May/June 2019

Calculate $\frac{4}{9} + \frac{2}{5}$.

Fractions, decimals and percentages

Factorise the expression $25x - 5$.

Algebraic manipulation

Express $0.0000845$ in standard form.

Standard form

The expression for the $r$th term of the sequence, $u_r$, is $u_r = 3r + 2$.

Sequences

Points $A, B, C, D$ and $E$ lie on the circumference of a circle with centre $O$. The tangent $AF$ is drawn so that it touches the circle at $A$. $O$ is the midpoint of $AD$. $\angle DOE = 138^\circ$ and $\angle BEO = 12^\circ$.

Circle theorems I

The diagram displays the layout of a garden, $ABCD$.

Geometrical constructions

Given $T = \begin{pmatrix} 2 & 7 \\ 1 & 5 \end{pmatrix}$, determine $T^{-1}$.

Algebraic manipulation

Starting with $c = \frac{3b^2 - 2a}{5}$, rewrite the equation so that $b$ is the subject.

Algebraic manipulation

The line segment $AB$ connects $A(3,2)$ with $B(-1,10)$.

Length and midpoint

Jim has a $0.7$ chance of taking part in the next match. If Jim does play, the probability that his team wins is $0.8$. If Jim does not play, the probability that his team wins is $0.6$.

Probability of combined events

The diagram depicts triangle $ABC$. Point $D$ lies on $AC$ so that $\angle BDC = \angle ABC$.

Similarity

For the grids shown below, each square contains the sum of the two numbers directly beneath it. One grid has already been finished for you. Complete the two grids below.

The four operations

On the Venn diagram, shade the area represented by $B' \cap A$.

Sets

Shapes $A$ and $B$ are shown on the grid.

Transformations

Solve these simultaneous equations: $2x + 3y = 4$ and $3x + 2y = 11$.

Equations

Solve the equation $(2x + 1)(x + 4) = 22$.

Equations

Arrange these numbers from smallest to largest: $\sqrt{17}$, $4$, $4.5$, $\sqrt[3]{63}$.

Ordering

Write $0.09$ in percentage form.

Fractions, decimals and percentages

A glass of this drink is prepared by mixing 20 millilitres of orange juice with water.

Ratio and proportion

Evaluate the value of $36^{\frac{1}{2}}$.

Indices II

Yasmin goes swimming at her nearby local pool.

Money

The diagram represents a triangular prism that is $12$ cm long. Its cross-section is a right-angled triangle with sides of $6$ cm, $8$ cm and $10$ cm. Using the grid, construct a net for this prism. Apply a scale where $1$ cm stands for $2$ cm. One face has already been drawn for you.

Surface area and volume

Students select one fruit: an apple, a banana, or an orange.

Statistical charts and diagrams

Evaluate the value of $\frac{4}{7}-\frac{5}{8}$.

Fractions, decimals and percentages

Express the shaded part of the Venn diagram in set notation.

Sets

Solve the simultaneous equations, and show the working you use. $9x+4y=-5$ $6x-2y=6$

Equations

Arrange these numbers from smallest to largest. $2.1\times10^{-3},\;4.2\times10^{-4},\;1.7\times10^{-5},\;3.5\times10^{-4}$.

Standard form

Expand then simplify $(x-3)^2$.

Algebraic manipulation

Express $x^2-7x+5$ in the format $(x-a)^2-b$.

Algebraic manipulation

Express $168$ as a product of its prime factors.

Types of number

Triangles $A$ and $B$ are shown on the grid.

Transformations

Nima is given these six cards. A shape appears on each card. She chooses two cards at random without replacement.

Probability of combined events

$r=\dfrac{4p+2}{3-p}$.

Algebraic manipulation

$y$ varies inversely as the square of $x$. If $x=4$, then $y=10$.

Ratio and proportion

For this question, use only a straight edge and a pair of compasses.

Geometrical constructions

Simplify the expression $\left(\dfrac{9x^7y}{x^5y^9}\right)^{-\frac{1}{2}}$.

Indices II

A cuboid has a square-shaped base. The base length of the cuboid is $y$ cm. Its height is double the length of the base. The cuboid’s total surface area is $360\,\text{cm}^2$.

Surface area and volume

The initial three dot-and-line patterns in the sequence are shown here.

Sequences

The diagram depicts two circles that share centre $O$. The smaller circle has radius $3$ cm, and the larger circle has radius $6$ cm. Minor sector $AOB$ subtends an angle of $60^\circ$. The combined area of the shaded parts is $k\pi\,\text{cm}^2$.

Circles, arcs and sectors

$A$, $B$ and $C$ lie on the circle with centre $O$, and $AB=BC$. $P$ is the midpoint of chord $AB$ while $Q$ is the midpoint of chord $BC$.

Circle theorems I

$P=\begin{pmatrix}4&0\\-2&3\end{pmatrix}$ and $Q=\begin{pmatrix}1&2\\0&-1\end{pmatrix}$. Evaluate the product $PQ$.

Algebraic manipulation

The water level, measured in cm, in a river is noted each week for 10 weeks. Relative to the normal level, the water heights are shown below. $-45, -30, -35, 0, 5, -10, -20, 40, 20, 25$

Averages and measures of spread

By replacing each number with its value to a single significant figure, estimate $\dfrac{71.8-32.4}{0.198^2}$.

Estimation

Lamps are manufactured in a factory. A random sample of 50 lamps is checked, and 4 are found to be faulty. In one day, 4000 lamps are produced.

Ratio and proportion

Each month, Daniel receives $\$760$. He gives $15\%$ of this income as tax. Calculate how much money Daniel has left each month after tax is paid.

Percentages

Determine the fraction that is exactly midway between $\frac{3}{5}$ and $\frac{3}{4}$.

Fractions, decimals and percentages

A beverage is prepared by combining fruit juice and water in a $3:5$ ratio. The mixture uses $2$ litres of water. Calculate how much fruit juice is needed. Give your answer in millilitres.

Ratio and proportion

A car sets off from rest. For the first $20$ seconds, it accelerates uniformly until its speed becomes $15\,\text{m s}^{-1}$. After that, it continues at a steady speed of $15\,\text{m s}^{-1}$ for $40$ seconds.

Graphs in practical situations

During 2017, Lauren’s monthly income came to $1800.

Percentages

The diagram represents a triangular prism, and every length is measured in centimetres.

Graphs of functions

The diagram indicates the locations of ports $A$ and $B$, together with lighthouse $L$. From $L$, the bearing of $B$ is $062^\circ$. $AB = 13\text{ km}$, $BL = 14\text{ km}$ and $AL = 8\text{ km}$.

Non-right-angled triangles

Ten boys took part in a $100\text{ m}$ race and a $200\text{ m}$ race. The table underneath gives their times in seconds.

Scatter diagrams

PBQ and RACS lie parallel to each other. BAE and BCD each form straight lines. $AB = AC$, $\angle QBC = 58^\circ$ and $\angle AED = 47^\circ$.

Angles

Express $\frac{6y}{35} \div \frac{10y^2}{7}$ as a fraction in simplest form.

Algebraic fractions

$ABCD$ forms a parallelogram. $AD = 6\text{ cm}$, $AB = 8\text{ cm}$ and $BD = 11\text{ cm}$.

Geometrical constructions

Solve the compound inequality $10 < 3(x + 1) \le 24$ for $x$.

Inequalities

$f(x) = 5x - 7$, $g(x) = \frac{x + 4}{3}$.

Functions

The table gives the distances, $d$ m, of the javelin throws made by $80$ women.

Cumulative frequency diagrams

The diagram shows that $\overrightarrow{PQ} = 4\mathbf{p}$, $\overrightarrow{QR} = 3\mathbf{q}$ and $\overrightarrow{PT} = \mathbf{p} + 2\mathbf{q}$. Also, $\overrightarrow{QU} = \frac{2}{3}\overrightarrow{QR}$ and $\overrightarrow{PT} = \frac{2}{3}\overrightarrow{PS}$.

Vector geometry

The table below gives the admission cost for a zoo visit.

Percentages

$\mathbf{f} = \begin{pmatrix}4\\-3\end{pmatrix}$ and $\mathbf{g} = \begin{pmatrix}1\\-5\end{pmatrix}$. Points $O, A$ and $B$ are given with $\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OB} = \mathbf{b}$. Point $P$ lies on $OA$ so that $OP = \frac{1}{3}OA$. The points $O, Q$ and $R$ are collinear, and $Q$ is the midpoint of $PB$.

Vector geometry

The functions are $f(x) = 3(x-2)$ and $g(x) = \frac{7x}{4} - 1$.

Functions

The table presents the mean monthly temperatures ($^\circ\text{C}$) for Tokyo and Sydney over one year.

Scatter diagrams

Represent the inequality $-3 < x \leq 2$ on the number line shown below.

Inequalities

The figure $ACD$ forms a triangle with $\angle ADC = 110^\circ$. Also, $EB$ is parallel to $DC$, and $AE = EB$.

Similarity

Write this as one fraction, then give it in simplest form: $\frac{3a}{4b} - \frac{a}{6b}$.

Algebraic manipulation

The diagram plots boats $A$ and $B$ on a scale of $1:m$. In reality, the separation between the two boats is $4\text{ km}$.

Scale drawings

The table lists several values of $y = 1 + \frac{2}{x}$, rounded to $2$ decimal places whenever that is necessary.

Sketching curves

The mass, measured in grams, of each of the $75$ oranges is recorded. The findings are set out in the table.

Cumulative frequency diagrams

The diagram depicts a triangular prism, with $AC = 15\text{ cm}$, $BC = 14\text{ cm}$, and $\angle ACB = 27^\circ$.

Non-right-angled triangles