May/June 2016

Evaluate the expression $12 - 6 \div 3 + 4$.

The four operations

Write $3xy - 20 + 5x - 12y$ in fully factorised form.

Algebraic manipulation

This function is given by $f(x) = 2x - 9$.

Functions

The map uses a scale of $2\text{ cm}$ representing $5\text{ km}$.

Scale drawings

Solve the pair of simultaneous equations $3x = 4y$ and $1 + 5x = 6y$.

Equations

For a sphere, the volume is $\frac{4}{3}\pi r^3$. For a cone, the volume is $\frac{1}{3}\pi r^2 h$. A cone is taken away from a solid wooden hemisphere with radius $3\text{ cm}$. The cone also has radius $3\text{ cm}$ and height $2\text{ cm}$. The volume of the wood left is $k\pi \text{ cm}^3$.

Surface area and volume

$y$ is directly proportional to $x^2$. When $y = 8$ and $x = 4$, determine $y$ for $x = 3$.

Ratio and proportion

From the diagram, $AB$, $BC$, $CD$ and $DE$ form four consecutive sides of a regular polygon. The polygon has each interior angle equal to $160^\circ$. $ABPQR$, $DCP$ and $EDQ$ are straight lines.

Angles

A chain of diagrams is built with counters.

Sequences

Henri conducted a survey on the lengths of the leaves of a plant. The results are shown in the table.

Statistical charts and diagrams

In the diagram, the two circles intersect at $P$ and $Q$. The line $AB$ touches both circles at $A$ and $B$. $AD$ and $BC$ are diameters. $BD$ meets the larger circle at $R$. $\angle DBC = 40^\circ$.

Circle theorems I

Evaluate the expression $\frac{2}{3} - \frac{5}{8}$.

Fractions, decimals and percentages

The number of goals scored in each of $50$ football matches was recorded, and the outcomes are shown in the table.

Averages and measures of spread

Write $500$ as the product of its prime factors.

Indices II

The diagram depicts triangle $ABC$.

Geometrical constructions

On the Venn diagram, shade the region that shows the subset $(P \cup Q)' \cap R$.

Sets

The diagram presents the speed-time graph of a train that slows from $20\text{ m/s}$ to rest in $T$ seconds.

Graphs in practical situations

The diagram shows $X$ on $AB$ with $AX = \frac{1}{4}AB$, $Y$ on $AC$ with $AY = \frac{1}{3}AC$, and $Z$ on the extension of $BC$ with $CZ = 2BC$. Also, $\overrightarrow{AY} = p$ and $\overrightarrow{AX} = q$.

Vector geometry

Box $1$ has $2$ white balls, while Box $2$ contains $4$ white balls and $3$ black balls.

Probability of combined events

An aircraft departs at $22\,35$ for a flight lasting $3$ hours and $50$ minutes. Find the time of arrival.

Limits of accuracy

A bottle filled with liquid has a combined mass of $1.27\text{ kg}$. If the bottle is only half-full of liquid, the combined mass is $900$ grams. Calculate the mass of the bottle.

Equations

Stella is heading to a park. She spends $4$ minutes walking at $80$ steps per minute, and then $1$ minute walking at $120$ steps per minute. Find the mean number of steps per minute she takes.

Averages and measures of spread

Write $0.034 \times 10^{-3}$ in standard form.

Standard form

Use rounding to $1$ significant figure to estimate $\frac{29.2 \times 8.17}{0.396}$.

Estimation

Complete the diagram so that it forms a quadrilateral $ABCD$ with $AC$ as the line of symmetry.

Symmetry

The shaded part of the diagram is described by three inequalities. One of them is $y > \frac{1}{2}x + 2$.

Inequalities

Evaluate the value of $(2.05 + 1.4) \times 0.2$.

Fractions, decimals and percentages

Solve the pair of simultaneous equations $6x + y = 1$ and $4x - y = 4$.

Equations

Simplify the expression $\frac{5x^{7}y}{15x^{3}y^{4}}$.

Indices II

The diagram shows the triangle labelled $ABC$.

Geometrical constructions

Write these values in ascending order, beginning with the smallest: $2^{5},\;5^{2},\;\sqrt[3]{1000},\;27^{0}$.

Powers and roots

The midpoint of line $AB$ is $(1, 2)$, and the length of line $AB$ is $10$ units.

Length and midpoint

The diagram displays the lines $x + y = 8$ and $2y = x + 4$.

Inequalities

Anil has a batch of sweets with a mass of $600\text{ g}$, rounded to the nearest $10$ grams.

Limits of accuracy

The diagram shows that the bearing of $B$ from $A$ is $170\degree$. The bearing of $A$ from $C$ is $060\degree$. The bearing of $C$ from $B$ is $x\degree$.

Angles

The diagram shows the speed-time graph for one section of a car journey. The car’s retardation from $t=8$ to $t=12$ is $4\text{ m s}^{-2}$.

Graphs in practical situations

The circle has centre $O$ and diameter $AB$. $PA$ and $QB$ are tangents to the circle at $A$ and $B$ respectively.

Circle theorems I

Fill in the blanks. A rectangle has rotational symmetry of order .......... and .......... lines of symmetry.

Symmetry

A bag has $10$ counters in total: $8$ are blue and $2$ are white. Two counters are chosen at random from the bag without replacement.

Probability of combined events

The table lists the values taken by the function $f(x)$ for certain values of $x$.

Functions

The table gives the populations of some African countries in $2014$, rounded to $2$ significant figures.

Standard form

The table together with the histogram gives some details about how long a group of students took to get to school on one day.

Histograms

The diagram represents a sector of a circle with radius $3r\text{ cm}$ and angle $a\degree$, together with a circle of radius $r\text{ cm}$. The area ratio of the sector to the circle with radius $r$ cm is $8:1$.

Circles, arcs and sectors

The $n$th term of a sequence is $n^{2}-5n$. Determine the $2$nd term of the sequence.

Sequences

Rearrange the formula $t=\frac{p+3}{p-4}$ to make $p$ the subject.

Algebraic manipulation

It is stated that $100$ dollars $(\$)$ is the same as $56$ pounds $(\pounds)$.

Graphs in practical situations

Complete the table by converting between fraction, decimal and percentage forms.

Fractions, decimals and percentages

The table gives some details of the temperatures recorded in a city.

Interpreting statistical data

Express $96$ as a product made from its prime factors.

Types of number

The table gives details of several flights from Dubai to Mumbai.

Time

$y$ varies directly as the square of $x$. For $x = 10$, $y = 20$. Determine the value of $y$ when $x = 6$.

Ratio and proportion

The preferences of 50 students are surveyed to find out what kind of movie they enjoy watching.

Sets

A retailer purchases some plates from a manufacturer at $12$ per plate.

Percentages

A sample of 100 electric light bulbs from Brand A was tested to measure the lifetime of each bulb. The findings are summarised in the table below.

Cumulative frequency diagrams

Triangle $ABC$ has vertices $A(2, 2)$, $B(3, 5)$ and $C(4, 1)$. Triangle $A'B'C'$ has vertices $A'(-4, 4)$, $B'(-3, 7)$ and $C'(-2, 3)$. Write down the column vector for the translation that takes triangle $ABC$ to triangle $A'B'C'$.

Vectors in two dimensions

Solve $ \dfrac{p - 1}{7 - p} = 5$.

Algebraic manipulation

The table shown below corresponds to $y = x^2 + x - 3$.

Sketching curves

ANB, BLC and CMA are straight lines. $NM$ runs parallel to $BC$, and $LN$ runs parallel to $CA$.

Similarity

AB is vertical, whereas CB is horizontal. $AB = 31\text{ m}$ and $CB = 115\text{ m}$. J and K are two points out at sea. The lighthouse has its base at L. J lies due East of L, while K lies due South of L. $KL = 354\text{ m}$ and $KJ = 1100\text{ m}$.

Right-angled triangles

The matrices are $A = \begin{pmatrix}4 & -1 \\ 1 & 3\end{pmatrix},\quad B = \begin{pmatrix}2 & 0 \\ 7 & -5\end{pmatrix}$.

Introduction to algebra

In Section B, $AC$ is a diameter of the circle, centre $O$, with radius $5\text{ cm}$. $\angle ACB = 64^\circ$. A baking tray is an open cylinder with radius $15.5\text{ cm}$ and a rim. The outer edge of the rim is a circle with radius $16.5\text{ cm}$.

Circles, arcs and sectors

$p = \dfrac{8 - 5q}{q}$. In trapezium $A$, the two parallel sides are $x\text{ cm}$ and $(x-2)\text{ cm}$. In trapezium $B$, the two parallel sides are $x\text{ cm}$ and $(x+3)\text{ cm}$. Each trapezium has area $15\text{ cm}^2$.

Algebraic manipulation

$ABCD$ is the sloping rectangular face of the triangular prism. $ABEF$ forms a horizontal rectangle, and $EC$ and $DF$ are vertical. $\angle CBE = 15^\circ$, $DC = 5\text{ m}$ and $AD = 2\text{ m}$.

Non-right-angled triangles

Every year, the Reds and the Blues face each other in a baseball match. In 2014, the match had 40 500 tickets sold. By 2015, ticket sales were 2.4% higher than they had been in 2014.

Percentages

Triangles $A$, $B$, $C$ and $D$ have been drawn on a centimetre square grid.

Transformations

Solid I takes the form of a cylinder with a smaller cylinder cut out from the middle.

Surface area and volume

The given vectors are $\vec{JK}=\begin{pmatrix}2\\5\end{pmatrix}$, $\vec{KL}=\begin{pmatrix}4\\-2\end{pmatrix}$, and $\vec{LM}=\begin{pmatrix}-1\\3\end{pmatrix}$.

Vectors in two dimensions

Steven asked 25 women how many children they have, and the results are shown in the table below.

Averages and measures of spread

Triangle $ABC$ is drawn with sides $AB=8\text{ cm}$, $AC=7\text{ cm}$ and $BC=12\text{ cm}$.

Geometrical constructions

Fully factorise $8x^2y-12x^5$.

Equations

$\mathcal{E}=\{2,3,4,5,6,7,8,9,10,11,12\}$, while $A=\{x:x\text{ is prime}\}$, $B=\{x:x\text{ is even}\}$, and $C=\{x:x\text{ is a multiple of }5\}$.

Sets

Garage A notes how much petrol was bought by the first 120 customers in one day.

Cumulative frequency diagrams

The table underneath gives several values of $x$ together with the matching values of $y$ for $y=\frac{1}{4}\times2^x$.

Graphs of functions

The figure depicts a vertical wind turbine whose blades measure 30 m in length.

Right-angled triangles