Mathematics 0580 · IGCSE

Feb/March 2023

72 questions from this paper, with worked solutions and instant marking.

Write twenty-five million in numeral form.

Types of number

The diagram depicts one face of a cuboid on a $1\text{ cm}^2$ grid. The cuboid’s volume is $24\text{ cm}^3$.

Surface area and volume

For a collection of six numbers, the median is $61$. The five given values are $24,\ 43,\ 58,\ 71$ and $85$.

Averages and measures of spread

Determine the size of one interior angle in a regular $9$-sided polygon.

Angles

Inside the universal set, a Venn diagram displays two sets named $A$ and $B$.

Sets

Factorise completely $8g-2g^2$.

Algebraic manipulation

The diagram represents a circle with centre $O$ and diameter $AC$. The points $A, B, C, D$ and $E$ are all on the circle's circumference. This sketch is not drawn to scale. At $C$, the angle between $CB$ and $CO$ is $65^\circ$. At $D$, the angle between $DO$ and $DE$ is $46^\circ$. At $A$, the angle between $AO$ and $AB$ is labelled $x^\circ$. At $E$, the angle between $EO$ and $EA$ is labelled $y^\circ$.

Circle theorems I

Show every step of your working and give your answer as a fraction in lowest terms.

Fractions, decimals and percentages

Over $40$ weeks, a school keeps track of how many calculators it sells in each week. The information is shown in the table.

Averages and measures of spread

The mass, $m$ kg, of a bag of sand is given as $12$ kg to the nearest kilogram.

Limits of accuracy

Qianna places $3000$ into an account earning compound interest at $4\%$ per year.

Percentages

Express $0.7$ as a fraction.

Fractions, decimals and percentages

Solve the equation $\frac{25-2u}{3}=2$.

Equations

Calculate $0.3^2$. Express your answer in standard form.

Standard form

The probability that a person passes a driving test is $0.36$. A total of $600$ people take this driving test.

Relative and expected frequencies

You need to show all your working.

Equations

The figure depicts a right-angled triangle. Its base is $74\text{ cm}$, its vertical height is $46\text{ cm}$, and the angle on the left is marked $y$. The figure is not drawn to scale.

Right-angled triangles

The figure shows two right-angled triangles, $ABC$ and $ACD$. For triangle $ABC$, $AB=39\text{ cm}$ and $BC=52\text{ cm}$. For triangle $ACD$, $CD=25\text{ cm}$. The side $AD$ is marked as $x$ cm. The figure is not drawn to scale.

Pythagoras' theorem

A circle’s area is $25\pi\text{ cm}^2$.

Circles, arcs and sectors

The numbers in the list are: $-7,\ 12,\ -3,\ 2,\ 8,\ -6,\ 15,\ -4,\ -8$.

Ordering

The exam begins at $11\,50$ and goes on for $2\frac{1}{4}$ hours.

Time

Give $56.17345$ correct to $1$ decimal place.

Limits of accuracy

Find how many seconds there are in $5$ hours.

Time

The numbers in the list are $12,\ 15,\ 27,\ 29,\ 91,\ 93$.

Types of number

Let $\mathbf{v}=\begin{pmatrix}-1\\3\end{pmatrix}$ and $\mathbf{y}=\begin{pmatrix}2\\5\end{pmatrix}$.

Coordinates

A suit is priced at $6500$ rupees. The exchange rate is $1$ rupee $=\ $0.013.

Rates

From the numbers in the list, write down 12 15 27 29 91 93

Types of number

From the diagram, AB and CD are parallel, and CB together with AD meet at X. AB = 3.0 cm, AX = 2.0 cm, BX = 2.7 cm and CD = 7.5 cm.

Similarity

Determine the highest common factor (HCF) of $12x^{12}$ and $16x^{16}$.

Algebraic manipulation

For a regular polygon, the interior angle and the exterior angle are related by the ratio interior : exterior = 11 : 1.

Angles

The speed-time graph for part of a car journey is shown in the diagram.

Graphs in practical situations

The diagram depicts a sector of a circle whose centre is O and whose radius measures 9 cm. The chord PQ has length 6 cm.

Circles, arcs and sectors

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

Powers and roots

y varies inversely with $x^2$. If $x = 3$, then $y = 2$.

Proportion

ABCD is a cyclic quadrilateral, ABX lies on a straight line and PQ is tangent to the circle at A. Angle CBX = 85^{\circ}, angle BAQ = 55^{\circ} and angle CAD = 42^{\circ}.

Circle theorems I

The two solids are mathematically similar, and their volumes are 81 cm$^3$ and 24 cm$^3$. The smaller solid has surface area 44 cm$^2$.

Similarity

Find the values of $x$ for which $6x + y = 10$ and $y = x^2 - 3x + 10$ both hold.

Equations

Vector v is $\begin{pmatrix}-1\\3\end{pmatrix}$ vector y is $\begin{pmatrix}2\\5\end{pmatrix}$

Vectors in two dimensions

Find the $n$th term for each sequence given below.

Sequences

A car covers 14 km, measured to the nearest kilometre. The journey lasts 12 minutes, measured to the nearest minute.

Limits of accuracy

The prism ABCDQP has a length of 7 cm and is triangular. Its cross-section is triangle PAB, with PA = 4 cm, AB = 5 cm and angle PAB = 90^{\circ}.

Pythagoras' theorem and trigonometry in 3D

Simplify the expression $\frac{5x^2 - 19x + 12}{x^2 - 9}$.

Algebraic fractions

The chance that Jamie hits a target is $\frac{1}{3}$. The chance that he lands a hit on the target for the first time on his $n$th attempt is $\frac{64}{2187}$.

Introduction to probability

The function is given by $f(x) = x^3 + 1$.

Functions

A Venn diagram displays two overlapping sets, A and B, within the universal set.

Sets

23, 17, 11, 5, ...

Sequences

Write $8g - 2g^2$ in fully factorised form.

Algebraic manipulation

Work out $\frac{4}{7} \div 8$!

Fractions, decimals and percentages

Find the solution.

Equations

Calculate $0.3^2$. Write your answer in standard form.

Standard form

Solve the simultaneous equations. You must present all of your working. $3x - 2y = 19$ $x + y = 3$

Equations

The table gives some details about a café's opening times. The café is open for 4 days each week. Day | Opening time | Closing time | Number of hours open Thursday | 8 am | 4.30 pm | Friday | 8.30 am | | 7 1/2 Saturday | 9.30 am | 5.30 pm | 8 Sunday | | 3.30 pm | 5 Total number of hours open: 29

Statistical charts and diagrams

Statistical charts and diagrams

To make 20 biscuits, the recipe needs 150 g flour, 125 g butter, and 50 g sugar.

Ratio and proportion

State 6479 correct to the nearest 100.

Limits of accuracy

The figure shows three triangles, A, B and C, set on a grid where each square represents 1 cm$^2$.

Transformations

Determine the equation of line L in the form $y = mx + c$.

Sketching curves

The scale diagram plots the locations of three towns, R, S and T, on a map. RS and ST are straight roads joining the towns. The scale is 1 centimetre represents 8 kilometres.

Scale drawings

State the order of rotational symmetry for the diagram.

Sequences

Write down an expression for the area of this rectangle. Give your answer in its simplest form.

Algebraic manipulation

Show that Alain gets $400$.

Percentages

The sketch depicts quadrilateral $ABCD$. Also, $AC = 12.3$ cm and $AD = 16.5$ cm. In addition, angle $BAC = 31^\circ$, angle $ABC = 90^\circ$ and angle $ACD = 90^\circ$.

Right-angled triangles

The functions are $f(x)=2x-1$, $g(x)=3x+2$, $h(x)=\frac{1}{x}$, $x\neq0$, and $j(x)=x^2$

Functions

Sketch y = \tan x for $0^\circ \leq x \leq 360^\circ$.

Trigonometric functions

A reaction test was taken by 100 students. The results are listed in the table. Reaction time in seconds: 6, 7, 8, 9, 10, 11 Frequency of students: 3, 32, 19, 29, 11, 6

Averages and measures of spread

A cylinder partly filled with water is shown. A solid metal sphere is resting on the base of the cylinder, and half of the sphere is submerged in the water. The cylinder has radius 12 cm and the sphere has radius 3 cm.

Surface area and volume

Enlarge triangle $T$ with scale factor 3 about centre $(0,2)$.

Transformations

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

Algebraic manipulation

A lies at $(0,4)$ and B lies at $(8,0)$. $L_1$ runs parallel to the $x$-axis. $L_2$ goes through A and B.

Equations of linear graphs

$\xi = \{\text{students in a class}\},\; P = \{\text{students who study Physics}\},\; C = \{\text{students who study Chemistry}\}$ Also, $n(\xi)=24,\; n(P)=17,\; n(C)=14,\; n(P\cap C)=9$

Sets

Calculate the triangle’s area.

Non-right-angled triangles

$f(x) = x^3 - 3x^2 - 4$

Differentiation