May/June 2022

At an activity centre, students select one from four activities. The bar chart displays some of their selections.

Statistical charts and diagrams

Jason begins a run at 10.05 am and ends at 1.02 pm. Work out the time Jason takes to finish the run.

Time

Calculate $\frac{1 - 0.7}{0.45 - 0.38}$, then give the answer correct to 4 significant figures.

Limits of accuracy

Kirsty converts $380.80$ into pounds (£) using the rate £1 = $1.19. Work out the amount that Kirsty gets.

Rates

A spinner with 4 sides is labelled 1, 2, 3 and 4. The table gives the probabilities that the spinner lands on 1, 2 and 4.

Introduction to probability

Without a calculator, calculate $\frac{3}{7} - \frac{2}{21}$. Show every stage of your working and write your answer as a fraction in lowest terms.

Fractions, decimals and percentages

The diagram represents a right-angled triangle, $ABC$. $AC = 15$ cm and angle $BAC = 38^{\circ}$.

Right-angled triangles

$v = 3 - 5t$

Algebraic manipulation

Kim has a 6-sided spinner numbered 1 to 6. She spins it 63 times, and the results are recorded in the table.

Averages and measures of spread

Write $14xy - 7y^2$ in fully factorised form.

Algebraic manipulation

Lin invests $16000$ at a simple interest rate of $r\%$ per year. After 5 years, the total amount is $17920$.

Rates

A straight line with the label L is drawn.

Geometrical constructions

The sequence shown is 22, 17, 12, 7, 2, …

Sequences

State an irrational number with a value between 10 and 20.

Types of number

The table presents the population figures and land area of three countries in 2020.

Rates

Calculate the area of this trapezium.

Area and perimeter

Triangle $ABC$ and triangle $PQR$ are mathematically similar.

Similarity

Draw a chord on the diagram.

Circles, arcs and sectors

The scale diagram indicates where town A and town B are located. On this scale, 1 cm stands for 15 km.

Scale drawings

Express 0.56 kilometres in metres.

Units of measure

Put these numbers into order, beginning with the least: $\frac{6}{17}$, $34\%$, $\frac{9}{25}$, $0.345$.

Ordering

The diagram depicts two parallel lines intersected by a straight line.

Angles

The following facts are given about six numbers: • The smallest number is 37. • The range equals 24. • The mode is 43. • The median is 46. • One of the numbers is a multiple of 11.

Averages and measures of spread

Calculate the value of $4^5 - 5^4$.

Indices I

Express the number six hundred and seven thousand five hundred and thirty-two in digits.

Types of number

$\mathbf{p} = \begin{pmatrix}2\\8\end{pmatrix}$ while $\mathbf{q} = \begin{pmatrix}-1\\4\end{pmatrix}$

Coordinates

Determine the total surface area of a cuboid with length 8 cm, width 6 cm and height 3 cm.

Surface area and volume

Give an expression for the price of one bag of flour.

Introduction to algebra

Find the numerical value of $\sqrt{68} \times \sqrt{153}$.

Powers and roots

Express the ratio $5 \times 10^{-1} : 2 : 3 \times 10^{1}$ in simplest form.

Ratio and proportion

A sequence has $n$th term given by $n^2 + 12$.

Sequences

$33\frac{1}{3}\% ,\; \pi ,\; \frac{1}{13} ,\; 343^{\frac{1}{3}} ,\; \sqrt{3} ,\; 5.6 \times 10^{-7}$

Types of number

$9^x \times 9^2 = 9^{12}$

Indices I

After rounding each number in the calculation to 1 significant figure,

Estimation

A length of rope is represented by $l$ metres, and this is given as 30.7 m, accurate to 1 decimal place.

Limits of accuracy

61, 62, 63, 64, 65, 66, 67, 68, 69

Types of number

Simplify the expression $3(2a - b) - b$.

Algebraic manipulation

Determine the lowest common multiple (LCM) of 24 and 28.

Types of number

A diagram depicts points A and B, with North shown. NOT TO SCALE. The bearing from A to B is 059^{\circ}.

Angles

Do not use a calculator to find $4\frac{1}{8} - 2\frac{5}{6}$. Show all your working clearly, and write your answer as a mixed number in its simplest form.

Fractions, decimals and percentages

The diagram illustrates two right-angled triangles, $ABD$ and $BCD$. $AD = 5$ m, $DC = 14$ m and angle $BAD = 53^{\circ}$. NOT TO SCALE.

Right-angled triangles

On the grid, construct a triangle congruent to triangle A.

Transformations

This stem-and-leaf diagram gives the school journey times of some students. 1 | 3 5 7 9 9 2 | 3 4 5 3 | 0 3 4 6 7 4 | 2 4 5 8 Key: 1|3 means 13 minutes

Statistical charts and diagrams

Arania’s approach for dividing 213 by $12\frac{1}{2}$ without using a calculator is shown here. $213 \div 12\frac{1}{2} = 426 \div 25$ $= 852 \div 50$ $= 1704 \div 100$ $= 17.04$

Fractions, decimals and percentages

Sammy notes down the favourite hot drink chosen by some students. He uses a bar chart to present the information. The vertical axis has the label Frequency. The categories shown are Hot chocolate, Coffee and Tea.

Statistical charts and diagrams

$6 \times 7 - 5 + 4 = 16$

The four operations

At noon, the temperature reads 4^{\circ}C. At midnight, the temperature reads -9^{\circ}C.

The four operations

Thibault counts how many cars of each colour are in a car park. Colour: Black, White, Silver, Red Number of cars: 8, 5, 4, 3

Statistical charts and diagrams

Express the number one hundred and three thousand eight hundred and six in figures.

Types of number

These 12 parcels have masses, in kg, of 0.3, 0.4, 1.2, 0.8, 1.1, 2.1, 1.7, 1.8, 1.2, 2.3, 0.7 and 1.1. A stem-and-leaf diagram is given with stems 0, 1 and 2. Key: $0|3$ represents $0.3\,\text{kg}$.

Statistical charts and diagrams

The diagram displays point P and point R on a coordinate grid whose x- and y-axes are labelled.

Coordinates

Simplify the expression $y^3 \div y^5$.

Indices I

The scatter diagram displays, for each of eight days, the number of visitors at a zoo and the total amount spent, measured in thousands of dollars.

Scatter diagrams

The diagram shows a triangle with two equal sides of $5\,\text{cm}$, a base measuring $4\,\text{cm}$, and a perpendicular height $h$ dropped from the top vertex to the base.

Pythagoras' theorem

Factorise completely $18px - 27p$.

Algebraic manipulation

In this sequence, the $n$th term equals $n^2 - 1$.

Sequences

The figure shows two right-angled triangles, ABC and PQR. In triangle ABC, AB = $7.3\,\text{cm}$ and BC = $9.2\,\text{cm}$. In triangle PQR, RQ = $9.2\,\text{cm}$ and PQ = $7.3\,\text{cm}$.

Right-angled triangles

Determine the lowest common multiple (LCM) of 32 and 40.

Types of number

Joe chooses a number, $n$, multiplies it by 3, and then takes away 5. The answer is 22.

Equations

A straight line AB is drawn, with A at one end and B at the opposite end.

Geometrical constructions

A coordinate grid contains a straight line labelled L, which slopes down from the top left towards the bottom right.

Gradient of linear graphs

Dominic asks 30 students in his class whether they are right-handed or left-handed. 7 students are left-handed.

Relative and expected frequencies

No calculator is to be used.

Fractions, decimals and percentages

Solve the simultaneous equations. Make sure that you show all your working.

Equations

Simplify $3x - 4x + 7x$.

Algebraic manipulation

Calculate the area of a rectangle measuring $9.5\,\text{m}$ in length and $6.8\,\text{m}$ in width.

Area and perimeter

Calculate the probability of not choosing a red sweet.

Introduction to probability

The diagram shows A and B, with north indicated at both positions. A straight line connects A to B.

Angles

Calculate the value of $\frac{mk^3}{\sqrt{3}}$.

Algebraic manipulation

A cuboid-shaped box has a volume of $357\,\text{cm}^3$. Its length is $8.5\,\text{cm}$ and its width is $6\,\text{cm}$.

Surface area and volume

PQRS is a quadrilateral. RST forms a straight line. The diagram marks the angles at R, Q and P as $73^{\circ}$, $129^{\circ}$ and $75^{\circ}$ respectively.

Angles

State a prime number in the interval from 30 to 40.

Types of number

22, 17, 12, 7, 2, ...

Sequences

Triangle $ABC$ is similar to triangle $PQR$ in the mathematical sense. Diagram labels: For triangle $ABC$, $AB = 8\text{ cm}$ and $BC = 6\text{ cm}$. For triangle $PQR$, $PQ = 6\text{ cm}$. Diagram marked NOT TO SCALE.

Similarity

A pentagon’s interior angles are given in the ratio $4 : 5 : 5 : 7 : 9$.

Angles

Calculate $2 \times 10^{100} - 2 \times 10^{98}$ and give the result in standard form.

Standard form

A train moves through a station at 108 km/h. The station measures 120 m in length. It needs 7 seconds to go right through the station.

Rates

$4^x=\dfrac{1}{64}$

Indices I

A coordinate grid displays triangle $T$ and triangle $P$, with triangle $T$ shaded. The axes are labelled $x$ and $y$.

Transformations

Find the radius of a hemisphere whose volume is $80\text{ cm}^3$. [The volume, $V$, of a sphere with radius $r$ is $V = \dfrac{4}{3}\pi r^3$]

Powers and roots

A, B, C and D lie on a circle. TU is a tangent to the circle at D, and DA is parallel to CB. The diagram marks angles at D of $38^\circ$ (between tangent TU and chord DC) and $60^\circ$ (inside the circle). Diagram NOT TO SCALE.

Circle theorems I

In the diagram, $AB$ runs parallel to $PQ$. The lines $AQ$ and $PB$ meet at $X$, and $AX = XQ$.

Angles

Calculate the value of $4^5 - 5^4$.

Powers and roots

A speed-time graph is given for a car journey lasting 24 seconds. The vertical axis is marked Speed (m/s) up to 12, while the horizontal axis is marked Time (s) from 0 to 24. The speed remains constant from 0 to 16 seconds, then drops in a straight line to 0 at 24 seconds. NOT TO SCALE.

Graphs in practical situations

Factorise completely, giving $1 - q - a + aq$.

Algebraic manipulation

Simplify completely $(216x^{216})^{\frac{2}{3}}$.

Indices II

$x^2+8x+10=(x+p)^2+q$

Algebraic manipulation

A cuboid has dimensions of 24 cm, 12 cm and 8 cm.

Pythagoras' theorem and trigonometry in 3D

The quantity $w$ varies as the square root of $y$, and $y$ varies inversely with $x$. If $x = 4$, then $y = 16$ and $w = 8$.

Proportion

The diagram depicts triangle $OAB$ together with parallelogram $OALK$. The position vector of $A$ is $\mathbf{a}$ and the position vector of $B$ is $\mathbf{b}$. Point $K$ lies on $AB$ so that $AK : KB = 1 : 2$. Sketch NOT TO SCALE.

Vectors in two dimensions

The graphs of $y = x + 1$ and $y = x^2 - 3x - 11$ meet at the points $A$ and $B$.

Equations

Jason begins his run at 10.05 am and completes it at 1.02 pm.

Time

Calculate $\dfrac{1 - 0.7}{0.45 - 0.38}$, and express your answer correct to 4 significant figures.

Fractions, decimals and percentages

Kirsty converts $380.80 into pounds (£), using the rate £1 = $1.19.

Money

Write 180 in prime-factor form.

Types of number

Work out $\dfrac{3}{7} \div \dfrac{2}{21}$ without using a calculator. Show every stage of your working and write your answer as a fraction in simplest form.

Fractions, decimals and percentages

$s=\dfrac{1}{2}at^2$

Algebraic manipulation

Factorise fully. $14xy - 7y^2$.

Algebraic manipulation

When it is noon, the temperature is $4^$. When it is midnight, the temperature is $-9^$.

The four operations

$p = \begin{pmatrix}2\\8\end{pmatrix}$ and $q = \begin{pmatrix}-1\\4\end{pmatrix}$.

Vectors in two dimensions

Determine $p$ in $6^p \times 6^4 = 6^{28}$.

Indices I

Annette rides her bicycle for a distance of $70\text{ km}$ from Midville to Newtown. After leaving Midville, she travels for $1$ hour $30$ minutes at a steady speed of $20\text{ km/h}$, then pauses for $30$ minutes. She then completes the trip to Newtown at a steady speed of $16\text{ km/h}$. A distance-time grid is given, with the vertical axis marked Distance (km) from $0$ to $80$ and the horizontal axis marked Time (h) from $0$ to $5$.

Graphs in practical situations

Without a calculator, calculate $4\frac{1}{8} - 2\frac{5}{6}$. Show every step of your working and present your answer as a mixed number in simplest form.

Fractions, decimals and percentages

Carlos puts $4540$ into an account that pays compound interest at $r\%$ per year. By the end of $10$ years, the interest accumulated is $1328.54$.

Exponential growth and decay

State the highest common factor (HCF) of $12a^3b$ and $20a^2b^2$.

Algebraic manipulation

A class of $40$ students is represented in a Venn diagram for physics ($P$), mathematics ($M$) and geography ($G$). The diagram contains $4$ in $P$ only, $11$ in $M$ only, $6$ in $G$ only, $2$ in $P \cap M$ only, $3$ in $P \cap G$ only, $5$ in $M \cap G$ only, and $9$ in $P \cap M \cap G$. The shaded area contains everything in $M$ and everything in $G$, apart from the region that belongs only to $P$.

Sets

The diagram shows a pair of axes, with $x$ marked from $0$ to $360$ and $y$ marked from $-1$ to $1$.

Trigonometric functions

Find the value of $y$ for $x = 124$.

Ratio and proportion

$f(x) = 7x - 8$, $g(x) = \frac{4}{x} + 5$, with $h(x) = 2^x + 1$.

Functions

Thibault counts how many cars of each colour are in a car park.

Statistical charts and diagrams

Factorise the expression $2m + 3p - 8km - 12kp$ fully.

Algebraic manipulation

The $n$th term in a sequence is given by $an^2 + bn - 4$. Its first term is $-3$ and its second term is $2$.

Sequences

A diagram includes the points $O$, $A$, $D$ and $B$. The diagram is not drawn to scale. $\vec{OA} = x$, $\vec{OB} = y$ and $\vec{OD} = \frac{3}{7}x + \frac{4}{7}y$.

Vector geometry

The diagram depicts a solid metal object formed from a cone and a hemisphere, each with radius $6.2\text{ cm}$. The solid’s total surface area is $600\text{ cm}^2$. For a sphere of radius $r$, the curved surface area, $A$, is $A = 4\pi r^2$. For a cone of radius $r$ and slant height $l$, the curved surface area, $A$, is $A = \pi r l$.

Surface area and volume

Each fig costs 43 cents. Lyra has $5 available to spend on figs.

Money

Calculate the value of $\sqrt{68} \times \sqrt{153}$.

Surds

Work out the total surface area of a cuboid whose length is $8\text{ cm}$, width is $6\text{ cm}$ and height is $3\text{ cm}$.

Surface area and volume

Each card shows either a square, a circle, or a triangle. Piet picks one card at random.

Introduction to probability

A coat costs $126. In a clearance sale, the price is cut by $18\%$.

Percentages

The $n$th term in this sequence is $n^2 + 12$.

Sequences