May/June 2018

Express 4647 correct to the nearest 100.

Limits of accuracy

A boy’s height, $h$ metres, is given as 1.72 m, accurate to the nearest centimetre.

Limits of accuracy

Expand then simplify $6(2y-3)-5(y+1)$.

Algebraic manipulation

Let $g = \begin{pmatrix}2\\5\end{pmatrix}$ and $h = \begin{pmatrix}-3\\4\end{pmatrix}$.

Coordinates

Determine the lowest common multiple (LCM) of 18 and 21.

Types of number

Calculate the size of one exterior angle in a regular octagon.

Angles

Calculate $\sqrt{2.38 + 6.4^2}$ and record the full calculator display.

Powers and roots

A rectangle is plotted on a grid, and the centre of enlargement $O$ is marked. The diagram appears on a grid.

Transformations

Find the probability that the pen is green.

Relative and expected frequencies

Simplify the expression $(x^3)^4$.

Indices I

The list is: $\pi$, $3^{-2}$, $3\frac{4}{7}$, $33.3\%$, $\sqrt{3}$, $0.3$, $3^{999}$.

Types of number

State 0.007 as a fraction.

Fractions, decimals and percentages

Write down the mathematical name for this quadrilateral.

Geometrical terms

A solid’s net is shown on a $1\text{ cm}^2$ grid, and the diagram displays that net.

Surface area and volume

Factorise $10 + 16w$ fully.

Algebraic manipulation

Work out $1\frac{3}{4} \times \frac{6}{35}$ without using a calculator. Show all your working, and give your answer as a fraction in its simplest form.

Fractions, decimals and percentages

Solve the simultaneous equations.

Equations

The figure represents a quadrilateral. The angles indicated are $95^\circ$, $82^\circ$, $47^\circ$ and $x^\circ$. The diagram is labelled NOT TO SCALE.

Angles

Write down the first three terms in the sequence.

Sequences

Write 0.00268 rounded to 2 significant figures.

Standard form

Calculate the value of $7x + 3y$.

Introduction to algebra

The diagram depicts two parallel lines $PAQ$ and $SBCT$. Also, $AB = AC$ and angle $QAC = 43^\circ$. The diagram is labelled NOT TO SCALE.

Angles

Solve $8x - 5 = 7$.

Equations

Express 6.54 kilometres in metres.

Units of measure

One morning, Marcia is at work from 08 20 until 11 15. Find how long she works for. Give your answer in hours and minutes.

Time

Factorise completely the expression $4xy^2 - 6y^3$.

Algebraic manipulation

The numbers below are given in standard form: $3.4 \times 10^{-1}, 1.36 \times 10^{6}, 7.9 \times 10^{0}, 2.4 \times 10^{5}, 5.21 \times 10^{-3}, 4.3 \times 10^{-2}$.

Standard form

$a=\begin{pmatrix}5 \\ -2\end{pmatrix},\;b=\begin{pmatrix}-7 \\ -3\end{pmatrix}$

Coordinates

Express $y$ as the subject in the equation $5x - 2y + 7 = 0$.

Algebraic manipulation

Convert 600 euros into dinars using the exchange rate 1 euro = 0.429 dinars. State your answer correct to the nearest dinar.

Money

If $w = [BLANK],\; 10w = 70$.

Equations

A right-angled triangle, shown NOT TO SCALE, has a base measuring 5 cm and a hypotenuse measuring 8 cm, and the angle at the left-hand end of the base is marked $x^\circ$.

Right-angled triangles

The diagram depicts a solid cuboid (NOT TO SCALE) whose base area is $7\text{ cm}^2$. Its volume is $21\text{ cm}^3$.

Surface area and volume

The figure depicts a rectangular playground $ABCD$ (NOT TO SCALE). $AB = 23.4\text{ m}$ and $AC = 35.1\text{ m}$.

Pythagoras' theorem

Friedrich takes out a loan of $1200 for 3 years at 5.6% per year compound interest. Calculate the total amount he repays after 3 years.

Percentages

Simplify $7g - g + 2g$.

Algebraic manipulation

A glass in the shape of a cylinder has radius 3.6 cm and height 11 cm, and it is filled with water.

Surface area and volume

For 12 days, hiring the car costs $167.90. The charge for the first day of hiring this car is $20.50.

The four operations

The table gives the number of customers in a restaurant on each day that it opens across one week.

Averages and measures of spread

The scatter diagram compares the values, in thousands of dollars, of eight houses in 1996 with the values of those same houses in 2016. The vertical axis carries the label 'Value in 2016 ($ thousands)' and the horizontal axis carries the label 'Value in 1996 ($ thousands)'.

Scatter diagrams

Without a calculator, calculate the following. Show every step of your working and write each answer as a fraction in its simplest form.

Fractions, decimals and percentages

Expand $7(x - 8)$ to simplify the expression.

Algebraic manipulation

Determine the value of $p$ when $5^p \div 5^8 = 5^{13}$.

Indices I

Calculate the difference between the two prime numbers in the list above.

Types of number

Find the values of $a$ and $b$.

Equations

The lighthouse lies on a bearing of $113^\circ$ from the coastguard station. Work out the bearing of the coastguard station from the lighthouse.

Angles

The diagram has concentric circular rings, with four evenly spaced radial gaps in the outer ring and a central circle marked with an 'X'.

Symmetry

Arrange these numbers from the smallest to the largest: $5^{-2}, \frac{1}{27}, \frac{2}{55}, 0.038$.

Ordering

Express $75\%$ as a fraction in simplest terms.

Fractions, decimals and percentages

Write $209\,802$ correct to the nearest thousand.

Limits of accuracy

Find the amount Soraya receives.

Ratio and proportion

Calculate an estimate for the number of games Kim will win.

Relative and expected frequencies

Write the ordinary-number form of $4.82 \times 10^{-3}$.

Standard form

Solve for $p$: $\dfrac{1 - p}{3} = 4$.

Equations

Complete the sentence relating to the value of $m$.

Equations

Calculate $x$.

Equations

Find the surface area of this cuboid.

Surface area and volume

Work out $\dfrac{2}{3} \div \dfrac{1}{5}$ without a calculator. Show every step of your working, and write the result as a fraction in simplest form.

Fractions, decimals and percentages

Calculate \begin{pmatrix}5\\-1\end{pmatrix} + \begin{pmatrix}2\\6\end{pmatrix}.

Coordinates

Factorise $w + w^3$ into a product.

Algebraic manipulation

State the gradient of line $L$.

Parallel lines

Express $568\,000$ cm in metres.

Scale drawings

Work out the area of the shape.

Area and perimeter

Solve the simultaneous equations. All your working must be shown. $3x - 2y = 23$ $2x + 5y = 9$

Equations

ABCD is a quadrilateral. The diagram displays vertices A, B, C, and D joined consecutively to make a quadrilateral.

Geometrical constructions

Calculate Liz’s average speed.

Rates

Calculate $sqrt{dfrac{18^2}{0.5 + 1.75}}$.

Powers and roots

Work out the value of $4^{-2}$.

Indices I

State the mathematical name for the marked angle.

Geometrical terms

Arrange the following numbers from smallest to largest: $\dfrac{4}{15}$, $26\%$, $0.24$, $\dfrac{1}{4}$.

Ordering

Complete the factor list for 36. $1,\; 2,\; \ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots,\; 36$

Types of number

Raise $\$22$ by $15\%$.

Percentages

Give one prime number in the range 20 to 30.

Types of number

Calculate $\sqrt{2.38 + 6.4^2}$, and show the full calculator display.

Using a calculator

Find the exact value for $8^{\frac{2}{3}} \times 49^{-\frac{1}{2}}$.

Powers and roots

Solve the inequality $3n - 5 > 17 + 8n$.

Inequalities

Without a calculator, determine $1\frac{3}{4} \times \frac{6}{35}$. Show every step of your working and present your answer as a fraction in simplest form.

Fractions, decimals and percentages

Triangle $ABC$ is drawn. $AC = 5.9\text{ cm}$ and $AB = 17.8\text{ cm}$. The angle at $C$ measures $84.6^{\circ}$. The diagram is labelled NOT TO SCALE.

Non-right-angled triangles

$y$ varies directly with $(x - 1)^2$. At $x = 5$, $y = 4$.

Ratio and proportion

A figure named $R$ is shown on a coordinate grid whose axes are marked $x$ and $y$.

Transformations

The speed-time graph gives details of a tram journey between two stations. Speed is shown in m/s and time is shown in seconds. The diagram is NOT TO SCALE.

Graphs in practical situations

The cumulative frequency graph gives data on the time, $m$ minutes, that 120 students needed to finish some homework.

Cumulative frequency diagrams

A triangle $LMN$ is drawn. The lengths are $LN = 14\text{ cm}$, $NM = 16\text{ cm}$ and $LM = 19\text{ cm}$. The figure is labelled NOT TO SCALE.

Non-right-angled triangles

Write $0.0000387$ in standard form.

Standard form

Determine the probability that the pen is green.

Probability of combined events

Four inequalities define the region $R$. One of them is $y \le x + 1$. On a coordinate grid, region $R$ is shaded.

Equations of linear graphs

The functions are $f(x) = 5 - 2x$ $g(x) = x^2 + 8$

Functions

40 people were asked how many times they went to the cinema in one month. The table lists the results. Number of cinema visits: 0, 1, 2, 3, 4, 5, 6, 7 Frequency: 5, 5, 6, 6, 7, 3, 6, 2

Averages and measures of spread

Calculate the angle made by line $AB$ with the $x$-axis.

Equations of linear graphs

Express the recurring decimal $0.6\dot{3}$ as a fraction.

Fractions, decimals and percentages

Find $7x + 3y$ for $x = 12$ and $y = -6$.

Introduction to algebra

The diagram depicts two parallel lines $PAQ$ and $SBCT$. It also gives $AB = AC$ and $ ngle QAC = 43^{\circ}$. The labelled points are $P, A, Q, C, T, S, B$. The angle at $A$ formed by $QA$ and $AC$ is marked $43^{\circ}$, while the angle at $A$ between $CA$ and $AB$ is marked $x^{\circ}$. NOT TO SCALE.

Angles

Calculate the area of a circle that has radius $5.1\text{ cm}$.

Circles, arcs and sectors

A right-angled triangle $ABC$ is displayed. $AB = 2.5\text{ cm}$ and $BC = 4.1\text{ cm}$. The right angle lies at $B$. The diagram is labelled NOT TO SCALE.

Pythagoras' theorem

Expand and simplify the expression $6(2y - 3) - 5(y + 1)$.

Algebraic manipulation

$3^{-q} \times \frac{1}{27} = 81$. Find the numerical value of $q$.

Indices I

Find the length of time she works for. Give your answer in hours and minutes.

Time

Factorise the expression $xy + 2y + 3x + 6$ completely.

Algebraic manipulation

Expand the expression $7(x-8)$.

Algebraic manipulation

The sequence is $a, 13, 9, 3, -5, -15, b, \ldots$.

Sequences

Finish these statements.

Equations

The list of numbers is: 22, 17, 25, 41, 39, 4.

Types of number

Work out $\frac{2}{3}-\frac{1}{12}$. Show all of your working, and present your answer as a fraction in simplest form.

Fractions, decimals and percentages

On a map, towns A and B are shown. The bearing of A from B is $140^\circ$.

Angles

The numbers below are expressed in standard form: $3.4 \times 10^{-1}$, $1.36 \times 10^{6}$, $7.9 \times 10^{0}$, $2.4 \times 10^{5}$, $5.21 \times 10^{-3}$, $4.3 \times 10^{-2}$.

Standard form

Use only a straight edge and compasses to construct the locus of all points equidistant from A and B. In the diagram, point A is shown on the left and point B on the right.

Geometrical constructions

State the temperature at midnight.

Graphs in practical situations

Write $2a+4b-ax-2bx$ in fully factorised form.

Algebraic manipulation

Rearrange the equation to make $x$ the subject.

Algebraic manipulation

The Venn diagram contains a universal set $\xi$ and two intersecting circles named $P$ and $Q$.

Sets

Use algebra to simplify $\frac{3+x}{9-x^2}$.

Algebraic fractions

A triangle diagram, drawn not to scale, marks points $O$, $P$ and $Q$. $O$ is the origin. The vector $\vec{OP}$ is $\mathbf{p}$, while $\vec{OQ}$ is $\mathbf{q}$. Point $T$ is on $QP$ so that $QT:TP=2:1$.

Vector geometry

Work out $\frac{2}{3} \div 1\frac{1}{5}$ without using a calculator. Show all of your working and give your answer as a fraction in its simplest form.

Fractions, decimals and percentages

Calculate the upper bound for the perimeter of the square.

Limits of accuracy

Determine the value of $\left(\frac{1}{81}\right)^{-\frac{3}{4}}$.

Powers and roots

Expand the brackets and simplify the expression $(2p+3)(3p-2)$.

Algebraic manipulation

Find the value of $y$ for $x=6$.

Functions

Factorise the expression $w + w^3$.

Algebraic manipulation

A circle sketch, drawn not to scale, has points $A$, $B$, $C$, $D$ and $E$ placed on the circumference. The angles marked are $47\degree$ at $C$ (angle $DCE$) and $85\degree$ at $E$ (angle $CEA$). Inside the circle, the angles are labelled $w\degree$, $x\degree$ and $y\degree$.

Circle theorems I

Express as a single fraction in its simplest form: $\frac{1}{y-1} - \frac{1}{y}$.

Algebraic fractions