IGCSE Mathematics 0580 Paper 22, Oct/Nov 2025

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0580/22 · Oct/Nov 2025

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The temperature at 4 am is $- 12^{\circ} C$ . The temperature at 4 pm is $21^{\circ} C$ . Find the increase in temperature from 4 am to 4 pm.

[1]

Find all the common factors of 8 and 12.

[1]

A cuboid has length $6 cm$ , width $5 cm$ and height $2.5 cm$ . Work out the volume of the cuboid.

[2]

A computer costs $560. In a sale, this cost is reduced by 20%. Find the cost of the computer in the sale.

[2]

The pie chart shows some information about the way 600 students travel to school.

Work out the number of students that walk to school.

[2]

120 of the students travel to school by car. The remaining students travel by bus. Complete the pie chart.

[3]

ABCD is a quadrilateral. E lies on CD and AE is parallel to BC. EA = ED. Find

angle ABC

[1]

angle AED

[1]

angle DAB.

[2]

Represent the inequality $- 4 < x ⩽ 3$ on the number line.

[2]

Kemi buys $p$ pens that each cost 40 cents. She pays with $20. Write an expression, in terms of $p$ , for the change, in cents, Kemi receives from the $20.

[2]

Rajid has a full bottle of juice. He drinks $\frac{1}{3}$ of the full bottle on Monday. He drinks $\frac{3}{7}$ of the full bottle on Tuesday. Find the fraction of the bottle of juice remaining.

[3]

$b = d m + 2 m k$

$d = 3.14$ , $m = 7.92$ and $k = 10.16$ . By rounding each value correct to 1 significant figure, work out an estimate for $b$ .

[3]

Rearrange the formula to make $m$ the subject.

[2]

In the diagram, $S$ lies on $P Q$ and $T$ lies on $P R$ . $S T$ is parallel to $Q R$ .

Explain why triangle $P Q R$ is mathematically similar to triangle $P S T$ . Give a geometrical reason for each statement you make.

[3]

$S T = 3 cm$ , $Q R = 9 cm$ and $P S = 5 cm$ . Work out $P Q$ .

[2]

The area of triangle $P S T$ is $2 k {cm}^{2}$ . Find, in terms of $k$ , the area of quadrilateral $Q R T S$ .

[2]

A fitness club has 100 members. 60 swim ( $S$ ). 70 cycle ( $C$ ). 25 do not swim or cycle.

Complete the Venn diagram.

[3]

b(i)

One member of the fitness club is chosen at random. For this member, find $P (C)$

[1]

b(ii)

One member of the fitness club is chosen at random. For this member, find $P (S \cap C)$

[1]

b(iii)

One member of the fitness club is chosen at random. For this member, find $P (S \cup C^{'})$

[1]

$M = 2^{7} \times 3^{3} \times 5^{2}$

Write $14 M$ as a product of its prime factors. Give your answer in index form.

[2]

$R$ is an integer. $\frac{M}{R}$ is a cube number. Find the smallest possible value of $R$ .

[2]

Find the value of

$3^{2} \div 3^{- 2}$

[2]

$16^{- \frac{3}{2}}$

[2]

Factorise.

$x^{2} - 64$

[1]

$5 x (x - 2 y) + 6 (x - 2 y)^{2}$

[2]

$A B$ and $B D$ are two sides of a regular 15-sided polygon. $A B$ and $B C$ are two sides of a regular $n$ -sided polygon. Angle $D B C = 14^{\circ}$ . Work out the value of $n$ .

[4]

$B$ is the point $(- 3, 1)$ and $D$ is the point $(- 5, 9)$ . $B D$ is a diagonal of the kite $A B C D$ .

The ratio of the lengths of the diagonals $B D : A C = 2 : 3$ . Work out the length of $A C$ . Give your answer as a surd in its simplest form.

[5]

Find the coordinates of the midpoint of $B D$ .

[2]

The diagonal $A C$ of the kite passes through the midpoint of $B D$ . Find an equation of $A C$ . Give your answer in the form $y = m x + c$ .

[4]

Rationalise the denominator and simplify. $\frac{20}{4 + \sqrt{6}}$

[3]

The diagram shows a box in the shape of a cuboid. Mala has a straight rod of length $10 cm$ . Show that this rod does not fit completely inside the box.

[3]

$f (x) = \frac{21}{2 x - 1}, x \neq \frac{1}{2} g (x) = 3 x + 4$

a(i)

Find $g (2)$

[1]

a(ii)

Find $g f (- 1)$

[2]

a(iii)

Find $f^{- 1} (x)$ .

[3]

Solve $f (x) = g (x)$ .

[5]

On the diagram, sketch the graph of $y = \cos x$ for $0^{\circ} ⩽ x ⩽ 360^{\circ}$ .

[2]

Solve the equation $2 \cos x + \sqrt{3} = 0$ for $0^{\circ} ⩽ x ⩽ 360^{\circ}$ .

[3]

A graph with equation $y = x^{2} + b x + c$ has a minimum point at $(- 5, 12)$ . Find the value of $b$ and the value of $c$ .

[3]

The diagram shows a major sector, $A O B$ , of a circle. The sector angle is $240^{\circ}$ and the radius is $12 cm$ .

Show that the length of the major arc $A B$ is $16 π cm$ .

[1]

$O A$ is joined to $O B$ to form a cone. Work out the volume of the cone. Give your answer in the form $\frac{(a \sqrt{b}) π}{3}$ where $a$ is an integer and $b$ is a prime number.

[6]

$27^{n x} = (9^{x})^{2}$ Find the value of $n$ .

[2]

Mark scheme

0580/22 · Oct/Nov 2025

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1 · 1 mark

1 mark
33

2 · 1 mark

1 mark
1, 2, 4

3 · 2 marks

2 marks
75

4 · 2 marks

2 marks
448

5a · 2 marks

2 marks
150

5b · 3 marks

3 marks
Correctly completed pie chart including labels of car [72] and bus [138]

6a · 1 mark

1 mark
105

6b · 1 mark

1 mark
80

6c · 2 marks

2 marks
125

7 · 2 marks

2 marks
Open circle at –4, closed circle at 3, connected by a line

8 · 2 marks

2 marks
2000 – 40p final answer

9 · 3 marks

3 marks
$\frac{5}{21}$

10a · 3 marks

3 marks
184 with 3, 8 and 10 shown

10b · 2 marks

2 marks
[m =] $\frac{b}{d + 2 k}$ final answer

11a · 3 marks

3 marks
Three correct pairs with correct reasons $P Q R = P S T$ and corresponding angles $P R Q = P T S$ and corresponding angles $Q P R = T P S$ and common angle OR Two correct pairs with correct reasons and correct similarity condition e.g. AAA

11b · 2 marks

2 marks
15

11c · 2 marks

2 marks
16k

12a · 3 marks

3 marks
Venn diagram with S only: 5, intersection: 55, C only: 15, outside: 25 or Venn diagram with S only: 0.05, intersection: 0.55, C only: 0.15, outside: 0.25

12b(i) · 1 mark

1 mark
$\frac{70}{100}$ oe

12b(ii) · 1 mark

1 mark
$\frac{55}{100}$ oe

12b(iii) · 1 mark

1 mark
$\frac{85}{100}$ oe

13a · 2 marks

2 marks
$2^{8} \times 3^{3} \times 5^{2} \times 7^{[1]}$ cao

13b · 2 marks

2 marks
50 or –86 400

14a · 2 marks

2 marks
81

14b · 2 marks

2 marks
$[\pm] \frac{1}{64}$

15a · 1 mark

1 mark
(x + 8)(x – 8) final answer

15b · 2 marks

2 marks
(x – 2y)(11x – 12y) oe final answer

16 · 4 marks

4 marks
36

17a · 5 marks

5 marks
$3 \sqrt{17}$ cao

17b · 2 marks

2 marks
(– 4, 5)

17c · 4 marks

4 marks
$y = \frac{1}{4} x + 6$ final answer

18 · 3 marks

3 marks
$2 (4 - \sqrt{6})$ or $8 - 2 \sqrt{6}$

19 · 3 marks

3 marks
$\sqrt{99}$ and states this is less than 10 oe

20a(i) · 1 mark

1 mark
10

20a(ii) · 2 marks

2 marks
–17

20a(iii) · 3 marks

3 marks
$\frac{21 + x}{2 x}$ or $\frac{21}{2 x} + \frac{1}{2}$ oe final answer

20b · 5 marks

5 marks
$\frac{5}{3}$ and $- \frac{5}{2}$ oe

21a · 2 marks

2 marks
Correct sketch to go through (0, 1), (180, –1) and (360, 1)

21b · 3 marks

3 marks
150 and 210

22 · 3 marks

3 marks
[b =] 10 [c =] 37

23a · 1 mark

M1
$\frac{240}{360} \times 2 \times π \times 12$ oe

23b · 6 marks

6 marks
$\frac{(256 \sqrt{5}) π}{3}$ cao

24 · 2 marks

2 marks
[n =] $\frac{4}{3}$

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