May/June 2017

Write the number seventy thousand and twenty in figures.

Types of number

The table sets out the temperatures at five places at 10 am on one day in January.

Interpreting statistical data

Expand the brackets and simplify $7(2x + 3y) - x(14 - y)$.

Algebraic manipulation

The elephant’s mass, $m$ kilograms, is given as 3570 kg, rounded to the nearest 5 kg.

Limits of accuracy

$\mathbf{a} = \begin{pmatrix}5\\-1\end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix}-3\\-4\end{pmatrix}$.

Coordinates

By manual conversion, 5000 Mexican Pesos are exchanged for dollars, and he gets $336.

Rates

Maria surveys 50 students at her school about the month of their birthday and enters the results in the table.

Relative and expected frequencies

Calculate $\dfrac{\sqrt[3]{512}}{6^2}$. Give the answer as a fraction in lowest terms.

Powers and roots

$A=4\pi r^2$.

Algebraic manipulation

Calculate $3 + 2 \times -4$.

The four operations

Without a calculator, work out $1\frac{2}{3} + \frac{5}{7}$. Show every stage of your working and give the answer as a mixed number in its simplest form.

Fractions, decimals and percentages

Write the decimal form of $5^{-3}$.

Indices I

Solve the simultaneous equations. Show all your working. $5x - 2y = 24$ and $7x + 4y = -14$.

Equations

A cuboid measures 6 cm in length, 5 cm in width and 3 cm in height. On the 1 $\text{cm}^2$ grid, complete the net for the cuboid. The base has already been drawn.

Surface area and volume

Six students revise for a test. The scatter diagram displays how many hours each student spent revising and the mark they achieved in the test. The axes are labelled Time (hours) and Mark.

Scatter diagrams

The diagram displays the locations of points $A$, $B$ and $C$.

Geometrical constructions

The diagram displays a notice board. NOT TO SCALE. It has the form of a semicircle attached to a square whose side measures 74 cm.

Area and perimeter

Work out the thickness of 250 sheets of paper.

Ratio and proportion

Simplify the expression $(x^2)^5$.

Indices I

Two pentagons, A and B, are drawn on a grid. Beneath the figure there is a word list: opposite, congruent, reflected, translated.

Transformations

The list given is 31, 33, 35, 37, 39.

Types of number

Write 23.4571 rounded to 4 significant figures.

Limits of accuracy

Factorise $12n^2 - 4mn$ completely.

Algebraic manipulation

State the highest common factor (HCF) obtained from 126 and 150.

Types of number

Write $0.07164$ correct to $2$ significant figures.

Limits of accuracy

Write $0.03$ in percentage form.

Fractions, decimals and percentages

Determine the value of $5a - 3b$.

Introduction to algebra

The diagram depicts a transversal cutting across two parallel lines. One angle is labelled $70^\circ$, another is labelled $p^\circ$, and a further angle is labelled $q^\circ$. The diagram is NOT TO SCALE.

Angles

Solve the equation $2 - x = 5x + 1$.

Equations

Write $0.0605$ in standard form.

Standard form

Finish the statement about the value of $m$.

Equations

A right-angled triangle is drawn with a vertical side of $4$ cm, a hypotenuse of $7$ cm, and the base angle labelled $x^\circ$. The diagram is NOT TO SCALE.

Right-angled triangles

The diagram shows axes marked $x$ and $y$, with a straight line extending from point $A$ at about $(0,1)$ to point $B$ at about $(3,10)$.

Gradient of linear graphs

$ABC$ and $DEF$ are similar triangles. For triangle $ABC$, $AB = 15$ cm and the base $BC = 16.5$ cm. In triangle $DEF$, $DE = 5$ cm. The diagrams are NOT TO SCALE.

Similarity

The exchange rate for dollars and euros (€) is set at €1 = $\$1.158$.

Rates

Find the probability that Stephanie will not win her next tennis match.

Introduction to probability

Points $A$, $B$ and $C$ lie on the circumference of a circle whose diameter is $AB$. A tangent has been drawn at $A$. The angle at $A$ between that tangent and line $AC$ is $42^\circ$. The diagram is NOT TO SCALE.

Circle theorems I

Without a calculator, work out $\frac{5}{6} - \frac{1}{2}$. Present every stage of your working and write your answer as a fraction in its simplest form.

Fractions, decimals and percentages

Each diagram displays the net of a solid.

Surface area and volume

Pablo has $\$16\,400$ available to put into one of these savings plans. Plan A earns compound interest at a rate of $4\%$ per year. Plan B earns simple interest at a rate of $4\%$ per year. Pablo keeps the $\$16\,400$ invested for $3$ years.

Percentages

Find the area of a circle with radius $6$ cm.

Surface area and volume

Calculate the value of $\sqrt{120} + 3.8^2 - 25$.

Powers and roots

Find $85$ cents expressed as a percentage of $\$2.03$.

Percentages

Express $6200\ \text{cm}^2$ in $\text{m}^2$.

Units of measure

Factorise $14x - 21y$.

Algebraic manipulation

The temperature at $3$ pm in a town, measured in $^\circ\text{C}$, is given below.

Interpreting statistical data

Arrange these numbers in ascending order, starting with the smallest: $\frac{7}{22}$, $0.3$, $33\%$, $\frac{1}{3}$.

Fractions, decimals and percentages

The shape shown above is a rhombus.

Symmetry

Convert $400$ to euros (€) with the exchange rate $1 = €0.935.

Money

Line $l$ is given by the equation $y = 4x - 6$.

Equations of linear graphs

Fill in the blank to make the calculation correct: $\frac{24 + 8}{4} = \ldots$

The four operations

A triangle has angles of $5x^\circ$, $6x^\circ$ and $7x^\circ$.

Angles

A, B and C lie on the circumference of a circle with centre O.

Geometrical terms

Express 16% as a decimal.

Fractions, decimals and percentages

Green Lane School finishes every day at 3.45 pm.

Time

Write 5367 rounded to the nearest hundred.

Limits of accuracy

25 students selected their favourite drink. The results are shown below. Tea, Hot chocolate, Coffee, Milkshake Hot chocolate, Coffee, Hot chocolate, Milkshake Lemonade, Tea, Milkshake, Lemonade Coffee, Hot chocolate, Lemonade, Tea Hot chocolate, Lemonade, Hot chocolate, Lemonade

Classifying statistical data

State the next term in this sequence.

Sequences

Find the value of $17^3$.

Powers and roots

Factorise completely $4x^2 - 8xy$.

Algebraic manipulation

Simplify the expression $\left(x^2\right)^5$.

Indices I

The diagram shows a graph whose axes are labelled $x$ and $y$.

Drawing linear graphs

The two barrels in the diagram are mathematically similar. The smaller barrel stands at a height of $h$ cm and has a capacity of 100 litres. The larger barrel measures 90 cm in height and has a capacity of 160 litres.

Similarity

The gradient of the line is 5. $M$ and $N$ are two points on this line. $M$ is at $(x, 8)$ and $N$ is at $(k, 23)$.

Gradient of linear graphs

Work out the total thickness of 250 sheets of paper.

The four operations

Write $23.4571$ to $4$ significant figures.

Limits of accuracy

The table lists the temperatures for five places at 10 am on one day in January.

Interpreting statistical data

Factorise completely $12n^2 - 4mn$.

Algebraic manipulation

For $2^r = \frac{1}{16}$, find the value of $r$.

Indices I

Work out $1\frac{2}{3} + \frac{5}{7}$ without using a calculator. Show every stage of your working and give your answer as a mixed number in simplest form.

Fractions, decimals and percentages

Simon has two boxes of cards. One box contains cards with a single shape on each one, and each shape is either a triangle or a square. The other box contains cards that are either red or blue. Simon randomly selects one card from each box. The probability of choosing a triangle card is $t$. The probability of choosing a red card is $r$.

Probability of combined events

Since $h$ is directly proportional to the square root of $p$, $h = 5.4$ when $p = 1.44$.

Ratio and proportion

Round $0.07164$ to $2$ significant figures.

Limits of accuracy

Solve for $x$ in $2 - x = 5x + 1$.

Equations

Express $0.0605$ in standard form.

Standard form

The diagram illustrates a right-angled triangle. One side measures $4\text{ cm}$, the hypotenuse measures $7\text{ cm}$, and the angle at the base is labelled $x^\circ$. The diagram is marked NOT TO SCALE.

Right-angled triangles

Solve this inequality: $3n - 11 > 5n - 18$.

Inequalities

Calculate $125^{\frac{2}{3}}$.

Powers and roots

Rearrange the formula $p = 2q^2$ so that $q$ is the subject.

Algebraic manipulation

Triangle $ABC$ is displayed, with points $A$, $B$ and $C$ labelled.

Geometrical constructions

The figure shows a triangle in which $CB = 8.15\text{ m}$, the angle at $A$ is $110^\circ$, and the angle at $B$ is $30^\circ$. It is labelled NOT TO SCALE.

Non-right-angled triangles

A rectangle measures $62\text{ mm}$ in length and $47\text{ mm}$ in width, and each dimension is accurate to the nearest millimetre. Its area is $A \text{ mm}^2$.

Limits of accuracy

For triangle $PQR$, $PQ = 8\text{ cm}$ and $QR = 7\text{ cm}$. Its area is $17\text{ cm}^2$.

Non-right-angled triangles

Find the probability that Stephanie fails to win her next tennis match.

Introduction to probability

Express it as one fraction in simplest form: $\frac{2x - 1}{3} - \frac{2}{x + 1}$.

Algebraic fractions

The quantity $y$ varies inversely with $\sqrt{1 + x}$. At $x = 8$, $y = 2$.

Ratio and proportion

Factorise completely the expression $9t^2 - u^2$.

Algebraic manipulation

$\mathcal{E} = \{ \text{students in a class} \}$, $P = \{ \text{students who study physics} \}$, $C = \{ \text{students who study chemistry} \}$. The Venn diagram gives the student counts. In $P$ only: $5$. In $P \cap C$: $11$. In $C$ only: $8$. Outside both circles: $7$.

Sets

Within the diagram, $AB$ and $CD$ run parallel to each other. $AD$ and $BC$ meet at $X$. $AB = 8\text{ cm}$, $CD = 4\text{ cm}$, $CX = 2\text{ cm}$ and $DX = 2.8\text{ cm}$. The figure is labelled NOT TO SCALE.

Similarity

Simplify $(16x^{16})^{\frac{3}{4}}$ into its simplest form.

Indices II

Points $A$, $B$, $C$, $D$ and $E$ are on the circle. $AB$ is produced to $F$. Angle $AED = 140^\circ$ and angle $CBF = 95^\circ$. The figure labels angles $w^\circ$, $x^\circ$ and $y^\circ$. The diagram is not to scale.

Circle theorems I

The point $A$ is $(-2, 0)$ and the point $B$ is $(0, 4)$. A coordinate grid with the $x$- and $y$-axes labelled is shown. The sketch is marked NOT TO SCALE.

Equations of linear graphs

Convert $6200 \text{ cm}^2$ into $\text{m}^2$.

Units of measure

Calculate $\sqrt{120} + 3.8^2 - 25$.

Powers and roots

Work out the percentage that $85$ cents is of $\$2.03$.

Percentages

Factorise the expression $14x - 21y$.

Algebraic manipulation

Determine the value of $5a - 3b$.

Introduction to algebra

A straight line intersects two parallel lines. The figure shows angles labelled $p^\circ$, $70^\circ$ and $q^\circ$. The figure is NOT TO SCALE.

Parallel lines

Calculate $\frac{5}{6} - \frac{1}{2}$ without using a calculator. Show every stage of your working and give your answer as a fraction in its simplest form.

Fractions, decimals and percentages

Calculate $\sqrt{\frac{1}{2}(1 - \cos 48^{\circ})}$.

Trigonometric functions

A triangle has angles of $5x^{\circ}$, $6x^{\circ}$ and $7x^{\circ}$. The diagram labels these angles, and it is not drawn to scale.

Angles

The diagram presents a coordinate grid with the axes marked $x$ and $y$, together with a number of straight lines.

Inequalities

The functions are given by $f(x) = 3 + 4x \quad g(x) = 6x + 7$.

Functions

The two bottles, along with their labels, are mathematically similar. The smaller bottle holds 0.512 litres of water and has a label with area $96\text{ cm}^{2}$. The larger bottle holds 1 litre of water.

Similarity

Write the recurring decimal $0.\dot{6}\dot{3}$ as a fraction in lowest terms, and show all your working.

Fractions, decimals and percentages

A triangle is drawn with sides measuring 24 cm and 39 cm, together with one angle of $71.8^{\circ}$ and the angle at the opposite vertex labelled $x^{\circ}$. The diagram is not drawn to scale.

Non-right-angled triangles

Solve for $x$ in the inequality $x + 13 > 3x + 7$.

Inequalities

The diagram depicts triangle $OPQ$. $O$ is the origin, $\vec{OP}=\vec{p}$ and $\vec{OQ}=\vec{q}$. $Z$ lies on $PQ$ so that $PZ:ZQ = 5:2$. The diagram is not drawn to scale.

Vector geometry

The diagram presents a speed-time graph for a car's journey. Speed is shown in m/s and time is measured in seconds. It rises from 0 to 12.5 m/s in 20 s, remains constant until 220 s, and then drops to 0 by 280 s. The diagram is not drawn to scale.

Graphs in practical situations

Calculate $\frac{11}{12} - \left(\frac{3}{4} - \frac{2}{3}\right)$ without using your calculator. You must show all your working and express your answer as a fraction in its simplest form.

Fractions, decimals and percentages

Factorise the expression $4x^{2} - 8xy$ completely.

Algebraic manipulation

Simplify the expression $6w^{0}$.

Indices I

Solve the equation $5x^{2} + 10x + 2 = 0$. You should show every step of your working and present your answers correct to 2 decimal places.

Equations

The diagram depicts a cube $ABCDEFGH$ with side length 26 cm. It is not drawn to scale.

Pythagoras' theorem and trigonometry in 3D

Simplify the expression $\frac{4(x - 6)^{2}}{(x - 6)}$.

Algebraic manipulation

Marcel puts $2500 into an investment for 3 years at 1.6% per year simple interest. Jacques puts $2000 into an investment for 3 years at $x\%$ per year compound interest. After the 3 years, Marcel and Jacques get the same amount of interest.

Exponential growth and decay

Determine the lowest common multiple (LCM) of 20 and 24.

Types of number

Rearrange $x = y + \sqrt{a}$ so that $a$ becomes the subject of the formula.

Algebraic manipulation

For a sphere whose radius is $r$, the volume $V$ is given by $V = \frac{4}{3}\pi r^{3}$.

Surface area and volume

Calculate the probability that Pedro scores a goal in each of the next two matches.

Probability of combined events