC, AB = 15 cm and BC = 9 cm. For triangle PQR, PQ = 18 cm. The diagram is NOT TO SCALE. Triangle ABC is similar to triangle PQR.

Similarity

Calculate the length of $AB$.

Pythagoras' theorem

Finish this sentence about the value of $m$.

Inequalities

A square grid diagram with a few squares already shaded.

Symmetry

Kai and Ava each have a 57 cm-long piece of wood.

Ratio and proportion

The diagram depicts triangle ABC, with ACD forming a straight line. The angle at A is $73^{\circ}$, the angle at B is $59^{\circ}$, and at C there are two neighbouring angles marked $x^{\circ}$ and $y^{\circ}$. The drawing is labelled NOT TO SCALE.

Angles

Find the temperature that lies $8^{\circ}C$ below $-5^{\circ}C$.

Types of number

Within this list, the prime numbers are 27, 47, 57, 61, 75 and 93, and there are two of them.

Types of number

Over ten days, Stefan notes how many minutes he waits for a train: 1, 3, 12, 5, 4, 23, 5, 24, 11, 8. A stem-and-leaf diagram is given with stems 0, 1 and 2. Key: 0|1 means 1 minute.

Statistical charts and diagrams

The scale drawing indicates where town A and town B are located. A north arrow marked "North" is drawn at A. The diagram is NOT TO SCALE.

Angles

Write 928 rounded to the nearest ten.

Limits of accuracy

Olga believes that 87 is a prime number. Is Olga right? Give a reason for your answer.

Types of number

A film has a duration of 2 hours 50 minutes. It finishes at 23 05.

Time

Triangle $ABC$ is isosceles, with $AB = AC$ and $\angle BAC = 38^\circ$. The diagram is not drawn to scale.

Angles

Using each number in the expression rounded correctly to 1 significant figure, determine an estimate for the value of $\dfrac{6.8 \times 10.6}{3.2 - 0.98}$. Show your working.

Limits of accuracy

The scatter diagram presents data on how long was spent in a shop and how many items were purchased. The horizontal axis is labelled "Time (min)" and the vertical axis is labelled "Number of items bought".

Scatter diagrams

Simplify the expression $d^8 \div d^2$.

Indices I

At an exchange rate of $\$1 = 0.913$ Swiss francs, Maddie converts 4000 Swiss francs into dollars.

Rates

Find the highest common factor (HCF) for 32 and 120.

Types of number

Tom’s probability of arriving late for school is 0.12. This year has 200 school days.

Relative and expected frequencies

Expand the product $(x - 5)(x + 8)$ and then simplify it.

Algebraic manipulation

Write down any fraction equivalent to $\frac{7}{9}$.

Fractions, decimals and percentages

The figure shows a circle with centre $O$ and diameter $AB$. Points $A$, $B$ and $C$ are on the circumference. The diagram is not drawn to scale.

Circle theorems I

The spinner has five sides, and the colours are red, blue, green, yellow and orange. The table gives some of the probabilities that the spinner will land on each colour.

Probability of combined events

Vanessa puts $8500 into an investment earning compound interest at 3.5% each year.

Percentages

The diagram illustrates three shapes, $A$, $B$ and $C$, drawn on a $1\,\text{cm}^2$ grid with the $x$- and $y$-axes labelled.

Transformations

Work out $5\frac{11}{12} + 2\frac{1}{4}$ without a calculator. Show all your working, and write your answer as a mixed number in its simplest form.

Fractions, decimals and percentages

All measurements in this question are in centimetres. The diagram shows two line segments, $AB$ and $CD$. $AB$ has length $10x - 12$. $CD$ has length $2x + 3$. $AB$ is 3 times the length of $CD$. The diagram is not drawn to scale.

Equations

Calculate $-4 + 6 \times 3$.

The four operations

Bobby counts the number of days that a shop is open over 30 days. The table gives tally marks for the open days.

Classifying statistical data

A $3 \times 3$ array of squares is shown. Three of the squares are shaded: the upper right square, the middle square, and the lower left square.

Symmetry

State the reciprocal of 16 in decimal form.

Fractions, decimals and percentages

The stem-and-leaf diagram gives the ages of 21 people. Key: 1 | 6 shows 16 years.

Averages and measures of spread

A cuboid is illustrated, and the diagram is not drawn to scale. The dimensions given are length 8 cm, width 3 cm and height 5 cm.

Surface area and volume

The diagram depicts a triangular prism. Each triangular face has side lengths of 4 cm, and the prism’s length is 2 cm. The diagram is not drawn to scale.

Surface area and volume

The diagram depicts a shape made by points B, A, C, D and E. The angle at B is $104^\circ$, the angle at A is $71^\circ$, and the angle at C is $56^\circ$. Points C, D and E are positioned on one straight horizontal line. The figure is labelled NOT TO SCALE. The note underneath says: CDE is a straight line.

Angles

Without using a calculator, work out $2\frac{1}{7} \div \frac{5}{9}$. Show all of your working and present your answer as a mixed number in its simplest form.

Fractions, decimals and percentages

A coordinate grid displays three congruent triangles, A, B and C, placed in different locations.

Transformations

Draw a ring round the correct criterion for this statement: SAS, RHS, SSS, ASA.

Angles

Calculate the value of $r$.

Equations

Find $y$ for $x = 1$.

Functions

Calculate the probability that one button is green and the other is not green.

Probability of combined events

Find the size of the vector $\begin{pmatrix}-4\\5\end{pmatrix}$.

Magnitude of a vector

Simplify the expression $(81x^{12})^{\frac{3}{4}}$.

Indices II

The diagram places three towns, $U$, $V$ and $W$, on a map. $U$ lies exactly to the west of $V$, and angle $UVW = 125^\circ$. The distances labelled are $UV = 27.3\text{ km}$ and $UW = 62.4\text{ km}$. At $W$ there is a north arrow. Diagram marked NOT TO SCALE.

Non-right-angled triangles

From the diagram, sketch the graph of $y = \cos x$ for $0^\circ \leq x \leq 360^\circ$.

Trigonometric functions

Determine the time at which the journey ends.

Time

The table gives several values of $y = 3x^2 - 2x - 1$.

Graphs of functions

Find the gradient of the curve at the point $(1, -11)$ on the curve.

Differentiation

A straight line crosses two parallel lines. At the upper crossing, the angle shown is $58^\circ$. At the lower crossing, the angles are labelled $a^\circ$ and $b^\circ$. The diagram is not drawn to scale.

Angles

Find an estimate for the value of $\dfrac{6.7 \times 2.1}{18 - 5.9}$ by first changing every number in the calculation to 1 significant figure. Show all your working.

Estimation

Eric has four paint colours. The table lists the chance that he uses each colour.

Introduction to probability

Calculate the volume for a sphere with diameter $4.8\text{ cm}$. [$\text{The volume, } V, \text{ of a sphere with radius } r \text{ is } V = \frac{4}{3}\pi r^3.$]

Surface area and volume

Calculate the island’s actual length, and give your answer in kilometres.

Scale drawings

The $n$th term is $10 - n^2$. Write down the first three terms of this sequence.

Sequences

Two right-angled triangles are illustrated. In triangle $ABC$, the base is $BC = 7.2\text{ cm}$ and the height is $AC = 5.6\text{ cm}$. In triangle $DEF$, the base is $EF = 8.1\text{ cm}$ and the height is $DF = h\text{ cm}$. The diagram is labelled NOT TO SCALE.

Similarity

Work out the temperature that is $8^{\circ}\text{C}$ below $-5^{\circ}\text{C}$.

The four operations

A coordinate grid displays shape A and shape B on axes marked $x$ and $y$.

Transformations

The diagram illustrates shape $ABCD$, which is created from sectors of two circles with the same centre $O$. Each sector has an angle of $140^{\circ}$, $OC = 3.2\text{ cm}$ and $CB = 2.6\text{ cm}$. The area of the shape is $k\pi\text{ cm}^2$.

Circles, arcs and sectors

For the equation $ax^2 + b = 181$, one root is $x = 8$. Both $a$ and $b$ are positive integers greater than 1.

Equations

$A$, $B$, $C$ and $D$ lie on a circle. $AB$ is parallel to $DC$, and $\angle ACD = 32^{\circ}$. The chords $AC$ and $DB$ cross at $E$.

Circle theorems I

The function can be expressed as $f(x) = 5x + 2$.

Functions

$C$ lies at $(5, -1)$, while $D$ lies at $(13, 15)$.

Perpendicular lines

Express $0.\dot{6}2\dot{1}$ as a fraction in lowest terms. Show all of your working.

Fractions, decimals and percentages

The diagram depicts a triangle in which an acute angle is labelled $x^{\circ}$. Its area is $2143\text{ cm}^2$. The sides of the triangle are marked $92.5\text{ cm}$ and $71\text{ cm}$.

Non-right-angled triangles

Rearrange the formula $c = \frac{3x}{2x - 5}$ so that $x$ is the subject.

Algebraic fractions

$m$ varies inversely as the square of $(t + 2)$. When $t = 3$, $m = 0.64$.

Ratio and proportion

In this list, there are two prime numbers: $27\quad 47\quad 57\quad 61\quad 75\quad 93$

Types of number

A Venn diagram presents universal set $\mathscr{E}$, together with three intersecting circles marked $A$, $B$ and $C$.

Sets

For $0^{\circ} \le x \le 360^{\circ}$, Solve the equation $5\sin x = -3$.

Trigonometric functions

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

Algebraic fractions

Bag $A$ and bag $B$ each hold red sweets and yellow sweets. Anna chooses one sweet at random from bag $A$, and Ben chooses one sweet at random from bag $B$. The chance that Anna selects a red sweet is $\frac{2}{5}$. The probability that Anna and Ben both select a yellow sweet is $\frac{1}{10}$.

Probability of combined events

Over ten days, Stefan notes the number of minutes that he waits for a train. $1\; 3\; 12\; 5\; 4\; 23\; 5\; 24\; 11\; 8$

Statistical charts and diagrams

On the map, the gap between town A and town B is $3.5\text{ cm}$. The map scale is shown as $1 : 250\,000$.

Scale drawings

A spinner is set spinning. The possible results are A, B, C or D. The probability of landing on A, C or D is given in the table. The table gives: Spinner letter: A, B, C, D Probability: 0.2, [blank], 0.05, 0.35

Introduction to probability

$\mathscr{E}=\{x:1\le x\le 20\}$ $E=\{\text{even numbers}\}$ $M=\{\text{multiples of }5\}$

Sets

Do not use a calculator. Find $\frac{4}{7} \div 1\frac{5}{21}$. Show every step of your working and express your answer as a fraction in simplest form.

Fractions, decimals and percentages

Solve for $x$ in $\frac{30}{x} = 6$.

Equations

$F$ is located at $(1, -4)$, while $\vec{FG} = \begin{pmatrix}8\\-3\end{pmatrix}$ and $\vec{GH} = \begin{pmatrix}-12\\35\end{pmatrix}$.

Vectors in two dimensions

A diagram displays a $3\times3$ array of equal squares. The grey-shaded squares are the top right square, the centre square, and the bottom left square.

Symmetry

Work out $5\frac{11}{12} + 2\frac{1}{4}$ without a calculator. Show every step of your working and present your answer as a mixed number in its lowest terms.

Fractions, decimals and percentages

For part (a), $\mathcal{E} = \{a, b, e, g, l, m, o, r, t, y\}$, $P = \{a, b, e, g, l, r\}$ and $Q = \{e, g, m, o, r, t, y\}$. The diagram shows two intersecting circles marked $P$ and $Q$ within a rectangle marked $\mathcal{E}$. For part (b), the figure shows two intersecting circles marked $A$ and $B$ inside a rectangle labelled $\mathcal{E}$.

Sets