Oct/Nov 2016

Write 30000000 in words.

Types of number

$\frac{2}{9}$ of an amount equals $48$.

Fractions, decimals and percentages

ELEPHANT. Francesca picks one letter from this word at random.

Introduction to probability

State the type of correlation between the number of litres of fuel used by a car and the distance it travels.

Scatter diagrams

Eleven children tried to solve a puzzle. The data below give the number of attempts made by each child: 7, 6, 8, 5, 6, 5, 7, 8, 3, 8, 1.

Averages and measures of spread

Calculate $\frac{4}{5}$ of $90$.

Fractions, decimals and percentages

Solve the simultaneous equations, and make sure you show all your working. The equations given are $2x + 3y = 13$ and $x + 2y = 9$.

Equations

The diagram depicts a triangle whose vertices are labelled $A$, $B$ and $C$.

Geometrical constructions

The diagram depicts a right-angled triangle. The vertical side measures $15\text{ cm}$, the hypotenuse is labelled $x\text{ cm}$, and the angle at the base is $37^{\circ}$. The diagram carries the note NOT TO SCALE.

Right-angled triangles

Find the $n$th term for the sequence $7, 13, 19, 25, 31, \ldots$.

Sequences

Work out $x$.

Angles

State the temperature that is $5^{\circ}\text{C}$ below $-2^{\circ}\text{C}$.

The four operations

The diagram displays a coordinate grid with points $A$ and $B$ marked on it. The axes are named $x$ and $y$.

Coordinates

Solve for $x$ in $4x + 3 = 11$.

Algebraic manipulation

Express $$0.70$ as a fraction of $$5.60$, and give the result in lowest terms.

Fractions, decimals and percentages

Write $0.0401907$ rounded to 3 significant figures.

Limits of accuracy

For triangle $ABC$, the lengths are $AB = 7\text{ cm}$, $BC = 4\text{ cm}$ and $AC = 6\text{ cm}$. With only a ruler and compasses, construct triangle $ABC$. The side $BC$ has already been provided. The diagram shows a vertical line with $B$ at the top and $C$ at the bottom.

Geometrical constructions

Arrange these in ascending order, beginning with the smallest.

Ordering

Calculate $2\mathbf{a} - \mathbf{b}$.

Introduction to algebra

Work out $\frac{2}{3} - \frac{1}{4}$ and give your answer as a fraction in its lowest terms. Do not use a calculator, and show all stages of your working.

Fractions, decimals and percentages

The pool is circular, and its radius is $8\text{ m}$.

Circles, arcs and sectors

The distance-time graph represents the cyclist’s first 10 minutes of travel. Its vertical axis is labelled "Distance (km)" and runs from 0 to 4, while its horizontal axis is labelled "Time (minutes)" and runs from 0 to 10. The graph contains a straight line rising from (0,0) to (6,2), then a horizontal segment from 6 to 7 minutes at a distance of 2 km, and finally another straight line rising from 7 minutes at 2 km to 10 minutes at 3.5 km.

Graphs in practical situations

A diagram of intersecting lines with angles labelled $67^\circ$, $42^\circ$ and $a^\circ$; the diagram is marked "NOT TO SCALE".

Angles

Solve for $k$. $6(k - 8) = 78$

Equations

The trapezium sketch has parallel sides measuring 13 cm and 16 cm, together with a 4 cm vertical right-hand side and a 5 cm sloping side on the left. It is labelled "NOT TO SCALE".

Area and perimeter

A right-angled triangle is drawn. Its hypotenuse measures 14 cm, one angle is $32^\circ$, and the vertical side is marked $x$ cm. The diagram carries the label "NOT TO SCALE".

Right-angled triangles

The diagram depicts a cuboid-shaped solid from which a small cube has been removed. The cuboid has dimensions 9 cm by 6 cm by 5 cm. The edge length of the small cube is 2 cm. The diagram is marked "NOT TO SCALE".

Surface area and volume

A factory produces cups. The chance that any one of these cups is faulty is 0.02.

Relative and expected frequencies

A coordinate grid is displayed, with the axes marked $x$ and $y$. Point $A$ appears at $(3,6)$, and point $B$ appears at $(7,1)$.

Gradient of linear graphs

Four triangles, $A$, $B$, $C$ and $D$, are illustrated with angle markings: triangle $A$ has angles $55^\circ$ and $75^\circ$; triangle $B$ has angles $55^\circ$ and $80^\circ$; triangle $C$ has angles $35^\circ$ and $80^\circ$; triangle $D$ has angles $45^\circ$ and $55^\circ$. The diagrams are marked "NOT TO SCALE".

Similarity

The scatter diagram presents the speaking test scores together with the writing test scores for 15 students. The vertical axis carries the label "Writing test score" and the horizontal axis carries the label "Speaking test score".

Scatter diagrams

Calculate the perimeter of a regular octagon with side length 4 cm.

Angles

Convert $9\%$ into a fraction.

Fractions, decimals and percentages

Without a calculator, calculate $\frac{3}{4} + \frac{2}{3} - \frac{1}{8}$. Show every step of your working and give the answer as a mixed number in its simplest form.

Fractions, decimals and percentages

Simplify this expression. $3p + 5p + p$

Equations

Insert the correct symbol, $>$, $=$ or $<$, in each statement. $500\text{ m} \ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots 5\text{ km}$ $50\text{ mm} \ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots 0.5\text{ cm}$ $5000\text{ cm} \ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots 0.05\text{ km}$

Units of measure

A bag holds 20 counters in total. Of these, 10 are red, 8 are blue and the remainder are yellow. A single counter is chosen from the bag at random.

Introduction to probability

Insert one pair of brackets into the calculation below so that it is correct. $6 + 12 \div 2 \times 3 = 8$

The four operations

With $m = \begin{pmatrix} 5 \\ -7 \end{pmatrix}$ and $n = \begin{pmatrix} 2 \\ 6 \end{pmatrix}$, work out

Coordinates

At midnight, the temperature measured $-8^\circ\text{C}$. By 5 am, it had risen to $-3^\circ\text{C}$.

Types of number

Give a common multiple of 8 and 12.

Types of number

Write $14835$ rounded to the nearest thousand.

Limits of accuracy

Write five thousand and thirty four in figures.

Types of number

Without a calculator, find the value of $\frac{3}{5} + \frac{1}{6}$. Show all of your working and give your answer as a fraction in simplest form.

Fractions, decimals and percentages

Triangles $ABC$ and $DEF$ are similar. In triangle $ABC$, one side measures 16 cm and another measures 12 cm. In triangle $DEF$, the matching sides are $x$ cm and 30 cm. The diagram is not drawn to scale.

Similarity

Express 0.183 metres in centimetres.

Units of measure

The heights, in centimetres, of 8 people are: 153, 175, 168, 158, 161, 172, 164, 172.

Averages and measures of spread

Write $\frac{3}{5}$ in decimal form.

Fractions, decimals and percentages

The dollar–Thai Baht exchange rate is $\$1 = 31.48$ Baht.

Rates

Work out the probability that the chosen counter is blue.

Introduction to probability

The table sets out the doctor’s surgery opening hours.

Time

Work out the sum $\begin{pmatrix}5\\-1\end{pmatrix} + \begin{pmatrix}-3\\2\end{pmatrix}$.

Coordinates

The shaded figure is formed by connecting a square with a rhombus. The square’s side length is 5 cm. The rhombus has a horizontal diagonal measuring 4.5 cm. The diagram is not drawn to scale.

Area and perimeter

Calculate $-2 + 7 - 8$.

The four operations

Triangle $ABC$ is an isosceles triangle, with $AB = CB$. The angle $ABC$ measures $44^\circ$. The diagram is not drawn to scale.

Angles

The diagram represents an equilateral triangle.

Symmetry

The radius of a circle is 6.4 cm.

Surface area and volume

Solve the pair of simultaneous equations. Show all of your working. $5x + 4y = 17$ $2x - 3y = 16$

Equations

Find the value of $V$ when $p = 3$.

Introduction to algebra

Simplify $n^2 \times n^5$.

Indices I

Complete the sentence that gives the value of $m$.

Equations

Write the standard form of 2 470 000.

Standard form

Arrange them by size, starting with the smallest.

Ordering

James works as a vet. The table gives some details about the cats he examined over one week.

Interpreting statistical data

The diagram gives the distance-time graph covering the first 65 minutes of a bicycle trip.

Graphs in practical situations

State the temperature that is $5^{\circ}\text{C}$ lower than $-2^{\circ}\text{C}$.

The four operations

Simplify $\sqrt{36x^{16}}$.

Powers and roots

Solve the simultaneous equations. You must show all your working. $2x + 3y = 13$, $x + 2y = 9$.

Equations

Express $\$0.70$ as a fraction of $\$5.60$, and give the result in its simplest form.

Fractions, decimals and percentages

Factorise $4p^2 - 9$ completely.

Algebraic manipulation

$y$ varies directly with the square root of $(x + 2)$. When $x = 7$, $y = 2$.

Ratio and proportion

Calculate $2\left(\frac{3}{5}\right) - \left(\frac{1}{2}\right)$.

Fractions, decimals and percentages

The two cups are mathematically similar. The bigger cup has a capacity of $0.5$ litres and a height of $8$ cm, whereas the smaller cup has a capacity of $0.25$ litres.

Similarity

Construct the locus of points that lie inside the triangle and are $5$ cm from $B$.

Geometrical constructions

The equation is $y = p^2 + qr$.

Equations

Find the $n$th term for the sequence $7, 13, 19, 25, 31, \ldots$

Sequences

Write $0.0401907$ rounded to $3$ significant figures.

Limits of accuracy

A train moves for $m$ minutes at a speed of $x$ metres per second.

Rates

The coordinate diagram displays an unshaded triangular region $R$. Its boundaries are the $y$-axis, the vertical line $x=1$, the $x$-axis, and a straight line descending from $(0,4)$ to $(4,0)$. The shaded parts are outside $R$.

Inequalities

$n(A) = 7$, $n(B) = 6$, $n(\mathcal{E}) = 10$, $n(A \cup B)' = 1$.

Sets

Solve the equation $2x^2 + 3x - 3 = 0$. Include all stages of your working and state your answers correct to $2$ decimal places.

Equations

The diagram depicts a cube with edge length $8$ cm.

Pythagoras' theorem and trigonometry in 3D

Calculate how much the two prices differ by. State your answer in euros.

Money

Calculate $\frac{2}{3} - \frac{1}{4}$ and give your answer as a fraction in its lowest terms. Do not use a calculator; show every stage of your working.

Fractions, decimals and percentages

Write $5^{-3}$ in fractional form.

Indices I

In the diagram, $PT$ touches the circle at $P$ as a tangent. $PW$ is a diameter, and angle $TPQ = 42^{\circ}$.

Circle theorems I

Simplify the expression $\dfrac{x^3y + 2xy^3}{x^2y^2}$.

Algebraic manipulation

Express $1 - \dfrac{2}{p} - \dfrac{3}{t}$ as a single fraction.

Algebraic fractions

Points $A, B, C$ and $D$ are located on the circle with centre $O$.

Circle theorems I

Write 14 835 rounded to the nearest thousand.

Limits of accuracy

A backpack with a capacity of $30$ litres has a length of $53\text{ cm}$.

Similarity

The diagram depicts angle $BAC$, with $A$ as the vertex and $B$ and $C$ placed on two rays starting from $A$.

Geometrical constructions

Ralf and Susie divide $57$ in the ratio $2:1$.

Ratio and proportion

Factorise $m^3+m$.

Algebraic manipulation

Without using your calculator, calculate $\frac{3}{4}+\frac{2}{3}-\frac{1}{8}$. Show every step of your working and state your answer as a mixed number in lowest terms.

Fractions, decimals and percentages

The Venn diagram indicates how many people enjoy films ($F$), music ($M$) and reading ($R$). The regions show these counts: $8$ in $F$ only, $4$ in $M$ only, $2$ in $R$ only, $2$ in $F\cap M$, $1$ in $F\cap R$, $0$ in $M\cap R$, $3$ in $F\cap M\cap R$, and $2$ outside all three sets.

Sets

The vectors are specified as $\vec{BC}=\begin{pmatrix}2\\3\end{pmatrix}$ and $\vec{BA}=\begin{pmatrix}-5\\6\end{pmatrix}$.

Vectors in two dimensions

The diagram presents the cross section of part of a park bench. It is formed from a rectangle with length $32\text{ cm}$ and width $8\text{ cm}$, together with a curved section. The curved section is formed by two concentric arcs with sector angle $125^{\circ}$. The inner arc has radius $40\text{ cm}$ and the outer arc has radius $48\text{ cm}$. The diagram is labelled "NOT TO SCALE".

Compound shapes and parts of shapes

A coordinate grid displays triangle $A$, shaded, with vertices at $(3,1)$, $(5,1)$ and $(5,5)$, and triangle $B$ with vertices at $(1,2)$, $(2,2)$ and $(2,4)$. The coordinate axes are marked $x$ and $y$.

Transformations

Find the inverse matrix of $\begin{pmatrix}2&-3\\5&-4\end{pmatrix}$.

Algebraic manipulation

The diagram shows two straight lines crossing, with the angles labelled $67^{\circ}$, $42^{\circ}$ and $a^{\circ}$. The diagram is labelled "NOT TO SCALE".

Angles

On a coordinate grid, point $A$ is located at $(3,6)$ and point $B$ appears at $(7,1)$.

Perpendicular lines

Solve the equation $6(k-8)=78$.

Equations

The trapezium is drawn with the top edge measuring $13\text{ cm}$, the bottom edge measuring $16\text{ cm}$, the left sloping side measuring $5\text{ cm}$, and the right side being vertical with a length of $4\text{ cm}$. A right angle is indicated at the bottom-right corner. The figure is labelled "NOT TO SCALE".

Area and perimeter

Simplify the expression $36y^5 \div 4y^2$.

Indices I

A square has side length $8\text{ cm}$, rounded to the nearest centimetre.

Limits of accuracy

Determine the positive integers that make the inequality $t+2>3t-6$ true.

Inequalities

Solve the simultaneous equations. You must show all your working. $\frac{1}{2}x+y=8$ and $x-2y=2$.

Equations

Viewed from the top of a building that is $300\text{ m}$ tall, the angle of depression to a car, $C$, is $52^{\circ}$. A right-angled triangle is drawn with a vertical side of $300\text{ m}$, and at the top the angle between the vertical and the sloping line of sight is $52^{\circ}$. The figure is labelled "NOT TO SCALE".

Right-angled triangles

Given $V = 4p^2$. Work out $V$ when $p = 3$.

Introduction to algebra

Calculate – $2^3 - \sqrt{10 + 4^2}$.

Trigonometric functions

Ahmed bought a car for $34000. By the close of the first year, its value had fallen by 40%. At the end of the second year, the value was 10% lower than it had been at the close of the first year.

Percentages

The diagram depicts a hemisphere with a diameter of 5 cm. For a sphere of radius r, the volume V is given by V = \frac{4}{3}\pi r^3. The diagram is NOT TO SCALE.

Surface area and volume

[0.2̇ denotes 0.222…]

Fractions, decimals and percentages

The shaded figure is formed by joining a square to a rhombus. The square has side length 5 cm, and the rhombus has a diagonal measuring 4.5 cm. The diagram is NOT TO SCALE.

Compound shapes and parts of shapes

In triangle ABC, the equal sides are $AB$ and $CB$, so the triangle is isosceles. Angle $ABC = 44^\circ$. The diagram is NOT TO SCALE.

Angles

The quantity $d$ varies inversely as $(w + 1)^2$. When $w = 4$, $d = 3.2$.

Proportion

The point $A$ is $(8, 3)$, while $B$ is $(12, 1)$.

Perpendicular lines

We have $f(x) = x^2\;\;\; g(x) = \dfrac{x - 3}{2}$.

Functions

The curve $y = x^3 + 2x^2 - 4x$ is displayed on the grid. The diagram includes the $x$-axis and $y$-axis, along with the cubic curve.

Differentiation