Oct/Nov 2016

Evaluate the expression $3\frac{1}{6} - 2\frac{3}{5}$.

Fractions, decimals and percentages

Write $45000000$ in standard form.

Standard form

The function is given by $f(x) = \frac{3 - x}{10}.$

Functions

Let $\mathcal{E} = \{84,85,86,87,88,89,90,91,92,93,94,95,96\}$, with $P = \{x : x \text{ is an even number}\}$ and $Q = \{x : x \text{ is a multiple of } 3\}$.

Sets

Over one day, the temperature, in $^\circ\text{C}$, was measured at intervals of $2$ hours. The twelve readings are: $-3, -2, -1, 1, 2, 4, 5, 4, 2, 0, -2, -2$.

Averages and measures of spread

From the diagram, triangles $APQ$ and $ABC$ are similar. Since $BC$ is parallel to $PQ$, and $AP = 4$ cm, $PB = 2$ cm and $PQ = 1.8$ cm.

Similarity

State the single transformation that carries triangle $A$ onto triangle $B$.

Transformations

Factorise completely the expression $5 - 20t^2$.

Algebraic manipulation

Points $A,B,C,D$ and $E$ are on circle centre $O$ in the diagram. $AD$ is a diameter, while $\angle DAC = 33^\circ$ and $\angle ACE = 70^\circ$.

Circle theorems I

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

Equations

The diagram displays the points $O$ and $R$, together with the vectors $\mathbf{a}$ and $\mathbf{b}$.

Vectors in two dimensions

A roll of paper is $4.5$ metres in length. Mary cuts out as many pieces as she can, with each piece measuring $60$ cm, from the roll.

Units of measure

The diagram presents quadrilateral $ABCD$.

Geometrical constructions

The matrix is $A = \begin{pmatrix}2 & -1 \\ 1 & 3\end{pmatrix}$.

Algebraic manipulation

The speed-time graph shown represents part of a car's journey.

Graphs in practical situations

The diagrams present a run of triangles constructed from identical sticks. Compared with the triangle before it, each one has two extra sticks on every edge.

Sequences

$A$ is the point $(5,2)$ while $B$ is the point $(9,6)$. $AC$ runs parallel to the $x$-axis, and $CB$ runs parallel to the $y$-axis. The equation for the line $AB$ is $x - y = 3$.

Coordinates

Express $32\frac{1}{2}\%$ as a fraction in its lowest terms.

Fractions, decimals and percentages

Tea is priced at $\$16$ for one kilogram. Calculate the cost of $300$ grams of tea.

Ratio and proportion

Write $357.864$ rounded to $2$ significant figures.

Limits of accuracy

$y$ varies inversely with $x$. Given that $y = -50$ when $x = 3$.

Ratio and proportion

For a regular polygon, each interior angle is $171^\circ$. Determine how many sides the polygon has.

Angles

Evaluate the value of $2^3 - 2^0$.

Algebraic fractions

The cumulative-frequency graph contains data on the reaction times of $60$ people.

Cumulative frequency diagrams

Evaluate $9.03 - (4.273 + 2.3)$.

Fractions, decimals and percentages

The function is defined by $f(x)=4+3x$.

Functions

The value of $y$ is inversely proportional to the square of $x$.

Proportion

A school kept a record of the number of absent students during a $50$-day period. The results are shown in the table.

Averages and measures of spread

Triangle $ABC$ is rotated about centre $O$ by $110^\circ$ clockwise to give triangle $A'B'C'$.

Transformations

On the Venn diagram, shade the part corresponding to the subset $(A \cap B)' \cup C$.

Sets

Solve the simultaneous equations $3x+y=9$ together with $2x+3y=-8$.

Equations

Evaluate the value of $3^2+3^1+3^0$.

Powers and roots

Money is divided among Ali, Ben and Carl in the ratio $5:3:2$. Ben gets $\$60$. What amount of money is shared altogether?

Ratio and proportion

Four cards are labelled $1,2,3$ and $4$. One card is picked at random. Then, from the three cards left, a second card is selected at random. The total of the numbers on the two selected cards is found.

Probability of combined events

A box has mass $1.7\text{ kg}$, rounded to the nearest $0.1\text{ kg}$.

Limits of accuracy

Since $192 \times 64.3 = 12\,345.6$, write down the values of

Fractions, decimals and percentages

Solve for $x$ in the equation $\dfrac{2x-1}{4} + \dfrac{x-2}{3} = 2$.

Equations

The volume of a sphere is $\frac{4}{3}\pi r^3$. In a storm, raindrops collect inside a cylinder that is standing on level ground. Before the storm began, the cylinder contained no water. The cylinder has radius $20\text{ mm}$. Each raindrop is a sphere with radius $2\text{ mm}$. When the storm has ended, the water depth in the cylinder is $16\text{ mm}$.

Surface area and volume

The diagram represents triangle $ABC$.

Geometrical constructions

The diagram depicts a square card with a triangle and two small squares cut away. Every length on the diagram is measured in centimetres.

Area and perimeter

In the diagram, $A,B,C,D$ and $E$ are placed on a circle with centre $O$. $BOE$ is a straight line. $\angle DAB = 34^\circ$.

Circle theorems I

In the diagram, the line $3y + 2x = 13$ cuts the coordinate axes at $A$ and $B$.

Equations of linear graphs

The first three terms of two sequences are $1,3,5$.

Sequences

The diagram depicts the speed-time graph of a car that reduces its speed from $30\text{ m s}^{-1}$ to $12\text{ m s}^{-1}$ over $20$ seconds, and then continues at $12\text{ m s}^{-1}$.

Graphs in practical situations

The diagram contains five shaded small squares. Shade one further small square so that there is exactly one line of symmetry in the diagram.

Symmetry

The cost of $3$ pencils is $\$1.23$. Work out the cost of $5$ pencils.

Ratio and proportion

From the diagram, $ABC$ lies on one straight line, and $BF$ is parallel to $DE$. Also, $\angle FBA = 74^\circ$ and $\angle DBF = 65^\circ$.

Angles

Using suitable approximations, estimate the value of $\dfrac{\sqrt{3.98} \times 602.3}{2.987}$. Make clear the approximations you choose.

Estimation

The diagram displays triangle $A$. An enlargement maps triangle $A$ onto triangle $B$. The enlargement is centred at $(3,3)$ and has scale factor $-2$.

Transformations

Write $513\,000$ in standard form.

Standard form

The diagram displays a section of the histogram that shows the spread of the times some people took to travel to work.

Statistical charts and diagrams

During 2016, a television costs $1995$.

Percentages

The times recorded for $135$ runners as they completed a cross-country course are given in the table.

Statistical charts and diagrams

The diagram shows $\overrightarrow{AB} = \begin{pmatrix} -6 \\ 11 \end{pmatrix}$ and $\overrightarrow{AC} = \begin{pmatrix} 12 \\ -5 \end{pmatrix}$.

Vectors in two dimensions

Evaluate the expression $\sqrt[3]{\dfrac{543}{28.6 - 1.35}}$.

Algebraic manipulation

Within the diagram, $ABCD$ forms a parallelogram. $P$ and $Q$ lie on $AB$ and $BC$ respectively, with $PB = BQ$ and $DPQ = 90^\circ$. $BPQ = a^\circ$.

Angles

For a sphere, the volume is $\frac{4}{3}\pi r^3$. For a sphere, the surface area is $4\pi r^2$. A hemispherical bowl is formed from material with thickness $0.8\text{ cm}$. The external rim radius of the bowl is $9\text{ cm}$. The bowl is fixed onto a base that is a solid cylinder, with radius $3.8\text{ cm}$ and height $1.5\text{ cm}$.

Surface area and volume

A semicircle is shown with radii $OP$ and $OQ$ drawn in. A circle with centre $C$ touches the radii at $A$ and $B$, and also meets the semicircle at $T$. The circle has radius $1.8\text{ cm}$. $\angle BCA = 120^\circ$.

Circles, arcs and sectors

The building’s four walls make up the faces of cuboid $ABCDEFGH$. Because $T$ is positioned vertically above $C$ and $G$, $\angle ABT = \angle ADT = 90^\circ$. The cuboid is $9\text{ m}$ long, $8\text{ m}$ wide and $5\text{ m}$ high. $TC = 6\text{ m}$.

Pythagoras' theorem and trigonometry in 3D

The sketch locates four islands at $A$, $B$, $C$ and $D$. $A$ lies due north of $B$. $\angle DAC = 48^\circ$, $\angle CAB = 55^\circ$ and $\angle BCA = 51^\circ$. $AC = 19\text{ km}$ and $AD = 27\text{ km}$.

Non-right-angled triangles

$y = \frac{3}{5} \times 2^x$. The table gives some $x$-values together with the matching $y$-values, correct to one decimal place where needed.

Graphs of functions

On Monday, Abdul sold $140$ boxes of matches for $30$ cents each.

Equations

The basic price for the 2016 version of a car is $21000. Sayeed and Rasheed both purchase this same model of car.

Percentages

Six hundred candidates sat a mathematics examination made up of two papers. Each paper was marked out of 100. The diagram presents the cumulative frequency curves for Paper 1 and Paper 2 on the same grid.

Cumulative frequency diagrams

Simplify the expression $\dfrac{3a^2}{10bc} \div \dfrac{9a}{5b^2c}$.

Algebraic manipulation

The function is $y=\dfrac{x}{20}(x^2-10)$.

Graphs of functions

Within the quadrilateral $ABCD$, $BD=3\text{ m}$. Also, $\angle BDA=27^\circ$ and $\angle BCD=41^\circ$. The angles $\angle DBC$ and $\angle DAB$ are right angles.

Non-right-angled triangles

In the diagram, $A$ and $B$ are the centres of two circles which are tangent at $P$. The line $ACT$ is tangent to the smaller circle at $T$ and cuts the larger circle at $C$. $D$ is the point on $AB$ for which $\angle CDA=90^\circ$.

Circle theorems I

$A=\begin{pmatrix}2&0\\3&1\end{pmatrix}$ together with $B=\begin{pmatrix}1&2\\-1&3\end{pmatrix}$.

Algebraic manipulation