May/June 2023

Calculate.

Fractions, decimals and percentages

Work out $1\frac{1}{3} \times \frac{8}{9}$. Write your answer as a mixed number in its simplest form.

Fractions, decimals and percentages

Solve the inequality given by $x - 5 > 3x + 7$.

Inequalities

Ali keeps a log of the computer games he plays. In the first $6$ games, Ali wins $4$. In the first $20$ games, Ali wins $13$. Use these results to find the best estimate for the probability that Ali will not win the next computer game he plays.

Relative and expected frequencies

The bearing of Mingfield from Lenton is $156^{\circ}$. Calculate the bearing of Lenton from Mingfield.

Scale drawings

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

Algebraic manipulation

The sequence starts with these four terms: $1, 7, 13, 19$.

Sequences

$OMN$ is a sector of a circle with centre $O$. $ON = 20\text{ cm}$, and the area of the sector is $30\pi\text{ cm}^2$.

Circles, arcs and sectors

The mass of the planet Saturn is $5.7 \times 10^{26}\text{ kg}$. The mass of the planet Venus is $4.9 \times 10^{24}\text{ kg}$. Calculate the difference in mass between Saturn and Venus. Give your answer in standard form.

Standard form

Starting from $y = \sqrt{\frac{x + 2}{3}}$, rearrange the equation so that $x$ is the subject.

Algebraic manipulation

Points $A$ and $B$ lie on the circle with centre $O$. The lines $TA$ and $TB$ touch the circle at $A$ and $B$.

Circle theorems I

Write down the fraction of the $3 \times 3$ square that is shaded.

Fractions, decimals and percentages

Starting with $f(x) = 10 + 7x$, determine $f^{-1}(x)$.

Functions

Rectangle $ABCD$ is similar to rectangle $PQRS$. The measurements are $AB = 12\text{ cm}$, $BC = 9\text{ cm}$ and $PQ = 8\text{ cm}$.

Similarity

Factorise $7y + 2xy - 6x - 21$.

Algebraic manipulation

The number of spectators at a cricket match is $36000$, rounded to the nearest thousand.

Limits of accuracy

100 batteries are checked to find out how long each one lasts. The table presents the outcomes.

Statistical charts and diagrams

For $(ax^b)^3 = 27x^4$, determine the value of $a$ and the value of $b$.

Indices I

$A$ has coordinates $(-2, 3)$, while $B$ has coordinates $(4, 7)$.

Length and midpoint

In the parallelogram $OABC$, $\overrightarrow{OA} = 2a$ and $\overrightarrow{OC} = 3c$. $M$ is the midpoint of $BC$, and $T$ lies on $OB$ with $OT : TB = 2 : 1$.

Vector geometry

The diagram depicts four straight lines intersecting at one point.

Angles

Benjamin's age is $t$ years.

Introduction to algebra

To make the calculation correct, place one pair of brackets: $3 + 5 \times 2 - 7 = 9$.

The four operations

Finish the pattern so that $AB$ is the only line of symmetry.

Symmetry

Five temperatures are shown here in $^{\circ}\text{C}$: $-18, -21, -2, 17, -10$. Put these temperatures in order from the coldest to the hottest.

Ordering

A rope is divided into three sections whose lengths are in the ratio $3:5:4$. The shortest section measures $180\text{ cm}$.

Ratio and proportion

The diagram presents triangles $A$ and $B$.

Transformations

Calculate.

Fractions, decimals and percentages

Solve the simultaneous equations: $x + 2y = 7$ and $3x + 4y = 11$. Show your working.

Equations

Using each number rounded to $1$ significant figure, estimate the value of $\dfrac{18.2^3}{0.395}$.

Estimation

Shapes $A$ and $B$ are shown on the grid.

Transformations

The initial four terms of a sequence are $1, 3, 9, 27$. Find the next term in the sequence.

Sequences

Write $325$ in the form of a product made from prime factors.

Types of number

A $1\text{ cm}$ square grid shows three lines together with a shaded region.

Inequalities

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

Graphs in practical situations

The functions are $f(x) = 2 - 3x$ and $g(x) = x - 4$.

Functions

Juan sells gift bags that contain soaps and candles. Matrix $C$ shows the contents of one large gift bag and one small gift bag: $C = \begin{pmatrix} 6 & 4 \\ 2 & 1 \end{pmatrix}$. A soap has mass $120\text{ g}$ and a candle has mass $60\text{ g}$. Matrix $M = \begin{pmatrix}120 \\ 60\end{pmatrix}$.

Introduction to algebra

Shade the part of the Venn diagram that corresponds to $(A \cap B)' \cup (B \cap C)'$.

Sets

The figure shows a circle with centre $O$. A straight line is touching the circle. Fill in each label with the correct mathematical term. One radius has already been named for you.

Geometrical terms

Expand then simplify $(4x - y)(2x + 5y)$.

Algebraic manipulation

Solve $rac{5x}{x-3}=x+4$.

Algebraic fractions

$y$ varies directly as $w^2$, while $x$ varies inversely as $w$. When $w = 10$, $y = 5$ and $x = 0.4$.

Proportion

The set contains $10$ cards altogether. Each card displays either a square or a triangle. On every card, the shape is coloured either green or red.

Probability of combined events

$A$ lies at $(3,11)$ and $B$ lies at $(-5,-5)$. Line $L$ has equation $2y + x = 5$.

Perpendicular lines

Put these numbers into ascending order, starting from the smallest: $0.65$, $\frac{5}{8}$, $62\%$, $\frac{11}{20}$, $0.595$.

Ordering

At midday the temperature is $8^{\circ}\text{C}$. By midnight it is $12^{\circ}\text{C}$ lower. Determine the temperature at midnight.

Averages and measures of spread

Maya places $\$480$ into an investment at $2\%$ per year simple interest.

Money

The scale sketch locates two villages, $A$ and $B$. The scale is $1\text{ cm to }2\text{ km}$.

Scale drawings

Ben exercises by walking. The scatter diagram gives the distance for $10$ walks and how long each walk lasts.

Scatter diagrams

Find $1\frac{3}{4} + \frac{5}{6}$. Write your answer as a mixed number in simplest form.

Fractions, decimals and percentages

A triangular prism is shown in the diagram. Its cross-section is a right-angled isosceles triangle.

Surface area and volume

A cuboid is given with side lengths of $x$ cm, $(5-x)$ cm and $15$ cm. The diagram depicts a pyramid whose square base has side length $9$ cm. The pyramid has height $x$ cm and volume $y$ cm$^3$.

Graphs of functions

Bags of sweets are placed into boxes.

Probability of combined events

The diagram depicts an open-top container resting on a horizontal plane. The container is a prism, and trapezium $ABCD$ forms its cross-section. $AB = 28$ cm, $DC = 24$ cm, $AD = 16$ cm and $BF = 29$ cm. Angle $ADC$ and angle $DAB$ are right angles.

Right-angled triangles

Filomena starts work at 10.45 am on Monday. She works for another 2 hours 50 minutes. Find the time she finishes work on Monday.

Percentages

$PQ$ runs parallel to $RS$. $ABCD$ lies on a straight line. Also, $BE = CE$ and $\angle ABE = 110^\circ$. Points $U, V, W, X$ and $Y$ are located on the circumference of a circle with centre $O$. $UY$ is a diameter of the circle, and $ZX$ is tangent to the circle at $X$. $\angle VUX = 35^\circ$, $\angle XZY = a^\circ$ and $\angle VWY = b^\circ$.

Circle theorems II

The cumulative frequency graph indicates the amount of fuel, $f$ litres, purchased by 100 customers at a service station on one day.

Cumulative frequency diagrams

Take $\mathcal{E} = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}$. Define $P = \{x : x \text{ is a multiple of } 3\}$, $Q = \{x : x \text{ is an odd number}\}$, and $R = \{x : x \text{ is a factor of } 24\}$.

Sets

Simplify the expression $3u - 6w - 5u + 9w$.

Algebraic manipulation

For triangle $ABC$, $AC = 8.3$ cm and $\angle BAC = 105^\circ$. The diagram also represents quadrilateral $PQRS$. Here $SQ = 15$ cm, $\angle SPQ = 67^\circ$ and $\angle PQS = 74^\circ$.

Non-right-angled triangles

The distance-time graph represents Maya’s trip from the office to the factory. The diagram gives the cyclist’s distance-time graph. The cyclist covers $d$ metres from home to a lake and then heads back home.

Graphs in practical situations

The Bukhari family and the Garcia family are heading off on holiday. In the Bukhari family there are 2 adults and 3 children, whereas in the Garcia family there are 4 adults and 1 child.

Equations

A shop purchases some fruit. The table displays the bill for this fruit.

Percentages

$F$ is located at $(6,1)$, $G$ is located at $(-2,4)$ and $\overrightarrow{GH} = \begin{pmatrix} -1 \\ -8 \end{pmatrix}$. Calculate $|\overrightarrow{FH}|$.

Vector geometry

Each chocolate bar is $p$ cents and each packet of sweets costs $75$ cents. Tanish spends $9.10$ on $5$ chocolate bars and $8$ packets of sweets. Set up an equation and solve it to determine $p$. Show your working.

Algebraic manipulation

The figure shows equilateral triangle $B$ together with part of regular polygon $A$, and they share one side. The interior angle of polygon $A$ is $165^\f$.

Angles

Fill in the values table for $y = \frac{2^x}{5}$.

Exponential growth and decay

The table lists the population figures and land areas for three countries in 2019.

Standard form

In the Trio Challenge, Sophia first walks, then cycles, and finally swims.

Limits of accuracy

On Monday, the amount of money each customer spent on a website was noted. The results are shown in the table.

Statistical charts and diagrams

Determine the length and width of rectangle $R$.

Surface area and volume

A, B and C are points on flat ground. The bearing of $B$ from $A$ is $072^\f$. The bearing of $C$ from $A$ is $205^\f$. $AB = 170\text{ m}$ and $AC = 95\text{ m}$.

Non-right-angled triangles