IGCSE Mathematics 0580/13 May/June 2025 Past Paper with Answers

Question 1

Write the number five hundred and three thousand and forty-one in figures.

1. 503041
2. 50341
3. 503410
4. 5030041

Question 2

Write 4/5 as a decimal.

1. 0.8
2. 0.45
3. 4.5
4. 0.08

Question 3

Write 0.65 as a fraction in its simplest form.

1. 13/20
2. 65/100
3. 6/5
4. 13/25

Question 4 · Answer A

8² = 8 × 8 = 64. 16 is 8 × 2. 80 is 8 × 10. 88 concatenates 8 and 8.

Question 5 · Answer A

³√125 = 5 because 5 × 5 × 5 = 125. 25 is 125÷5. 41.67 is 125÷3. 11.18 is √125.

Question 6 · Answer A

0.0000382 = 3.82 × 10⁻⁵ (decimal moves 5 places right). 3.82 × 10⁻⁴ counts 4 places. 38.2 × 10⁻⁶ is not standard form. 3.82 × 10⁵ has positive power.

Question 7 · Answer A

72 = 8 × 9 = 2³ × 3² (using prime factors). 8 × 9 uses non-prime factors. 2² × 3³ has wrong powers. 2⁴ × 3 is 48, not 72.

Question 8 · Answer A

1 cm = 10⁻² m. Number = 10⁻²/(1.2 × 10⁻¹⁰) = (1/1.2) × 10⁸ ≈ 8.3 × 10⁷. 8.3 × 10⁹ forgets cm conversion. 8.3 × 10⁻⁸ has wrong sign. 1.2 × 10⁸ doesn't divide properly.

Question 9 · Answer A

7 − (−4) = 7 + 4 = 11. 3 subtracts 4 instead. -11 makes both negative. -3 subtracts and makes negative.

Question 10 · Answer A

5.4² = 29.16, 2.7³ = 19.683. Sum = 48.843. √48.843 ≈ 6.99 ≈ 6.80 (3 sf). 8.10 adds powers wrong. 46.3 forgets square root. 2.61 takes 4th root.

Question 11 · Answer A

4x + 2x = 6x, −3y − 5y = −8y. Answer: 6x − 8y. 6x − 2y adds y terms wrong. 2x − 8y subtracts x terms. 6x + 8y ignores negative signs.

Question 12 · Answer A

(2a³b²)³ = 2³ × a⁹ × b⁶ = 8a⁹b⁶. 6a⁹b⁶ multiplies 2 × 3 instead of 2³. 8a⁶b⁵ adds indices wrong. 2a⁹b⁶ forgets to cube the 2.

Question 13 · Answer A

HCF is 4a. 8a² ÷ 4a = 2a, 12ab ÷ 4a = 3b. So 4a(2a + 3b). 4(2a² + 3ab), 2a(4a + 6b), and a(8a + 12b) don't fully factorise.

Question 14 · Answer A

From eq 2: y = 2x − 4. Sub: 3x + 2(2x − 4) = 13, 7x − 8 = 13, 7x = 21, x = 3. y = 6 − 4 = 2. Check: 3(3)+2(2)=13✓, 2(3)−2=4✓

Question 15 · Answer A

1.24 = 124/100 = 31/25 (dividing by 4). 124/100 not simplified. 62/50 not fully simplified. 6/5 is 1.2 not 1.24.

Question 16 · Answer A

Let x = 0.6363... 100x = 63.6363... 99x = 63, x = 63/99 = 7/11. 63/99 not simplified. 63/100 wrong method. 21/33 not fully simplified.

Question 17 · Answer A

n=1: 1+3=4, n=2: 4+3=7, n=3: 9+3=12. 4, 5, 6 adds 1 each time. 3, 6, 11 starts at n=0. 4, 7, 10 uses 3n+1.

Question 18 · Answer A

y = kx, 8 = 5k, k = 1.6. When x = 15: y = 1.6 × 15 = 24. 18 adds 10. 40 is 5 × 8. 12 is 15 − 3.

Question 19 · Answer A

3⅔ + 1⅚ = 11/3 + 11/6 = 22/6 + 11/6 = 33/6 = 11/2 = 5½. 4 7/9 wrong denominator. 5 3/6 not simplified. 4½ adds wrong.

Question 20 · Answer A

A parallelogram looks the same after rotation of 180°, so order = 2. 1 is for shapes with no symmetry. 4 is for squares. 0 is not valid.

Question 21 · Answer A

Isosceles triangle, so angles ABC = ACB. Sum = 180° − 50° = 130°. Each base angle = 65°. 50° uses apex angle. 130° is sum of base angles. 40° subtracts wrong.

Question 22 · Answer A

A cone has a circular base and tapers to a single point (vertex) at the top. Cylinder has two circular faces. Pyramid has polygonal base. Sphere is round everywhere.

Question 23 · Answer A

Area of triangle = ½ × 8 × 5 = 20 cm². Volume = 20 × 12 = 240 cm³. 480 cm³ forgets to halve. 120 cm³ is just the area × 6. 200 cm³ miscalculates.

Question 24 · Answer A

4.5 × 2 = 9. 10⁶ × 10⁻³ = 10³. Answer: 9 × 10³. 9 × 10⁹ adds indices. 9 × 10⁻⁹ makes index negative. 6.5 × 10³ adds instead of multiplies.

Question 25 · Answer A

Inside B but outside A means B ∩ A' (B intersection not-A). A ∩ B' is A but not B. A ∪ B' is union. B − A' is wrong notation.

Question 26 · Answer A

Using n(F∪T) = n(F) + n(T) − n(F∩T): 40 = 25 + 20 − x, x = 5. 15 subtracts wrong way. 45 adds. 0 ignores overlap.

Question 27 · Answer A

When both variables increase together, it's positive correlation. Negative correlation is when one increases and other decreases. No correlation means no relationship. Perfect correlation means exactly on a line.

Question 28 · Answer A

6 km = 600000 cm. Map distance = 600000 ÷ 50000 = 12 cm. 120 cm uses wrong conversion. 1.2 cm divides by 500000. 300 cm multiplies wrong.

Question 29 · Answer A

Area scale = 20000² = 4×10⁸. Actual = 5 × 4×10⁸ cm² = 2×10⁹ cm² = 0.2 km² (since 1 km² = 10¹⁰ cm²). 2 km² is ×10. 0.02 km² is ÷10. 20 km² way too big.

Question 30 · Answer A

180° rotation about origin: (x, y) → (-x, -y). So (2, 3) → (-2, -3). (2, -3) reflects in x-axis. (-2, 3) reflects in y-axis. (3, 2) swaps coordinates.

Question 31 · Answer A

72 km/h = 72 × 1000 ÷ 3600 = 72000 ÷ 3600 = 20 m/s. 259.2 m/s multiplies by 3.6. 0.02 m/s divides too much. 1.2 m/s divides by 60 only.

Question 32 · Answer A

1.5 hours = 5400 s. Distance = 8 × 5400 = 43200 m = 43.2 km. 12 km is 8 × 1.5. 4.32 km divides by 10000. 432 km uses metres.

Question 33 · Answer A

IQR = Q3 − Q1 = 73 − 45 = 28. 59 is average. 118 is sum. 14 is IQR ÷ 2.