May/June 2017

Evaluate the product $0.2 \times 0.08$.

Fractions, decimals and percentages

Write $248.367$ rounded to $2$ decimal places.

Limits of accuracy

Solve the simultaneous equations below: $2x + 3y = 5$ $3x - y = -9$.

Equations

The ages of the guests at a family party were collected and organised in the table below. Age ($b$ years): $5 < b \leq 10$, $10 < b \leq 20$, $20 < b \leq 30$, $30 < b \leq 50$. Frequency: $p$, $18$, $14$, $q$. Some of these results are shown in the histogram below.

Statistical charts and diagrams

Evaluate the value of $\frac{3}{5} - \frac{1}{8}$.

Fractions, decimals and percentages

Write $0.000186$ in standard form by expressing it as a power of ten.

Standard form

A pyramid’s volume = $\frac{1}{3} \times$ base area $\times$ perpendicular height. The diagrams depict a solid pyramid $L$ split into two sections, $M$ and $N$, by a plane parallel to the base. Pyramid $L$ has a rectangular base measuring $9\text{ cm}$ by $12\text{ cm}$. Its perpendicular height is $30\text{ cm}$.

Surface area and volume

Evaluate $2a - b$ when $a = 3$ and $b = -7$.

Algebraic manipulation

The diagram displays triangle $A$.

Transformations

Evaluate the value of $3^{-2}$.

Indices I

Carpet covering $6$ square metres costs $258$. Work out how much $10$ square metres will cost.

Ratio and proportion

Every angle is a right angle. Every length is given in centimetres.

Area and perimeter

This distance-time graph represents a red bus’s trip from a village to a town.

Graphs in practical situations

In a sports club $24$ members play basketball ($B$), $28$ play cricket ($C$), $16$ play football ($F$), $9$ play basketball and cricket, $11$ play cricket and football and $6$ play basketball and football. Five members take part in all three games, and eight members do none of these games.

Sets

The sketch depicts line segment $AB$, connecting $A(2,3)$ with $B(5,4)$.

Transformations

$OPRQ$ forms a parallelogram, and $S$ lies on $PR$ so that $PS : SR = 1 : 3$. $\vec{OP} = \mathbf{p}$ and $\vec{OQ} = \mathbf{q}$.

Vector geometry

Solve the equation $\frac{6}{x+1} = \frac{5}{x-3}$.

Equations

The diagram shows $AB$ parallel to $DC$, while $AC$ and $BD$ meet at $E$. Triangle $ADE$ is right-angled and isosceles, with $AD = DE$. $\angle ABD = 25^\circ$.

Angles

Express $36$ in prime factor form.

Types of number

Carl spent $t$ minutes doing his English homework. His Mathematics homework took three times as long as his English homework. Altogether, he spent $2$ hours $20$ minutes on his English and Mathematics homework.

Equations

Complete the sentences that describe two distinct types of quadrilateral.

Symmetry

The diagram shows $AB$ parallel to $DE$. $\angle ABC = 114^\circ$ and $\angle CDE = 143^\circ$.

Angles

A car is moving at $84\text{ km/h}$. Calculate how many metres the car covers in one minute.

Limits of accuracy

A bag holds red and blue pegs. There are $40$ pegs altogether in the bag. The probability of picking out a red peg from the bag is $0.4$.

Introduction to probability

Evaluate the value of $\frac{4}{5} - \frac{1}{3}$.

Fractions, decimals and percentages

The masses, measured in grams, of 8 apples are $189, 175, 185, 192, 202, 161, 174, 196$. Find the median mass.

Averages and measures of spread

Solve the pair of simultaneous equations $5x - 2y = 16$ and $3x + 4y = 7$.

Equations

$y$ varies inversely with the square of $x$. The table gives some values of $x$ and $y$.

Ratio and proportion

Rani has $\$240$. She uses $\frac{5}{8}$ of this to buy a new phone. Work out how much the phone costs.

Fractions, decimals and percentages

The function is given by $f(x) = \frac{3x - k}{4}$.

Functions

The figure shows triangles $A$ and $B$.

Transformations

$A$ has coordinates $(0, 3)$, $B$ has coordinates $(1, 5)$ and $C$ has coordinates $(p, -1)$.

Equations of linear graphs

In the diagram, $P, Q, R, S$ and $T$ are points on the circle. $QT$ is a diameter of the circle with centre $O$. $X$ is the point where $PS$ and $QT$ cross. $\angle PXT = 125^\circ$ and $\angle PSQ = 35^\circ$.

Circle theorems II

The speed-time graph shown represents 25 seconds of a car journey. Starting at a speed of $v\,\text{m s}^{-1}$, the car decreases uniformly to $12\,\text{m s}^{-1}$ in 15 seconds. It then keeps a constant speed for another 10 seconds.

Graphs in practical situations

In a pentagon, four of the angles are $2x^\circ$ and the remaining angle is $x^\circ$. Calculate the value of $x$.

Angles

Add one further small triangle to the shape below so that the pattern has one line of symmetry.

Symmetry

Write $54$ as a product of prime factors.

Indices I

$OACB$ forms a parallelogram. $\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OB} = \mathbf{b}$. $P$ and $Q$ are points on $OC$ such that $OP = PQ = QC$.

Vector geometry

This container is formed from thin material and has the shape of an open-topped cuboid. Its length measures $15\text{ cm}$ and its width measures $8\text{ cm}$. The container's volume is $720\text{ cm}^3$.

Surface area and volume

Solve the equation $\frac{7x}{4 - 3x} = 3$.

Algebraic fractions

A bag contains $n$ balls. 3 of the balls are white. Two balls are selected at random from the bag without replacement.

Probability of combined events

Using each number rounded correctly to one significant figure, estimate the value of $\frac{58.7 \times 4.08}{19.7^3}$.

Limits of accuracy

A group of students was asked whether they would prefer the school day to begin later. The pie chart shows the results. 200 students replied no.

Statistical charts and diagrams

The diagram illustrates where the two villages $A$ and $B$ are located.

Angles

A thermometer reads temperature to the closest degree. The temperature outside is recorded as $-8^\circ\text{C}$.

Limits of accuracy

Describe the shaded region in the Venn diagram by using set notation.

Sets

A film begins at $22\ 35$ and ends at $01\ 20$. For how many hours and minutes does the film run?

Time

List all integers that satisfy the inequality $-\frac{3}{2} \leq x < 2$.

Types of number

Trevor owns 54 toy vehicles altogether. Among them, 24 are cars, 12 are vans, and the remainder are trucks.

Ratio and proportion

ABC forms a triangle. B and D lie on opposite sides of the line AC. DA = 9 cm and CD = 7 cm.

Geometrical constructions

Amira’s income in 2016 was $\$36\,720$.

Percentages

A student must choose one humanities subject and two different science subjects. Humanities: Geography (G), History (H), Religious studies (R). Science: Physics (P), Chemistry (C), Biology (B).

Probability of combined events

The matrices $A=\begin{pmatrix}2 & 0 \\ 4 & -1\end{pmatrix}$ and $B=\begin{pmatrix}2 & -1 \\ 6 & -1\end{pmatrix}$ are provided.

Algebraic manipulation

This table is based on $y=x^3-3x-1$.

Sketching curves

A, B, C and D are four towns. B lies $12\text{ km}$ directly north of A, C lies $8\text{ km}$ directly east of A, and D lies $15\text{ km}$ directly west of B.

Scale drawings

Factorise $12a^2b-15ab^3$ completely.

Algebraic manipulation

The cost details provided cover flights to Sydney, accommodation and insurance cover. A family made up of 2 adults and 3 children goes to Sydney for a holiday of 14 nights.

Money

The diagram indicates where point $A$ is positioned.

Geometrical constructions

Information on running-time data is provided.

Cumulative frequency diagrams

$OAB$ forms a sector of a circle with centre $O$ and radius $10\text{ cm}$. $\angle AOB=72^\circ$.

Circles, arcs and sectors

The table presents the populations, rounded to the nearest thousand, for a selection of countries.

Percentages

Rowena spins two fair spinners, and each spinner is labelled 1 to 4. Her score is the product of the two numbers shown.

Introduction to probability

Matrices $A$, $B$ and $C$ are provided.

Algebraic manipulation

Express as a single fraction, in its simplest form, $\frac{1}{2x}+\frac{2}{5x}$.

Equations

Triangle $PQR$ is right-angled at $P$, with $\angle PRQ=38^\circ$ and $RQ=12\text{ cm}.$

Right-angled triangles

The diagrams illustrate designs built from crosses (X) and circles (O).

Sequences

$ABCDE$ represents the building’s cross-section. Every length is measured in metres.

Area and perimeter

A value of $x$ is chosen at random, where $x$ may be any real number.

Sketching curves