May/June 2018

Evaluate the expression $\frac{3.5 - 1.9}{0.8}$.

Fractions, decimals and percentages

The scale sketch represents a barn $ABCD$. Here, $AB = 7\text{ m}$ and $BC = 4\text{ m}$. The scale used is $1\text{ cm}$ to represent $1\text{ m}$. A horizontal rail is fixed to the outside wall of the barn from $A$ to $B$. Jasper is a dog on a $3\text{ m}$ rope. One end of the rope is fastened to the rail, and that end may move along the rail.

Geometrical constructions

$\begin{pmatrix}2 & a\\ -3 & 1\end{pmatrix}\begin{pmatrix}-4 & b\\ 3 & 2\end{pmatrix} = \begin{pmatrix}7 & 10\\ 15 & 2\end{pmatrix}$!

Algebraic manipulation

Basia notes the colours of $100$ cars that go past the school gate. The outcomes are listed in the table.

Relative and expected frequencies

The table gives the birth masses of $10$ mothers and their babies.

Scatter diagrams

Factorise completely the expression $2ax - 3by + 6bx - ay$.

Algebraic manipulation

Let $f(x) = 3 - 2x$ and $g(x) = 4x^3 - 1$.

Functions

Evaluate $3^3 - 3^0$ to find the result.

Powers and roots

The diagram illustrates a square $ABCD$ attached to a right-angled triangle $BEC$. $BE = 6\text{ cm}$ and $EC = 7\text{ cm}$.

Compound shapes and parts of shapes

The graph of speed against time shows how a car moves.

Graphs in practical situations

On one day in $2016$, Nepal’s population was $28\,795\,701$. Write this number correct to three significant figures.

Standard form

Work out $15\%$ of $80$.

Fractions, decimals and percentages

A paving slab is a cuboid whose length is $40\text{ cm}$, width is $20\text{ cm}$ and depth is $h\text{ cm}$. The volume of the slab is $2400\text{ cm}^3$.

Surface area and volume

Vector $p$ is $\begin{pmatrix}3\\4\end{pmatrix}$, while vector $q$ is $\begin{pmatrix}-4\\3\end{pmatrix}$.

Vectors in two dimensions

The relation is $s = \sqrt[3]{t + 4}$.

Algebraic manipulation

From the figures shown, state which one could represent the graph of the equations given.

Graphs of functions

Demonstrate that $\frac{12}{x + 2} + \frac{10}{x - 1} = \frac{7}{2}$ can be reduced to the equation $7x^2 - 37x - 30 = 0$.

Equations

List these numbers by increasing size, with the least first: $\frac{1}{3}$, $0.32$, $\frac{15}{40}$, $0.3$, $\frac{9}{31}$.

Ordering

The figure shows one half of a shape that is symmetrical about the line $L$. Complete the figure.

Symmetry

The plane departs from London bound for Dubai.

Time

What fraction of the $4 \times 4$ square is shaded?

Fractions, decimals and percentages

Usama noted how many items each of $20$ customers bought in a shop. The data are shown in the table.

Averages and measures of spread

$A$, $B$ and $C$ lie on the circumference of a circle with centre $O$. $O$ is the midpoint of $BC$ and $\angle ABC = 38^{\circ}$. From $T$, tangents are drawn so that they touch the circle at $A$ and $B$.

Circle theorems I

Find which integers satisfy $1 < 3x + 5 \leq 11$.

Inequalities

Evaluate $\frac{4}{11}-\frac{2}{7}$.

Fractions, decimals and percentages

By rewriting each number correct to $2$ significant figures, calculate an estimate of $\frac{596\times\sqrt{16.12}}{0.2984}$.

Limits of accuracy

Let $f(x)=\frac{1}{3x+2}$.

Functions

The die was tossed $400$ times. The outcomes are listed in a table showing the number thrown $1,2,3,4,5,6$ and the corresponding frequencies $65,80,70,75,50,60$.

Relative and expected frequencies

In a school with $270$ children, the distance each child could swim was recorded. These distances are summarised in a table.

Histograms

A $22$-sided irregular polygon is given. Calculate the combined measure of all its interior angles.

Angles

Across a two-week span, a shopkeeper notes how many packets of two distinct kinds of tea he sells and the profit earned from them. Matrices may be used to represent this information.

Algebraic manipulation

The masses of $200$ beetles were recorded. The findings are shown in a cumulative frequency table, and a section of the cumulative frequency curve has been drawn.

Cumulative frequency diagrams

The diagram represents a pyramid. The square base, $ABCD$, has an edge length of $3$ cm. The base lies horizontally, and vertex $E$ is directly above $D$, where $ED=4$ cm.

Surface area and volume

The inequalities $2\le x\le 8$, $5\le y\le 10$, $x+y\ge10$ define region $R$. On the diagram, shade and label region $R$.

Graphs in practical situations

On the diagram, construct the perpendicular bisector for $AB$.

Geometrical constructions

Cecil purchased a camera for $120$ and, after two years, sold it for $90$. Calculate the percentage loss.

Ratio and proportion

Thus, $N=2\times10^8$.

Standard form

The first four terms in the sequence are $u_1,u_2,u_3,u_4$.

Sequences

The diagram depicts a circle, with centre $O$, passing through $A,B,C$ and $D$. The tangents drawn at $A$ and $B$ intersect at $T$. $\angle ATB=62^\circ$ and $\angle DAB=53^\circ$.

Circle theorems I

$A=\begin{pmatrix}4&-1\\2&0\end{pmatrix}$ and $B=\begin{pmatrix}6&-3\\0&-2\end{pmatrix}$.

Equations

Triangle $A$ is first translated, and then an enlargement with centre $(10,-4)$ maps it to triangle $B$. The translation sends triangle $A$ to triangle $C$, while the enlargement sends triangle $C$ to triangle $B$.

Transformations

This diagram shows the speed-time graph for $60$ seconds of a train’s trip. At the start of this section, the train is moving at $u\ \text{m s}^{-1}$.

Graphs in practical situations

The five values listed are $0.05, -0.3, 1.3, -1.2, 0.2$.

Averages and measures of spread

$y$ is inversely proportional to the square of $x$. Since $y=10$ when $x=3$, determine $y$ when $x=\frac{1}{2}$.

Ratio and proportion

Factorise the expression $25t^2-4$.

Algebraic manipulation

A rectangle has a length of $64\text{ mm}$ and a width of $37\text{ mm}$, with both measurements given correct to the nearest millimetre. Write down the lower bound for the length.

Limits of accuracy

Triangle $OPQ$ is part of a figure with rotational symmetry of order $2$ about point $O$. Complete the figure.

Symmetry

Solve $\frac{4}{x-11}=\frac{1}{3x}$.

Algebraic fractions

Write $rac{2}{3a}+rac{5}{2a}$ as one fraction in lowest terms.

Algebraic fractions

Use set notation to state which part of the Venn diagram is shaded.

Sets

A boat departs from $A$ and covers $12\text{ km}$ to reach $B$.

Right-angled triangles

On the grid, the line $4y=x+2$ is shown.

Length and midpoint

Sami puts $2000 into an account that earns compound interest at $1.8\%$ per year. Calculate the total interest paid to Sami after $3$ years.

Exponential growth and decay

Solve the equation $4(p-3) = 2p + 7$.

Equations

Inside a bag there are twelve lettered tiles that spell TRIGONOMETRY.

Probability of combined events

Find an expression, in terms of $n$, for the $n$th term in this sequence $1,\,7,\,13,\,19,\,25,\ldots$

Sequences

$ABC$ is a triangle with $AC=6\text{ cm}$ and $BC=9\text{ cm}$. $AB$ is shown below. Using only a ruler and a pair of compasses, construct triangle $ABC$.

Geometrical constructions

The diagram represents the net of an open box with height $3\text{ cm}$. The base of the box has area $15\text{ cm}^2$. The rectangular base has length $x\text{ cm}$. The net’s total area is $A\text{ cm}^2$.

Equations

The grid contains triangles $A$ and $B$ together with rectangle $R$.

Transformations

The figure presents a sector of a circle with radius $8\text{ cm}$ and central angle $70^\circ$.

Circles, arcs and sectors

Leah works for 5 days each week and gets a total of $682$. On each day, she works from 0845 to 1215 and then again from 1315 to 1730. Calculate Leah’s hourly rate of pay.

Percentages

$A$ has coordinates $(-4, -1)$, $B$ is located at $(2, 2)$ and $\overrightarrow{BC} = \begin{pmatrix}4 \\ -8\end{pmatrix}$.

Length and midpoint

Jenny collected the time, in minutes, for 40 movies. Her results are shown in the table. On the grid, construct a frequency polygon to display this information.

Statistical charts and diagrams

Solve the equation $3(x + 10) = 12 - 7x$.

Equations

$\mathcal{E} = \{x : x \text{ is an integer } 1 \le x \le 18\}$, $A = \{x : x \text{ is a prime number}\}$, and $B = \{1, 2, 3, 4, 6, 9, 12, 18\}$.

Sets

The diagram depicts sector $OAB$ of a circle with centre $O$ and radius $11\text{ cm}$, and the angle $\angle AOB$ is $134^\circ$.

Circles, arcs and sectors

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

Sketching curves

A yacht travels around the triangular path shown. The bearing of $B$ from $A$ is $135^\circ$. $BC = 3.7\text{ km}$, $AC = 2.8\text{ km}$ and $\angle ABC = 42^\circ$.

Non-right-angled triangles

The diagram shows triangle $OYC$. Point $A$ lies on $OY$ and point $B$ lies on $CY$. $AB$ is parallel to $OC$. The lines $AC$ and $OB$ intersect at $X$.

Vector geometry

On Monday, Ravi goes for a $20\text{ km}$ run. His mean speed over the first $12\text{ km}$ is $x\text{ km/h}$. Write down an expression, in terms of $x$, for the time taken to cover the first $12\text{ km}$. Give your answer in minutes.

Rates