May/June 2022

State the value of the digit 5 in the number 253624.

Limits of accuracy

The table below gives the monthly rent for nine apartments together with how far each apartment is from the city centre.

Scatter diagrams

100 adults were asked about the colour of their car. Write down the relative frequency that one of these cars is blue.

Relative and expected frequencies

Using each number correct to $1$ significant figure, estimate the value of $\dfrac{0.28\times37.4}{77.8}$.

Limits of accuracy

Expand, then simplify.

Algebraic manipulation

State $0.000863$ in standard form.

Standard form

Evaluate $7^{-3}\div7^{-4}$.

Indices I

The vectors are $\mathbf{p}=\begin{pmatrix}2\\3\end{pmatrix}$ and $\mathbf{q}=\begin{pmatrix}-3\\2\end{pmatrix}$.

Vectors in two dimensions

The map is drawn using a scale of $2\text{ cm}$ for every $1\text{ km}$. The wood’s area on the map is $6\text{ cm}^2$.

Scale drawings

On the Venn diagram, shade the area represented by $P\cap Q'$.

Sets

The points $B,D,E,F$ and $G$ lie on the circumference of a circle whose centre is $O$. $AC$ is tangent to the circle at $B$. Angle $DFG=75^{\circ}$ and angle $ABG=48^{\circ}$.

Circle theorems I

Write down the number of lines of symmetry shown by this diagram.

Symmetry

The function is specified by $f(x)=\dfrac{6x+2}{5}$.

Functions

$y$ varies inversely with $(x+1)^2$.

Ratio and proportion

Factorise the expression.

Algebraic manipulation

The equation is $y=\dfrac{3x+2}{2x-1}$.

Algebraic manipulation

Let $M=\begin{pmatrix}1&0\\4&3\end{pmatrix}$ and $N=\begin{pmatrix}k&0\\1&4\end{pmatrix}$. If $MN=NM$, determine the value of $k$.

Algebraic manipulation

In triangle $ACD$, $B$ lies at the midpoint of $AC$, and $E$ lies at the midpoint of $AD$. $\overrightarrow{AB}=6\mathbf{a}+3\mathbf{b}$ and $\overrightarrow{DC}=5\mathbf{a}+2\mathbf{b}$.

Vector geometry

The table gives the mean monthly temperatures, in $^{\circ}\text{C}$, at Vladivostok.

The four operations

The combined volume of two cubes is $152\text{ cm}^3$. One cube has an edge length of $5\text{ cm}$.

Surface area and volume

The diagram presents the net of a solid on a $1\text{ cm}$ grid.

Surface area and volume

State

Types of number

20 students were questioned about how many pets they owned. Their replies appear in the table.

Averages and measures of spread

Find the value.

Fractions, decimals and percentages

Order these lengths by magnitude, with the least first: $32000\text{ cm}$, $3300\text{ mm}$, $3.1\text{ km}$, $34\text{ m}$.

Units of measure

Find the answer.

Fractions, decimals and percentages

A bag contains red, blue and green balls. The ratio of red to blue is $3:8$. The ratio of green to blue is $2:5$.

Ratio and proportion

Construct the bisector of angle $PSR$ using only a straight edge and compasses.

Geometrical constructions

Give $0.00203561$ correct to $3$ significant figures.

Limits of accuracy

Calculate the value of $(\sqrt{9}\times\sqrt[3]{64})^2$.

Powers and roots

Arrange these numbers from the smallest to the largest: $2000$, $0.002$, $2\times10^{-4}$, $2\times10^{-2}$.

Standard form

$y$ varies in direct proportion to $(x-1)^2$. For $x=5$, the value of $y$ is $32$.

Ratio and proportion

The graph of $y=\frac{x^3}{5}-3x+2$ has been plotted on the grid.

Graphs of functions

Ryan states that each diagonal of quadrilateral $Q$ splits it into two congruent isosceles triangles. Draw a ring around every quadrilateral in the list for which Ryan’s statement is always true: Square, Rectangle, Rhombus, Parallelogram, Trapezium, Kite.

Geometrical terms

The function is defined by $f(x)=3x-7$.

Functions

$\mathcal{E}=\{a,b,c,d,e,f,g,h,i,j\}$, $P=\{i,a,e\}$, $Q=\{j,i,h,g,f\}$, $R=\{g,f,e,d,c\}$.

Sets

Asha asks a group of students which fruit they prefer most. The table and pictogram display some of the results.

Statistical charts and diagrams

$A$, $B$, $C$ and $D$ lie on a circle with centre $O$.

Circle theorems I

Factorise the expression $4x^2+5x-6$.

Algebraic manipulation

Inside the bag are these $9$ letter tiles: I, S, O, S, C, E, L, E, S.

Probability of combined events

The diagram represents the major sector of a circle with centre $O$ and radius $3\,\text{cm}$.

Circles, arcs and sectors

Solve for $x$: $\frac{2-5x}{3x+10}=3$.

Algebraic fractions

$OABC$ and $OPQR$ are parallelograms. $A$ lies on $OP$ and $C$ lies on $OR$. Also, $\overrightarrow{OA}=\mathbf{a}$ and $\overrightarrow{OC}=\mathbf{c}$. The ratio $OA:OP=1:4$ and $OC:CR=2:3$.

Vector geometry

Points A and B are corners of a quadrilateral. Line $L$ is the quadrilateral’s line of symmetry.

Transformations

The temperature inside Luke’s house is $18^{\circ}\text{C}$. The temperature outside his house is $-3^{\circ}\text{C}$. Find the difference between these temperatures.

Limits of accuracy

The scale diagram shows where villages $A$ and $B$ are located. The scale given is $1\,\text{cm}$ for $2\,\text{km}$.

Scale drawings

Kabir places $\$250$ in a savings account. The account earns simple interest at a rate of $1.5\%$ each year.

Rates

The rectangle has area $9\,\text{cm}^2$. The triangle has area $85\,\text{mm}^2$.

Units of measure

The diagram depicts a pentagon.

Angles

Shani creates a run of patterns with counters.

Sequences

In 2020, Frederick’s car had running costs of $5200. Insurance took up 28% of the running cost. Maintenance accounted for $\frac{3}{25}$ of the running cost. $740 of the running cost was used for tax. Whatever was left from the running cost was spent on petrol.

Percentages

Triangles $A$, $B$ and $C$ are shown in the diagram.

Transformations

$P$ is located at $(3, -3)$, while $Q$ is located at $(1, 5)$.

Length and midpoint

Shown below are the first four patterns in a sequence formed from grey tiles and white tiles.

Sequences

The graph gives the cost, in dollars, of buying a length of fabric $t$ metres long.

Graphs in practical situations

A cuboid measures $x\text{ cm}$ by $x\text{ cm}$ by $10\text{ cm}$. Its volume is $62.5\text{ cm}^3$. Determine the value of $x$.

Surface area and volume

Fill in the value table for $y = x^3 - 4x + 3$.

Graphs of functions

The diagram displays a field, $ABCD$, drawn at a scale of $1\text{ cm}$ to $50\text{ m}$.

Geometrical constructions

Yasir commutes to work by car, bus, train or bike.

Probability of combined events

If apples are priced at $x$ per kilogram and oranges at $y$ per kilogram, the cost of $5\text{ kg}$ of apples plus $10\text{ kg}$ of oranges is $\$40$. Show that $x + 2y = 8$.

Equations

The diagram indicates where the three towns, $A$, $B$ and $C$, are located.

Non-right-angled triangles

The price of mailing a letter was 84 cents in 2021. In 2022, it increases to 96 cents. Mailing a letter costs 96 cents. A 15% discount is given when monthly postage is more than $1000.

Percentages

$D$ is located at $(4,6)$, while $E$ has coordinates $(e,e)$. $F$ is at $(-f,5f)$.

Length and midpoint

Using $A = 3p + q$, find $q$ when $A = 23$ and $p = 5$.

Algebraic manipulation

The spinner has 5 sides labelled 1, 2, 3, 4 and 5. The table records the outcomes after 200 spins of the spinner.

Relative and expected frequencies

The diagram depicts a pentagon. Every length is given in centimetres.

Surface area and volume

The table lists the time, $t$ seconds, needed by each student to finish a puzzle. A different group, made up of adults, also completed the puzzle. Their times are represented on a cumulative frequency diagram.

Cumulative frequency diagrams

The graph includes the line $y = x + 2$.

Drawing linear graphs

The diagram presents the cyclist’s journey as a speed-time graph.

Graphs in practical situations

Matrix $A$ obeys an equation. Shapes $A$ and $B$ are displayed.

Transformations

The pole $PQ$ is vertical. A rope goes from the top of the pole, $P$, to a point on the ground, $R$. Two more ropes are attached too.

Non-right-angled triangles