Oct/Nov 2024

Jacob is 10 years 8 months old. Amy is 15 months younger than Jacob.

Time

The sequence has these first four terms: 10, 16, 22, 28

Sequences

A wheel's circumference measures 198.55 cm.

Circles, arcs and sectors

The grid shows one half of a shape, and $L$ is its line of symmetry.

Symmetry

Find the value obtained for $6c + 7d$ when $c = 3$ and $d = -4$.

Equations

Express 34 as a percentage of 80.

Percentages

On a journey, a bus makes 25 stops. The table gives the number of passengers who board the bus at each stop.

Averages and measures of spread

The diagram displays a trapezium whose parallel sides measure 12 cm and $w$ cm. Its height is 8 cm. The area of the trapezium is $78\text{ cm}^2$.

Area and perimeter

The distance, $d$ metres, is recorded as 34.6 m to the nearest 0.1 m.

Limits of accuracy

On a $1\text{ cm}^2$ grid, the diagram displays a rectangle together with two points, $P$ and $C$.

Transformations

Jo asks a group of people whether they own a car ($C$) and whether they own a motorbike ($M$). There are 86 people who own a car. There are 39 people who own a motorbike. 7 people own neither a car nor a motorbike. A Venn diagram is displayed with $C$ and $M$, and 74 appears in region $C$ only.

Sets

Convert 6.7 kilometres into metres.

Units of measure

Josh purchases a car for $7800 and later sells it for $5265.

Percentages

Factorise the expression $28x - 35$.

Algebraic manipulation

Solve the simultaneous equations below. You must show all your working. $5x + 6y = 9$ $3x - 2y = -17$

Equations

Calculate $5\frac{1}{3} - 3\frac{4}{7}$ without a calculator. Show all your working, and write your answer as a mixed number in its simplest form.

Fractions, decimals and percentages

The diagram shows a section of a regular polygon. Its interior angle is $132^{\circ}$ greater than its exterior angle. In the diagram, the exterior angle is marked $x^{\circ}$ and the interior angle is marked $(x + 132)^{\circ}$. The diagram is not drawn to scale.

Angles

The diagram depicts an angle marked x.

Angles

A concert begins at 1950 and ends 2 hours 42 minutes later.

Time

Choose one symbol from <, > or = so that each statement is correct. $\frac{2}{7}$ ............. 0.2861 $\frac{99}{900}$ ............. 11% $1^3$ ............. $4^0$

Fractions, decimals and percentages

The stem-and-leaf diagram summarises the number of cars a company sold each day. Stem 1: 0 3 4 5 6 7 Stem 2: 1 2 2 4 7 7 7 Stem 3: 0 0 1 2 2 5 6 8 Stem 4: 0 1 4 6 Stem 5: 1 2 4 Key: 3|2 stands for 32

Averages and measures of spread

Find the value of the reciprocal of $1\frac{1}{4}$.

Fractions, decimals and percentages

The diagram represents a quadrilateral whose interior angles are $49^{\circ}$, $28^{\circ}$, $138^{\circ}$ and $y^{\circ}$. The figure is not drawn to scale.

Angles

Edith puts $3000$ into a savings account. This account earns simple interest at $2.6\%$ each year.

Percentages

Express half a million in figures.

Fractions, decimals and percentages

The numbers in the list are $0.25$, $3.142$, $\sqrt{3}$, $-3$, $24$, $-0.4$, $1.2$, $-\frac{1}{4}$.

Types of number

Town $A$ has a temperature of $-8^\circ\text{C}$, while town $B$ is at $16^\circ\text{C}$.

The four operations

The scale diagram gives the positions of the two ships, $A$ and $B$. The scale is $1\text{ cm}$ represents $6\text{ km}$. A North arrow points straight up from $A$. The segment from $A$ to $B$ slopes down to the right.

Scale drawings

The scatter diagram displays the number of rooms against the number of people for eight buildings. The vertical axis is labelled \"Number of rooms\" from 0 to 70. The horizontal axis is labelled \"Number of people\" from 0 to 80.

Scatter diagrams

The diagram depicts a trapezium. Its lower base measures $6\text{ cm}$. The right-hand vertical side measures $9.5\text{ cm}$. The left-hand vertical side is marked $x\text{ cm}$. Right angles appear at each end of the lower base. NOT TO SCALE. The area of the trapezium is $42\text{ cm}^2$.

Area and perimeter

In a league competition, a team earns 4 points for every win, 2 points for every draw, and bonus points. Suppose the team has $x$ wins, $y$ draws and $b$ bonus points.

Introduction to algebra

Dana places $\$3600$ into an investment earning compound interest at $3.8\%$ each year.

Percentages

The diagram depicts a parallelogram. One interior angle is marked $(132 - 2x)^\circ$ and the neighbouring interior angle is marked $(15 + 5x)^\circ$. The parallel sides are indicated. NOT TO SCALE.

Angles

Simplify the expression $\dfrac{18x^6}{3x^2}$.

Indices II

The diagram includes a coordinate grid, and point $B$ is marked at $(4, 2)$. The vector $ \vec{AB} = \begin{pmatrix} 3 \\ -2 \end{pmatrix}$.

Coordinates

Anton notes the colour of every car in a car park. His findings are presented in the table.

Statistical charts and diagrams

Points $A$ and $B$ are located on a circle with centre $O$ and radius $r$. The diagram depicts triangle $AOB$ with a right angle at $O$. Since both $OA$ and $OB$ are radii of the circle, the circle has area $120\text{ cm}^2$.

Circles, arcs and sectors

Work out $2\frac{3}{4} \times 1\frac{1}{2}$ without a calculator. Show every step of your working and express your answer as a mixed number in simplest form.

Fractions, decimals and percentages

Show all your working to Solve the simultaneous equations. $2x + 7y = 34$ and $3x + 5y = 18$.

Equations

Find $x$ in $5x = 14$.

Equations

The diagram depicts a regular polygon.

Symmetry

Write $53\,683.588$ to the nearest hundred.

Limits of accuracy

Triangle $ABC$ has side lengths $AC = 4.2\text{ cm}$ and $BC = 5.6\text{ cm}$. With a ruler and compasses only, construct triangle $ABC$. Keep the construction arcs visible. The side $AB$ has already been drawn for you.

Geometrical constructions

Insert one pair of brackets into $15 + 12 - 3 \times 4 = 51$ so that it becomes correct.

The four operations

Reduce $8c - d - 3c + 3d$.

Algebraic manipulation

The figure is an isosceles triangle. The base angle on the right is $43^\circ$. The two congruent sides are indicated. The angle at the apex is marked $x^\circ$. NOT TO SCALE.

Angles

Write 6475 rounded to the nearest ten.

Limits of accuracy

A cube has a surface area of 121.5 $\text{cm}^2$.

Surface area and volume

These diagrams are not drawn to scale.

Angles

Write down an equation for a line that is parallel to the line $y = 2x$.

Parallel lines

In a class containing 20 students, each one notes down how many coins are in their pockets. The table gives the outcomes. Number of coins: 0, 1, 2, 3, 4, 5, 6. Frequency: 3, 1, 7, 8, 0, 0, 1.

Averages and measures of spread

Expand the expression $4(x - 3)$.

Algebraic manipulation

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

Angles

Rio purchases some pens. He sells 63 pens, and that is $\frac{7}{9}$ of the pens he buys.

Fractions, decimals and percentages

Ed has $n$ books. Sam owns 3 times as many books as Ed does. Jane has 2 books less than Sam. Altogether, there are 54 books.

Equations

Do not use a calculator to find $2\frac{1}{4} - 1\frac{11}{12}$. Show every step of your working, and present your answer as a fraction in its simplest form.

Fractions, decimals and percentages

$\mathcal{E} = \{\text{office workers}\}$, $C = \{\text{workers who drink coffee}\}$, $T = \{\text{workers who drink tea}\}$. 47 people work in the office. 32 people drink tea. A Venn diagram is displayed. The region outside both circles contains 8. The T-only region contains 6.

Sets

Express 0.75 as a fraction.

Fractions, decimals and percentages

The weight, $w$ grams, of a box rounds to 463.9 grams, correct to 1 decimal place.

Limits of accuracy

Calculate $(6.4 \times 10^5) \div (2.5 \times 10^{-7})$. Present your answer in standard form.

Standard form

Mia places $1270 into an investment for 5 years at an annual compound interest rate of 2.1%.

Percentages

The figure depicts a right-angled triangle. Its hypotenuse measures 9.7 cm. One acute angle is $42^{\circ}$. The side drawn vertically is marked $x$ cm.

Right-angled triangles

Shapes A and B are mathematically similar. In shape A, the base length is 6.75 cm and the matching height is $x$ cm. In shape B, the base length is 2.7 cm and the matching height is 3.2 cm.

Similarity

A string measures 65.1 cm in length. It is then divided into 7 equal sections.

Fractions, decimals and percentages

The numbers given are 7, 19, 8, 12, 3, 12, 9, 7, 12.

Averages and measures of spread

A bag has 6 red balls, 4 green balls and 2 blue balls. Zia selects a ball from the bag at random. A probability scale is displayed from 0 to 1, with points marked A, B, C, D, E, F, G.

Introduction to probability

The first four terms of a sequence are 19, 26, 33, 40.

Sequences

Simplify $3p - t - p - 4t$.

Algebraic manipulation

The numbers shown are 61, 62, 63, 64, 65, 66, 67, 68 and 69.

Types of number

The scale diagram indicates where town K and town L are placed. The scale is 1 cm represents 10 km. North arrows are shown at each town.

Scale drawings

A concert begins at 19 50 and ends 2 hours 42 minutes later. Work out the time at which the concert comes to an end.

Time

Jacinda is playing a game with a friend. The game can end in a win, a loss or a draw for her. The chance that she wins is $0.28$.

Relative and expected frequencies

Work out $5\frac{1}{3} - 3\frac{4}{7}$ without a calculator. Show every step of your working and write your answer as a mixed number in simplest form.

Fractions, decimals and percentages

Solve the simultaneous equations. You should show every step of your working. $5x + 6y = 9$ $3x - 2y = -17$

Equations

For a sequence with $n$th term $3n^2 - 1$, determine the second term.

Sequences

Two solid steel statues are mathematically similar. The smaller statue is 12 cm tall and the larger statue is 15 cm tall. The mass of the larger statue is 2.5 kg. The density of steel is $8\,\text{g cm}^{-3}$. Calculate the volume of the smaller statue. [Density = mass \div volume.]

Similarity

Pupils in class $P$ sit a test. The figures below summarise their marks. • lower quartile = 38 • median = 53 • interquartile range = 28 • range = 81 • highest mark = 96

Interpreting statistical data

The figure depicts a sphere with radius 6 cm and a cylinder with height 18 cm and radius $R$ cm. NOT TO SCALE. The sphere has the same volume as the cylinder. [For a sphere of radius $r$, the volume is $V = \frac{4}{3}\pi r^3$]

Surface area and volume

Solve. $3x^2 - 7x - 16 = 0$ You need to show all stages of your working and give your answers accurate to 2 decimal places.

Equations

The function is given by $g(x)=4^{x+3}$.

Exponential growth and decay

$\mathcal{E}=\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ $P=\{\text{odd numbers}\}$ $Q=\{\text{multiples of } 3\}$ $R=\{\text{square numbers}\}$

Sets

Find the reciprocal for $1\frac{1}{4}$.

Fractions, decimals and percentages

The diagram depicts two right-angled triangles $PQS$ and $RQT$. $PQR$ and $QTS$ lie on straight lines. The diagram gives: $PS = 18$ cm; angle at $P$ is $28^\circ$; $SQ$ is vertical with a right angle at $Q$; $ST = 4$ cm; $QR = 9$ cm. NOT TO SCALE.

Trigonometric functions

Solve the equation $3\tan x + 5 = 1$ for values of $x$ in the interval $0^\circ \leq x \leq 360^\circ$.

Trigonometric functions

The diagram on the grid shows the graph of $y = (x + 2)(x - 1)^2$.

Algebraic manipulation

From $(x - 5)^2 + k = x^2 - px - 21$, determine the value of $p$ and the value of $k$.

Algebraic manipulation

Choose one of the symbols $<$, $>$ or $=$ to complete each statement correctly. $\frac{2}{7}$ ............... $0.2861$ $\frac{99}{900}$ ............... $11\%$ $1^3$ ............... $4^0$

Fractions, decimals and percentages

Safia has a length of fabric measuring 5.6 m. She divides it into two sections whose lengths are in the ratio $3:4$. Calculate the length of the longer part.

Ratio and proportion

Find $3\begin{pmatrix}6\\-4\end{pmatrix}$.

Vectors in two dimensions

The diagram depicts right-angled triangle $ABC$ alongside quadrilateral $AEDC$. It shows a right angle at $A$; angle $B$ is $34^\circ$; at $C$ there is an angle marked $y^\circ$ with an adjacent angle of $17^\circ$; at $E$ the angles are $122^\circ$ and $x^\circ$; and at $D$ there is an angle marked $z^\circ$. NOT TO SCALE.

Angles

Factorise the expression. $28x - 35$

Algebraic manipulation

Edith places $3000 in a savings account. Simple interest is paid on the account at a rate of $2.6\%$ per year. Calculate the total interest earned after 3 years.

Percentages

The diagram shows a section of a regular polygon. The interior angle is marked $(x + 132)^\circ$, while the exterior angle is marked $x^\circ$. NOT TO SCALE.

Angles

The first eight terms of a sequence are: $c,\; -3,\; -9,\; -15,\; -21,\; -27,\; -33,\; k$.

Sequences

Calculate $\sqrt[3]{1 + 10.9 \times 0.4^2}$.

Powers and roots

Factorise fully the expression $24x^2 - 9xy$.

Algebraic manipulation

$y$ varies directly with the square root of $x + 1$. $y = 10.5$ when $x = 8$.

Ratio and proportion

A coordinate grid is displayed, with the $x$-axis and $y$-axis labelled from $-5$ to $5$. Region $R$ is defined by the inequalities $-3 < y \le 2$ and $y \le x - 1$.

Inequalities

The diagram gives the speed-time graph for the first 17 seconds of a car journey. NOT TO SCALE. The speed rises in a straight line from $0\,\text{m s}^{-1}$ at $0$ seconds to $20\,\text{m s}^{-1}$ at $10$ seconds, then stays at $20\,\text{m s}^{-1}$ until $17$ seconds.

Graphs in practical situations

At the beginning of the experiment, there are $40\,000$ bacteria. The bacteria population rises by $15\%$ each hour.

Exponential growth and decay

75 people are surveyed about whether they own a car, $C$, and whether they have a job, $J$. The Venn diagram displays the outcomes. There are 9 in $C$ only. There are 51 in the overlap of $C$ and $J$. There are 13 in $J$ only. There are 2 outside both circles.

Conditional probability

Points $A$, $B$ and $C$ lie on the circumference of a circle with centre $O$. $DA$ and $DC$ are tangents to the circle. Angle $ABC = 64^\circ$. NOT TO SCALE.

Circle theorems II

$\mathcal{E} = \{8 \times 10^{-1},\; 0.8,\; 8\%,\; \sqrt{0.08}\}$ $A = \{a : 0.08 < a \le 0.8\}$ $B = \{b : b \ge 0.8\}$ The Venn diagram for sets $A$ and $B$ within the universal set $\mathcal{E}$ is displayed.

Sets

Made up of a cylinder and a hemisphere, each having radius $4.3\,\text{cm}$. The cylinder has a length of $11.9\,\text{cm}$. The figure represents the solid. NOT TO SCALE. The volume, $V$, of a sphere with radius $r$ is $V = \frac{4}{3}\pi r^3$. The surface area, $A$, of a sphere with radius $r$ is $A = 4\pi r^2$.

Surface area and volume

The diagram depicts an isosceles triangle. NOT TO SCALE. The triangle has two sides marked equal. The top angle is labelled $x^\circ$. The angle at the bottom right is labelled $43^\circ$.

Angles

The terms in the sequence are: $\frac{1}{7},\; 1,\; 7,\; 49,\; 343,\; 2401,\; \ldots$

Sequences

Expand and simplify $(x + 3)(x + 5)(2x + 1)$

Algebraic manipulation

$A$ lies at $(17,9)$, while $B$ lies at $(23,39)$.

Perpendicular lines

The large box and the small box are mathematically similar. The volume of the large box is $72.8\%$ greater than the volume of the small box. The small box has length $3.5\,\text{cm}$ and the large box has length $x\,\text{cm}$. Diagrams of the small and large boxes are shown. NOT TO SCALE.

Similarity

Inside the box, the numbers are listed as: $0.25,\; 3.142,\; \sqrt{3},\; -3,\; 24,\; -0.4,\; 1.2,\; \frac{1}{4}$.

Types of number

The scatter diagram plots the number of rooms against the number of people for eight buildings. The horizontal axis runs from 0 to 80 and is labelled "Number of people". The vertical axis runs from 0 to 70 and is labelled "Number of rooms". A number of points have been plotted.

Scatter diagrams

Convert $7.51\,\text{m}^2$ to $\text{cm}^2$.

Units of measure

The diagram depicts a trapezium. It is not drawn to scale. The lower base measures $6\,\text{cm}$. The right vertical side measures $9.5\,\text{cm}$. The left vertical side is marked $x\,\text{cm}$. Right-angle markers are shown at the two bottom corners. The area of the trapezium is $42\,\text{cm}^2$.

Area and perimeter

Without using a calculator, calculate $\frac{2}{7} \div \frac{6}{11}$. Show all of your working and write your answer as a fraction in its simplest form.

The four operations

The figure depicts a parallelogram. NOT TO SCALE. The interior angle at the top left is marked $(132 - 2x)^\circ$. The interior angle at the bottom left is marked $(15 + 5x)^\circ$. Arrow symbols indicate the parallel sides.

Angles

Points $A$, $B$, $C$ and $D$ are positioned on a circle. $ABCD$ makes a square with area $72\,\text{cm}^2$. The diagram below depicts the square within a circle. NOT TO SCALE.

Circles, arcs and sectors

Using the numbers in the list, 61 62 63 64 65 66 67 68 69

Types of number

Calculate $2\tfrac{1}{4} - 1\tfrac{11}{12}$ without a calculator. Show all of your working and write your answer as a fraction in simplest form.

Fractions, decimals and percentages

Solve the pair of simultaneous equations $3p - 2q = 7$ and $p + 2q = 1$.

Equations

The relation can be written as $V = \sqrt[3]{\frac{x}{y}}$.

Algebraic manipulation