Oct/Nov 2023

Calculate $6 + 4 \div 2$.

The four operations

A pentagon has exterior angles of $150^\circ$, $100^\circ$, $45^\circ$ and $35^\circ$. Calculate the size of the missing exterior angle.

Angles

Evaluate the expression $4^2 + \sqrt[3]{27}$.

Powers and roots

The scale diagram indicates the locations of two boats $A$ and $B$. The scale used is $1 : 20000$.

Geometrical constructions

Calculate $1\frac{3}{5} \div 1\frac{2}{3}$.

Fractions, decimals and percentages

Write $36$ in prime-factor form.

Types of number

Points $A$, $B$ and $C$ lie on the circle with centre $O$. The lines $AB$ and $AC$ are tangents to the circle. Angle $BAC = 38^\circ$.

Circle theorems I

The region $R$ is bounded by the inequalities $1 \le x \le 3$, $2 \le y \le 3$, and $y \ge \frac{x}{2} + 1$.

Drawing linear graphs

$y$ varies directly with the square root of $x$. If $x = 16$, then $y = 2$.

Proportion

In a sports club with $40$ members, $22$ are runners ($R$), $24$ are cyclists ($C$), $14$ sail ($S$), $3$ cycle and sail but do not run, $9$ run and cycle but do not sail, $5$ run and sail but do not cycle, and $6$ run only.

Sets

The diagram presents the speed-time graph for one section of a car journey.

Graphs in practical situations

Arrange these numbers by size, beginning with the smallest: $\frac{1}{5}$, $\frac{3}{25}$, $13\%$, $0.1$.

Fractions, decimals and percentages

Take $A = \begin{pmatrix}-2 & 1 \\ 4 & 3\end{pmatrix}$ and $B = \begin{pmatrix}3 & 2 \\ -1 & 1\end{pmatrix}$.

Algebraic manipulation

Factorise the expression $6a - 9$.

Algebraic fractions

The functions are given by $f(x) = \frac{x}{4} + 3$ and $g(x) = 2(x - 1)$.

Functions

The diagram shows parallelogram $OABC$. The vector $\vec{OA}$ is $\vec{a}$ and $\vec{OC}$ is $\vec{c}$. $X$ lies halfway along $AC$. $Y$ is on $AB$ such that $AY : YB = 2 : 1$.

Vector geometry

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

Algebraic fractions

Find the temperature that is $20$ degrees above $-12^\circ\text{C}$.

The four operations

Kasia purchases $12$ apples. The cost of each apple is $65$ cents.

Money

Yasmin asks $20$ people how many pets they each own. The outcomes are displayed in the bar chart.

Averages and measures of spread

The diagram depicts one straight line intersecting two parallel lines.

Angles

Estimate the value of $\dfrac{53.7}{2.61 + 7.48}$ by first writing each number correct to $1$ significant figure.

Limits of accuracy

Change $78\text{ mm}$ into cm.

Units of measure

The scatter diagram plots the ages of ten people against the time each one needs to finish a task.

Scatter diagrams

Find the answer.

Fractions, decimals and percentages

Using each number rounded to $1$ significant figure, estimate the value of $\sqrt{\dfrac{1240 \times 3.8}{11.2}}$.

Estimation

Solve the inequality $7m - 13 \leq 8$.

Inequalities

Solve the simultaneous equations, and show each stage of your working.

Equations

A set of eight numbers has an average of $12$. The mean of the first five numbers is $9$.

Averages and measures of spread

Determine angle $ABC$.

Geometrical constructions

A linear sequence has second term $28$ and fifth term $16$. Find the first term, the third term and the fourth term of the sequence.

Sequences

The relation is $T = \sqrt{P - 4}$.

Algebraic manipulation

The heights of $80$ plants are recorded. The table gives the results.

Cumulative frequency diagrams

The speed-time graph in the diagram represents part of the journey for cyclists $A$ and $B$.

Graphs in practical situations

Express the expression below as a single fraction in its simplest form: $\dfrac{x+1}{8} + \dfrac{3x}{4} - \dfrac{5x}{16}$.

Algebraic fractions

This rectangle has been divided into squares of two different sizes.

Fractions, decimals and percentages

Factorise this expression.

Algebraic manipulation

Diagram $A$ depicts a sector of a circle, with centre $D$ and radius $\dfrac{3}{4}y$ cm. The obtuse angle $EDF = 6x^{\circ}$. Diagram $B$ depicts a sector of a circle, with centre $P$ and radius $y$ cm. The sector angle is $x^{\circ}$.

Circles, arcs and sectors

$\begin{pmatrix} x & 3 \\ 2 & x+1 \end{pmatrix} \begin{pmatrix} x-1 \\ 2 \end{pmatrix} = \begin{pmatrix} 2x+6 \\ y \end{pmatrix}$.

Equations

A shop offers hats ($H$), scarves ($S$) and gloves ($G$). A set of $40$ people are asked which of these items they purchase in the shop. Part of the information is displayed in the Venn diagram.

Sets

$OAB$ forms a triangle. $P$ is on $AB$ and $AP : PB = 2:3$. $\overrightarrow{OA} = 4a$ and $\overrightarrow{OP} = 3a + 2b$.

Vector geometry

Find the decimal that is exactly midway between $\frac{3}{5}$ and $68\%$.

Powers and roots

Sonu notes the temperature, in $^{\circ}\text{C}$, at midnight on each of 12 days. The readings, listed from the coldest upward, are $-6, -5, -3, -2, -1, -1, T, 5, 5, 6, 6, 7$

Averages and measures of spread

Anna and Ria divide some money in the ratio $5:9$. Ria gets $\$8$ more than Anna.

Ratio and proportion

$AB$ and $CD$ are parallel. $EC$ and $FB$ are also parallel. Angle $ABF = 73^{\circ}$.

Angles

On a centimetre square grid, shape $A$ and triangles $P$ and $Q$ are shown.

Transformations

Express $0.00493$ in standard form.

Standard form

Write $180$ as a product made from prime factors.

Types of number

The population of a town is $36\,400$. $23\%$ of the population are aged $18$ and under. Work out the number of people in the town aged over $18$.

Percentages

The diagram depicts a circle with centre $O$, and $AC$ and $BD$ are diameters.

Circle theorems I

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

Transformations

Line $L$ is represented by $4y = x - 5$. Find the gradient of line $L$.

Equations of linear graphs

These are the first five terms in a sequence: $5,\ 12,\ 19,\ 26,\ 33$. Find the next term in the sequence.

Sequences

This question comes from a bus timetable. Work out the time taken for the bus to travel from the town square to the business park.

Averages and measures of spread

Rectangle $ABCD$ has area $30\text{ cm}^2$. $AB=x$ cm. Rectangle $DEFG$ is cut away from one corner of rectangle $ABCD$. $AE = CG = 2$ cm.

Graphs of functions

Simplify this expression. $7a - 4b - 2a + b$.

Algebraic manipulation

A set of $40$ children is each asked how many books they read in the previous month. Write down the mode.

Probability of combined events

A triangular prism is shown in the diagram. Calculate the volume of the prism. State the units for your answer.

Surface area and volume

Idris repairs computers. He works out the fee for a repair like this. $56 for the first hour$ Then $12.25$ for each extra $15$ minutes.

Exponential growth and decay

Volume of a cone $= \frac{1}{3}\pi r^2 h$. Curved surface area of a cone $= \pi r l$. A solid is made by removing a smaller cone from the middle of a larger cone. The small cone is mathematically similar to the large cone. The vertex of the large cone lies vertically below the vertex of the small cone. The height of the large cone is $21$ cm and the diameter of the top is $18$ cm. The height of the small cone is $14$ cm.

Surface area and volume

The functions provided are $f(x) = 4x + 1$ alongside $g(x) = 2x - 3$.

Functions

Determine the size of one interior angle of a regular $15$-sided polygon.

Circle theorems I

Laila asks a group of people which kind of exercise they prefer. The results are displayed in the pie chart.

Statistical charts and diagrams

Solve the equation $5x + 6 = 3x$.

Equations

State the probability that the ball is green.

Probability of combined events

Complete the entries in the table for $y = \frac{4^x}{10}$.

Graphs of functions

Points $A$ and $B$ are plotted on a centimetre square grid.

Length and midpoint

The diagram contains a small rectangle positioned inside a larger rectangle. The larger rectangle has height $x$ cm. Its length is $4$ times its height. The shaded border has width $3$ cm. The area of the small rectangle is $80\,\text{cm}^2$.

Equations

$ABCD$ forms a rectangular field. $C$ lies directly south of $A$. $AB = 450\,\text{m}$ and $BC = 210\,\text{m}$.

Non-right-angled triangles