May/June 2015

January has 31 days. 21st January 2015 fell on a Wednesday.

Time

Simplify the expression $6uw^{-3} \times 4uw^{6}$.

Indices II

A right-angled triangle $ABC$ is drawn. Angle $ABC$ is a right angle, angle $ACB = 37^{\circ}$, and $AC = 11.2\,\text{cm}$. The diagram is labelled NOT TO SCALE.

Right-angled triangles

Give the coordinates of the point at which the line $y = 3x + 5$ cuts the $y$-axis.

Parallel lines

Factorise the expression $3w^{2} - 2w$.

Algebraic manipulation

Each of the six donkeys is given two 5 ml spoons of medicine, three times every day.

Units of measure

The volume of a cuboid is $288\,\text{cm}^3$.

Surface area and volume

Without a calculator, calculate $1\frac{4}{5} \div \frac{3}{7}$. Include all your working and write your answer as a fraction in simplest form.

Fractions, decimals and percentages

Write, in standard form, 82 600.

Standard form

Solve the equation $5(3y - 2) = 35$.

Equations

Use a ruler and compasses for this question. The rectangle $ABCD$ is shown.

Geometrical constructions

The temperature in Berlin is $-7\,^{\circ}\text{C}$, while the temperature in Istanbul is $-3\,^{\circ}\text{C}$.

The four operations

The terms are $2,\ 3,\ 6,\ 11,\ 18,\ \ldots$

Sequences

Write 30 in the form of a product of its prime factors.

Types of number

Triangles $ABC$ and $DEF$ are similar. For triangle $ABC$, $AB = 8\,\text{cm}$, $BC = 10\,\text{cm}$ and $AC = y\,\text{cm}$. For triangle $DEF$, $DE = 6\,\text{cm}$, $EF = x\,\text{cm}$ and $DF = 9\,\text{cm}$. The diagrams are labelled NOT TO SCALE.

Similarity

Find the mass increase.

Percentages

The combined mass of 38 spoons is 1824 g.

Ratio and proportion

Prince Charming places $3000$ for 5 years at an annual simple interest rate of 4%.

Percentages

Draw a triangle with side lengths 5 cm, 6 cm and 7 cm.

Geometrical constructions

Here are the shapes in the list: equilateral triangle, square, regular pentagon, parallelogram, regular hexagon, circle.

Symmetry

Take $\mathbf{a} = \begin{pmatrix}3\\5\end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix}-8\\7\end{pmatrix}$.

Coordinates

The scatter diagram plots the daily numbers of sun hats and ice creams sold by a shop over a two-week period. The horizontal axis carries the label "Number of sun hats sold" and the vertical axis is labelled "Number of ice creams sold".

Scatter diagrams

A doctor begins work at 2040 and ends work at 0610 the following day. How long does the doctor work for? Give your answer in hours and minutes.

Time

Factorise fully $3x^2y-5xyz$.

Algebraic manipulation

The figure contains two straight lines, $AE$ and $BD$, that cross at $C$. Angle $ABC$ = angle $EDC$. Triangles $ABC$ and $EDC$ are congruent.

Geometrical terms

Work out, without using a calculator, $\frac{4}{5}\div 2\frac{2}{3}$. Show every stage of your working and give your answer as a fraction in its simplest form.

Fractions, decimals and percentages

Find what $x$ is.

Pythagoras' theorem

Calculate the amount he gets after 3 years.

Money

A random sample of 200 families was selected from the city’s families. For each family, the number of children was noted. The results are displayed in the table below.

Relative and expected frequencies

The coordinate grid displays two points, $A$ and $B$.

Coordinates

The diagram shows $ABF$ parallel to $EDC$. Angle $EDA=37^\circ$, angle $ADB$ is a right angle, and $BC=BD$.

Angles

Write down the next two terms for the sequence shown below.

Sequences

The diagram depicts the barn's front elevation. The barn has a width of 12 m and a height of 8 m. Each side of the barn is 5 m high.

Surface area and volume

Express $53\,400\,000$ in standard form.

Standard form

ABCDE makes a pentagon.

Angles

The table below lists the average monthly temperatures ($^\circ$C) for Silvas, Turkey. Month: Jan, Feb, Mar, Apr, May, Jun, Jul, Aug, Sep, Oct, Nov, Dec Temperature ($^\circ$C): -4, -3, 2, 8, 13, 17, 19, 20, 16, 11, 8, -1

Averages and measures of spread

State the gradient of the line $y=-3x+4$.

Gradient of linear graphs

Simplify $5x^0$.

Indices I

What kind of correlation does the scatter diagram show?

Scatter diagrams

Write $64\%$ in decimal form.

Fractions, decimals and percentages

Expand the brackets, then simplify $5(x-3)-3(x-5)$.

Algebraic manipulation

Arrange the following by size, beginning with the smallest.

Ordering

A loaded 4-sided dice is thrown. The outcomes can be 1, 2, 3 or 4. The table gives the probability of getting a 1, 3 or 4.

Introduction to probability

What value does the digit $7$ have in $43\,782$?

Types of number

Write $270\,000$ as standard form.

Standard form

An isosceles triangle $ABE$ and a quadrilateral $BCDE$ are shown. $ABC$ is a straight line. The angle at $E$ inside triangle $ABE$ is $82^\circ$. The angle at $D$ is $102^\circ$. The angle at $C$ is $64^\circ$. It is marked NOT TO SCALE.

Angles

The shape shown is symmetrical.

Symmetry

James pays $2$ euros (€) for a drink.

Rates

Do not use a calculator to find $1\frac{7}{8} \div \frac{5}{9}$. Include your working, and write your answer as a fraction in its simplest form.

Fractions, decimals and percentages

Solve $3(x + 4) = 2(4x - 1)$.

Equations

During a sale, the price of a coat is cut from $\$85$ to $\$67.50$.

Percentages

A coordinate grid is displayed, and points $A$ and $B$ are marked on it. The axes are named $x$ and $y$.

Coordinates

Calculate $\dfrac{6.4 + 7.3}{19.56 - 3.51}$. Round your answer to $2$ significant figures.

Fractions, decimals and percentages

Write down the prime number that lies between $62$ and $70$.

Types of number

A sketch contains two horizontal parallel lines that are intersected by one sloping line. The angle at the top intersection is labelled $124^\circ$, while the angle at the lower intersection is labelled $x^\circ$. The sketch is marked NOT TO SCALE.

Angles

Calculate the cylinder's volume.

Surface area and volume

Write $0.88$ as a fraction in lowest terms.

Fractions, decimals and percentages

Ahmed and Babar split $240\text{ g}$ of sweets in the ratio $7 : 3$.

Ratio and proportion

Factorise completely the expression $9x^2 - 6x$.

Algebraic manipulation

The mass of a car is stated as $1400\text{ kg}$, to the nearest hundred kilograms.

Limits of accuracy

The sketch represents a right-angled triangle. Its base measures $2\text{ cm}$, the hypotenuse measures $5\text{ cm}$, and the angle at the end of the base is labelled $x^\circ$. The sketch is marked NOT TO SCALE.

Right-angled triangles

Calculate $(-6) - (-8)$.

The four operations

With a calculator, find $\sqrt{4.2^2 + 5.8^2}$.

Powers and roots

At noon, the temperature measured 4^{\circ}C. By midnight, it had dropped to -5.5^{\circ}C.

The four operations

A tram departs from a station, speeds up for 2 minutes until its speed is 12 metres per second, then travels at that speed for 1 minute. After that, it slows down for 3 minutes and comes to rest at the next station. The diagram gives the speed-time graph for this trip. Speed (metres per second) is shown on the vertical axis, while Time (minutes) is shown on the horizontal axis.

Graphs in practical situations

Find the $n$th term for the sequence $4, 8, 12, 16, 20, \ldots$.

Sequences

$p$ varies inversely with the square of $(q + 4)$. When $q = 2$, $p = 2$.

Ratio and proportion

At an average speed of 64 kilometres per hour, a car covers 1280 metres.

Rates

PQRS is a quadrilateral, and M lies at the midpoint of PS. $\overrightarrow{PQ} = \mathbf{a}$, $\overrightarrow{QR} = \mathbf{b}$ and $\overrightarrow{SQ} = \mathbf{a} - 2\mathbf{b}$. Diagram labels: points P, Q, R, S and M, together with vectors marked $a$ and $b$. The diagram is NOT TO SCALE.

Vectors in two dimensions

A graph is shown with the $x$-axis horizontal and the $y$-axis vertical. Two straight lines cross, and a dashed horizontal line is included. The shaded area lies outside the central unshaded area.

Inequalities

Georg puts $5000 into an investment for 14 years at 2% compound interest per year.

Exponential growth and decay

Write 30 as a product formed from its prime factors.

Types of number

Solve the simultaneous equations below. You must show all your working. $5x + 2y = -2$ $3x - 5y = 17.4$

Equations

Triangle ABC and triangle DEF are similar. Diagram information: For triangle ABC, AB = 8 cm, BC = 10 cm, and AC = $y$ cm. For triangle DEF, DE = 6 cm, EF = $x$ cm, and DF = 9 cm. Diagram is NOT TO SCALE.

Similarity

Use a calculator to determine $\sqrt{10 + 0.6 \times (8.3^2 + 5)}$.

Powers and roots

Factorise $yp + yt + 2xp + 2xt$ fully.

Algebraic manipulation

The diagram depicts a toy. The toy has the form of a cone with radius 4 cm and height 9 cm, placed on a hemisphere with radius 4 cm. Diagram is NOT TO SCALE. Formulae given: The volume, $V$, of a cone with radius $r$ and height $h$ is $V = \frac{1}{3}\pi r^2 h$. The volume, $V$, of a sphere with radius $r$ is $V = \frac{4}{3}\pi r^3$.

Compound shapes and parts of shapes

Calculate the product $\begin{pmatrix} 3 & 7 \\ -1 & 4 \end{pmatrix} \begin{pmatrix} -2 & 1 \\ 4 & 2 \end{pmatrix}$.

Transformations

The rule is $f(x) = 5 - 3x$.

Functions

In standard form, write 270 000.

Standard form

Expand and simplify $x(2x + 3) + 5(x - 7)$.

Algebraic manipulation

Paul and Sammy compete in a race. The probability of Paul winning the race is $\frac{9}{35}$. The probability of Sammy winning the race is 26%.

Introduction to probability

Rice is sold in packs of 75 grams and packs of 120 grams. The mass of each pack is given correct to the nearest gram.

Limits of accuracy

Simplify $6uw^{-3} \times 4uw^6$.

Indices II

Point A is at co-ordinates (-4, 6), and point B is at co-ordinates (7, -2).

Length and midpoint

Calculate $1\frac{4}{5} \div \frac{3}{7}$. Show all your working, and write your answer as a fraction in its lowest terms.

Fractions, decimals and percentages

Express 5 340 000 in standard form.

Standard form

A year earlier, Ahmed’s height measured 114 cm. At present, his height measures 120 cm. Each reading is accurate to the nearest centimetre.

Limits of accuracy

The matrix can be written as $M = \begin{pmatrix}3 & 1 \\ -11 & -2\end{pmatrix}$.

Algebraic manipulation

Without using a calculator, calculate $\frac{4}{5} \div 2\frac{2}{3}$. Show each stage of your working and give your answer as a fraction in simplest form.

Fractions, decimals and percentages

ABCD forms a square with $AB = (x - 4)$ cm and $BC = 7$ cm. EFG is an isosceles triangle with the equal sides indicated, and each of those equal sides is $(x - 1)$ cm.

Area and perimeter

The diagram represents a water channel on level ground. The channel has a fixed rectangular cross-section with area $0.95\,m^2$. It is completely full, and the water moves through it at 4 metres per minute.

Rates

Express $\frac{3}{x+2} - \frac{4}{2x-5}$ as one fraction in simplest form.

Algebraic fractions

Find the value of $(\frac{1}{4})^{0.5}$.

Powers and roots

The straight line $l$ is drawn passing through $(0,3)$ and $(4,11)$.

Equations of linear graphs

The sketch presents the barn's front elevation. The barn is 12 m wide and 8 m tall. Each side of the barn has a height of 5 m.

Surface area and volume

The diagram indicates the positions of the three points, $A$, $B$ and $C$.

Geometrical constructions

A doctor begins work at 2040 and ends work at 06 10 on the following day. How long is the doctor at work? Give your answer in hours and minutes.

Time

Let $\xi = \{x: 1 \le x \le 12,\ x\text{ is an integer}\}$, with $M$ as the odd numbers and $N$ as the multiples of 3. The Venn diagram below places sets $M$ and $N$ inside $\xi$.

Sets

The function is $f(x) = x^2 + 4x - 6$.

Algebraic manipulation

The cumulative frequency graph provides data on the distances, measured in kilometres, travelled by 60 people.

Cumulative frequency diagrams

The diagram presents the speed-time graph for 120 seconds of a car journey.

Graphs in practical situations

The functions given are $f(x) = 3x + 5$ and $g(x) = x^2$.

Functions

$8^{\frac{1}{x}} = 3$. Determine the value of $x$.

Indices II

The list contains the numbers 7, 9, 20, 3, 9.

Averages and measures of spread

A biased 4-sided dice is rolled. The outcomes can be 1, 2, 3 or 4. The probability of getting a 1, 3 or 4 is listed in the table.

Introduction to probability

Solve for $w$ in $5(w + 4 \times 10^3) = 6 \times 10^4$.

Equations

The diagram illustrates two straight lines, $AE$ and $BD$, crossing at $C$. Angle $ABC$ = angle $EDC$. Triangles $ABC$ and $EDC$ are congruent.

Geometrical terms

The sequence runs as follows: 5, 11, 21, 35, 53, ...

Sequences

Express the recurring decimal $0.2\dot{5}$ in fraction form. [$0.2\dot{5}$ denotes $0.2555\ldots$]

Fractions, decimals and percentages

Calculate how much Ahmed gets.

Money

Calculate the percentage reduction in the coat’s cost.

Percentages

A triangle $ABC$ is drawn, not to scale. The angle at $A$ measures $30^{\circ}$, the angle at $C$ measures $100^{\circ}$, and $AB = 24$ cm.

Non-right-angled triangles

A speed-time graph (not to scale) depicts a car beginning from rest, increasing its speed for $u$ seconds until it reaches $10\,\text{m s}^{-1}$, and then continuing at $10\,\text{m s}^{-1}$ for $2u$ seconds. The distance covered in the first $3u$ seconds is 125 m.

Graphs in practical situations

Simplify $12x^{12} \div 3x^3$.

Indices I

Solve the equation $2x^2 + x - 2 = 0$. Show all your working and give the answers correct to 2 decimal places.

Equations

Calculate the radius of the circle.

Circles, arcs and sectors

A Venn diagram is displayed. In a class of 30 students, 25 take French (F) and 18 take Spanish (S). One student takes neither French nor Spanish.

Sets

The cumulative frequency diagram gives the reaction times, measured in seconds, for 200 students.

Cumulative frequency diagrams

The diagram shows a solid pyramid on a square horizontal base $ABCD$ (not drawn to scale). The diagonals $AC$ and $BD$ meet at $M$. $P$ is directly above $M$. $AB = 20$ cm and $PM = 8$ cm.

Surface area and volume