Oct/Nov 2021

Calculate $\frac{7}{8} - \frac{1}{4}$.

Fractions, decimals and percentages

One angle in a triangle measures $55^\circ$. The remaining two angles are in the ratio $3:2$.

Angles

Estimate the value of $\dfrac{58.24}{32.5 \times 0.126}$ by first rounding every number to $1$ significant figure.

Limits of accuracy

Find the simultaneous equations and show every step of your working.

Equations

The diagram depicts triangle $A$ alongside triangle $B$.

Transformations

Express $60$ in prime-factor form.

Types of number

The position vector of point $A$ is $\begin{pmatrix}3 \\ -7\end{pmatrix}$, and $\overrightarrow{AB}=\begin{pmatrix}-5 \\ 12\end{pmatrix}$.

Vectors in two dimensions

A spinner with $4$ sides, labelled from $1$ to $4$, is spun on many occasions. The table records the outcomes of the spins.

Relative and expected frequencies

Factorise $4b^2 - c^2$.

Algebraic manipulation

Region $R$ is specified by the inequalities $1 \le x \le 5$, $0 \le y \le 4$, and $y \ge 3 - x$.

Drawing linear graphs

Given $\mathcal{E} = \{1,2,3,4,5,6,7,8,9,10,11,12\}$, $X = \{2,3,5,7,11\}$ and $Y = \{1,2,3,4,5,6\}$.

Sets

Arrange these numbers from smallest to largest. $\frac{3}{4}$, $0.83$, $\frac{17}{20}$, $82\%$, $0.8$

Ordering

The sequence begins with the five terms $4, 8, 16, 32, 64$.

Sequences

The function is $f(x) = \dfrac{6}{2 - x}$.

Functions

The diagram shows that $ACD$ and $BCE$ are straight lines, with $AB$ parallel to $ED$.

Similarity

Expand and simplify the expression $(x + 5)(x - 2)$.

Algebraic fractions

Using $A = \begin{pmatrix}-6 & 2 \\ 1 & 4\end{pmatrix}$, find $A^2$.

Algebraic manipulation

Calculate $45\%$ of $30$.

Percentages

The diagrams show shaded squares.

Symmetry

Simplify the expression $3a - a + 2a$.

Algebraic manipulation

$ABC$ forms a triangle, with $AC = 5\,\text{cm}$ and $BC = 7\,\text{cm}$.

Geometrical constructions

Evaluate the value of $4^0$.

Indices I

Write the value of $6300\,\text{m}$ in kilometres.

Units of measure

A regular polygon has an interior angle of $156^\circ$.

Angles

Evaluate the value of $\sqrt{4900}$.

Powers and roots

The diagram displays triangle $A$ and triangle $B$.

Transformations

The scale diagram depicts a garden with two trees $P$ and $Q$. The scale is $1$ centimetre represents $6$ metres.

Geometrical constructions

In the diagram, rectangle $ABCD$ is shown. Point $E$ lies on the diagonal $AC$ so that $\angle DEC = 90^\circ$.

Similarity

The average of five numbers is $17$. Written from smallest to largest, the three least numbers are the same. The three middle numbers add up to $35$. The greatest number is four times the smallest number.

Averages and measures of spread

The diagram presents the cyclist’s speed-time graph for the opening part of the journey.

Graphs in practical situations

Over one year, the bicycle’s value fell from $\$200$ to $\$160$. Calculate the percentage decrease in the bicycle’s value.

Percentages

Find the solution to the inequality $23 + 2n > 5 - 6n$.

Inequalities

Factorise the expression $3xy - qy + 6px - 2pq$.

Algebraic manipulation

The diagram displays a shaded region $ABC$. The equation for line $AC$ is $y = -\frac{1}{2}x + 5$.

Equations of linear graphs

The points $A$, $B$ and $C$ are on a circle with centre $O$. The straight line $PBQ$ is tangent to the circle at $B$, and $O$, $C$ and $Q$ lie on one straight line. $\angle BQO = 36^\circ$ and $\angle BAC = x^\circ$.

Circle theorems I

Calculate $-8 + 7 \times (-5)$.

The four operations

Find the inverse of $\begin{pmatrix} 3 & -2 \\ 1 & 2 \end{pmatrix}$.

Algebraic manipulation

The cumulative frequency diagram gives the masses, $m$ grams, of $120$ eggs.

Cumulative frequency diagrams

Solve for $k$ in $27^k = 9$.

Indices I

$y$ varies inversely with $(x+1)^2$. If $x = 1$, then $y = 5$.

Ratio and proportion

$f(x) = 2x^2 + 7x + 4$ and $g(x) = 2x + 6$.

Functions

Forty students are available for three activities: Art $(A)$, Dancing $(D)$ and Gardening $(G)$. Five students do not join any of the activities. Twelve take Art only. Four take both Dancing and Gardening, but not Art. One student takes part in all three activities.

Sets

Shade one extra small triangle so that the shape has rotational symmetry of order $3$.

Symmetry

Write down the name of the solid that each net makes.

Surface area and volume

In the diagram, $ABCD$ and $EFGH$ are parallel lines. The lines $CF$ and $BG$ cross at $X$. $\angle CFG = 53^\circ$, $\angle BGF = 46^\circ$ and $\angle BXC = 81^\circ$.

Angles

Calculate $69 \div 0.3$.

Fractions, decimals and percentages

Using each number rounded to $1$ significant figure, estimate the value of $\dfrac{8230 \times 0.64}{18.7}$.

Estimation

Express $0.06\text{ km}$ in metres.

Units of measure

Write $216$ in prime-factor form.

Types of number

In October, Sara pays $84.25$ for water. An added tax of $8\%$ is charged on this amount. Calculate the total amount Sara pays for water in October including tax.

Percentages

A cuboid has dimensions $6.2$ cm by $4.8$ cm by $2.5$ cm. Each dimension is stated correct to the nearest millimetre. Calculate the upper bound of the cuboid's surface area.

Limits of accuracy

Determine the mode.

Interpreting statistical data

Explain why triangle $AOB$ has two equal sides and is isosceles.

Circle theorems I

Calculate angle $PRQ$.

Angles

Find the probability that Khalid's first card has an even number on it.

Probability of combined events

Show that the equation reduces to $5x^2 + 30x - 39 = 0$.

Equations

Draw the graph of $y = 2^x$ on the grid below for $0 \le x \le 4$.

Sketching curves

$A$ has coordinates $(-2, 3)$ and $B$ has coordinates $(4, 5)$.

Perpendicular lines

Solve the equation $3x-8=7$.

Algebraic fractions

Jasmine purchases a family holiday in India. The information below gives the cost.

Percentages

The pie chart gives a summary of the ages of the people who attended a science fair.

Statistical charts and diagrams

Use $y = \frac{x^3}{2} - 3x - 1$ to complete the table.

Sketching curves

The first three patterns in a counter-based sequence are shown here.

Sequences

A mass of $4$ cards together with $3$ envelopes is $85\,g$. A mass of $2$ cards together with $5$ envelopes is $67\,g$. Set up a pair of simultaneous equations and solve them to determine the mass of one card and the mass of one envelope.

Equations