IGCSE Mathematics (0580) topical questions with answers
Every IGCSE Mathematics topic, with free past-paper-style questions and full mark schemes. Pick a chapter to practise its questions, or generate a complete 0580-format paper in seconds, from easy warm-ups to A*-level "brutal" papers.
Paper 1 · Non-calculator (Core · 80 marks · 1h 30m); Paper 2 · Non-calculator (Extended · 100 marks · 2h); Paper 3 · Calculator (Core · 80 marks · 1h 30m); Paper 4 · Calculator (Extended · 100 marks · 2h).
Sample IGCSE Mathematics questions
A mixed set from across the syllabus. Questions marked A* challenge come from the hardest bank.
Question 1Structured · 9 marksA* challenge
A bag contains 3 red counters and n blue counters. Two counters are taken at random, without replacement. The probability that the two counters are the same colour is 37.
(a)
Form an equation in n.
[3]
(b)
Solve the equation to find n.
[3]
(c)
The bag is returned to its original contents. Three counters are now taken without replacement. Find the probability that exactly two of them are red.
[3]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
7(6 + n2 − n) = 3(n + 3)(n + 2) or equivalent
3n + 3 × 2n + 2 + nn + 3 × n − 1n + 2M1 combined over the common denominator M1 equated and cross-multiplied correctly A1
3
(b)
n = 4
expanded to a quadratic = 0 M1 factorise / formula M1 n = 4 (reject the other root) A1
3
(c)
1235
37 × 26 × 45M1 × 3 for the arrangements M1 1235A1
3
Question 2Structured · 5 marksA* challenge
Two points A and B are on horizontal ground, 80 m apart, in a straight line with the foot of a vertical tower. B is between A and the tower. The angle of elevation of the top of the tower from A is 21° and from B is 38°. Calculate the height of the tower.
[5]
Show answers and mark scheme
Part
Answer and marking points
Marks
60.4 m
h = (80 + x) tan 21 and h = x tan 38, where x = distance from B to tower M2 equate and solve for x M1 x = 77.3 A1 60.4 A1
5
Question 3Structured · 5 marks
The vertices of triangle ABC are A(1, 0), B(7, 4) and C(13, 0).
(a)
Show that the triangle is isosceles.
[3]
(b)
Find the area of the triangle.
[2]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
AB = BC
AB = √(62 + 42) M1 BC = √(62 + 42) M1 AB = BC so isosceles A1
3
(b)
24
½ × 12 × 4 M1 24 A1
2
Question 4Structured · 3 marks
x = 7220.072
(a)
Write x correct to 2 significant figures.
[1]
(b)
Write x correct to 1 decimal place.
[1]
(c)
Write x in standard form, correct to 3 significant figures.
[1]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
7200
7200 B1
1
(b)
7220.1
7220.1 B1
1
(c)
7.22 × 103
7.22 × 103B1
1
Question 5Structured · 4 marks
The equation of a line is 4x + 3y = 12.
(a)
Find the gradient of the line.
[2]
(b)
Find the coordinates of the points where the line crosses the x-axis and the y-axis.
[2]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
−43
rearranges to y = … M1 −43A1
2
(b)
(3, 0) and (0, 4)
(3, 0) B1 (0, 4) B1
2
Question 6Structured · 4 marks
Triangle PQR is similar to triangle XYZ. PQ = 8 cm, QR = 12 cm and XY = 12 cm.
(a)
Find YZ.
[2]
(b)
The area of triangle PQR is 48 cm². Find the area of triangle XYZ.
[2]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
18 cm
scale factor 1.5 M1 18 A1
2
(b)
108 cm²
× 1.52M1 108 A1
2
Frequently asked questions
Are these real 0580 past paper questions?
No. They are original questions written to match the IGCSE Mathematics 0580 syllabus and the style of real Cambridge papers, so you can practise beyond the past papers you have already done.
Do the Mathematics questions have answers?
Yes. Every question on these pages has its answer, and every generated paper comes with a full Cambridge-style mark scheme you can download as a PDF.
Can I make a full IGCSE Mathematics practice paper?
Yes. Choose the paper (Paper 1 Non-calculator, Paper 2 Non-calculator, Paper 3 Calculator, Paper 4 Calculator), the topics, the marks and the difficulty, and the generator builds a fresh paper with the same number and style of questions as the real exam.
Want a mentor, not just a paper?
Riyaz Shaikh has mentored IGCSE, IB and A Level students for 15+ years, one-on-one, online and in Mumbai. The hardest questions on this site are the ones he uses to find a student's weak points.
Original Cambridge-style practice questions written for mentorriyaz.in. These are not Cambridge past paper questions, and this site is not affiliated with or endorsed by Cambridge University Press & Assessment. Updated 2026-11-04. Paper generator · mentorriyaz.in