May/June 2021

Zachary asks the 30 students in his class to name their favourite sport. The results are shown in the table. Netball: 7, Football: 12, Hockey: 6, Tennis: 5. A pictogram is displayed with rows labelled Netball, Football, Hockey and Tennis. In the Netball row, there is one full circle split into four equal sections and one circle that is half shaded. Key: one full circle stands for 4 people.

Statistical charts and diagrams

The figure shows two parallel lines crossed by two straight lines. The angles indicated are 59^{\circ}, 37^{\circ}, $a^{\circ}$, $b^{\circ}$ and $c^{\circ}$. The figure is not drawn to scale.

Angles

State the mathematical name of a polygon that has 5 sides.

Geometrical terms

For this sequence, the nth term is $6n-4$.

Sequences

A circle has a radius of 42 cm.

Circles, arcs and sectors

Convert $680\,000\text{ cm}^3$ to $\text{m}^3$.

Units of measure

The rope length, $l$ metres, is 5.67 m, rounded to the nearest centimetre.

Limits of accuracy

Without a calculator, calculate $1\frac{3}{8}-\frac{5}{6}$. Show every step of your working and write your answer as a fraction in simplest form.

Fractions, decimals and percentages

Write $\dfrac{1}{2\times2\times2\times2\times2}$ in the form of a power of 2.

Indices I

Annie places $8300 in an account earning compound interest at 5.6% each year.

Money

State an irrational number, $n$, for which $31 \le n \le 32$.

Types of number

State the value of $\sqrt{225}$.

Fractions, decimals and percentages

Estimate the value of $\dfrac{38.7\times3.115}{20.3-4.1^2}$ by rounding every number in the calculation to 1 significant figure. Show all your working.

Estimation

Solve the simultaneous equations $2x+y=3$ and $x-5y=40$. Show all your working.

Equations

The towns of A and B are joined by a straight road that is 130 km long. Maxi sets off from town A towards town B, while Pippa goes from town B towards town A. Each one moves at a steady speed of 40 km/h, and Maxi starts 30 minutes ahead of Pippa.

Rates

The diagram shows a square grid, with several of the corner squares shaded.

Symmetry

The stem-and-leaf diagram gives the number of hours studied last week by each of the 16 students. Stem 1 shows leaves 2 5 6 8 Stem 2 shows leaves 0 1 1 7 9 Stem 3 shows leaves 2 3 4 5 Stem 4 shows leaves 4 5 7 Key: $1|2$ means 12 hours.

Averages and measures of spread

A cuboid has volume $24\text{ cm}^3$. Its base measures 3 cm by 2 cm. A 1 cm$^2$ grid is supplied.

Surface area and volume

The travel graph below plots distance (km) against time from 12:00 to 16:00.

Graphs in practical situations

The chance that a train arrives late is 0.15.

Introduction to probability

At a bank, Nazaneen exchanges $6500 for 5798 euros.

Rates

Calculate $\begin{pmatrix}6\\-5\end{pmatrix}+\begin{pmatrix}8\\-1\end{pmatrix}$.

Coordinates

Draw the line of symmetry for this figure.

Symmetry

Jane's chance of winning the game is $\frac{7}{10}$.

Introduction to probability

Calculate $\sqrt[4]{0.0256}$.

Powers and roots

Emma needs to answer 15 mathematics questions. A stem-and-leaf diagram gives the time, in minutes, that she spends on each question. The stems and leaves are: 0 | 3 5 6 7 7 8 8 1 | 1 2 2 3 6 6 6 2 | 0 Key: 2 | 0 = 20 minutes.

Averages and measures of spread

Complete the statements below. The reciprocal of $0.2$ equals [BLANK]. One prime number within the range 90 to 100 is [BLANK].

Types of number

The equation $7^{x} \div 7^{4} = 7^{9}$ is provided.

Indices I

Circle the single correct statement about this scatter diagram.

Scatter diagrams

The cuboid shown has a square base, with volume $867\text{ cm}^3$ and height $12\text{ cm}$.

Surface area and volume

A rhombus with side length $6.5\text{ cm}$ can be formed by constructing two triangles.

Geometrical constructions

Without a calculator, calculate $\frac{2}{3} \div 1\frac{3}{7}$. Show every step of your working, and write your answer as a fraction in simplest form.

Fractions, decimals and percentages

Inside a bag are 5 red balls and 3 blue balls. Sophie chooses a ball at random, records its colour, and then returns it to the bag. She repeats this for a second draw.

Probability of combined events

List all the factors of 42.

Types of number

The diagram displays two cylinders. The smaller cylinder is 5 cm in diameter and 13 cm high. The larger cylinder is 12.5 cm in diameter and 32.5 cm high. The diagram is not drawn to scale.

Similarity

Write $0.00654$ as a number in standard form.

Standard form

A quadrilateral is shown in a diagram. Its interior angles are marked as $4x^{\circ}$, $(3x+75)^{\circ}$, $(87-x)^{\circ}$, and one right angle ($90^{\circ}$). The drawing is not to scale.

Angles

Find the lowest common multiple (LCM) of 24 and 54.

Types of number

Expand and simplify the expression $5(2x-7) - 3(x-5)$.

Algebraic manipulation

The diagram depicts a sector of a circle, with centre O and radius 9 cm. Its sector angle is $72^{\circ}$. The points are marked O, A and B. The drawing is not to scale.

Circles, arcs and sectors

From point P, draw a line that is perpendicular to this line.

Geometrical constructions

Calculate the mean for these values.

Averages and measures of spread

To convert a temperature from Celsius (^{\circ}C) into Fahrenheit (^{\circ}F), use $F = \frac{9C}{5} + 32$.

Units of measure

Work out $9 + 5 \times 7 - 4 \div 2$ without a calculator. You should show all your working.

The four operations

The vectors are given as $\mathbf{a} = \begin{pmatrix}5\\-7\end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix}-2\\6\end{pmatrix}$.

Coordinates

State the number you add to $-4$ so that the result is $9$.

The four operations

The bar chart displays the marks achieved by a set of 55 students in an examination. The vertical axis is labelled 'Number of students'. The horizontal axis is labelled 'Marks' and has class intervals: 1 to 20, 21 to 40, 41 to 60, 61 to 80, 81 to 100.

Interpreting statistical data

The diagram depicts two straight lines meeting at a vertex where the angle is marked $a$.

Angles

The vectors are given by $\mathbf{a} = \begin{pmatrix}3 \\ -4\end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix}5 \\ 2\end{pmatrix}$.

Coordinates

Maria purchases $n$ pencils, each priced at $p$ cents. She uses a $\$y$ note to pay.

Introduction to algebra

Francesca uses a spinner with four equal sections, labelled 1, 2, 3 and 4. The table gives some of the probabilities for each possible result.

Introduction to probability

Alex converts 190 euros (€) into pounds (£) when $£1 = €1.1723$.

Rates

A regular polygon has an exterior angle of $36^\circ$.

Angles

Carry out the expansion and then simplify $6(t - q) - 2(t - 3q)$.

Algebraic manipulation

Work out $1\frac{2}{3} \div 7\frac{1}{2}$ without using a calculator. Show every step of your working and present your answer as a fraction in simplest form.

Fractions, decimals and percentages

The first four members of a sequence are 7, 11, 15, 19.

Sequences

Calculate the volume of a cylindrical vase whose radius is $14.2\text{ cm}$ and whose height is $18\text{ cm}$.

Surface area and volume

Express $0.00074$ in standard form.

Standard form

Points $A$ and $B$ are situated on a circle with centre $O$. The sketch is labelled NOT TO SCALE. The line $AB$ is shown inside the circle.

Geometrical terms

A cohort of 120 students sit two tests, mathematics ($M$) and English ($E$). The following information is given about how many students pass mathematics and how many pass English: • 61 students pass mathematics. • 27 students pass both mathematics and English. • 19 students do not pass mathematics and do not pass English. A Venn diagram is displayed with two overlapping circles labelled $M$ and $E$ inside a rectangle.

Sets

A coordinate plane is displayed with a straight line marked $L$. The x-axis runs from $-4$ to $6$, and the y-axis runs from $-2$ to $5$.

Equations of linear graphs

The diagram shows two triangles. Triangle $ABC$ has sides $AB$, $BC = 34\text{ cm}$ and $AC = 28\text{ cm}$. Triangle $DEF$ has sides $DE$, $EF = 85\text{ cm}$ and $DF$. The diagram is not drawn to scale. Triangle $ABC$ is similar to triangle $DEF$.

Similarity

Simplify $3x^3 \times 4x^4$ by applying index laws.

Indices I

The diagram depicts a right-angled triangle. One side has length $18\text{ cm}$, the hypotenuse measures $30\text{ cm}$ and the angle at the base is marked $x^\circ$. The diagram is labelled NOT TO SCALE.

Right-angled triangles

State the number that is 23 below $-1.6$.

The four operations

Write $72\%$ as a fraction in its simplest form.

Fractions, decimals and percentages

The diagram depicts two parallel lines cut by a straight line. One angle is labelled $40^\circ$, and a second angle on the top line is marked $x$. The diagram is NOT TO SCALE.

Angles

The list of numbers is: 18, 28, 7, 15, 41, 19, 31, 53.

Averages and measures of spread

The diagram depicts a box in the form of a cuboid. Its top is open. The cuboid measures length $5\text{ cm}$, width $2\text{ cm}$, and height $4\text{ cm}$. The diagram is labelled NOT TO SCALE.

Surface area and volume

The figure contains three straight lines that intersect at a single point. Two of the angles are marked $y^\circ$ and the remaining angle is marked $100^\circ$. The figure is NOT TO SCALE.

Angles

The numbers in the list are 12, 18, 29, 49, 91 and 125.

Types of number

A square grid is displayed, and several corner squares are shaded.

Symmetry

A coordinate grid shows three triangles, marked A, B and C.

Transformations

Express $(4ab^5)^4$ in simplest form.

Indices II

A company’s profit falls exponentially by 0.9% each year. The profit in 2014 was $9500.

Exponential growth and decay

A lake’s area on a map is $32\text{ cm}^2$. The map is drawn to the scale $1:24000$.

Scale drawings

$y$ varies directly with the square root of $(x-3)$. If $x=28$, then $y=20$.

Ratio and proportion

Rearrange $2mh = g(1-h)$ so that $h$ is the subject of the formula.

Algebraic manipulation

A straight-line graph of $l$ is shown on the coordinate axes.

Equations of linear graphs

A bag holds 3 blue buttons, 8 white buttons and 5 red buttons. Two buttons are selected at random from the bag, without replacement.

Probability of combined events

The figure shows triangle $O$, $P$ and $Q$. Vector $\vec{OP}=a$ and vector $\vec{OQ}=b$. Point $S$ lies on $PQ$ so that $PS:SQ = 4:5$. The diagram is not drawn to scale.

Vector geometry

Sketch $y=\tan x$ over the domain $0^\circ \leq x \leq 360^\circ$.

Trigonometric functions

The chance that a train arrives late is 0.15.

Introduction to probability

The separation between two towns is 600 km, correct to the nearest 10 km. A car needs 8 hours 40 minutes, correct to the nearest 10 minutes, to cover this distance.

Limits of accuracy

The stem-and-leaf diagram gives the number of hours studied last week by each of 16 students. Stem 1: leaves 2, 5, 6, 8 Stem 2: leaves 0, 1, 1, 7, 9 Stem 3: leaves 2, 3, 4, 5 Stem 4: leaves 4, 5, 7 Key: 1|2 represents 12 hours.

Averages and measures of spread

The sketch depicts two parallel lines crossed by two straight lines. The angles shown are 59^{\circ}, 37^{\circ}, and angles labelled a^{\circ}, b^{\circ}, and c^{\circ}. The sketch is not drawn to scale.

Angles

Calculate $\begin{pmatrix}6\\-5\end{pmatrix}+\begin{pmatrix}8\\-1\end{pmatrix}$.

Vectors in two dimensions

Work out the first three terms of this sequence.

Sequences

Solve the simultaneous equations below. Show all your working. $2x+y=3$ $x-5y=40$

Equations

Without using a calculator, calculate $1\frac{3}{8} \div \frac{5}{6}$. Show all of your working and present your answer as a fraction in its simplest form.

Fractions, decimals and percentages

$A$ is located at $(5,-5)$, and $B$ is located at $(9,3)$.

Length and midpoint

The chance that Jane wins a game is $\frac{7}{10}$.

Relative and expected frequencies

Write $0.00654$ in standard form by expressing it as a number between $1$ and $10$ multiplied by a power of ten.

Standard form

Write $0.0\dot{4}$ as a fraction in lowest terms.

Fractions, decimals and percentages

$\mathcal{E} = \{\text{integers exceeding }2\}$ $A = \{\text{primes}\}$ $B = \{\text{odd integers}\}$ $C = \{\text{perfect squares}\}$

Sets

A, B and C are points on a circle with centre O. DA and DC are tangents, and angle ADC = $44^\circ$.

Circle theorems I

The diagram depicts a trapezium with one line of symmetry. Its top base measures $15.4$ cm, the marked height is $18.2$ cm, and the bottom-left angle is $62^\circ$.

Area and perimeter

Complete the table that shows whether each pair of triangles is congruent. The first two rows have already been completed for you. None of the diagrams is drawn to scale.

Similarity

The point $A$ is $(5, 7)$, while the point $B$ is $(9, -1)$.

Length and midpoint

Find the gradient of the line perpendicular to $3y = 4x - 5$.

Gradient of linear graphs

Take $f(x) = x^2 - 25$  $g(x) = x + 4$.

Functions

Calculate the perimeter of the shape.

Circles, arcs and sectors

Calculate the value of $\sqrt[4]{0.0256}$.

Powers and roots

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

Algebraic manipulation

Between two magnets, the attractive force $F$ Newtons varies inversely with the square of the distance $d$, cm, separating the magnets. If $d = 1.5$, then $F = 48$.

Proportion

Simplify the expression $\frac{2x^2 - 5x - 12}{3x^2 - 12x}$.

Algebraic fractions

Find every solution of $4\sin x = 3$ for $0^\circ \leq x \leq 360^\circ$.

Trigonometric functions

Solve $\frac{1}{x + 1} + \frac{9}{x + 9} = 1$.

Equations

Emma is to complete 15 mathematics questions. The stem-and-leaf diagram shows how many minutes she needs for each question. Stem 0: leaf values 3, 5, 6, 7, 7, 8, 8 Stem 1: leaf values 1, 2, 2, 3, 6, 6, 6 Stem 2: leaf value 0 Key: $2|0 = 20$ minutes.

Averages and measures of spread

Write down an expression showing the range for $k$ consecutive integers.

Sequences

Henrik draws this scatter diagram, using crosses to show the plotted points on the axes.

Scatter diagrams

A rhombus has a side length of $6.5$ cm. It may be formed by drawing two triangles. One diagonal of the rhombus has already been drawn.

Geometrical constructions

Complete the following statements: The reciprocal of $0.2$ is ............... . A prime number between $90$ and $100$ is ............... .

Types of number

The relation is $a = \frac{b^2}{5c}$.

Algebraic manipulation

Without a calculator, calculate $\frac{2}{3} \div 1\frac{3}{7}$. Show every stage of your working, and write your answer as a fraction in simplest form.

Fractions, decimals and percentages

State the number that is $23$ less than $-1.6$.

The four operations

The coordinate grid places triangle $T$ in quadrant I and triangle $A$ in quadrant IV.

Transformations

Simplify $3x^3 \times 4x^4$.

Indices I

The integer $x$ satisfies $-3 \leq 2x - 1 < 3$.

Inequalities

Expand the brackets, then simplify $6(t - q) - 2(t - 3q)$.

Algebraic manipulation