May/June 2019

Write $\frac{3}{4}$ in decimal form.

Fractions, decimals and percentages

The truck has length $L$ metres, and this is given as $8.2$ m to 1 decimal place.

Limits of accuracy

Simplify the expression $t^{21} \div t^{7}$.

Indices I

The diagram depicts a right-angled triangle. Its hypotenuse is 12 cm, one angle measures $35^\circ$, and the vertical side has been labelled $x$ cm. NOT TO SCALE.

Right-angled triangles

The equation is $p = \dfrac{1.6 + 9.6^2}{5.9 - 4.3}$.

Estimation

The numbers in the list are 27, 28, 29, 30, 31, 32 and 33.

Types of number

Without a calculator, calculate $\frac{5}{6} + \frac{2}{3}$. Show every step of your working and present your answer as a mixed number in its lowest terms.

Fractions, decimals and percentages

The sequence starts with these four terms: 5, 8, 11, 14.

Sequences

Two triangles are illustrated. In triangle $ABC$, the angles measure $65^\circ$, $75^\circ$, and $40^\circ$, and the side $AB = 6$ cm. In triangle $PQR$, the matching angles are $65^\circ$, $75^\circ$, and $40^\circ$, with base $PR = 27$ cm and side $PQ = 18$ cm. NOT TO SCALE.

Similarity

A car is moving with a steady speed of $20\text{ m s}^{-1}$.

Rates

The diagram shows two figures: (i) a double-headed arrow, and (ii) a circle with a rotated square inside it.

Symmetry

Work out how much $1.20$ is as a percentage of $16$.

Percentages

Change 3670 centimetres into metres.

Scale drawings

Michael’s route from his home to the beach is represented by the travel graph. The vertical axis gives Distance (km), marked from Home (0) up to Beach (12). The horizontal axis shows Time running from 08:00 to 11:00. A straight line climbs from 08:00 to around 09:15 and reaches 8 km, followed by a flat section up to 10:00, and then another straight line rises to 12 km at 11:00.

Graphs in practical situations

The diagram depicts a point $P$ and a line $L$ on a coordinate grid whose axes are labelled $x$ and $y$. Point $P$ is shown at about $(-3, -1)$. Line $L$ slopes upwards from the lower left to the upper right.

Equations of linear graphs

Factorise the expression $5y - 6py$.

Algebraic manipulation

A bag contains only green balls and red balls. One ball is selected at random from the bag. The probability of selecting a green ball is 0.38.

Introduction to probability

Calculate the difference between these two temperatures.

Graphs in practical situations

A cuboid has a volume of $180\text{ cm}^3$, and its base is a square with side length $6\text{ cm}$.

Surface area and volume

Write $640\,000$ in standard form using powers of 10.

Standard form

Find $\begin{pmatrix}4\\-2\end{pmatrix} - \begin{pmatrix}1\\5\end{pmatrix}$.

Coordinates

Asif and Ben divide $2100 in the ratio Asif : Ben = 3 : 7.

Ratio and proportion

Write 30682 in words, using the place values of the digits.

Types of number

Write 0.047 883 rounded to 2 significant figures.

Limits of accuracy

Calculate the highest common factor (HCF) of 90 and 48.

Types of number

A triangle is drawn with a base measuring 8.4 cm and a perpendicular height of 3.5 cm. The diagram is not drawn to scale.

Area and perimeter

The diagram depicts a right-angled triangle with a side measuring 6.2 m, a hypotenuse measuring 23 m, and an angle $x^\circ$. The diagram is not drawn to scale.

Right-angled triangles

Work out how many cubes are in the cuboid.

Geometrical terms

There are 22 coloured pencils in a box. Of these, 6 are pink, 9 are blue and 7 are yellow.

Ratio and proportion

Expand the bracketed expression $x^2(x-7)$.

Algebraic manipulation

Demonstrate that no square number lies between 50 and 60.

Types of number

This machine consistently needs 5 minutes to paint an 80 metre white line on a road.

Rates

Simplify the expression $5m^2 \times 2m^3$.

Indices I

Convert 4365 metres to centimetres.

Units of measure

Without a calculator, work out $2\frac{1}{4} \div \frac{3}{7}$. Show every step of your working and give your answer as a mixed number in its simplest form.

Fractions, decimals and percentages

The diagram depicts a semicircle whose diameter measures 9 cm. The diagram is not drawn to scale.

Circles, arcs and sectors

Gerry and Alain each lap a running track. Gerry always needs 90 seconds for one complete circuit, whereas Alain always needs 105 seconds. They begin together from the same point.

Ratio and proportion

Rearrange this formula so that $x$ is the subject. $5x^2 - 3y = 4y + 8$

Algebraic manipulation

Triangle $ABC$ is illustrated.

Geometrical constructions

Add one pair of brackets so that the equation becomes true. $4 \times 6 - 2 + 1 = 17$

The four operations

Tommy has probability 0.4 of bringing his calculator to his mathematics lesson. In one year, there are 120 mathematics lessons.

Relative and expected frequencies

Take 123 away from 1 million.

The four operations

A quadrilateral with just one pair of parallel sides is known as a [BLANK].

Geometrical terms

Shade an amount equal to five-eighths of the rectangle.

Symmetry

The figure displays two parallel lines cut by one straight transversal. The angles are marked a, b, c, d, e, f, g and X.

Angles

Each of 50 students selects a favourite colour from six options. The tally chart shows the results for Red, Orange, Yellow, Green and Blue.

Classifying statistical data

Write 3.058 rounded to 3 significant figures.

Limits of accuracy

The numbers in the list are 21, $\frac{2}{3}$, $\sqrt{13}$, 31, $\sqrt{121}$, 51 and 0.7.

Types of number

The vectors are $p = \begin{pmatrix}5\\0\end{pmatrix}$ together with $q = \begin{pmatrix}1\\6\end{pmatrix}$.

Coordinates

State the type of correlation you would expect between the average speed of a train and the time taken for a journey.

Scatter diagrams

The scale diagram depicts a rock, R. The scale means 1 centimetre represents 30 metres. A lighthouse, L, is 210 m from R, on a bearing of 125^{\circ}. The diagram includes a North arrow and point R. Scale: 1 cm to 30 m.

Scale drawings

Rearrange $2(w + h) = P$ so that $w$ is the subject.

Algebraic manipulation

Genaro records the length, $l$ cm, of his desk as 120 cm when it is rounded to the nearest centimetre.

Limits of accuracy

Solve the equation $7x - 5 = 16$.

Equations

Without a calculator, calculate $\frac{12}{35} \times \frac{7}{9}$. Show every step of your working and express your answer as a fraction in its lowest terms.

Fractions, decimals and percentages

The shapes ABCD and PQRS are displayed. The marked lengths are 5.5 cm, $a$ cm, 7.7 cm and 9.8 cm, and the diagram is NOT TO SCALE.

Similarity

Harry puts $800 into an account for 2 years at 3% per year compound interest.

Percentages

Convert 0.45 into a fraction in its simplest form.

Fractions, decimals and percentages

Solve the simultaneous equations $5x - 2y = 26$ and $7x + 6y = 10$. All working must be shown.

Equations

State the next term in the sequence: 12, 7, 2, -3, -8, ...

Sequences

A closed cuboid-shaped box measures 5 cm in length, 4 cm in width and 2 cm in height.

Surface area and volume

The figure depicts two regular pentagons. Pentagon FGHIJK is an enlargement of pentagon ABCDE about centre O. The labelled points are A, B, C, D, E, O, together with F, G, H, J and K. Diagram is not drawn to scale.

Transformations

A vertical flagpole, BD, is set on horizontal ground and supported by two ropes, AD and CD. AD = 15 m, BC = 10.7 m and angle DAB = 38^{\circ}. The diagram is marked NOT TO SCALE.

Right-angled triangles

Jason is at school for 480 minutes each day. The pie chart illustrates how that time is divided among three lessons. The sectors are named Maths, English and Science. The angle of the English sector is shown as 45^{\circ}.

Statistical charts and diagrams

Factorise $2x^2 - x$.

Algebraic manipulation

Determine the co-ordinates of the point where the line $y = 3x - 8$ meets the y-axis.

Equations of linear graphs

Find how much the two reaction times differ.

The four operations

Find the probability that Alex does not win a prize.

Introduction to probability

The table presents the various ways 20 people travel to work.

Averages and measures of spread

Calculate the value of $-12 \div -2$.

Powers and roots

Simplify the expression $4x - 12y + 10x + 25y$.

Algebraic manipulation

Calculate $\$1.20$ as a percentage of $\$16$.

Percentages

The function is defined as $f(x) = 2x + 3$.

Functions

The diagram presents five cards numbered 1, 2, 3, 4 and 5. Two cards are then selected at random, without replacement.

Introduction to probability

Starting from the list of numbers 27, 28, 29, 30, 31, 32, 33, write down

Types of number

$x^2 + 4x - 9 = (x + a)^2 + b$.

Algebraic manipulation

Without a calculator, calculate $\frac{5}{6} + \frac{2}{3}$. Show every step of your working and write your answer as a mixed number in its simplest form.

Fractions, decimals and percentages

Expand then simplify $(x + 1)(x + 2) + 2x(x - 3)$.

Algebraic manipulation

The variable $y$ is inversely proportional to the square root of $(x + 1)$. When $x = 8$, $y = 2$.

Ratio and proportion

Factorise the expression $p^2 - q^2$.

Algebraic manipulation

Simplify the expression $(81y^{16})^{\frac{3}{4}}$.

Indices I

A scale model of a car is drawn at $1 : 20$. The actual car has volume $12\text{ m}^3$.

Ratio and proportion

Factorise $5y - 6py$.

Algebraic manipulation

Rewrite the expression as one fraction in simplest form: $\frac{1}{x+2} - \frac{2}{3x-1}$.

Algebraic fractions

The figure is a pyramid with a square base $ABCD$, where each side measures 8 cm. The diagonals of the square, $AC$ and $BD$, meet at $M$. $V$ is vertically above $M$ and $VM = 10\text{ cm}$. The diagram is labelled NOT TO SCALE.

Pythagoras' theorem and trigonometry in 3D

Write down what the next term is.

Sequences

Hence, $P = \begin{pmatrix} 3 & 1 \\ 2 & 4 \end{pmatrix}$.

Algebraic manipulation

The diagram gives data for the final 70 seconds of a car journey. The vertical axis is labelled Speed (m/s) and runs up to 20, while the horizontal axis is labelled Time (seconds) and extends from 0 to 70. The speed remains steady at 20 m/s from 0 to 60 seconds, then falls linearly to 0 m/s at 70 seconds. The diagram is labelled NOT TO SCALE.

Graphs in practical situations

OABC is a parallelogram with O at the origin. $CK = 2KB$ and $AL = LB$. $M$ is the midpoint of $KL$. $\overrightarrow{OA} = \mathbf{p}$ and $\overrightarrow{OC} = \mathbf{q}$. The diagram has been drawn NOT TO SCALE.

Vectors in two dimensions

The line $L$ goes through the points $(0, -3)$ and $(6, 9)$.

Perpendicular lines

Calculate $\sqrt[3]{8.1^2 - 1.3^{0.8}}$.

Powers and roots

An equilateral triangle has each side measuring 15 cm, to the nearest centimetre.

Limits of accuracy

The volume of a cuboid is $180\text{ cm}^3$. Its base is a square with side length 6 cm.

Surface area and volume

Simplify the expression $t^{21} \div t^{7}$.

Indices I

A right-angled triangle is drawn. One side measures 12 cm, the base angle is $35^{\circ}$, and the vertical side is marked $x$ cm. The figure is labelled NOT TO SCALE.

Right-angled triangles

Points A, B and C lie on a circle with centre O. The angle at B between BA and BC is $130^{\circ}$. The diagram is labelled NOT TO SCALE.

Circle theorems I

Write the recurring decimal $0.4\dot{7}$ in fraction form. Show all your working.

Fractions, decimals and percentages

State a prime number from 50 to 60.

Types of number

For a map drawn at scale $1 : 25\,000$, the lake covers an area of $33.6$ square centimetres.

Scale drawings

State the matrix for an enlargement with scale factor $3$ and centre $(0,0)$.

Transformations

Simplify this expression.

Indices I

Without a calculator, work out $2\frac{1}{4} \div \frac{3}{7}$. You must show every step of your working and give your answer as a mixed number in its simplest form.

Fractions, decimals and percentages

Solve the simultaneous equations, and make sure that all of your working is shown. $5x + 8y = 4$ $\frac{1}{2}x + 3y = 7$

Equations

Shona purchases a chair in a sale for $435.60. This amount is a $12\%$ reduction from the original price.

Percentages

The diagram displays the inequalities $y \leq -\frac{1}{2}x + 6$, $y \geq 3x - 4$, $x + y \geq 5$.

Inequalities

Rewrite as a single fraction in its simplest form: $\frac{2x}{x+3} + \frac{x+3}{x-5}$.

Algebraic fractions

The table gives the numbers of people in several age bands at a cinema. Dexter makes a histogram to represent the data. The bar he sketches for the class $15<y\leq25$ has height $7\text{ cm}$.

Statistical charts and diagrams

Rewrite the formula so that $m$ is the subject: $P = \frac{k + m}{m}$.

Algebraic manipulation

Use a calculator to calculate $\sqrt{1-(\sin 33^{\circ})^2}$.

Trigonometric functions

Solve the equation. Show all your working and give your answers accurate to 2 decimal places.

Equations

The Venn diagrams are displayed. For part (a), a rectangle is used to show the universal set, with two overlapping circles marked $X$ and $Y$. For part (b), the Venn diagram contains these values: In $M$ only: 3. In $M \cap P$: 6. In $P$ only: 12. In $M \cap C$: 5. In $M \cap P \cap C$: 2. In $P \cap C$: 10. In $C$ only: 7. Outside all circles: 23.

Sets

The matrices are $A = \begin{pmatrix}2 & 7\\ 1 & 3\end{pmatrix}$ and $B = \begin{pmatrix}3 & 4\\ 0 & 1\end{pmatrix}$.

Algebraic manipulation

The diagram depicts parallelogram $ABCD$, where $\overrightarrow{AB}=q$ and $\overrightarrow{AD}=p$. $ABM$ is collinear, with $AB:BM = 1:1$. $ADN$ is collinear, with $AD:DN = 3:2$.

Vector geometry

A cuboid $ABCDEFGH$ is displayed, and it is NOT TO SCALE. $AB = 18\text{ cm}$, $BC = 7\text{ cm}$ and $CG = 12\text{ cm}$.

Pythagoras' theorem and trigonometry in 3D

The coordinate grid contains triangle $A$ in the first quadrant, triangle $B$ in the fourth quadrant, and triangle $C$ in the second quadrant.

Transformations

Write the recurring decimal $0.\dot{7}$ in fraction form.

Fractions, decimals and percentages

Finish every statement.

Geometrical terms

Prague and Vienna are 254 kilometres apart. Prague has the same local time as Vienna. A train departs from Prague at 15 20 and reaches Vienna at 19 50 on the same day.

Rates

Solve the equation $9f + 11 = 3f + 23$.

Equations

A triangle is drawn and the diagram is NOT TO SCALE. Its base measures $8.4\text{ cm}$, and the perpendicular height is $3.5\text{ cm}$.

Area and perimeter