Mathematics 0580 · IGCSE
Oct/Nov 2023
120 questions from this paper, with worked solutions and instant marking.
The angle is shown in a diagram.
Angles
The diagram depicts two parallel lines $AB$ and $CD$. At the top-left crossing, the angle is $52^\circ$ and the interior angle is labelled $x^\circ$. At the bottom-right crossing, the angle is $135^\circ$. The diagram is not drawn to scale.
Angles
Write $0.03682$ correct to 2 significant figures.
Limits of accuracy
The table provides some details about Amir's shopping. Fruit | Cost per kilogram | Number of kilograms Amir buys | Cost Oranges | $2.35 | 3.2 | $[BLANK] Bananas | $[BLANK] | 2.8 | $[BLANK] Total cost: $13.54
Fractions, decimals and percentages
The scatter diagram compares, for each of 10 people in an office, salary and car value. The x-axis is Salary ($), ranging from 0 to 80000. The y-axis is Value of car ($), ranging from 0 to 10000.
Scatter diagrams
Factorise completely to obtain: $42mk - 35m$.
Algebraic manipulation
Find the highest common factor (HCF) shared by 140 and 126.
Types of number
Simplify this expression.
Indices II
A circle has a circumference of 59 cm.
Circles, arcs and sectors
After rounding each number in the calculation to 1 significant figure, find an estimate for the value of $\frac{36.9 + 24.2}{3.8 - 1.2}$. You are required to show all your working.
Limits of accuracy
Indira places $6000$ into an investment that earns simple interest at a rate of $r\%$ each year. After 4 years, her investment is worth $6840$.
Percentages
State the value of the 8 in the number 58 317.
Types of number
The diagram shows a trapezium with the upper parallel side measuring 8 cm, the lower parallel side measuring 14 cm, and a perpendicular height of 6 cm. It is not drawn to scale.
Area and perimeter
Write the following numbers in standard form.
Standard form
The ship’s length, $s$ metres, is given as 287 m when rounded to the nearest metre.
Limits of accuracy
The table gives the counts of people in a town who are left-handed and who are right-handed. Number of people: Left-handed: 8400 Right-handed: 48600 Total: 57000
Relative and expected frequencies
Express $1.2\text{ m}^2$ in $\text{mm}^2$.
Units of measure
Calculate the area of a semicircle whose radius is 10 cm.
Circles, arcs and sectors
Finish the statements below.
Equations
Determine the value of $\sqrt[3]{5832}$.
Powers and roots
A watch has a price of $12400. During a sale, a 16% discount is applied.
Percentages
State the five integers.
Types of number
Arjun is in Delhi, while Haru is in Tokyo. They are both playing an online computer game at the same moment. They begin at 14 45 Tokyo local time. The game takes 3 hours 50 minutes. Delhi local time is 3 hours 30 minutes earlier than Tokyo local time.
Time
The figure depicts an isosceles triangle. One angle is $41^\circ$. The angle at the base is marked $x^\circ$. This diagram is not drawn to scale.
Angles
The stem-and-leaf diagram gives the times, in minutes, that each of 15 people took to finish a race. Stem 1: leaves 6, 6, 7 Stem 2: leaves 1, 3, 3, 4, 5, 6, 7, 7, 7 Stem 3: leaves 0, 1, 1 Key: $1|6$ shows 16 minutes.
Averages and measures of spread
Express $\frac{8}{10}$ as a decimal.
Fractions, decimals and percentages
Express $62000$ millimetres in kilometres.
Units of measure
The diagram illustrates two straight lines intersecting two parallel lines. The angles marked are $50^\circ$, $114^\circ$ and $x^\circ$. The sketch is not drawn to scale.
Angles
Explain why 111 cannot be a prime number.
Types of number
A diagram that is NOT TO SCALE shows a line from point Q heading East. A second line from Q reaches point P and makes an angle of $39^\circ$ above East. North is shown at both Q and P.
Angles
When a car gets older, its selling price goes down. What kind of correlation is this?
Scatter diagrams
Determine the interior angle of a regular 9-sided polygon.
Angles
Filip places $4000$ in an investment for 3 years at a simple interest rate of 2.5\% per year.
Percentages
A, B and C lie on a circle with centre O. NOT TO SCALE.
Circle theorems I
Expand the brackets, then simplify $2(t + w) + 3(w - t)$.
Algebraic manipulation
Calculate $3\frac{1}{8} - 1\frac{3}{4}$ without a calculator. Show every step of your working and give your answer as a mixed number in its simplest form.
Fractions, decimals and percentages
Asha is employed at a café. Her wage is found by the formula wage = hourly rate \times number of hours + bonus. Her hourly rate comes to $11.52. In one week, Asha works 25 hours and is paid a bonus of $5.40.
Money
$\mathcal{E} = \{\text{students in a class}\}$ $C = \{\text{students who play cricket}\}$ $F = \{\text{students who play football}\}$ Altogether, the class has 36 students. 15 students play cricket. 20 students play football. The Venn diagram has 11 in $C$ only and 5 outside both sets.
Sets
$ABC$ forms a right-angled triangle. NOT TO SCALE. $BC = 18\text{ cm}$, and the angle at $C$ is $34^\circ$.
Right-angled triangles
Point A and line $L$ are displayed on the grid.
Equations of linear graphs
Bell A sounds every 22 minutes. Bell B sounds every 14 minutes. Both bells sound at 09 00.
Time
Triangle $ABC$ is similar to triangle $DEF$ mathematically. The measurements are $AC = 5.5$ cm, $BC = 7.3$ cm, $DF = 4.4$ cm and $EF = x$ cm. The diagram is not drawn to scale.
Similarity
The sequence starts with these first four terms: $-3\quad 4\quad 11\quad 18$
Sequences
Calculate $\frac{2}{5}$ of 180.
Fractions, decimals and percentages
Arrange these numbers from least to greatest. $\frac{3}{16}\quad 18.7\%\quad 0.19\quad \frac{9}{50}$
Ordering
State the number that is $9$ more than $-23$.
The four operations
During 16 days, Safia keeps a record of how many dresses she sells. $24\ 6\ 18\ 14\ 27\ 37\ 9\ 16$ $22\ 17\ 16\ 16\ 24\ 20\ 15\ 32$
Averages and measures of spread
Write 24.07839 rounded to 2 decimal places.
Limits of accuracy
$v = u + at$
Introduction to algebra
The diagram contains five arrow shapes labelled A, B, C, D and E, and the shaded one points downwards.
Transformations
Here, $\mathbf{a} = \begin{pmatrix}4\\9\end{pmatrix}$, $\mathbf{b} = \begin{pmatrix}-6\\1\end{pmatrix}$, and $\mathbf{c} = \begin{pmatrix}13\\-2\end{pmatrix}$.
Coordinates
Factorise completely the expression $15v^2 - 3v$.
Algebraic manipulation
Rama asks a group of students how they get to school. The table gives the probability that a student picked at random travels to school by each method. Bus: 0.40, Walk: 0.32, Car: 0.17, Other: (blank).
Relative and expected frequencies
Show every step of your method and give your answer as a mixed number in simplest form.
Fractions, decimals and percentages
The diagram depicts triangle $ABC$. $M$ lies at the midpoint of $AC$. Triangle $ABC$ is turned through $180^\circ$ about centre $M$. The image and the original triangle combine to make a quadrilateral $ABCD$. The diagram gives angle $ABC = 112^\circ$ and angle $BCM = 44^\circ$. Diagram not to scale.
Transformations
Shubhu places $750 into a savings account for 5 years. The account pays simple interest at 1.8% per year.
Percentages
Find the value of $x$ in the equation $5x + 7 = 9x - 3$.
Equations
A coordinate grid displays the straight line labelled $L$.
Equations of linear graphs
A chocolate bar is priced at $3$, whereas a bag of sweets is priced at $5$.
Introduction to algebra
The bag has these cards: 1, 7, 3, 9, 4, 5, 2.
Probability of combined events
List every factor of 32.
Types of number
The diagram illustrates a sector of a circle with radius $r$ cm and sector angle $72^\circ$. The arc length is 9.35 cm. Diagram is not drawn to scale.
Circles, arcs and sectors
The universal set $\mathcal{E} = \{2, 4, 8, 9, 10, 12\}$ is given, with $Q = \{\text{square numbers}\}$ and $R = \{\text{multiples of }4\}$. The diagram of the Venn diagram for sets $Q$ and $R$ within $\mathcal{E}$ is shown.
Sets
Find the highest common factor (HCF) for 48 and 80.
Types of number
All your working must be shown.
Equations
The diagram depicts a rhombus.
Symmetry
The numbers listed are 61, 63, 64, 66, 68 and 69.
Powers and roots
Tara travels by train. Her train departs at 06:48. The trip lasts 12 hours and 35 minutes.
Time
Jamie measures the masses of two orange samples, type A and type B. The stem-and-leaf diagram displays the mass, in grams, of each of 30 oranges of type A. Stem-and-leaf diagram: 17 | 6 8 8 9 18 | 0 1 2 2 4 7 19 | 1 2 2 3 6 7 8 20 | 0 2 5 5 5 6 7 7 8 21 | 1 5 6 8 In the key, 17|6 stands for 176 grams. The table gives: Mean (g): Type A = 195.7, Type B = 215.8 Range (g): Type A = (blank), Type B = 35.
Averages and measures of spread
For triangle $LMN$, the lengths are $LN = 7.5\text{ cm}$ and $MN = 8\text{ cm}$. The side $LM$ is already drawn.
Geometrical constructions
A cube has a surface area of $73.5\text{ cm}^2$.
Surface area and volume
A diagram, not drawn to scale, depicts an equilateral triangle. A line extending beyond one vertex creates two neighbouring angles, each marked $x^\circ$.
Angles
The diagram depicts an isosceles triangle. The angle at the top is $41^{\circ}$. The angle at the bottom is labeled $x^{\circ}$. The two equal sides are indicated.
Angles
Simplify the expression $n^5 \times n$.
Indices II
Solve the inequality $4(2x-3) \ge 43 + 3x$.
Inequalities
Express $0.\dot{4}2$ as a fraction in lowest terms. Show all your working.
Fractions, decimals and percentages
By the close of 2021, there were $27\,000$ rhinos living in the wild. The rhino population is predicted to fall exponentially by $3\%$ each year.
Exponential growth and decay
Determine $n(A \cap B)$.
Sets
A coordinate grid is displayed, with the x-axis running from $-5$ to $7$ and the y-axis running from $-3$ to $8$.
Inequalities
Given $P = 2w + 2h$. $w = 11$ and $h = 9.5$, each correct to 2 significant figures.
Limits of accuracy
Points A, B and C lie on the circumference of a circle, with centre O. The tangent DE meets the circle at C. Angle $BCE = 53^{\circ}$ and angle $ACO = 20^{\circ}$.
Circle theorems I
The diagram shows a triangle with sides of $8.3\,$cm and $16.2\,$cm, together with an angle of $105^{\circ}$. The angle left unknown is marked $y^{\circ}$.
Non-right-angled triangles
The diagram shows axes with $0^{\circ}$ to $360^{\circ}$ on the $x$-axis and values from $-1$ to $1$ on the $y$-axis.
Trigonometric functions
The stem-and-leaf diagram displays the time, in minutes, that each of 15 people needs to finish a race. Stem 1: leaves 6, 6, 7 Stem 2: leaves 1, 3, 3, 4, 5, 6, 7, 7, 7 Stem 3: leaves 0, 1, 1 Key: $1|6$ stands for 16 minutes.
Averages and measures of spread
Express as one fraction in lowest terms: $\frac{10x^2-60x}{x^2-x-30}$.
Algebraic fractions
The diagram depicts a cuboid $ABCDEFGH$. $AB = 15.1\,$cm, $BC = 4.5\,$cm and $CG = 9.2\,$cm.
Pythagoras' theorem and trigonometry in 3D
$ABCD$ forms a rhombus whose side length is $13.6\,$cm. The angle $ABC$ is $41^{\circ}$. $BAC$ is a sector of a circle centred at $B$, and $DAC$ is a sector of a circle centred at $D$.
Circles, arcs and sectors
Complete the statement: When $x = [BLANK]$, then $x + 3 = 8$.
Equations
The table provides some details about Amir’s shopping. Fruit: Oranges, cost per kilogram $2.35$, number of kilograms $3.2$, cost $[BLANK]$. Bananas, cost per kilogram $[BLANK]$, number of kilograms $2.8$, cost $[BLANK]$. Total cost comes to $13.54$.
Money
Factorise completely the expression $42mk - 35m$.
Algebraic manipulation
For the 10 office workers, the scatter diagram displays their salary alongside the value of their car. The vertical axis is labelled Value of car ($), while the horizontal axis is labelled Salary ($).
Scatter diagrams
The exchange rate for Singapore dollars against euros is 1 Singapore dollar = 0.62 euros.
Money
Calculate $7\frac{3}{11} \times 3\frac{3}{10}$.
Fractions, decimals and percentages
Find the highest common factor (HCF) for $140$ and $126$.
Types of number
Write $24.07839$ rounded to $2$ decimal places.
Limits of accuracy
Expand and then simplify $2(t + w) + 3(w - t)$.
Algebraic manipulation
The diagram shows two shapes that are mathematically similar. The larger shape measures $9\text{ cm}$ in width and $6.3\text{ cm}$ in height. The smaller shape measures $5\text{ cm}$ in width and $h\text{ cm}$ in height. The diagrams are marked NOT TO SCALE.
Similarity
The diagram gives a speed-time graph for 16 seconds of a car journey. The vertical axis is Speed (m/s), while the horizontal axis is Time (seconds). From 0 to 12 seconds the speed stays constant at 10 m/s. From 12 to 16 seconds the speed drops linearly to 0. The diagram is marked NOT TO SCALE.
Graphs in practical situations
Using the equation $3^{3p} \times 3^{2p} = 729$, find the value of $p$.
Indices I
$y=2w^2-x$
Algebraic manipulation
The figure shows two Venn diagrams, each one placed inside a rectangle that represents the universal set.
Sets
Determine the lowest common multiple (LCM) for $12x^8$ and $8x^{12}$.
Indices II
For part (a), a circle with centre $O$ is drawn, and $A$, $B$ and $C$ are points on it. The angle $OBA$ is labelled $28^{\circ}$. For part (b), $P$, $Q$ and $R$ are points on a circle. $TU$ is a tangent to the circle at $P$. The angle $TPR$ is $47^{\circ}$ and the angle $PRQ$ is $52^{\circ}$. Both diagrams are marked NOT TO SCALE.
Circle theorems II
The cylinder is solid, with a radius of $5\text{ cm}$ and a height of $8\text{ cm}$.
Surface area and volume
Determine the $n$th term of the sequence.
Sequences
State the number that is 9 greater than $-23$.
The four operations
A rectangle has an area of $55.2\text{ cm}^2$, accurate to 1 decimal place. Its length is $9\text{ cm}$, rounded to the nearest cm.
Limits of accuracy
The straight line $y = x + 1$ cuts the curve $y = x^2 + x - 3$ at two points.
Graphs of functions
$x$ varies inversely with the square root of $w$. If $w = 16$, then $x = 3$.
Proportion
The reaction times recorded by some students are listed in the table. Reaction time $t$ (seconds): $0 < t \leq 6$ has frequency 18; $6 < t \leq 10$ has frequency 16.
Histograms
Simplify the expression $\dfrac{ax - 2a - x + 2}{a^2 - 1}$.
Algebraic fractions
The derivative of $2ax^7 + 3x^k$ is $42x^6 + 15x^{k-1}$.
Differentiation
The figure depicts a parallelogram $OPQT$. The position vector of $P$ is $\mathbf{a}$ and the position vector of $T$ is $\mathbf{b}$. $K$ lies on $PQ$ such that $PK : KQ = 3 : 1$. The straight lines $OK$ and $TQ$ are produced until they intersect at $X$. The sketch is marked NOT TO SCALE.
Vector geometry
Use $v = u + at$.
Introduction to algebra
Express 62 000 millimetres in kilometres.
Units of measure
The diagram depicts two straight lines that intersect and cut across two parallel lines. It is labelled NOT TO SCALE. An angle on the upper parallel line is shown as $50^{\circ}$. Another angle on the lower parallel line is shown as $114^{\circ}$. The interior angle between the intersecting lines is marked $x^{\circ}$.
Angles
Explain why 111 cannot be a prime number.
Types of number
A diagram labelled NOT TO SCALE shows points $Q$ and $P$. At $Q$, a vertical arrow marked North appears, and at $P$, another vertical arrow marked North appears. From $Q$, a horizontal arrow marked East appears. The line joining $Q$ to $P$ is drawn at an angle of $39^{\circ}$ above the East direction at $Q$.
Angles
Work out $3\frac{1}{8} - 1\frac{3}{4}$ without a calculator. Show all of your working, and give your answer as a mixed number in simplest form.
Fractions, decimals and percentages
Express 90 as a product of prime factors.
Types of number