May/June 2016

A train departs Zurich at 22 40 and reaches Vienna at 07 32 the following day. Work out the time taken.

Time

Omar exchanges 2000 Saudi Arabian riyals (SAR) for euros (€) at the rate €1 = 5.087 SAR. Calculate the amount Omar gets, giving your answer correct to the nearest euro.

Rates

Determine the lowest common multiple (LCM) of 36 and 48.

Types of number

Using $y = mx + c$, determine the value of $y$ when $m = -2$, $x = -7$ and $c = -3$.

Introduction to algebra

Using $y = \frac{qx}{p}$, write $x$ in terms of $p$, $q$ and $y$.

Algebraic manipulation

Triangle $ABC$ is isosceles, and $AC$ runs parallel to $BD$. In the diagram, angle $C$ is $40^\circ$, while the angles at $B$ are marked $a^\circ$ and $b^\circ$.

Angles

Triangles $ABC$ and $DEF$ are similar. In triangle $ABC$, $AC = 6\,$cm and $BC = 8.4\,$cm. In triangle $DEF$, $DF = 15\,$cm.

Similarity

Without a calculator, calculate $\frac{6}{7} \div 1\frac{2}{3}$. Include your full working and write your answer as a fraction in its simplest form.

Fractions, decimals and percentages

Find the next term in the sequence: 3, 7, 11, 15, …

Sequences

The scatter diagram plots house selling prices against how far they are from the city centre. The vertical axis reads "Price of house (thousands of dollars)" while the horizontal axis reads "Distance from the city centre (km)".

Scatter diagrams

Using only a straight edge and compasses, construct the bisector of angle $ABC$.

Geometrical constructions

In a batch of 80 batteries, 3 are faulty. Calculate the percentage of faulty batteries.

Percentages

Solve the pair of simultaneous equations. You must show every step of your working.

Equations

The line $AB$ is plotted on the grid, with point $A$ positioned at about $(0.5, 1)$ and point $B$ at $(3, 7)$.

Equations of linear graphs

Calculate the volume for this cuboid.

Surface area and volume

Among a group of students, the probability that a student is left-handed is 0.28. A student is picked at random from the group. Find the probability that this student is not left-handed.

Introduction to probability

Write $1.27 \times 10^{-3}$ in ordinary form.

Standard form

Convert 60 000 metres to kilometres.

Units of measure

Calculate $(2.1 - 0.078)^{17}$, and give your answer correct to 4 significant figures.

Indices I

Write down the mathematical name used for an angle that measures less than $90^\circ$.

Geometrical terms

Calculate $\begin{pmatrix}-2\\-3\end{pmatrix} + \begin{pmatrix}-4\\7\end{pmatrix}$.

Coordinates

The diagram shows a shape.

Symmetry

Arrange these numbers from the smallest to the largest. 0.304  0.2  0.008  0.57

Ordering

Triangles $A$ and $B$ are similar. For triangle $A$, one side measures 12 cm, and the matching side in triangle $B$ measures 20 cm. Another side of triangle $A$ measures 9 cm, and the matching side in triangle $B$ is $x$ cm. The diagram is not drawn to scale.

Similarity

Express $2\,600\,000$ in standard form.

Standard form

Finish the table. The table contains two columns headed Fraction and Decimal. Row 1: $\frac{1}{2} = 0.5$ Row 2: [blank] $= 0.25$ Row 3: $\frac{3}{10} = [blank]$ Row 4: $\frac{2}{25} = [blank]$

Fractions, decimals and percentages

Draw an arrow to represent the probability that the counter is blue.

Introduction to probability

Here is a graph used to convert dollars ($) into pounds (£). The vertical axis is marked Pounds (£). The horizontal axis is marked Dollars ($). The line is straight and runs through the origin.

Graphs in practical situations

Let $p = \begin{pmatrix}4\\-2\end{pmatrix}$ and $q = \begin{pmatrix}-3\\0\end{pmatrix}$.

Coordinates

The equation for line $L$ is $y = 4x - 3$.

Equations of linear graphs

A regular polygon has an interior angle of $172^\circ$.

Geometrical terms

Work out $2\frac{5}{8} \times \frac{3}{7}$ without a calculator. Include all steps of your working and present your answer as a mixed number in its simplest form.

Fractions, decimals and percentages

Solve the simultaneous equations. Show every step in your working. $3x + 4y = 14$ $5x + 2y = 21$

Equations

Calculate the value of $\dfrac{3.07 + 2^4}{5.03 - 1.79}$

Fractions, decimals and percentages

The diagram depicts triangle $ABC$ with the vertices marked $A$, $B$ and $C$.

Geometrical constructions

The table presents the temperature recorded on each night for a week. Monday: -3^{\circ}C Tuesday: 1^{\circ}C Wednesday: -4^{\circ}C Thursday: -2^{\circ}C Friday: 5^{\circ}C Saturday: 3^{\circ}C Sunday: -1^{\circ}C

Averages and measures of spread

A trapezium is drawn and marked NOT TO SCALE. Its parallel sides measure 7 cm and 10 cm. The perpendicular distance between the parallel sides is 6 cm.

Surface area and volume

Sonali travels by bicycle from home to the sports centre. The travel graph traces her trip. The vertical axis gives distance in km and includes the labels Home and Sports centre. The horizontal axis gives time from 15:30 to 17:45.

Graphs in practical situations

Write the value of $3.5897$ correct to $4$ significant figures.

Limits of accuracy

State the mathematical name for this quadrilateral.

Geometrical terms

Calculate how much she earns each week.

Money

Calculate $216$ as a percentage of $600$.

Fractions, decimals and percentages

Simplify $3f + 4f - 2f$.

Algebraic manipulation

David travels to college by bus. The bus is late on 6 mornings out of 45. During one year, David travels to college by bus 180 times.

Ratio and proportion

The first five terms of a sequence are shown below. 4  11  18  25  32

Sequences

Give the numeral for nine million eighty two thousand five hundred and seven.

Types of number

Lanying sells potatoes in bags. The potatoes in each bag have a mass of $5\text{ kg}$, accurate to the nearest $0.1\text{ kg}$.

Limits of accuracy

Do not use a calculator. Work out $\dfrac{1}{12} \times 1\dfrac{1}{5}$. Show all your steps, and give the answer as a fraction in lowest terms.

Fractions, decimals and percentages

A ramp, $AB$, is $23\text{ m}$ in length and rises at an angle of $16^{\circ}$ above the horizontal, $AC$.

Right-angled triangles

Jamal converts $800$ Chinese Yuan into dollars, using an exchange rate of $\$1 = 6.24$ Chinese Yuan.

Rates

Cheng put $\$4500$ into an investment earning compound interest at $3.5\%$ each year.

Percentages

The diagram consists of $5$ congruent kites.

Angles

Solve the simultaneous equations below. You must show all of your working. $3x+7y=-21$, $6x+4y=3$.

Equations

The table gives the counts of screws with different lengths in a box containing $100$ screws.

Relative and expected frequencies

On a grid, the straight line $L$ has been drawn.

Equations of linear graphs

The diagram depicts a rectangular field, $ABCD$, together with a straight path, $BD$. Here $BD = 93.5\text{ m}$ and $BC = 82.5\text{ m}$.

Pythagoras' theorem

Round $71\,496$ to $2$ significant figures.

Limits of accuracy

Alice departs from home at $08\,15$ and moves at $80$ metres per minute, reaching her friend’s house half an hour afterwards.

Graphs in practical situations

Forty-five athletes in a club were asked to select a colour for their club vests. The options were red, blue and green. The pie chart shows the sector representing the members who picked red.

Statistical charts and diagrams

Determine the cube root of $4913$.

Powers and roots

A scatter diagram is displayed, and the plotted points slope downward from left to right.

Scatter diagrams

Calculate the value of $\dfrac{17.85 - 7.96}{18 - 3.5^2}$.

Fractions, decimals and percentages

Using the numbers $2$, $3$, $-4$, $-6$, $-8$.

The four operations

Solve the equation $6(y+1)=9$ for $y$.

Equations

Express $3\begin{pmatrix}-2 \\ 1\end{pmatrix}$ as one vector.

Coordinates

This parallelogram has an area of $51.5\text{ cm}^2$. In the diagram, the base is $x$ cm, the perpendicular height is $5$ cm and the sloping side measures $6$ cm.

Area and perimeter

A train departs Zurich at 22 40 and reaches Vienna at 07 32 on the following day. Work out the time taken.

Time

An equilateral triangle has side lengths of 9.4 cm, given correct to the nearest millimetre.

Limits of accuracy

A, B, P and Q are on a circle with centre $O$. The angle $APB$ is $56^{\circ}$. The diagram shows the circle with $A$, $B$, $P$ and $Q$ labelled. The angle at centre $O$ between $OA$ and $OB$ is marked $x^{\circ}$. The angle at $Q$ between $QA$ and $QB$ is marked $y^{\circ}$. Diagram not to scale.

Circle theorems I

Determine the simplified form of $(16p^{16})^{\frac{1}{4}}$.

Powers and roots

Solve this inequality: $n + 7 < 5n - 8$.

Inequalities

The diagram illustrates a rectangular garden split into several sections. $FG$ acts as the perpendicular bisector of $BC$. The arc $HJ$ is centred at $D$ and has a radius of 20 m. $CE$ is the bisector of angle $DCB$. Diagram NOT TO SCALE.

Geometrical constructions

The sequence shown is $7, 5, 3, 1, -1, \ldots$

Sequences

Without a calculator, work out $\frac{6}{7} \div 1\frac{2}{3}$. Show your full working and express your answer as a fraction in lowest terms.

Fractions, decimals and percentages

Each of the five angles in the hexagon measures $115^{\circ}$.

Angles

A car measuring 4.3 m in length is moving at 105 km/h and travels across a bridge that is 36 m long.

Rates

The tree diagram displays the chances of a cricket team either winning or losing its first two matches. In the first match, win $\frac{1}{3}$, lose $\frac{2}{3}$. In the second match, when the first match is a win: win $\frac{3}{4}$, lose $\frac{1}{4}$. In the second match, when the first match is a loss: win $\frac{3}{4}$, lose $\frac{1}{4}$.

Probability of combined events

A sample containing 80 batteries includes 3 faulty ones.

Percentages

AB is an arc of a circle with centre $O$ and radius 9 cm. The length of arc $AB$ is $6\pi$ cm. The area of sector $AOB$ is $k\pi$ cm$^2$. The diagram shows sector $AOB$ with $OB = 9$ cm. The diagram is NOT TO SCALE.

Circles, arcs and sectors

The quantity $y$ varies directly with the positive square root of $x$. If $x = 9$, then $y = 12$.

Ratio and proportion

The Venn diagram indicates how many elements fall in each section. Set $A$ only has 3 elements, $A \cap B$ has 7 elements, $B$ only has 12 elements, and the part outside both sets has 5 elements.

Sets

The diagram depicts a trapezium with $AD$ drawn vertically, $AB = 12$ cm, $DC = 8$ cm, $CB = 8$ cm, and a right angle at $A$. The diagram is labelled NOT TO SCALE.

Area and perimeter

Factorise fully: $2a + 4 + ap + 2p$.

Algebraic manipulation

The points are $A=(4, 1)$ and $B=(10, 15)$.

Perpendicular lines

The diagram displays triangle $ABC$ with $AB = 7$ cm, $BC = 10$ cm and angle $ABC = 35^{\circ}$. It is not drawn to scale.

Non-right-angled triangles

Express $1.27 \times 10^{-3}$ as a decimal number.

Standard form

Calculate $(2.1 - 0.078)^{17}$, and give your answer accurate to 4 significant figures.

Indices I

Omar converts 2000 Saudi Arabian riyals (SAR) into euros (€) using an exchange rate of €1 = 5.087 SAR.

Rates

Work out the lowest common multiple (LCM) of 36 and 48.

Types of number

Work out $y$ when $m = -2$, $x = -7$ and $c = -3$.

Introduction to algebra

Express $x$ in terms of $p$, $q$ and $y$.

Algebraic manipulation

The diagram depicts triangle $ABC$, which is isosceles, with $AC$ parallel to $BD$. The angle at $C$ is $40^{\circ}$. The points $A$, $B$, $C$, $D$ are labelled. At $B$, the angle between $AB$ and $AC$ is marked $a^{\circ}$, and the angle between $AC$ and $BD$ is marked $b^{\circ}$. The diagram is marked NOT TO SCALE.

Angles

Write $0.0000574$ as standard form.

Standard form

Make $p$ the subject in the formula $rp + 5 = 3p + 8r$.

Algebraic manipulation

Shahruk takes part in four golf games. The mean of his four scores is $75$, the mode is $78$ and the median is $77$.

Averages and measures of spread

[$0.\overline{36}$ represents $0.3666\ldots$]

Fractions, decimals and percentages

The triangle has a base of $9\text{ cm}$, rounded to the nearest cm, and its area is $40\text{ cm}^2$, rounded to the nearest $5\text{ cm}^2$.

Limits of accuracy

Work out $2\frac{5}{8} \times \frac{3}{7}$ without a calculator. Show all of your working, and write your answer as a mixed number in its lowest terms.

Fractions, decimals and percentages

The expression $y = x^2 + 7x - 5$ may also be expressed in the form $y = (x + a)^2 + b$.

Algebraic manipulation

Solve the simultaneous equations. Provide all the working. $3x + 4y = 14$, $5x + 2y = 21$.

Equations

The figure shows triangle $ABC$.

Geometrical constructions

Find the $n$th term for the sequence $16, 19, 22, 25, 28, \ldots$.

Sequences

The world’s population is estimated to be increasing by $1.14\%$ each year. On January $1$st $2014$, the population stood at $7.23$ billion.

Exponential growth and decay

Calculate the value of $\frac{3.07 + 2^4}{5.03 - 1.79}$.

Fractions, decimals and percentages

Deborah keeps a tally of the number of minutes late, $t$, for trains as they arrive at a station. This histogram presents the data. The horizontal axis is marked “Number of minutes late” and runs from $0$ to $25$. The vertical axis is marked “Frequency density”.

Statistical charts and diagrams

In the diagram, $A$, $B$ and $C$ are points on the circle’s circumference, with centre $O$. The diagram is shown NOT TO SCALE. The angle at $B$ between $OB$ and $AB$ is labelled $28^\circ$.

Circle theorems I

The matrix $M$ is defined as $\begin{pmatrix} 5 & 1 \\ -3 & -2 \end{pmatrix}$.

Transformations

Region $R$ is determined by the inequalities $y \leq 2x$, $3x + 4y \geq 12$, and $x \leq 3$. The coordinate grid shown has $x$ ranging from $0$ to $5$ and $y$ ranging from $0$ to $7$.

Drawing linear graphs

In the diagram shown, $O$ is the origin, $\overrightarrow{OA} = \mathbf{a}$, $\overrightarrow{OC} = \mathbf{c}$ and $\overrightarrow{AB} = \mathbf{b}$. $P$ lies on the line $AB$ such that $AP : PB = 2 : 1$. $Q$ is the midpoint of $BC$. The diagram is marked NOT TO SCALE.

Vectors in two dimensions

Write $3.5897$ rounded to 4 significant figures.

Limits of accuracy

A quadrilateral has rotational symmetry of order 2 and no lines of symmetry. State the mathematical name of this quadrilateral.

Symmetry

The numbers in the list are 8, 9, 10, 11, 12, 13, 14, 15 and 16.

Powers and roots

Simplify the expression \(\left(\frac{1}{2}x^{\frac{2}{3}}\right)^3\).

Indices II

The map is drawn at a scale of $1 : 1\,000\,000$. On the map, the forest covers an area of $4.6\text{ cm}^2$.

Scale drawings

Solve the inequality $\frac{x}{3} + 5 > 2$.

Inequalities

The interior angle of a regular polygon is $172^\circ$.

Angles

Determine the cube root of 4913.

Powers and roots

Determine the highest common factor (HCF) of 56 and 70.

Types of number

Hattie has a box of coloured pens. She chooses one pen at random from the box. The probability that she chooses a red pen is 0.4.

Introduction to probability

The diagram depicts a cyclic quadrilateral $ABCE$. $AED$ and $BCD$ are straight lines. $AC=CD$, $\angle ABC=45^{\circ}$ and $\angle ACE=20^{\circ}$. The diagram is labelled NOT TO SCALE.

Circle theorems I