Oct/Nov 2022

Express the number two million eight hundred and forty thousand three hundred and twenty-seven as figures.

Types of number

The birth weights of 11 babies, measured in kg, are as follows: 2.1, 1.6, 2.7, 4.2, 4.0, 2.2, 3.1, 1.7, 2.6, 3.3, 3.7.

Statistical charts and diagrams

Victoria notes the colour of every one of the 240 cars leaving a car park. Part of the data is displayed in the pie chart. The chart shows a blue sector of a right angle, together with a green sector measuring $27^{\circ}$.

Statistical charts and diagrams

A computer costs $\$520$. In the sale, the cost is lowered by $15\%$.

Percentages

Without a calculator, calculate $\frac{1}{3} + \frac{5}{6}$. Show all your working and write your answer as a mixed number in simplest form.

Fractions, decimals and percentages

Mario examines new cars. The probability that a car is faulty is $0.04$.

Introduction to probability

A café sells 330 sandwiches. That amounts to $\frac{11}{14}$ of the sandwiches it produces.

Fractions, decimals and percentages

Fill in the values table for $y = \frac{5}{x}$.

Graphs of functions

State the term to term rule for this sequence.

Sequences

The Venn diagram illustrates the elements of the sets $\mathcal{E}$, $P$ and $Q$. The items in $P$ only are $a$, $b$, $c$. The common region contains $d$. The items in $Q$ only are $e$ and $f$. The element lying outside both sets in $\mathcal{E}$ is $g$.

Sets

From $B$, the bearing to $A$ is $137^{\circ}$.

Angles

Write down the mathematical name for this kind of angle.

Angles

Express $0.00273$ in standard form.

Standard form

Triangles $ABC$ and $DEF$ are mathematically similar. In triangle $ABC$, $AB = 51\text{ cm}$. In triangle $DEF$, $DE = 6\text{ cm}$ and the base $EF = 4\text{ cm}$. The sketch is not drawn to scale.

Similarity

The figure depicts a right-angled triangle. One edge has length $17.5\text{ cm}$, an angle of $48^{\circ}$ is marked, and the hypotenuse is named $x\text{ cm}$. The drawing is not to scale.

Right-angled triangles

Natalie purchases 4 tomato plants and 3 pepper plants for $\$9.35$. Samir purchases 2 tomato plants and 11 pepper plants for $\$16.55$.

Equations

Determine the length of this line in millimetres.

Geometrical constructions

In triangle $PQR$, $PR = 5\text{ cm}$ and $QR = 4\text{ cm}$. With only a ruler and compasses, construct triangle $PQR$. Keep the construction arcs visible. The side $PQ$ has already been drawn for you.

Geometrical constructions

Give a common multiple of 18 and 24.

Types of number

Express $32\text{ cm}$ as a fraction of $2\text{ m}$, and simplify your result fully.

Units of measure

Temperatures, in $^{\circ}\text{C}$, are recorded simultaneously for six cities. London 6, Helsinki -2, Oslo -5, Paris 7, Madrid 9, Berlin 2.

Ordering

The diagram depicts two straight lines crossing two parallel lines. One angle is labelled $71^{\circ}$, another is labelled $55^{\circ}$, and the unknown angle is called $x^{\circ}$. The diagram is not drawn to scale.

Angles

Share $\$200$ in the ratio $7:3$.

Ratio and proportion

Write eighty thousand and eighty in numerals.

Types of number

Do the calculation of $\frac{5}{9} - \frac{1}{6}$ without a calculator. Show every step of your working and give the result as a fraction in simplest form.

Fractions, decimals and percentages

A dice with 4 faces is labelled 1 to 4. The table gives some of the probabilities of obtaining each number.

Introduction to probability

Determine the following two terms in the sequence.

Sequences

Daryl has noted the number of hours per week spent exercising by 8 people. 5 2 1.5 3 18 4.5 2 4

Averages and measures of spread

Calculate:

Fractions, decimals and percentages

Jenna purchases 2.4 m of ribbon together with 4.8 m of fabric, and the combined price is $33.48. The ribbon price is $0.85 per metre.

Rates

Expand the expression $x(x + 8)$

Algebraic manipulation

$\xi = \{\text{people in a group}\}$ $B = \{\text{people who own a bicycle}\}$ $C = \{\text{people who own a car}\}$ A group contains 120 people. 21 people own a bicycle. 15 people own both a bicycle and a car. 35 people own neither a bicycle nor a car.

Sets

The prism is a right-angled triangular prism with height 6 cm, width $w$ cm and length 18 cm. Its volume is $810\text{ cm}^3$. A labelled diagram of the prism is shown and it is marked NOT TO SCALE.

Surface area and volume

In a survey of 1200 people, 150 were left-handed.

Ratio and proportion

Determine the value of $\sqrt{53.29}$.

Powers and roots

$5^8 \div 5^x = 5^2$. Determine the value of $x$.

Indices I

A right-angled triangle $ABC$ is shown. The hypotenuse measures 18 cm and the angle at $C$ is $42^\circ$. The diagram is marked NOT TO SCALE.

Right-angled triangles

A football squad of 16 players is at a training session. Kim notes the colour of each shirt they are wearing. Blue Silver Green Green Silver Silver Red Silver Green Red Silver Silver Blue Green White Blue Complete the frequency table below. You may use the tally column to help you.

Classifying statistical data

A cuboid is shown with dimensions marked 5 cm, 4 cm and 2 cm. The diagram is marked NOT TO SCALE.

Surface area and volume

Draw every line of symmetry on this shape.

Symmetry

Insert a single pair of brackets into each statement so that it becomes correct.

The four operations

The diagram depicts an isosceles triangle. One angle on the base is marked $34^\circ$. The angle at the top is marked $x^\circ$. The diagram is marked NOT TO SCALE.

Angles

Simplify the following. $6a + 3b - 2a - 5b$

Algebraic manipulation

Find the answer.

Coordinates

Fill in this shopping bill. 2.25 kg apples at $2.80 per kg = $ [BLANK] 3.2 kg carrots at $0.65 per kg = $ [BLANK] Total = $ [BLANK]

Money

The diagram displays angles at a point, with each one marked $a^\circ$.

Angles

Expand the brackets before simplifying $3(2d - 3) + 4(d + 1)$.

Algebraic manipulation

Without using a calculator, calculate $\frac{5}{7} - \frac{2}{3}$. Show all your working and give your answer as a fraction in its simplest form.

Fractions, decimals and percentages

$x$ is an integer subject to $x > -4$ and $x \leq 2$.

Inequalities

The diagram represents a straight line $L$ on axes marked $x$ and $y$, and it goes through the origin.

Equations of linear graphs

The diagram represents a trapezium with parallel sides measuring $8.6\text{ cm}$ and $14.2\text{ cm}$, together with a perpendicular height of $7.5\text{ cm}$. NOT TO SCALE.

Area and perimeter

A circular pond has a circumference of $130\text{ cm}$.

Circles, arcs and sectors

Put $0.007$ into standard form.

Standard form

A right-angled triangle is illustrated. The hypotenuse measures $14\text{ cm}$, the base is $10\text{ cm}$ and the perpendicular side is $h\text{ cm}$. NOT TO SCALE.

Pythagoras' theorem

The diagrams illustrate Venn diagrams. In (a), the two sets $A$ and $B$ overlap within the universal set. In (b), the elements are arranged as follows: In $X$ only: $a, h, f$ In $X \cap Y$: $c, s, d$ In $Y$ only: $p, b, g$ Outside both sets: $m, e$.

Sets

Calculate her age in months.

Time

Solve the pair of simultaneous equations $3x - 2y = 21$ and $5x + 2y = 51$.

Equations

The diagram represents a right-angled triangle. One of the angles is $56^\circ$, the hypotenuse measures $21.8\text{ cm}$ and the base measures $x\text{ cm}$. NOT TO SCALE.

Right-angled triangles

For $120 = 2^m \times 3 \times 5$, find $m$.

Indices I

Rearrange the formula $q = \frac{m}{7} + r$ so that $m$ is the subject.

Algebraic manipulation

The diagram illustrates two sides of a regular polygon. The interior angle of the polygon is $(7x + 44)^\circ$ and the exterior angle is $(x + 8)^\circ$. NOT TO SCALE.

Angles

120, 121, 149, 164, 216

Types of number

The diagram depicts a figure with four sides that are all the same length.

Symmetry

The stem-and-leaf diagram gives the marks obtained by each of 35 students in a science test. Stem 2: 7 8 8 9 Stem 3: 0 2 2 4 5 7 8 9 Stem 4: 1 1 2 3 4 5 7 8 9 9 Stem 5: 0 2 3 5 5 5 6 7 7 8 Stem 6: 1 2 4 Key: 2 | 7 represents 27

Averages and measures of spread

Marco begins work at 2045 and ends at 0208 on the following day.

Time

Jo is counting how many of each nut type are in a bag. The pie chart is used to display the results. The sectors on the pie chart are named: Peanuts, Hazelnuts, Almonds, Brazil nuts. Two of the sectors have central angles of $108^\circ$ and $120^\circ$.

Statistical charts and diagrams

Calculate the value of $\sqrt{15}+\frac{4.8}{2.2}$.

Powers and roots

Nerina places $5400 in a simple interest account at an annual rate of $r\%$. After 3 years, the overall interest earned is $429.30$.

Rates

Write down one common multiple of 18 and 24.

Types of number

Simplify: $18x^{18} \div 9x^{9}$.

Indices I

The simultaneous equations are $x - 3y = 7$ and $2x - 3y = 11$.

Equations

Triangle $PQR$ is similar to triangle $ABC$, with $\frac{PR}{AC} = \frac{2}{3}$. Also, $AB = 9\text{ cm}$ and the area of triangle $ABC$ is $18\text{ cm}^2$. Diagrams are not drawn to scale.

Similarity

This diagram presents the speed-time graph for the first 15 seconds of a car journey. The speed rises at a constant rate from 0 to $14\text{ m/s}$ in the opening 5 seconds, and then it stays unchanged until 15 seconds.

Graphs in practical situations

On a coordinate grid, triangle $A$ has vertices located at about $(2,1)$, $(4,1)$ and $(4,2)$. Triangle $B$ has vertices located at about $(3,4)$, $(4,6)$ and $(4,4)$.

Transformations

A sector of a circle has radius $8\text{ cm}$ and perimeter $26\text{ cm}$. Calculate the angle of this sector.

Circles, arcs and sectors

This diagram depicts a circle with eight chords. The angles marked are $20^{\circ}$ and $72^{\circ}$ at the upper-right point on the circumference, $56^{\circ}$ at an internal intersection, $u^{\circ}$ at the lower-right point, $w^{\circ}$ at the lowest point, and $x^{\circ}$ and $v^{\circ}$ at the left point.

Circle theorems II

Simplify the expression $(3125x^{3125})^{\frac{1}{5}}$.

Powers and roots

In triangle $ABC$, $AB = 7\text{ cm}$, angle $A = 115^{\circ}$ and angle $B = 35^{\circ}$. The diagram is not drawn to scale.

Non-right-angled triangles

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

Algebraic manipulation

A train leaves at 2340 and arrives at 0650. Calculate the time taken for this journey.

Time

Factorise the expression $1 + x - y - xy$ completely.

Algebraic manipulation

This cubic graph has two turning points. The gradient is positive for $x < 0$ and also for $x > 4$. The gradient is negative for $0 \le x \le 4$. The curve goes through the origin.

Sketching curves

A pair of axes is displayed, with the $x$-axis extending from $0^{\circ}$ to $360^{\circ}$ and the $y$-values spanning from $-1$ to $1$.

Trigonometric functions

$y$ varies inversely with $\sqrt{x}$, and $x$ varies directly with $w^2$. If $w = 12$, then $y = 12$.

Proportion

Violet and Wilfred each gave their running times for 200 m, rounded to the nearest second. Violet's time was 36 seconds, while Wilfred's time was 39 seconds.

Limits of accuracy

The bag contains 5 red balls, 4 blue balls and 3 green balls.

Probability of combined events

Express 32 cm as a fraction of 2 m, and give the result in its simplest form.

Units of measure

Divide $200$ so that it is shared in the ratio $7 : 3$.

Ratio and proportion

Two straight lines cross two parallel lines. At the top intersection, one angle is $71^{\circ}$ and the neighbouring angle is labelled $x^{\circ}$. At the bottom intersection, an angle of $55^{\circ}$ is shown. The diagram is not drawn to scale.

Angles

A computer costs $520. During a sale, this price is lowered by $15\%$. Work out the sale price.

Percentages

The Venn diagram displays the members of the sets $\xi$, $P$ and $Q$. The items shown are: in $P$ only: $a, b, c$; in $P \cap Q$: $d$; in $Q$ only: $e, f$; and, outside both circles but still inside $\xi$: $g$.

Sets

Write down the next term for this sequence.

Sequences

Calculate $\frac{1}{3} + \frac{5}{6}$ without a calculator. Show all of your working and write the answer as a mixed number in its simplest form.

Fractions, decimals and percentages

The diagram depicts an isosceles triangle. The two sloping sides on the left and right are shown as equal. The angle at the top of the triangle is marked as $x^\circ$. The angle at the base on the right is marked $34^\circ$. The diagram is labelled NOT TO SCALE.

Angles

Calculate the value of $2000 \times 1.2^3$.

Fractions, decimals and percentages

The graph of $y=(x-3)(x+b)(x+2)$ cuts the y-axis at $-30$.

Algebraic manipulation

The prime-factor form of $x$ is $x = 3^2 \times 5^2 \times 7 \times 199^{57}$.

Types of number

The table gives details of the mass of each of 50 children.

Cumulative frequency diagrams

The jump distances, measured in centimetres, are recorded for 136 girls and 144 boys. Box-and-whisker plots are then used to display the distributions of these distances.

Statistical charts and diagrams

The figure depicts a quadrilateral whose interior angles are marked $101^\circ$, $95^\circ$, $85^\circ$ and $79^\circ$. It is labelled NOT TO SCALE.

Circle theorems I

In each Venn diagram, shade the region specified by $G \cap H'$ and $(J \cup K') \cap L$.

Sets

$f(x)=x^2$, $g(x)=\frac{x+5}{2}$, $h(x)=7x-3$.

Functions

Express $0.4\dot{9}$ as a fraction in simplest form. You must show all your working.

Fractions, decimals and percentages

Katy chooses one number at random from 2, 3 and 5, and then selects one number at random from 5, 6, 7 and 9. The sum of the two chosen numbers is even.

Conditional probability

Simplify $y \times 27 - y \times 77$ to its simplest form.

Algebraic manipulation

Write $(81x^{16})^{\frac{3}{4}}$ in simplest form.

Indices II

A solid is produced by removing a smaller cone from a larger cone, as shown in the diagram. The larger cone has a height of 12.5 cm. The height of the solid is 5.5 cm. The diameter of the base of the larger cone is 9.2 cm. The diagram is marked NOT TO SCALE. The volume $V$ of a cone with radius $r$ and height $h$ is $V=\frac{1}{3}\pi r^2h$.

Surface area and volume

The volumes of two mathematically similar objects are $56\text{ cm}^3$ and $875\text{ cm}^3$. The smaller object has a height of 18 cm.

Similarity

Solve $\frac{4}{x+1}+\frac{2}{2x-5}=3$. You must present all your working.

Algebraic fractions

Determine the total of $3^2$ and $-3^2$.

Powers and roots

Expand the expression $x(3+x^2)$.

Algebraic manipulation

Jenna purchases 2.4 m of ribbon and 4.8 m of fabric. Altogether, the cost comes to $33.48. The ribbon is priced at $0.85 per metre.

Rates

Find the following two terms in the sequence.

Sequences

You need to show every step of your working and present your answer as a fraction in simplest form.

Fractions, decimals and percentages

Daryl records how many hours in a week 8 people spend exercising: 5, 2, 1.5, 3, 18, 4.5, 2, 4.

Averages and measures of spread

The diagram displays three triangles A, B and C. On each triangle, angles of $60^\circ$ and $25^\circ$ are marked, and a side measuring 6 cm is labelled. The diagram is labelled NOT TO SCALE.

Geometrical terms

Find how long, in hours and minutes, he works.

Time

A right-angled triangle is drawn in the diagram. Its base measures 10 cm, the hypotenuse measures 14 cm, and the vertical side is marked $h$ cm. The figure is labelled NOT TO SCALE.

Pythagoras' theorem

The figure shows two sides of a regular polygon. Its interior angle is $(7x + 44)^{\circ}$, and its exterior angle is $(x + 8)^{\circ}$. The figure is NOT TO SCALE.

Angles

Find the interest gained on the investment after 3 years.

Exponential growth and decay