Oct/Nov 2022

The temperature was $-2\,^{\circ}\text{C}$. It then falls by $8\,^{\circ}\text{C}$. Find the temperature after the change.

The four operations

Write the value $320\,000\,000$ in standard form.

Standard form

Express $120$ as a product of prime factors.

Types of number

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

Algebraic manipulation

A sequence has $n$th term $3n^2 - 1$. Find the first three terms of the sequence.

Sequences

Points $B$, $C$ and $D$ are located on the circumference of a circle with centre $O$. $AB$ is a tangent to the circle at $B$. $BD$ is a diameter, and $OCA$ lies on a straight line. $\angle CDB = x^{\circ}$.

Circle theorems I

Triangle $ABC$ is similar in the mathematical sense to triangle $DEC$. $AB = 12\,\text{cm}$, $BC = 27\,\text{cm}$, $CD = 7\,\text{cm}$ and $DE = 3\,\text{cm}$.

Similarity

Determine the gradient of line $L$.

Gradient of linear graphs

The diagram illustrates Sam’s speed-time graph for the trip from home to work.

Graphs in practical situations

$b$ varies directly with the square of $a$. When $a = 3$, $b = 18$.

Ratio and proportion

$ABD$ forms an equilateral triangle. $C$ is located on $DB$ and $AC$ is perpendicular to $DB$.

Geometrical terms

Work out $45\%$ of $\$1.20$.

Percentages

A farmer measures the mass of each sheep. Some of the results are summarised in the table and shown on the histogram.

Statistical charts and diagrams

$A = \begin{pmatrix} 3 & 1 \\ -4 & 2 \end{pmatrix}$ and $A + 2B = \begin{pmatrix} 1 & 5 \\ 10 & 12 \end{pmatrix}$.

Algebraic manipulation

Express $x^2 - 6x - 7$ in the form $(x + a)^2 + b$. Determine the value of $a$ and the value of $b$.

Equations

Volume of a cone $= \dfrac{1}{3}\pi r^2 h$, curved surface area of a cone $= \pi r l$. Surface area of a sphere $= 4\pi r^2$. A solid cone has radius $y\,\text{cm}$. The slant height of the cone is $25\%$ greater than its radius. A solid sphere has radius $R\,\text{cm}$. The sphere's surface area is the same as the cone's total surface area.

Surface area and volume

Arrange these fractions in increasing order, beginning with the least: $\frac{11}{12}$, $\frac{4}{5}$, $\frac{27}{30}$, $\frac{13}{15}$.

Ordering

The diagram indicates the positions of ships $A$ and $B$. On this diagram, $1\,\text{cm}$ stands for $30\,\text{m}$.

Scale drawings

Write $306.248$ to $2$ decimal places.

Limits of accuracy

Write $4 \times 4 \times 4 \times 4 \times 4$ in index form using base $4$.

Powers and roots

Work out the value of $\frac{7}{8} - \frac{3}{4}$.

Fractions, decimals and percentages

Factorise $3a^2 + 12a$.

Algebraic manipulation

Shade the part of the Venn diagram that represents $A \cap B$.

Sets

Work out the value of $80 \div 0.02$.

Powers and roots

Write $420$ as a product of prime factors.

Types of number

Azra owns a spinner. Its sectors are coloured red, blue, yellow or green. The table shows the relative frequency with which the spinner lands on red, blue or yellow.

Relative and expected frequencies

Plot the inequality $-4 \le x < 2$ on the number line.

Inequalities

Sophie travels $2600$ metres in $12$ minutes by bicycle. Calculate Sophie’s average speed in kilometres per hour.

Rates

The scale diagram represents a plot of land, $PQRS$. The scale is $1\text{ cm}$ to $20\text{ m}$.

Geometrical constructions

The diagram displays the points $A(0,6)$, $B(p,0)$ and $C(p,6)$. The line $AB$ has equation $3y + 4x = 18$.

Equations of linear graphs

The point $P$ is at $(-2,1)$ and the point $Q$ is at $(6,13)$. The midpoint of line $PQ$ is $M$.

Length and midpoint

Simplify the expression $(x^2)^3$.

Powers and roots

$x$ varies directly as the square of $(y+1)$. If $y=2$, then $x=45$.

Ratio and proportion

Solve $\dfrac{3x-1}{6} + \dfrac{x+2}{4} = \dfrac{5}{3}$.

Equations

Insert a single pair of brackets into this calculation so that it becomes correct: $4 + 4 \times 4 - 4 = 4$.

The four operations

The table gives some details about how long each of $100$ children spent reading during one day.

Statistical charts and diagrams

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

Functions

Factorise the expression $9p^2 - q^2$.

Algebraic manipulation

Adam and Ben purchase cinema and theatre tickets.

Algebraic manipulation

$\sin x^\circ = \sin 50^\circ$ with $90 \le x \le 180$. Find $x$.

Non-right-angled triangles

Simplify the expression $\dfrac{x^2 - 4x}{x^2 - x - 12}$.

Algebraic fractions

$OAC$ forms a triangle, and $B$ lies on $AC$ with $AB : BC = 3 : 2$. $\overrightarrow{OA} = \vec{a}$ and $\overrightarrow{OB} = \vec{b}$.

Vector geometry

Express $7.54 \times 10^{-4}$ as a decimal number.

Standard form

Sam is given six square tiles labelled $A, B, C, D, E$ and $F$.

Symmetry

A regular hexagon has the same perimeter as a regular octagon. Each side of the octagon measures $9\text{ cm}$. Determine the length of one side of the hexagon.

Area and perimeter

Calculate $\frac{11}{15} - \frac{2}{3}$.

Fractions, decimals and percentages

The diagram shows that $AD$, $AB$ and $BC$ are three edges of a regular pentagon, while $DC$ is one diagonal of that pentagon. Also, $AB$ is parallel to $DC$.

Angles

$ABC$ is an isosceles triangle, with $AB = BC$. The ratio $\angle ABC : \angle BAC = 2 : 5$.

Angles

Estimate the value of $\dfrac{47.5 + 36.1}{64.9 \div 17.7}$ by first converting each number to $1$ significant figure.

Estimation

Abid is employed in an office for 5 days every week.

Percentages

The diagram indicates the locations of the three towns $P$, $Q$ and $R$.

Non-right-angled triangles

The diagram depicts a regular pentagon $ABCDE$ with centre $O$.

Surface area and volume

The table gives the ages and heights of 10 boys.

Scatter diagrams

Examine the graph of $y = \frac{x^3}{2} - 3x + 2$.

Sketching curves

Take $f(x) = x^2 - 7$ and $g(x) = \frac{4 - 3x}{2}$.

Functions

George has two bags, and each bag contains black balls as well as white balls. A separate experiment uses counters in three colours.

Probability of combined events

The position vectors for points $A$ and $B$ are provided.

Vectors in two dimensions

Shapes $A$, $B$ and $C$ are displayed on a coordinate grid.

Transformations

Lara and Marco both cycle $50\text{ km}$ along a cycle trail.

Rates

Kate is thinking of a number $n$.

Equations

Hala goes by train from London to Marseille. She has to change trains in Paris. The trip from London to Paris lasts 2 hours 28 minutes. The trip from Paris to Marseille lasts 3 hours 30 minutes. Marseille and Paris are 1 hour ahead of London local time.

Money

Marco cultivates two kinds of tomato plant, type A and type B. He records how many tomatoes are on each tomato plant. The table shows the findings for type A plants.

Cumulative frequency diagrams

A cuboid is of height $h$ cm and has a square base with edge $x$ cm. Its volume is $60\text{ cm}^3$.

Sketching curves

Let the sets be defined as follows: $\mathcal{E} = \{x : x \text{ is an integer } 10 \leq x \leq 40\}$ $P = \{x : x \text{ is a multiple of } 6\}$ $Q = \{x : x \text{ is a square number}\}$

Sets

A bag has $x$ five-cent coins and $y$ ten-cent coins. Altogether, there are $130$ coins in the bag. Write down an equation, in terms of $x$ and $y$, for the total number of coins in the bag.

Equations

Reflect triangle $A$ in the line $x = 1$. Give the image the label $B$.

Transformations