Oct/Nov 2019

Evaluate the expression $3 \times 1\frac{4}{7}$.

Fractions, decimals and percentages

Amir purchases a camera for $250 and later sells it for $200. Calculate his percentage loss.

Percentages

Simplify the expression $7 - 3(5x - 2)$.

Equations

Calculate $3^{-2} \times 3^4$.

Indices II

Express the number $0.00023$ in standard form.

Standard form

Here, $p = 2^3 \times 3 \times 5^2$ while $q = 2 \times 3^2 \times 5$.

Types of number

In the diagram, three small triangles are already shaded. Shade one additional small triangle so that the diagram has exactly one line of symmetry.

Symmetry

In the Venn diagram, shade the area that represents $C \cap (A \cup B)'$.

Sets

The diagram displays the lines $x + y = 8$, $y = \frac{1}{2}x$, $x = 0$ and $y = 0$. The spaces enclosed by the lines are marked using letters.

Inequalities

Measurements were taken for 120 cereal packets, and the outcomes are shown in the cumulative frequency diagram.

Cumulative frequency diagrams

The function is defined by $f(x) = \dfrac{5 - x}{x}$.

Functions

The scatter diagram displays the marks obtained by 12 students in test A and test B.

Scatter diagrams

The table presents the outcomes from 300 throws of a dice. The relative frequency for a 4 is $0.2$.

Relative and expected frequencies

In the diagram, the points $A,B,C,D$ and $E$ are on the circle with centre $O$. The points $B,O$ and $E$ are collinear. $AB$ is parallel to $ED$ and $\angle DEO = 53^\circ$.

Circle theorems I

The diagram depicts triangle $ABC$.

Geometrical constructions

This diagram shows the train’s journey on a speed-time graph for part of the trip.

Graphs in practical situations

Here $A = \begin{pmatrix}3 & 1 \\ -2 & 0\end{pmatrix}$ while $B = \begin{pmatrix}-2 & 1 \\ 3 & 0\end{pmatrix}$.

Algebraic manipulation

In the sketch, $B$ lies at the midpoint of $OD$, and the ratio $OA : AC = 1 : 3$. Also, $\overrightarrow{OA} = \vec{a}$ and $\overrightarrow{OB} = \vec{b}$.

Vector geometry

This diagram presents the net of a solid.

Surface area and volume

Factorise $1 - 36p^2$

Algebraic manipulation

A television programme lasts for 2 hours 40 minutes.

Time

Write these values in ascending order, beginning with the smallest: $\frac{1}{30},\;0.03,\;\frac{1}{10},\;5\%,\;\frac{2}{25}$.

Ordering

$y$ is directly proportional to $x$. For $x = 4$, $y = t$.

Ratio and proportion

Use each number correct to 1 significant figure to estimate the value of $\dfrac{59.843^2}{20.13 \times 0.9024}$.

Estimation

Find the simultaneous equations $x + 4y = 1$ and $3x + 2y = 8$.

Equations

Evaluate the expression $1\frac{1}{8} - \frac{1}{4}$.

Fractions, decimals and percentages

$y$ is inversely proportional to $x$. When $x = 2$, $y = t$. Find a formula for $y$, in terms of $t$, when $x = 3$.

Ratio and proportion

Express the value of $4500 \times 1000^2$ in standard form.

Standard form

A polygon with $12$ sides is given. Calculate the sum of its interior angles.

Angles

Write in simpler form.

Indices I

The temperatures, in $^\circ\text{C}$, of $120$ people were recorded. The findings are shown on the cumulative frequency diagram.

Cumulative frequency diagrams

The length of time each student in a group needed to solve a problem was noted down. A summary of some of the results appears in the table and is shown in the histogram.

Statistical charts and diagrams

In the diagram, $AP$, $BQ$ and $CR$ bisect the angles of triangle $ABC$. These bisectors meet at $O$. $\angle OBP = 31^\circ$, $\angle OCQ = 22^\circ$ and $\angle OAQ = 37^\circ$.

Angles

There are exactly two white beads and one black bead in the bag. Two beads are then chosen at random without replacing them.

Introduction to probability

The triangle has vertices at $A(7,0)$, $B(-1,6)$ and $C(-1,-4)$.

Length and midpoint

Point $C$ lies above $AB$, with $AC = 5\text{ cm}$ and $BC = 7\text{ cm}$. Use a pair of compasses and a ruler only to construct triangle $ABC$.

Geometrical constructions

Write these values in order, starting from the smallest. $\frac{7}{200},\ 4\%,\ \frac{3}{50},\ 0.03,\ \frac{1}{20}$

Ordering

On the circle with centre $O$, the points $A$, $B$, $C$ and $D$ are all on the circumference. Also, $\angle ACB = 69^\circ$ and $\angle DCA = 34^\circ$.

Circle theorems I

The diagram displays the points $O$ and $C$, along with the vectors $\mathbf{p}$ and $\mathbf{q}$.

Vectors in two dimensions

A tap lets water fall drop by drop into a container resting on a horizontal surface. The container is a cuboid whose base measures $5\text{ cm}$ by $4\text{ cm}$. The volume of one drop of water is $0.08\text{ cm}^3$.

Surface area and volume

From the diagram, $\vec{OB} = \begin{pmatrix}12\\6\end{pmatrix}$.

Vectors in two dimensions

Express $3\begin{pmatrix}1 & -1 \\ 0 & 2\end{pmatrix} - 2\begin{pmatrix}-1 & 1 \\ 2 & 1\end{pmatrix}$ in a single matrix.

Algebraic manipulation

The diagram shows the train’s motion on a speed-time graph for part of its journey.

Graphs in practical situations

Here is the set of numbers: $\sqrt{35},\ \sqrt{36},\ \frac{36}{37},\ 37,\ \frac{37}{36},\ 3.7$

Types of number

Solve for $x$ in $6 + 8x = 7 - 2x$.

Equations

Factorise this expression.

Algebraic manipulation

A movie runs for 1 hour 48 minutes and ends at 10.15 pm. Determine its starting time.

Limits of accuracy

The diagram contains a circle, its centre and two chords of equal length. There is only one line of symmetry in the diagram. Draw that line of symmetry.

Symmetry

Use each number rounded to $1$ significant figure to estimate $\dfrac{39.864 \times \sqrt{8.987}}{0.6013}$.

Estimation

A map uses a scale of $5\text{ cm}$ for every $2\text{ km}$.

Ratio and proportion

Neema is going on a business trip.

Limits of accuracy

On the grid, the points $P$ and $Q$ together with line $L$ are displayed.

Perpendicular lines

During one afternoon, the library had 200 visitors. The table below shows how long, in minutes, each visitor stayed in the library.

Statistical charts and diagrams

On the grid, shape $A$ and shape $B$ are shown.

Transformations

[Volume of cone $= \frac{1}{3}\pi r^2 h$] [Curved surface area of a cone $= \pi r l$] The sketch shows a gate post. It is formed as a cylinder with a cone placed on top. The cylinder and the cone each have diameter $8\,\text{cm}$. The cylinder has height $95\,\text{cm}$ and the cone has height $15\,\text{cm}$.

Surface area and volume

$x^2 + 7x - 13 = (x+a)^2 + b$ Work out the value of $a$ and the value of $b$.

Algebraic manipulation

The first four patterns in a counters sequence are displayed.

Sequences

$P$, $Q$, $R$ and $S$ lie on a circle. $PXR$ and $QXS$ are straight lines.

Circle theorems I

The rectangle $ABCD$ has an area of $80\,\text{cm}^2$. Triangle $PCQ$ is taken away from one corner of the rectangle. $BQ = DP = 4\,\text{cm}$. $AB = x\,\text{cm}$.

Graphs of functions

The diagram indicates the locations of three farms $A$, $B$ and $C$ on level ground. Farm $B$ lies on a bearing of $125^{\circ}$ from farm $A$. $AB = 950\,\text{m}$, $BC = 680\,\text{m}$ and $AC = 520\,\text{m}$.

Non-right-angled triangles

Tanya is the owner of a small business.

Percentages

The circle has centre $O$, and $AC$ and $BD$ are its diameters.

Circle theorems I

On the grid, shapes $A$, $B$ and $C$ are plotted.

Transformations

The table gives the running times, in seconds, for 12 athletics club members over 400 metres and 800 metres.

Scatter diagrams

Fill in the table for $y = 3 + 2x - \frac{x^3}{5}$.

Sketching curves

The universal set is $\mathcal{E} = \{x : x \text{ is an integer } 1 \le x \le 16\}$ Let $A = \{x : x \text{ is an even number}\}$ Let $B = \{x : x \text{ is a square number}\}$ Let $C = \{x : x \text{ is a factor of } 100\}$

Sets

Rearrange $v = \frac{3}{p+5}$ so that $p$ is the subject.

Algebraic manipulation

A quadrilateral field is labelled $PQRS$. A path runs across the field from $P$ to $R$. $PQ = 280$m, $RS = 146$m and $PR = 325$m. $S$ lies on a bearing of $042^\circ$ from $P$, $\angle PSR = 108^\circ$ and $\angle RPQ = 38^\circ$.

Non-right-angled triangles

The diagram presents a bowl with a circular base. Its curved outer face is produced by cutting away a cone of radius $12$ cm and height $45$ cm from a larger cone, as the diagram indicates. The radius across the top of the bowl is $16$ cm and its height is $15$ cm.

Surface area and volume

$ABCD$ forms a rectangle with $AB = 10$ cm and $AD = 12$ cm. $PQ$ runs parallel to $AB$ and $RS$ runs parallel to $AD$. The intersection of $PQ$ and $RS$ is $Y$. $BR = DP = x$ cm. A smaller rectangle and a square are shaded.

Area and perimeter