Oct/Nov 2015

Put these numbers into size order, beginning with the smallest. 5.024 0.524 5.204 5.0204

Ordering

Using each number rounded correctly to 1 significant figure, estimate the value of $\dfrac{\sqrt{3.9 \times 29.3}}{8.9 - 2.7}$. Show all your working.

Estimation

Without a calculator, calculate $\dfrac{2}{5} \div \dfrac{3}{4}$. Write your answer as a fraction. Show all the steps in your working.

Fractions, decimals and percentages

Leah is laying a gravel path in her garden. The diagram illustrates the path. Its measurements are width $0.8\,\text{m}$, horizontal length $6\,\text{m}$ and vertical length $3\,\text{m}$. One bag of gravel covers an area of $0.5\,\text{m}^2$.

Area and perimeter

The diagram represents quadrilateral $ACDE$. Line $AC$ is parallel to $ED$, and point $B$ lies on $AC$. Also, $\angle EAB = 120^{\circ}$, $\angle ABE = 32^{\circ}$ and $\angle CBD = 64^{\circ}$.

Angles

Find the size of one interior angle in a regular $15$-sided polygon.

Angles

Chico has a bag of sweets, and one sweet is chosen from it at random. The table gives the probabilities of selecting each sweet flavour.

Introduction to probability

The diagram shows that $AP$ is a tangent to the circle at $P$. $O$ marks the centre of the circle, $\angle PAO = 37^{\circ}$ and $AP = 11\,\text{cm}$.

Circle theorems I

Amir examines advertisements for the same car model. The scatter diagram gives the age and the price for each car.

Scatter diagrams

The protractor has the form of a semicircle with radius $6.1\,\text{cm}$.

Circles, arcs and sectors

The relationship is given by $s = 4t + 3u$.

Equations

At midnight, Newtown's temperature was $-8^{\circ}\text{C}$. By noon on the following day, it had reached $9^{\circ}\text{C}$.

The four operations

How many minutes does Maria take to travel from home to the shopping mall?

Graphs in practical situations

Calculate her earnings for one week.

Money

What is the next term in the sequence $5, 9, 13, 17, \ldots$?

Sequences

Simplify $\dfrac{r^6}{r^2}$.

Indices I

Work out the value of $\dfrac{5}{12}$ of $168$.

Fractions, decimals and percentages

Calculate the value of $3.2 \times (5.7 - 1.3) + 4.8$.

Fractions, decimals and percentages

The vectors are $p = \begin{pmatrix}4 \\ -2\end{pmatrix}$ and $q = \begin{pmatrix}-1 \\ 3\end{pmatrix}$.

Coordinates

A coordinate grid is displayed, with shape $A$ plotted on it.

Transformations

Pip and Ali divide $785$ in the ratio Pip : Ali = $4 : 1$.

Ratio and proportion

Jim obtains the following marks in 8 tests: $7, 8, 8, y, 6, 9, 10, 5$. His mean mark is $7.5$.

Averages and measures of spread

State the temperature difference between $8\degree\text{C}$ and $-9\degree\text{C}$.

The four operations

Triangle $ABC$ is drawn. $AB = 14.5\text{ m}$, $BC = 19.3\text{ m}$, and angle $A$ is a right angle. The diagram is marked NOT TO SCALE.

Right-angled triangles

Solve the equation $3x^2 = 108$.

Equations

Express $\vec{AB}$ in column-vector form.

Coordinates

Rewrite the formula with $y$ as the subject.

Algebraic manipulation

Write down the item in the list below that has the same value as $\frac{5}{8}$.

Fractions, decimals and percentages

A side measuring $85$ mm has already been drawn for you.

Geometrical constructions

The price of fruit juice is $\$1.27$ for each litre, whereas rice is priced at $\$1.68$ per kilogram.

Money

Jason is given some birthday money. He uses $\frac{11}{15}$ of it and is left with $\$14.40$.

Ratio and proportion

The table presents data on the number of pets owned by $24$ students.

Averages and measures of spread

A right-angled triangle is drawn with one side $5\text{ cm}$, the hypotenuse measures $8\text{ cm}$ and the base is $x\text{ cm}$. The diagram is marked NOT TO SCALE.

Pythagoras' theorem

The figures shown are a Parallelogram, a Trapezium and a Rhombus.

Symmetry

Without a calculator, calculate $2\frac{1}{4} - \frac{11}{12}$. Show every step of your working and write your answer as a fraction in simplest form.

Fractions, decimals and percentages

Give a five-number set with mode $3$, median $6$ and range $5$.

Averages and measures of spread

The diagram shows a circle whose centre is $O$.

Area and perimeter

Expand the expression $-4(2w - 5)$.

Algebraic manipulation

The scale drawing shows where the Police station, $P$, and the Fire station, $F$, are located in a town. The scale means $1$ centimetre represents $40$ metres.

Scale drawings

Write down which number in this list is irrational.

Types of number

Show that $0.\overline{3} \neq \frac{1}{3}$.

Fractions, decimals and percentages

Write $1426.3075$, rounded to $2$ decimal places.

Limits of accuracy

A sum of $2600 is placed for 5 years at a simple interest rate of $4\%$ per year.

Rates

Carlos exchanged $950 for euros (€) at an exchange rate of $€1 = \$1.368$.

Rates

The diagram presents two jugs that are mathematically similar. The shorter jug measures $15\text{ cm}$ in height and $7.2\text{ cm}$ across the base. The taller jug measures $25\text{ cm}$ in height and has base width $x\text{ cm}$. The diagram is marked NOT TO SCALE.

Similarity

The sequence shown is $-1, 3, 7, 11, \ldots$.

Sequences

Write the number six thousand and fifty four in figures.

Types of number

At 9 am, the temperature stood at -3^{\circ}C. By 1 pm, it had increased by 5^{\circ}C.

The four operations

A right-angled triangle is pictured. The vertical side is 2 cm long, the hypotenuse is 5 cm long, and the angle at the bottom is marked $x^{\circ}$. The diagram is marked NOT TO SCALE.

Right-angled triangles

Express 72 as a product of its prime factors.

Types of number

A right-angled triangle is drawn. One side measures 18 cm, the base measures 26 cm, and the hypotenuse is marked $x$ cm. The diagram is labelled NOT TO SCALE.

Pythagoras' theorem

For this question, use only a ruler and compasses, and make sure that every construction arc is shown. On a scale where 1 centimetre represents 50 metres, construct a triangle whose sides are 550 m, 450 m and 300 m. The 300 m side has already been drawn.

Geometrical constructions

A cuboid is drawn with edges measuring 5 cm, 12.5 cm and 9 cm. The illustration is labelled NOT TO SCALE.

Surface area and volume

Calculate $\frac{2}{3} + \frac{1}{6} - \frac{1}{4}$, and give your answer as a fraction in lowest terms. Do not use a calculator and show all working.

Fractions, decimals and percentages

Expand $3(x + 7)$.

Algebraic manipulation

This scale diagram gives the locations of towns A and B on a map, and a north-pointing arrow is included. Town A is joined to town B by a straight line.

Angles

Express $1.7 \times 10^{-4}$ as an ordinary number.

Standard form

A straight line is drawn at an angle.

Units of measure

The diagram depicts a quadrilateral with one side extended. The angles shown are 98^{\circ}, 66^{\circ}, 112^{\circ} and an exterior angle marked $x^{\circ}$. The diagram is NOT TO SCALE.

Angles

The figure displays two triangles, $ABC$ and $PQR$, that are similar. In triangle $ABC$, $AB = 10$ cm, $BC = 6$ cm and $AC = x$ cm. In triangle $PQR$, $PQ = 6.25$ cm, $PR = 7.5$ cm and $QR = y$ cm. The figure is labelled NOT TO SCALE.

Similarity

Find the solution of the simultaneous equations. You must show all your working. $5x + 2y = 8$, $2x - 3y = 26$.

Equations

The pie chart together with the table gives information about how students travel and how many children there are in each household.

Statistical charts and diagrams

The diagram shows a geometric figure with a shaded central region and three triangular parts arranged around it.

Symmetry

Give 168.9 correct to 2 significant figures.

Limits of accuracy

Calculate $\dfrac{2.07 - 1.89}{5.71 - 3.92}$.

Fractions, decimals and percentages

On any given day, the chance that it will rain is $\frac{1}{5}$.

Relative and expected frequencies

The numbers in the list are 11, 12, 13, 14, 15, 16.

Types of number

The first four terms of a sequence are 21, 17, 13, 9.

Sequences

Simplify the expression $1 - 2u + u + 4$.

Algebraic manipulation

At midnight, Newtown's temperature was $-8^\circ\text{C}$. At noon on the next day, Newtown's temperature was $9^\circ\text{C}$.

The four operations

[$0.15$ represents $0.1555\ldots$]

Fractions, decimals and percentages

A protractor has the shape of a semicircle with radius $6.1$ cm. The diagram is not drawn to scale.

Circles, arcs and sectors

$V$ varies directly as the cube of $(r + 1)$. When $r = 1$, $V = 24$.

Ratio and proportion

Find $x$ as the subject in the formula $y = ax^2 + b$.

Algebraic manipulation

A car is moving at $56$ km/h.

Rates

Simplify $\frac{x^2 - 16}{x^2 - 3x - 4}$.

Algebraic fractions

Hazel puts $1800$ into an account for $7$ years, earning compound interest at $1.5\%$ each year.

Exponential growth and decay

The coordinate plane displays triangles $S$ and $T$.

Transformations

Work out $\begin{pmatrix} 1 & -2 \\ 3 & 4 \end{pmatrix} \begin{pmatrix} -5 & -3 \\ 2 & 1 \end{pmatrix}$.

Algebraic manipulation

A wooden prism has a height of $5$ cm. Its cross section is a sector of a circle with sector angle $25^\circ$. The radius of the sector measures $15$ cm. The diagram is not drawn to scale.

Surface area and volume

The Venn diagram presents a universal set containing two overlapping circles labelled $A$ and $B$.

Sets

The table gives the chance that a person has blue, brown or green eyes.

Probability of combined events

The functions are $f(x) = x^3$, $g(x) = 3x - 5$, and $h(x) = 2x + 1$.

Functions

The coordinate grid below includes shape $A$.

Transformations

Pip and Ali divide $785 in the ratio Pip : Ali = $4 : 1$.

Ratio and proportion

Jim achieves the following marks in 8 tests: $7, 8, 8, y, 6, 9, 10, 5$. His average mark is $7.5$.

Averages and measures of spread

Use each number rounded to 1 significant figure to estimate the value of $\frac{\sqrt{3.9 \times 29.3}}{8.9 - 2.7}$. Show all your working.

Estimation

Find the highest common factor (HCF) of 36 and 90.

Types of number

Here, $AP$ touches the circle at $P$, so $AP$ is a tangent. $O$ is the circle’s centre, $PAO = 37^\circ$ and $AP = 11$ cm. The diagram is not drawn to scale.

Circle theorems II

Factorise completely the expression $ax + ay + 3cx + 3cy$.

Algebraic manipulation

Write down the temperature difference between $8\degree\text{C}$ and $-9\degree\text{C}$.

The four operations

Calculate the total amount of money he was given for his birthday.

Money

A right-angled triangle is drawn here, and it is not to scale. One vertical side measures $5\text{ cm}$, the hypotenuse measures $8\text{ cm}$, and the base is labelled $x\text{ cm}$.

Pythagoras' theorem

Work out $2\frac{1}{4} - \frac{11}{12}$ without a calculator. Show every step of your working and write your answer as a fraction in lowest terms.

Fractions, decimals and percentages

A triangle is drawn and it is not to scale. Its base measures $12.4\text{ cm}$. The side on the left is $y\text{ cm}$. The angle at the top is $74\degree$, while the angle at the bottom right is $39\degree$.

Non-right-angled triangles

Calculate how many stamps Jasjeet had at the beginning.

Equations

Factorise the expression $9w^2 - 100$.

Algebraic manipulation

The diagram depicts a sector of a circle with radius $15\text{ cm}$ and angle $26\degree$, and it is not drawn to scale.

Circles, arcs and sectors

Find $y$ when $x = 6$.

Introduction to algebra

Calculate the lower bound and the upper bound for the perimeter of this rectangle.

Limits of accuracy

Solve $5x^2 - 6x - 3 = 0$, showing all working and rounding your answers to 2 decimal places.

Equations

The three shapes displayed are: Parallelogram, Trapezium and Rhombus.

Symmetry

A car goes past a checkpoint at $t = 0$ seconds, moving at $8\text{ m/s}$. It keeps this speed for 10 seconds. It then slows down at a constant rate and comes to rest when $t = 55$ seconds.

Graphs in practical situations

(a) The diagram depicts two jugs that are mathematically similar. The shorter jug has height $15\text{ cm}$ and base width $7.2\text{ cm}$. The taller jug has height $25\text{ cm}$ and base width $x\text{ cm}$. (b) The diagram depicts two glasses that are mathematically similar. The larger glass has height $16\text{ cm}$ and volume $375\text{ cm}^3$. The smaller glass has height $y\text{ cm}$ and volume $192\text{ cm}^3$.

Similarity

The table presents data on how many pets are kept by 24 students.

Averages and measures of spread

A box holds 6 red pencils together with 8 blue pencils. One pencil is selected at random and is not put back. Then a second pencil is selected at random. In the tree diagram, several probabilities have been left blank. The probabilities already shown are $\frac{6}{14}$ for choosing red first and $\frac{8}{13}$ for choosing blue second given that red was chosen first.

Probability of combined events

Calculate how many euros Carlos obtained.

Money

Find the value of $|\overrightarrow{AB}|$.

Magnitude of a vector

[For a sphere of radius $r$, the volume is given by $V = \frac{4}{3}\pi r^3$.]

Surface area and volume

The Venn diagram indicates how many students take French ($F$), Spanish ($S$) and Arabic ($A$). The regions show: $7$ in $F$ only, $5$ in $S$ only, $8$ in $A$ only, $4$ in $F \cap S$, $2$ in $F \cap A$, $3$ in $S \cap A$, $1$ in $F \cap S \cap A$, and $0$ outside all sets.

Sets

The matrices $M = \begin{pmatrix}3 & -4\\-2 & 4\end{pmatrix}$ and $N = \begin{pmatrix}5 & 0\\1 & 2\end{pmatrix}$ are provided.

Algebraic manipulation

Calculate the value of the car after 7 years.

Exponential growth and decay

Calculate the field's actual area in square kilometres.

Scale drawings

Write 168.9 rounded to 2 significant figures.

Limits of accuracy

Find the value of the following:

Powers and roots

Express each of the following as one fraction.

Algebraic fractions

The Venn diagram gives the number of elements in each set. In set $P$, there are 3 in the part only in the left circle, the overlap contains 5, set $Q$ has 10 in the part only in the right circle, and 9 lie outside both sets but still within the universal set.

Sets

Matrix $M = \begin{pmatrix}7 & u \\ 2 & 3\end{pmatrix}$ has $|M| = 1$.

Algebraic manipulation

The two containers are mathematically similar. Their volumes are $54\text{ cm}^3$ and $128\text{ cm}^3$. The smaller container has a height of $4.5\text{ cm}$.

Similarity

Find $\frac{2}{3} + \frac{1}{6} - \frac{1}{4}$ and give your answer as a fraction in its simplest form. Do not use a calculator, and show every stage of your working.

Fractions, decimals and percentages