May/June 2021

Calculate it.

The four operations

Using each number rounded correctly to one significant figure, estimate the value of $\frac{362.4 - 187.2}{52.3}$.

Estimation

A survey found that $3$ out of every $100$ women were taller than $1.9\text{ m}$. One woman is selected at random from these $100$ women. Calculate the probability that she is not taller than $1.9\text{ m}$.

Relative and expected frequencies

Bernard purchased a game in the USA for $\$15$. Alice obtained the same game in Zambia and paid the equivalent amount in Zambian kwacha (ZK). Exchange rate: $1\text{ ZK} = \$0.075$.

Rates

The two numbers $x$ and $y$ satisfy $x:y = 5:11$ and have the sum $x+y = 112$.

Ratio and proportion

For this sequence, the term-to-term rule is: multiply by $2$ and add $3$. The first three terms are $1, 5$ and $13$. Write down the next term in this sequence.

Sequences

Foxby and Glanton are two villages linked by both a footpath and a road.

Rates

Triangles $A$, $B$ and $C$ have been plotted on the grid.

Transformations

The diagram depicts an isosceles triangle $ABC$ with $AB = AC$. Point $D$ lies on $AC$ so that angle $ADB = 90^\circ$. Point $E$ lies on $AB$ so that angle $AEC = 90^\circ$.

Right-angled triangles

For the simultaneous equations below, Solve and Show your working. $x + 6y = 0$ and $3x - 2y = 10$.

Equations

The quantity $y$ is proportional to $(x-1)^2$.

Ratio and proportion

The list of numbers is $15, 125, \sqrt{8}, 11, \sqrt{25}, 14, 60$. Using the numbers shown above, write down

Types of number

The straight line $2y = x$ is shown on the grid.

Drawing linear graphs

Factorise the expression.

Algebraic manipulation

The mass of a car is $2400\text{ kg}$, rounded to the nearest hundred kilograms. The mass of a caravan is $1460\text{ kg}$, rounded to the nearest ten kilograms.

Limits of accuracy

This gives $a = \frac{b^2 + c}{d}$.

Standard form

With $M = \begin{pmatrix}5 & 1 \\ 2 & 3\end{pmatrix}$ and $N = \begin{pmatrix}4 & -2 \\ 3 & 0\end{pmatrix}$, find the matrix $M - N$.

Introduction to algebra

The diagram displays a circle with centre $C(0,1)$. $P(3,5)$ lies on the circle's circumference.

Equations of linear graphs

Calculate $\frac{3}{7} + \frac{2}{5}$.

Fractions, decimals and percentages

A harbour's water level is recorded. In the morning, it measures $5\text{ m}$. By afternoon, it is $-2\text{ m}$. Work out the difference between the morning level and the afternoon level.

The four operations

The diagram illustrates quadrilateral $ABCD$, with $AD$ extended to $E$. The angles given are $\angle BCD = 135^\circ$, $\angle BAD = 83^\circ$ and $\angle CDE = 122^\circ$.

Angles

Express $308$ as a product of its prime factors.

Types of number

The diagram represents trapezium $ABCD$. The lengths $AB = 7\text{ cm}$ and $DC = 10\text{ cm}$ are given. The area of $ABCD$ is $85\text{ cm}^2$. The perpendicular height of the trapezium is $h\text{ cm}$.

Area and perimeter

Simplify the expression $6x + 15 - 2x + 8$.

Algebraic manipulation

Select the right symbol $=$, $>$ or $<$ so that each statement becomes correct.

Units of measure

Express $0.45$ as a fraction in simplest form.

Fractions, decimals and percentages

Write $270$ in the form of a product of prime factors.

Types of number

Solve these simultaneous equations. Show your working.

Equations

Lara takes a cycle ride, and her journey is represented on the distance-time graph.

Graphs in practical situations

Express $0.000053$ in standard form.

Standard form

A path length is recorded as $62\text{ m}$, correct to the nearest metre. State the upper bound for the length of the path.

Limits of accuracy

Use only a ruler and compasses in this question.

Geometrical constructions

From $(y^k)^{-2} = y^5$, Find the value of $k$.

Indices I

During a sale, a coat's price is cut by $25\%$, and the sale price becomes $\$120$.

Percentages

$y$ varies inversely with the cube of $x$. When $x = \frac{1}{2}$, $y = 24$.

Ratio and proportion

A museum is visited by $40$ adults and $20$ children on Monday. On Tuesday, $30$ adults and $35$ children go there. An adult ticket costs $\$2.50$ and a child ticket costs $\$2$. The matrices $M$ and $N$ may be used to show this information.

Introduction to algebra

Complete the symmetry description for each shape.

Symmetry

The sequence begins with the four terms $\frac{12}{16}, \frac{17}{25}, \frac{22}{36}, \frac{27}{49}$.

Sequences

Express $x^2 + 10x + 6$ in the form $(x + a)^2 + b$.

Equations

Write as one fraction in simplest form: $\dfrac{3}{x - 7} + \dfrac{2}{x + 5}$.

Algebraic fractions

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

Transformations

A set of office employees is each asked to note the distance, $d$ kilometres, that they travel to work. The histogram shows the results for some of their journeys. There were $20$ workers in the $0 < d \le 5$ group.

Statistical charts and diagrams

The algebraic fraction $\dfrac{2x^2 - 5x + a}{x^2 - 16}$ may be reduced to $\dfrac{2x + b}{x + 4}$.

Algebraic fractions

Each of the 60 students was asked to name their favourite fruit. The results are displayed in the table.

Statistical charts and diagrams

Write $64\,785\,491$ rounded to the nearest million.

Limits of accuracy

Omar and Jamil divide $\$540$ in the ratio $7:2$. Work out Omar’s share.

Ratio and proportion

The cuboid has a square base with side length $4\text{ cm}$. Its volume is $48\text{ cm}^3$.

Surface area and volume

A bag has coloured counters inside. One counter is chosen at random from the bag. The table gives the probabilities of selecting a counter of each colour.

Introduction to probability

Work out $3\frac{2}{5} - 1\frac{3}{4}$. Provide your answer as a fraction.

Fractions, decimals and percentages

The diagram indicates the locations of three villages, $A$, $B$ and $C$. $A$ lies directly east of $B$.

Scale drawings

In 2019, Nicole earned $22000$ as her yearly income.

Exponential growth and decay

The displacement vector is $\vec{AB} = \begin{pmatrix} -3 \\ 5 \end{pmatrix}$.

Vectors in two dimensions

A survey counted how many people lived in each of 50 houses. The bar chart presents the findings.

Averages and measures of spread

$p = \frac{3q + 5}{r^2}$. Calculate the value of $p$ for $q = 15$ and $r = -4$.

Algebraic manipulation

The diagram depicts a sketch of quadrilateral $ABCD$.

Pythagoras' theorem and trigonometry in 3D

Shade the region $A' \cap B \cap C$.

Sets

The functions are $f(x) = 2x + 3$ and $g(x) = \frac{12 - 3x}{5}$.

Functions

The equations are $y = 2x + 1$, $y = 2x - 1$, $y = -2x + 1$ and $y = -2x - 1$. The diagrams below contain sketches of two of these lines. Write the correct equation under each diagram.

Sketching curves

Points P, Q and R lie on the circumference of a circle with centre $O$. The angle $POQ$ is $8x^{\circ}$, the angle $RPO$ is $x^{\circ}$ and the angle $OQR$ is $24^{\circ}$.

Similarity

The cumulative frequency graph displays the marks scored by 80 students in a Maths test.

Cumulative frequency diagrams

An electric drill costs $78$. During a sale, its price is cut by $15\%$. Calculate the sale price.

Percentages

The table records the midday temperature and the number of cups of hot chocolate Natcha sells on each of ten days.

Scatter diagrams

Simplify the expression $4a - b + 6b - 7a$.

Algebraic manipulation

A survey of 100 adults in a town asked how many emails each person received on one day. The table presents the findings.

Averages and measures of spread

Write, in set notation, the subset shown shaded in the Venn diagram.

Sets

The triangle $PQR$ is isosceles, with $PR = QR$. $P$ is $(1,5)$ and $Q$ is $(5,1)$. $ ngle PRQ$ is not a right angle. Determine the coordinates of one possible location for $R$.

Perpendicular lines

A rectangular field has dimensions of $30\text{ m}$ and $45\text{ m}$. Calculate the perimeter.

Pythagoras' theorem

The functions are defined by $f(x)=3x-5$ and $g(x)=\frac{4x+4}{3}$.

Functions