Oct/Nov 2018

Evaluate $\frac{2}{7} + \frac{1}{5}$.

Fractions, decimals and percentages

The diagram represents a closed box. It is a cuboid. The measurements are given in centimetres.

Surface area and volume

Here, $N = 4 \times 10^{5}$.

Standard form

Obtain an estimate for the value of $\frac{614.2 \times 0.0304}{19.88}$ by first writing each number correct to $1$ significant figure.

Estimation

The table gives a summary of the lengths of Ellie’s telephone call times over one week.

Statistical charts and diagrams

Evaluate the expression $\left(\frac{1}{3}\right)^{0} - \left(\frac{1}{3}\right)^{2}$.

Powers and roots

Ashraf spun a five-sided spinner, labelled $1,2,3,4$ and $5$, $200$ times.

Relative and expected frequencies

A boat goes from $P$ to $Q$. On reaching $Q$, it turns through $90^{\circ}$ and continues to $R$ as the diagram shows. It then makes the return trip from $R$ to $Q$, and then back to $P$, retracing the same path in reverse. $PQ = 6\,\text{km}$ and $QR = 9\,\text{km}$. The outward part of the trip, from $P$ to $Q$ to $R$, lasts $3$ hours. The return leg, from $R$ to $Q$ to $P$, lasts $2$ hours.

Rates

Write $17\frac{1}{2}\%$ as a fraction in lowest terms.

Percentages

$y$ varies directly as the square of $x$.

Ratio and proportion

Over the course of one year, a child’s mass rose from $25\,\text{kg}$ to $30\,\text{kg}$.

Percentages

Factorise $15a + 3ab$ completely.

Algebraic manipulation

The function is defined by $f(x) = \frac{3}{x+4}$.

Functions

Each term in the sequence is $4$ greater than the one immediately before it. The third term is $17$, and the fourth term is $21$.

Sequences

State an irrational number that lies between $4$ and $5$.

Types of number

Within the diagram, $AFGB$, $BGDC$ and $FEDG$ are parallelograms. When extended, $EF$ and $BA$ meet at $H$. $\angle BGF = 130^{\circ}$ and $\angle DGF = 120^{\circ}$.

Angles

Evaluate the sum $\frac{1}{8} + \frac{2}{5}$.

Fractions, decimals and percentages

Use 2 significant figures for each number, then estimate the value of $\dfrac{1212.3}{299.35 \times \sqrt{24.73}}$.

Limits of accuracy

The first five terms of a sequence are given as $\frac{3}{4}, \frac{7}{8}, \frac{11}{12}, \frac{15}{16}, \frac{19}{20}$.

Sequences

Each interior angle of a regular polygon measures $175^\circ$. Find how many sides the polygon has.

Angles

The scale sketch shows two islands labelled $A$ and $B$. Scale: $2\text{ cm to }1\text{ km}$.

Scale drawings

The diagram represents a triangular prism. All dimensions are given in centimetres.

Surface area and volume

Each week, Henri notes the number of units of gas used in his house together with the mean temperature outdoors. Ten of his results are shown in the table.

Scatter diagrams

Find the values that $x$ can take, given that $x$ is an integer and $15 < 2x - 3 < 22$.

Inequalities

Bag A has 3 black and 2 white beads. Bag B has 2 black and 4 white beads. A bead is selected at random from Bag A and put into Bag B. Then a bead is selected at random from Bag B.

Probability of combined events

On the circle, $A, B, C, D$ and $E$ are points. $AB$ runs parallel to $ED$. $\angle ABE = 62^\circ$, $\angle CDE = 127^\circ$ and $\angle BEC = 40^\circ$.

Circle theorems I

In the diagram, $PQ$ crosses $LM$ at $O$.

Geometrical constructions

A spice costs 85 cents for 200 grams. Find the cost, in dollars, of 1 kilogram of this spice.

Ratio and proportion

$P$ has coordinates $(-3, 4)$, while $Q$ has coordinates $(5, 1)$.

Length and midpoint

Evaluate the expression $9^1 + 9^0$.

Indices II

$\mathcal{E} = \{0,1,2,3,4,5,6\}$, $P = \{x : x = 0,1,2\}$, and $Q = \{y : y = 0,2\}$.

Sets

Express $\begin{pmatrix}2\\1\end{pmatrix} - 3\begin{pmatrix}-1\\2\end{pmatrix} + 2\begin{pmatrix}0\\-2\end{pmatrix}$ in the form of a single vector.

Vectors in two dimensions

A train runs between two stations and is at rest both when it sets off and when it arrives. For the first $30$ seconds, it speeds up uniformly from rest until its speed is $20\text{ m s}^{-1}$; during the following $60$ seconds it continues at a steady $20\text{ m s}^{-1}$; over the final $20$ seconds it decelerates uniformly until it comes to rest.

Graphs in practical situations

The diagram shows that $ADB$ and $ACF$ are straight lines. $BC$ meets $DF$ at $E$. The ratio $AC : CF = 2 : 1$. Also, $\overrightarrow{DB} = \mathbf{p}$, $\overrightarrow{BE} = 3\mathbf{q}$, $\overrightarrow{EC} = 2\mathbf{q}$ and $\overrightarrow{AC} = 3\mathbf{p} + 5\mathbf{q}$.

Vector geometry

$y$ varies inversely with $x$.

Ratio and proportion

Put these fractions into order, beginning with the smallest: $\frac{4}{5}, \frac{9}{10}, \frac{17}{20}, \frac{21}{25}, \frac{41}{50}$. Show your working.

Ordering

Simplify the expression $4c - 3(2c - 5)$.

Algebraic manipulation

The function is given by $f(x) = \frac{2x + 5}{3x}$.

Functions

Out of a group of 35 people, 22 are wearing spectacles, 10 are wearing a hat, and 6 are wearing both spectacles and a hat.

Sets

The diagram contains part of a figure with rotational symmetry of order $4$ about point $O$. Complete the figure.

Symmetry

Here, $p = 8 \times 10^{-6}$ and $q = 2 \times 10^{11}$.

Standard form

The table lists the distances 10 people travel to work and the times they spend doing so.

Scatter diagrams

A prism is shown with a rectangular base measuring $15\text{ cm}$ by $x\text{ cm}$. Its cross-section is a right-angled triangle. The prism's height is $4\text{ cm}$ less than its width. The volume comes to $440\text{ cm}^3$.

Non-right-angled triangles

Express the expression as a single fraction in simplest form: $\frac{4}{2x-3} - \frac{3}{x-2}$.

Algebraic fractions

Dina runs a clothes shop.

Percentages

The diagram depicts a triangular field $ABC$. The lengths are $AB = 95\text{ m}$, $BC = 132\text{ m}$ and $AC = 174\text{ m}$.

Non-right-angled triangles

In this question, $\mathcal{E} = \{x : x \text{ is an integer } 1 \le x \le 10\}$, $A = \{x : x \text{ is a factor of } 20\}$, and $B = \{x : x \text{ is a multiple of } 4\}$.

Sets

Work with the function $y = \frac{x}{5}(6 + 2x - x^2)$.

Sketching curves

Points $A$, $B$, $C$ and $D$ lie on the circle with centre $O$. $TA$ and $TC$ are tangents to the circle. $T$, $D$, $O$ and $B$ are on one straight line. $\angle ATO = x^\circ$.

Circle theorems I

For point A, the position vector $\vec{OA}$ is $\begin{pmatrix}-4 \\ 7\end{pmatrix}$, and $\vec{AB} = \begin{pmatrix}6 \\ -3\end{pmatrix}$.

Vectors in two dimensions

The diagram shows part of a number grid. A vertical rectangle that surrounds three numbers can be placed anywhere on the grid. The grid extends further downward.

Introduction to algebra

The diagram depicts lamp $A$. It takes the form of a cylinder with a hemisphere placed on top. The radius of the hemisphere and the radius of the cylinder are both $3\text{ cm}$. The lamp’s total height is $24\text{ cm}$.

Surface area and volume

Kamal’s pay, savings and currency conversions.

Percentages

The diagram contains two circles that touch at $C$. $A$, $B$ and $C$ lie on the smaller circle, whose centre is $X$. $C$, $D$ and $E$ lie on the larger circle, whose centre is $Y$. $AXCYE$ and $BCD$ are straight lines, and $Y\hat{D}E = x^\circ$.

Similarity

Lim cultivates tomatoes. The masses, $m$ grams, of $200$ of her tomatoes are measured. The cumulative frequency table gives the data.

Cumulative frequency diagrams

Combine as a single fraction in simplest form $\dfrac{3}{y-1} - \dfrac{5}{y+6}$.

Algebraic fractions

Anna’s trips between home and work.

Rates

The circle is centred at $O$, and AC and BD are diameters. Here, $AC = 12\,\text{cm}$ and $\angle A\hat{O}B = 130^\circ$.

Circles, arcs and sectors

Zara encloses a plot of land beside a wall in order to create a vegetable garden. The garden is rectangular, with the wall forming one of its sides. The garden has an area of $18$ square metres. The width of the garden is $x$ metres.

Graphs of functions

On the grid, triangle $A$ has been plotted.

Transformations

A straight line for $2y = x + 4$ has been plotted on the grid.

Equations of linear graphs

The diagram depicts a pyramid with a rectangular base lying horizontally. Vertex $F$ of the pyramid is directly above the centre of the base, $E$. $AB = 6.2\,\text{cm}$ and $BC = 4.3\,\text{cm}$. Each sloping edge of the pyramid measures $9.5\,\text{cm}$.

Pythagoras' theorem and trigonometry in 3D