May/June 2015

Evaluate $1\frac{1}{3}+\frac{3}{8}$.

Fractions, decimals and percentages

$P=(3\ 4\ -5)$ and $Q=\begin{pmatrix}-1&1\\1&0\\0&-1\end{pmatrix}$.

Vectors in two dimensions

Put $\frac{1}{x}+2-\frac{3}{x+1}$ into a single fraction in simplest form.

Algebraic fractions

Since $6^x=9$.

Indices I

The figure shows triangle $ABC$ and triangle $XYZ$ are similar. The side lengths are $AB=3$ cm, $BC=8$ cm, $YZ=12$ cm and $ZX=15$ cm.

Similarity

A pie chart shows the counts of adults, girls and boys in a group of people. The angles for the adults and girls are $80^\circ$ and $120^\circ$ respectively.

Statistical charts and diagrams

$A$ lies at $(1,7)$ and $B$ lies at $(6,7)$. By the translation $\begin{pmatrix}0\\-5\end{pmatrix}$, the line $AB$ is carried onto the line $PQ$.

Transformations

Write this as one matrix expression: $\begin{pmatrix}-1&-3\\1&0\end{pmatrix}-\begin{pmatrix}1&-2\\2&-5\end{pmatrix}$.

Introduction to algebra

Factorise $3-12a^2$ completely.

Algebraic manipulation

The universal set is $\mathcal{E}=\{x:x\text{ is an integer},\ 40\le x\le50\}$, and $P=\{x:x\text{ is a prime number}\}$ while $Q=\{x:x\text{ is a multiple of }6\}$.

Sets

On the diagram, $A,B,C,D$ are on the circle with centre $O$. $AD$ is a diameter. The tangent at $B$ to the circle intersects the extension of line $DA$ at $T$. $\angle A\hat{O}B=68^\circ$ and $\angle C\hat{A}O=43^\circ$.

Circle theorems II

There are six small triangles shaded in the diagram. Shade one further small triangle so that the diagram will then have one line of symmetry.

Symmetry

The oil-rigs sit at the four vertices of rectangle $ABCD$. The bearing of $B$ from $A$ is $040^\circ$.

Angles

On the grid, shade region $R$, defined by the inequalities $1\le x\le5$, $2\le y\le4$.

Equations of linear graphs

The diagram illustrates the lines $AB$ and $BC$.

Geometrical constructions

Kim and Lee are running a 2000 metre cross-country route that begins at $P$ and finishes at $Q$. Lee sets off 1 minute after Kim. Their distance-time graphs are shown in the diagram.

Graphs in practical situations

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

Transformations

A bag holds 5 balls, with 3 red balls and 2 blue balls. A ball is chosen at random from the bag and is not put back. If that ball is red, a second ball is then chosen at random from the bag and is not put back. This continues until a blue ball is drawn from the bag.

Probability of combined events

The numerical pattern below is given: Row 1 $\frac{1}{1\times2}=\frac12$, Row 2 $\frac{1}{2\times3}=\frac12-\frac13$, Row 3 $\frac{1}{3\times4}=\frac13-\frac14$, Row 4 $\frac{1}{4\times5}=\frac14-\frac15$.

Sequences

Express $3\frac{3}{4}\%$ as a fraction in its simplest terms.

Percentages

To 6 decimal places, $\sqrt{10}=3.162278$, whereas $3\frac{1}{6}=3.166667$.

Limits of accuracy

Here, $p=4\times10^5,\ q=7\times10^6$.

Standard form

A car manufacturer reports that one model consumes 5 litres of fuel to travel 100 km, and emits 110 grams of CO$_2$ for each kilometre driven.

Rates

The time each person in a group needed to run one kilometre was recorded. The findings are presented in the table.

Statistical charts and diagrams

Find the integers $n$ which make $20<4n-3<30$ true.

Inequalities

$y$ varies inversely with the square of $x$.

Ratio and proportion

Evaluate $\dfrac{1.3 + 2.9}{0.2}$.

Fractions, decimals and percentages

The two triangles shown are congruent, and their side lengths are given in centimetres to the nearest $0.1$ cm.

Geometrical terms

The graph illustrates the line $y = 2x + 1$. Point $P$ has coordinates $(a, b)$, where $a$ and $b$ are both positive integers. The values of $a$ and $b$ obey the inequalities $a < 2$, $b < 7$ and $b > 2a + 1$.

Coordinates

Omar has a set of number cards. He chooses these five cards: $-2, -4, -2, 4, 1$.

Averages and measures of spread

Express $60$ as a product formed from its prime factors.

Types of number

Within triangle $ABC$, the lengths are $AB = 5$ cm and $AC = 6$ cm.

Geometrical constructions

The equation can be written as $c = \sqrt{8a - 3b}$.

Algebraic manipulation

Calculate $2^0 + 2^3$.

Indices II

The matrix $\begin{pmatrix}4 & 0 \\ 0 & 1\end{pmatrix}$ is the matrix for the transformation $T$.

Transformations

Factorise $p^2q - pq$ completely.

Algebraic manipulation

Luis works in an office. He gets $\$8$ per hour for normal time. For overtime, he is paid the normal rate plus an extra $50\%$. In one month, he works $140$ hours of normal time and $10$ hours of overtime. Work out how much he is paid for that month's work.

Percentages

Put these numbers into size order, beginning with the smallest: $\dfrac{13}{20},\; 0.7,\; \dfrac{7}{12},\; 0.64,\; \dfrac{5}{8}$.

Ordering

The finishing times for $200$ runners in a $5$ km race were recorded, and the data are shown in the cumulative frequency diagram.

Cumulative frequency diagrams

Express as a single matrix $3\begin{pmatrix}1 & 3 \\ -2 & 5\end{pmatrix} - \begin{pmatrix}4 & 0 \\ -1 & 2\end{pmatrix}$.

Algebraic manipulation

On the map, the scale is $1 : 25\,000$.

Scale drawings

Solve the inequality $-4 \leq 2x - 5 < 7$.

Inequalities

A hemisphere with radius $r$ cm is attached to a cone of radius $r$ cm and height $h$ cm to make the solid. The hemisphere's volume is one third of the volume of the whole solid.

Surface area and volume

From the diagram, $A$ lies midway along $OC$, and $B$ is the point on $OD$ for which $OB = \dfrac{1}{3}OD$. Also, $\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OB} = \mathbf{b}$.

Vector geometry

The first four terms in the sequence $S$ are $89, 83, 77, 71$.

Sequences

The diagram depicts a trapezium with side lengths measured in centimetres. Its area is $120\text{ cm}^2$.

Area and perimeter

A bag contains red counters, blue counters and yellow counters. Altogether, there are 60 counters in the bag. The probability that a counter chosen at random from the bag is red is $\dfrac{2}{5}$. The probability that a counter chosen at random from the bag is blue is $\dfrac{5}{12}$.

Introduction to probability

Fariza is travelling from London to Astana. Astana is 5 hours ahead of London, so if it is 1000 in London, the local time in Astana is 1500. She goes by plane from London to Moscow, and then from Moscow to Astana. The plane departs London at 1225 and needs 4 hours to get to Moscow. Fariza then stays in Moscow for $4\dfrac{1}{2}$ hours before the next flight to Astana. She reaches Astana at 0525 local time.

Time

Using numbers rounded to one significant figure, estimate the value of $\dfrac{29.3^2}{2.04 \times 0.874}$.

Estimation

$y$ varies inversely with the square of $x$.

Ratio and proportion

The Venn diagram displays the sets $A$, $B$ and $C$.

Sets

Write $0.00000521$ in standard form.

Standard form

Last year, a furniture salesman received $36200$.

Percentages

A farmer picked $300$ eggs from his chickens in one day. The table gives the mass distribution of the eggs.

Cumulative frequency diagrams

$ABCDE$ is a pentagon. $AFB$, $AHE$ and $BGC$ lie on straight lines. $F$ lies halfway along $AB$. $H$ lies halfway along $AE$. $G$ divides $BC$ in the ratio $1:2$. $\overrightarrow{AH}=a$, $\overrightarrow{AF}=a-b$, $\overrightarrow{BG}=\overrightarrow{ED}=c$.

Vector geometry

Q has coordinates $(-1,2)$, R has coordinates $(3,10)$, and S has coordinates $(-4,2)$.

Length and midpoint

In trapezium $ABCD$, $AB$ runs parallel to $DC$. $DB$ and $AC$ are straight lines.

Area and perimeter

A solid is made by attaching a cone of radius $4.5\text{ cm}$ and height $7.6\text{ cm}$ to a hemisphere of radius $4.5\text{ cm}$.

Surface area and volume

In the figure labelled $ABCDEF$, $BCD$ lies on one straight line, and $CA$ is parallel to $DF$. The angles $\angle ABD$, $\angle BDE$ and $\angle DEF$ are all right angles. Also, $AB = 4\text{ m}$, $DE = 11\text{ m}$ and $EF = 4\text{ m}$.

Right-angled triangles

Expand the brackets and simplify $(x-1)(x^2 + x + 1)$.

Equations

For triangle $ABC$, $BC = 16\text{ cm}$ and $\angle BAC = 60^\circ$. The length of $AB$ is $x\text{ cm}$, and $AC$ measures $2x + 3\text{ cm}$.

Non-right-angled triangles

The diagram depicts a sector $AOB$ of a circle, centred at $O$, with radius $9.3\text{ cm}$. The sector angle is $260^\circ$.

Circles, arcs and sectors

The function is $f(x) = x^3$.

Graphs of functions

Simplify the expression $\frac{4x-1}{3}+\frac{3x+5}{2}$.

Equations of linear graphs

Describe fully the one transformation that takes triangle $A$ to triangle $B$.

Transformations

A group of people were asked which continent they had visited on their most recent holiday. The results are shown in the table below.

Probability of combined events

The diagram shows a train’s trip between two stations on a speed-time graph.

Graphs in practical situations

In a survey, $50$ students were asked how much time they spent exercising in a single week. The outcomes are set out in the table.

Averages and measures of spread

A, B, C, D and E are five points lying on the circumference of a circle. $EB$ is parallel to $DC$. $\angle E\hat{A}C = 72^\circ$ and $\angle A\hat{E}B = 25^\circ$. $X$ is the point where $AC$ and $EB$ intersect.

Circle theorems I

The diagram shows $AB=8\text{ cm}$, $AC=11\text{ cm}$ and $DC=6.5\text{ cm}$. Also, $\angle B\hat{A}D=26^\circ$ and $\angle D\hat{A}C=30^\circ$.

Non-right-angled triangles

Yuvraj changes $20500$ rupees and gets $£250$. Sachin changes $26650$ rupees into pounds at the same exchange rate. How many pounds does Sachin receive?

Rates

Part B

Graphs of functions

Ports $A$ and $B$ are $15\text{ km}$ apart, with $B$ lying directly south of $A$. A boat sets off from $A$ on a bearing of $141^\circ$.

Right-angled triangles

Factorise $4x^3-10xy$ completely.

Algebraic manipulation