Oct/Nov 2024

At midday, the temperature reads $-2\degree\text{C}$. By 6 pm, it has risen to $4\degree\text{C}$.

The four operations

Solve the simultaneous equations $3a+b=-4$ and $2a+3b=9$. Show all working.

Equations

A straight line joins point $A(2,4)$ to point $B(5,-2)$.

Length and midpoint

Express $0.000257$ in standard form.

Standard form

Work out $2\dfrac{1}{5} \div \dfrac{3}{4}$. Present your answer as a mixed number in its simplest form.

Fractions, decimals and percentages

Write $360$ as a product of primes.

Powers and roots

A sector of a circle with angle $60\degree$ has an arc length of $4\pi$ cm. Determine the area of the sector. State your answer, in its simplest form, in terms of $\pi$.

Circles, arcs and sectors

Express as a single matrix: $2\begin{pmatrix}3 & -1 \\ 2 & 4\end{pmatrix} - \begin{pmatrix}1 & 3 \\ -2 & 5\end{pmatrix}$.

Algebraic manipulation

The diagram depicts the lines $y = 2x + 1$ and $x + y = 2$.

Inequalities

A total of 50 adults each sit a quiz. The cumulative frequency diagram plots their scores.

Cumulative frequency diagrams

Because $x$ is inversely proportional to the square root of $y$, if $x = 2$, then $y = 16$.

Ratio and proportion

Amber and Pablo divide $\$280$ in the ratio $2 : 5$.

Ratio and proportion

From $(ax^{n})^{\frac{2}{3}} = 4x^{10}$, determine the value of $a$ and the value of $n$.

Indices I

The diagram presents a speed-time graph for the journey.

Graphs in practical situations

The function is $f(x) = 3x^{2} + 5$.

Functions

A rectangle measures $32$ cm in length and $15$ cm in width. Both measurements are stated correct to the nearest centimetre.

Limits of accuracy

Simplify $\dfrac{2x^{2} + 5x - 3}{2x^{2} + 6x}$.

Algebraic fractions

A cuboid is shown in the diagram. $EH = 6$ cm, $HG = 2$ cm and $EC = 7$ cm.

Pythagoras' theorem and trigonometry in 3D

Solve $\dfrac{x}{x-1} - \dfrac{3}{2x-1} = 1$

Algebraic fractions

These eight integers are $-1, -5, -1, -3, -3, 2, -1, -7$.

Averages and measures of spread

Shade one additional small square so that the diagram has a single line of symmetry.

Symmetry

Simplify the expression $2a - 3b + 4b - 5a$.

Algebraic manipulation

The table gives the time taken for a homework task and how many errors were made by some students in a class. A scatter diagram grid is shown.

Scatter diagrams

Using numbers rounded correctly to $1$ significant figure, calculate an estimate of $\dfrac{3.1 \times 26.7}{6.9 - 2.3}$.

Estimation

The sequence begins with these four terms: $2, 8, 14, 20$.

Sequences

The grid displays triangle $A$ alongside triangle $B$.

Transformations

Write $43.07862$ correct to $3$ decimal places.

Limits of accuracy

The scale sketch shows field $ABCDE$. Inside the field is a stage, which is a sector of a circle with centre $C$. The scale used is $1\text{ cm}$ to $2.5\text{ m}$.

Geometrical constructions

Factorise the expression $4m^{2}-14m$.

Algebraic manipulation

The numbers in the list are: $\frac{1}{3}$, $\sqrt{4}$, $2^{0}$, $\sqrt{5}$, $\frac{10}{8}$, $2^{-1}$.

Types of number

Evaluate

Standard form

For $120$ and $126$, the least common multiple (LCM) comes to $2520$.

Types of number

A regular polygon has an interior angle of $160^{\circ}$.

Angles

Plot the image of shape $A$ after it has been translated by vector $\begin{pmatrix}3\\-5\end{pmatrix}$.

Transformations

A shaded triangle $ABC$ is drawn on a $1$ cm square grid. For the side $AC$, the equation is $y=-\frac{3}{2}x+1$.

Equations of linear graphs

Lin has recorded the masses, in grams, of $60$ onions. Her results are displayed on the cumulative frequency diagram.

Cumulative frequency diagrams

The diagram depicts two mugs that are mathematically similar. The smaller mug has width $w$ cm and, when full, contains $270$ ml. The larger mug has width $8$ cm and, when full, contains $640$ ml.

Similarity

At midnight, the temperature is $-7^{\circ}\text{C}$. By 11 am the following day, the temperature has risen to $12^{\circ}\text{C}$.

The four operations

Simplify this expression.

Algebraic manipulation

The inverse of matrix $A$ is $\frac{1}{20}\begin{pmatrix}m & 7\\-1 & k\end{pmatrix}$. Here, $m$ and $k$ are positive integers with $m<k$. The determinant of matrix $A$ equals $20$.

Algebraic manipulation

Let $f(x)=\frac{3x-1}{2}$ and $g(x)=5^{x}$.

Functions

A theatre provides a singing lesson $(S)$, a dancing lesson $(D)$ and an acting lesson $(A)$. A group of $40$ people are surveyed about which lessons they attend.

Sets

$P$ denotes the point $(-1,4)$, while $Q$ denotes the point $(-3,-2)$.

Perpendicular lines

The table gives the times taken by each of $110$ students to travel to school on one day.

Statistical charts and diagrams

The diagram shows $\overrightarrow{OA}=a$, $\overrightarrow{OB}=2b$ and $AB:BC=1:3$. $OBD$ is a straight line.

Vector geometry

Place these numbers in order of size, beginning with the smallest: $\frac{2}{3}$, $66\%$, $0.6$, $\frac{16}{25}$, $0.606$.

Fractions, decimals and percentages

Simplify this expression.

Indices II

A number of people are asked which kind of holiday they prefer. The table presents the results.

Statistical charts and diagrams

A laptop is priced at $\$800$. In a sale, its price is cut by $15\%$.

Percentages

Calculate $\frac{3}{4}+\frac{5}{6}$. Express your result as a mixed number in simplest form.

Fractions, decimals and percentages

Sophia travels at an average speed of $4\text{ km h}^{-1}$.

Rates

A series of patterns is formed using crosses and circles.

Sequences

Basma is the owner of a toy shop.

Money

Bag $A$ has red balls and green balls. Altogether, the bag contains $x$ balls. The number of green balls is $6$ greater than the number of red balls.

Equations

Mia has 25 shapes altogether. She sorts them into groups according to their properties. The table gives the number of shapes in each group.

Probability of combined events

A traffic survey records details of the vehicles that go through a checkpoint.

Statistical charts and diagrams

Complete the value table for $y = 4 + 2x - \frac{x^2}{2}$.

Graphs of functions

$\mathcal{E} = \{x \mid x \text{ is an integer}, 1 \le x \le 15\}$ $A = \{x \mid x \text{ is a multiple of } 3\}$ $B = \{x \mid x \text{ is a factor of } 30\}$

Sets

Solve $\frac{y}{4}=8$ for $y$.

Algebraic manipulation

The diagram depicts a sphere enclosed by a cube. The sphere is touching all 6 faces of the cube. The cube’s volume is $343\text{ cm}^3$. Calculate the volume of the sphere. (Volume of sphere $= \frac{4}{3}\pi r^3$)

Surface area and volume

$ABCD$ is a parallelogram whose sides are $AB$, $BC$, $CD$ and $DA$. The point $A$ is $(-3,7)$ and the point $B$ is $(2,5)$. $\vec{AD} = \begin{pmatrix}-1\\-6\end{pmatrix}$.

Vectors in two dimensions

$ABCD$ is a field. Its sides are $AB = 320\text{ m}$, $BC = 250\text{ m}$, $CD = 132\text{ m}$ and $AD = 365\text{ m}$. Angle $BCD = 90^\circ$.

Rates

$ABD$ forms a triangle. $C$ lies on $BD$, and $E$ lies on $AD$. $AB$ runs parallel to $EC$, and $CD = DE$.

Circle theorems II

This bag contains a mix of fruit.

Ratio and proportion

Apples cost $x$ cents per kilogram, and Mina pays $\$9$ altogether for apples.

Equations

The table gives the age and value of $10$ cars of the same model.

Scatter diagrams

The diagram represents a tank. It is a cuboid with length $1.2\,\text{m}$, width $0.6\,\text{m}$ and height $h\,\text{m}$. Its volume is $1.8\,\text{m}^3$.

Surface area and volume

Anya purchases $4$ shirts and $3$ hats. She hands over $\$100$ and gets $\$21.50$ back as change. Every shirt has the same price. Each hat is priced at $\$13.50$. Find the price of one shirt.

Exponential growth and decay

These are Mandeep’s $9$ number cards.

Probability of combined events

Finish the table for $y = \frac{x}{4}(2x^2 - x - 10)$.

Graphs of functions

Solve for $x$ in $4x + 7 = 16$.

Algebraic manipulation

A, D and B lie on the same straight line, and A, E and C lie on another straight line. BC is parallel to DE. BC = $9.8\,\text{cm}$, BD = $2.7\,\text{cm}$, DE = $5.6\,\text{cm}$ and CE = $3.9\,\text{cm}$.

Similarity

On a circle with centre $O$, A, B, C and D are points on the circumference. AOD and OCE are straight lines. DE is tangent to the circle at D. Given that $\angle ABC = 126^\circ$ and $DE = 10.5\,\text{cm}$, Calculate the radius of the circle.

Circles, arcs and sectors