May/June 2025 Paper 12 Worked Answers (IGCSE Maths 0580 Core)
47 questions · 47 marks · 90 minutes
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Worked answers for 47 questions
- Twenty-four thousand = 24000, thirty-five = 35. Combined: 24035. 24350 incorrectly places the 35 in the hundreds position. 2435 drops the thousands. 240035 adds an extra zero.Examiner tips
- Read the number in parts: 'twenty-four thousand' and 'thirty-five' separately before combining.
- 2/5 = 2 ÷ 5 = 0.4. 0.25 is 1/4. 2.5 puts 2 in the wrong place. 0.04 divides incorrectly.Examiner tips
- Remember that 1/5 = 0.2, so 2/5 = 0.4.
- Check your answer by multiplying back: 0.4 × 5 = 2.
- 2/5 = 0.4 = 40%. 25% is 1/4. 4% is 0.04 as percentage. 52% adds 5 and 2 then uses 7.Examiner tips
- Use the relationship: fraction → decimal → percentage.
- Remember to multiply by 100 when converting decimal to percentage.
- √81 = 9 because 9 × 9 = 81. 8 is close but 8² = 64. 40.5 is 81÷2. 18 is 81÷4.5.Examiner tips
- Memorize perfect squares up to 12² = 144.
- Check by squaring your answer: 9 × 9 = 81.
- 10⁴ = 10 × 10 × 10 × 10 = 10000. 1000 is 10³. 40 is 10 × 4. 100000 is 10⁵.Examiner tips
- For powers of 10, the exponent equals the number of zeros.
- 10⁴ has 4 zeros: 10000.
- 1 cm = 10 mm, so 6.5 cm = 6.5 × 10 = 65 mm. 0.65 mm divides by 10 instead. 650 mm multiplies by 100. 6.5 mm doesn't convert.Examiner tips
- Memorize key conversions: 1 cm = 10 mm, 1 m = 100 cm.
- When converting to smaller units, multiply.
- Perpendicular lines meet at right angles, which is 90°. 180° is a straight line. 45° is half a right angle. 60° is a common angle but not for perpendicular lines.Examiner tips
- The word 'perpendicular' always means 90°.
- Look for the small square symbol indicating a right angle.
- 1 week = 7 days, so 8 weeks = 8 × 7 = 56 days. 48 uses 6 days per week. 64 uses 8 days. 80 uses 10 days.Examiner tips
- Memorize: 1 week = 7 days.
- Use your times tables: 8 × 7 = 56.
- 3/5 of 20 = 3/5 × 20 = 12 squares. 15 is 3/4 of 20. 8 is 2/5 of 20. 4 is 1/5 of 20.Examiner tips
- Find the unit fraction first (1/5 = 4), then multiply by the numerator.
- Check: 12 out of 20 = 12/20 = 3/5.
- The reciprocal of 1/5 is 5/1 = 5. 0.2 is 1/5 as a decimal (not the reciprocal). -5 uses negative sign incorrectly. 1/5 is the original number.Examiner tips
- Reciprocal means 'flip the fraction'.
- The reciprocal of a/b is b/a.
- Check: 1/5 × 5 = 1.
- 3⁻² = 1/3² = 1/9. 9 ignores the negative index. -9 confuses negative index with negative number. 1/6 calculates 1/(3×2).Examiner tips
- Negative index means reciprocal: a⁻ⁿ = 1/aⁿ.
- First calculate the positive power, then take the reciprocal.
- (8 − 6) ÷ 2 + 1 = 2 ÷ 2 + 1 = 1 + 1 = 2. 8 − (6 ÷ 2) + 1 = 2 gives 8 - 3 + 1 = 6. 8 − 6 ÷ (2 + 1) = 2 gives 8 - 2 = 6. 8 − (6 ÷ 2 + 1) = 2 gives 8 - 4 = 4.Examiner tips
- Work systematically through each possible bracket position.
- Remember BIDMAS: Brackets, Indices, Division, Multiplication, Addition, Subtraction.
- (2 + 3) × 4 − 5 = 5 × 4 − 5 = 20 − 5 = 15. 2 + 3 × (4 − 5) = 15 gives 2 + (-3) = -1. 2 + (3 × 4) − 5 = 15 gives 2 + 12 - 5 = 9. (2 + 3 × 4) − 5 = 15 gives 14 - 5 = 9.Examiner tips
- Brackets change what gets calculated first.
- Test each option by calculating step by step.
- Convert to 24ths: 7/24, 8/24 (1/3), 9/24 (3/8), 10/24 (5/12). In order: 7/24, 1/3, 3/8, 5/12. 1/3, 7/24, 3/8, 5/12 swaps first two. 7/24, 3/8, 1/3, 5/12 has wrong middle order. 5/12, 3/8, 1/3, 7/24 is reversed.Examiner tips
- Find the LCM of all denominators.
- Convert each fraction to equivalent fractions with the common denominator.
- Compare numerators to order.
- Surface area = 2(lw + lh + wh) = 2(4×3 + 4×2 + 3×2) = 2(12 + 8 + 6) = 2(26) = 52 cm². 24 cm³ is volume with wrong units. 26 cm² only adds once. 48 cm² uses wrong formula.Examiner tips
- Remember there are 3 pairs of opposite faces.
- Use formula: SA = 2(lw + lh + wh).
- Check units: surface area uses cm², volume uses cm³.
- Volume = l × w × h = 6 × 4 × 2 = 48 cm³. 48 cm² has wrong units. 24 cm³ is 6×4 only. 12 cm³ is 6×2 only.Examiner tips
- Volume = length × width × height.
- Always use cubic units for volume.
- Check you've used all three dimensions.
- The mode is the most frequent value. 5 appears 3 times, 3 appears 2 times, others once. Mode = 5. 3 is the second most frequent. 4 is the median. 7 is the range.Examiner tips
- Mode = most frequent.
- Count the frequency of each value.
- There can be more than one mode, or no mode if all values appear equally.
- Range = highest − lowest = 18 − 3 = 15. 12 is the middle value. 11 is the mean. 18 is just the highest value.Examiner tips
- Range = largest − smallest.
- The range measures spread, not an average.
- Always subtract the smallest from the largest.
- 6 values, so median is average of 3rd and 4th: (8 + 11)/2 = 9.5. 8 is just the 3rd value. 10.33 is the mean. 11 is just the 4th value.Examiner tips
- Order the data first.
- For even count, average the two middle values.
- For odd count, the median is the exact middle value.
- Total = mean × count = 8 × 5 = 40. Sum of known = 5 + 6 + 9 + 12 = 32. Fifth = 40 − 32 = 8. 10 adds incorrectly. 7 subtracts wrong. 40 is just the total.Examiner tips
- Mean = Total ÷ Count, so Total = Mean × Count.
- Work backwards: find total first, then subtract known values.
- 67 × 99 = 67 × 100 − 67 = 6700 − 67 = 6633. 6700 is 67 × 100. 6767 is 67 × 101. 6600 is 67 × 100 − 100.Examiner tips
- Follow the given method exactly.
- n × 99 = n × 100 − n is based on n × (100 − 1).
- Check by estimating: 67 × 99 ≈ 67 × 100 = 6700, so answer should be just below 6700.
- A trapezium has exactly one pair of parallel sides. A parallelogram has 2 pairs. A rhombus is a special parallelogram. A kite has no parallel sides.Examiner tips
- Trapezium = exactly 1 pair of parallel sides.
- Parallelogram, rectangle, rhombus, square all have 2 pairs.
- Learn the properties of each quadrilateral type.
- A rectangle does NOT have all sides equal (that would be a square). All angles are 90°, Opposite sides are parallel, and Diagonals are equal in length are all true properties of rectangles.Examiner tips
- A square is a special rectangle with all sides equal.
- All rectangles have 4 right angles and equal diagonals.
- Read the question carefully: look for NOT.
- Area = ½(a + b) × h = ½(8 + 6) × 5 = ½ × 14 × 5 = 35 cm². 70 cm² forgets to halve. 40 cm² is 8 × 5. 30 cm² is 6 × 5.Examiner tips
- Formula: A = ½(a + b) × h where a and b are parallel sides.
- Add the parallel sides FIRST, then multiply by height, then halve.
- The ½ is essential - don't forget it!
- y = (3 + 2)(3 − 1) = 5 × 2 = 10. 4 is (3+1). 6 is (3+3). 8 is 5+3.Examiner tips
- Substitute carefully into each bracket.
- Calculate each bracket separately first.
- Then multiply the two bracket values together.
- The x-intercepts occur when y = 0, so (x + 3)(x − 2) = 0. This gives x = −3 or x = 2. x = 3 and x = −2, x = −3 and x = −2, and x = 3 and x = 2 have signs confused.Examiner tips
- Set y = 0 to find x-intercepts.
- If (x + a) = 0, then x = −a.
- If (x − b) = 0, then x = b.
- The x-coordinate of the vertex is halfway between the roots. Roots are x = 1 and x = 5. Vertex x = (1 + 5)/2 = 3. 2 and C are not midpoint. 5 is a root.Examiner tips
- The vertex x-coordinate = (root1 + root2) / 2.
- Use the line of symmetry, which is halfway between the roots.
- The vertex is the turning point of the parabola.
- The line of symmetry is x = (−1 + 3)/2 = 1. y = 1 is wrong axis. x = −1 and x = 3 are the x-intercepts, not the line of symmetry.Examiner tips
- Line of symmetry is always vertical (x = ...) for a parabola.
- It passes through the vertex.
- Find the midpoint of the two roots.
- The minimum of y = (x + 2)(x − 4) is at x = 1, y = 3 × (−3) = −9. Since y = 5 is above the minimum, the horizontal line will cross the parabola twice.Examiner tips
- A horizontal line above a U-shaped parabola's vertex intersects twice.
- A horizontal line below the vertex intersects zero times.
- A line at the vertex level intersects exactly once (tangent).
- n² + 20 = 45, so n² = 25, n = 5 (positive). 25 is n². 12.5 is 25/2. 6.7 is √45.Examiner tips
- Set up an equation from the words.
- Work backwards: undo operations in reverse order.
- Remember to take the square root at the end.
- Area = ½ × base × height = ½ × 5 × 4 = 10 cm². 20 cm² forgets to halve. 9 cm² is 5+4. 18 cm² is 9×2.Examiner tips
- Area of triangle = ½ × base × height.
- The ½ is essential - triangles are half of rectangles.
- Make sure to use perpendicular height.
- Moving right adds to x: −2 + 3 = 1. Moving down subtracts from y: 5 − 4 = 1. New coordinates: (1, 1). (−5, 9), (1, 9), and (−5, 1) have sign errors.Examiner tips
- Right = +x, Left = −x.
- Up = +y, Down = −y.
- Apply changes to each coordinate separately.
- (3, −2) + (5, −7) = (3 + 5, −2 + (−7)) = (8, −9). (−2, 5) subtracts instead of adds. (8, 5) and (−2, −9) have mixed errors.Examiner tips
- Translation = add the vector to coordinates.
- Be careful with negative numbers.
- Work out x and y separately.
- Point (3, 4) is 2 units above y = 2. Reflected point is 2 units below y = 2, at (3, 0). (−3, 4) reflects in y-axis. (3, −4) reflects in x-axis. (1, 4) reflects in x = 2.Examiner tips
- For reflection in y = k, only the y-coordinate changes.
- Distance from point to mirror = distance from mirror to image.
- The x-coordinate stays the same for horizontal mirror lines.
- For enlargement SF 3 from origin: multiply each coordinate by 3. (2, 1) → (6, 3). (5, 4) adds 3. (2/3, 1/3) divides by 3. (6, 1) only multiplies x.Examiner tips
- For enlargement from origin: new = original × scale factor.
- Both x and y must be multiplied by the same scale factor.
- Centre at origin makes calculation simpler.
- For 90° anticlockwise about origin: (x, y) → (−y, x). So (3, 1) → (−1, 3). (1, −3) is 90° clockwise. (−3, −1) is 180°. (1, 3) swaps coordinates only.Examiner tips
- 90° anticlockwise: (x, y) → (−y, x).
- 90° clockwise: (x, y) → (y, −x).
- 180°: (x, y) → (−x, −y).
- 38.7 ≈ 40, 4.2 ≈ 4, 19.8 ≈ 20. Estimate = (40 × 4) ÷ 20 = 160 ÷ 20 = 8. 160 forgets to divide. 0.8 divides by 200. 6 uses wrong values.Examiner tips
- Round to 1 s.f. first, then calculate.
- 1 s.f. means one non-zero digit.
- Check your answer is reasonable compared to the original calculation.
- 48 = 2⁴ × 3, 72 = 2³ × 3². HCF = 2³ × 3 = 8 × 3 = 24. 12 is a factor but not highest. 144 is LCM. 6 is too small.Examiner tips
- Use prime factorization for both numbers.
- HCF uses lowest powers of common primes.
- LCM uses highest powers of all primes.
- Primes between 20 and 30: 23 and 29. 21 = 3 × 7, 25 = 5², 27 = 3³ are not prime.Examiner tips
- Check divisibility by small primes (2, 3, 5, 7).
- Odd numbers ending in 5 are divisible by 5.
- Numbers whose digits sum to a multiple of 3 are divisible by 3.
- Open circle at 1 means 'not including 1' so >. Filled circle at 6 means 'including 6' so ≤. Thus 1 < x ≤ 6. 1 ≤ x < 6, 1 < x < 6, and 1 ≤ x ≤ 6 have wrong inequality symbols.Examiner tips
- Open circle = strict inequality (< or >).
- Filled circle = inclusive inequality (≤ or ≥).
- Check both ends of the inequality.
- The overlap is the intersection, A ∩ B (elements in both A AND B). A ∪ B is union (in A or B or both). A' is complement of A. A − B is A without B.Examiner tips
- ∩ means AND (intersection) - elements in both sets.
- ∪ means OR (union) - elements in either set.
- ' means NOT (complement).
- 1 2/3 = 5/3, 2 1/4 = 9/4. (5/3) × (9/4) = 45/12 = 15/4 = 3 3/4. 2 1/6 adds instead of multiplies. 3 11/12 and 4 1/2 are calculation errors.Examiner tips
- Convert to improper fractions: a b/c = (ac + b)/c.
- Multiply numerators, multiply denominators.
- Simplify and convert back to mixed number.
- Nearest mm = nearest 0.1 cm. Half of 0.1 = 0.05. Bounds: 5.8 − 0.05 = 5.75, 5.8 + 0.05 = 5.85. 5.7 cm ≤ length < 5.9 cm uses nearest cm. 5.79 cm ≤ length < 5.81 cm uses 0.01. 5.8 cm ≤ length ≤ 5.8 cm gives no range.Examiner tips
- Lower bound = value − half the rounding unit.
- Upper bound = value + half the rounding unit.
- 1 mm = 0.1 cm, so half = 0.05 cm.
- HCF of 12a and 8ab is 4a. 12a ÷ 4a = 3, 8ab ÷ 4a = 2b. So 12a + 8ab = 4a(3 + 2b). 4(3a + 2ab), 2a(6 + 4b), and a(12 + 8b) don't fully factorise.Examiner tips
- Find the HCF of coefficients and variables separately.
- Check by expanding: 4a × 3 + 4a × 2b = 12a + 8ab.
- The bracket should be fully simplified.
- (3x + 2)(x − 5) = 3x² − 15x + 2x − 10 = 3x² − 13x − 10. 3x² − 15x − 10 doesn't combine middle terms. 3x² − 13x + 10 and 3x² + 17x − 10 have sign errors.Examiner tips
- Use FOIL or grid method.
- Be careful with negative signs.
- Always collect like terms at the end.
- Check: 2(3) + 3(2) = 6 + 6 = 12 ✓. 4(3) − 2 = 12 − 2 = 10. Using elimination or substitution methods to solve simultaneously.Method:Using elimination or substitution to solve: 2x + 3y = 12, 4x − y = 5. Find x = 3, y = 2.Examiner tips
- Use elimination: multiply equations to match coefficients.
- Or use substitution: express one variable in terms of the other.
- Always check your answer in BOTH equations.
- Large arc = π × 10 = 10π. Small arc = π × 6 = 6π. Straight edges = 2(10 − 6) = 8. Perimeter = 10π + 6π + 8 = 16π + 8. 16π cm forgets straight edges. (10π + 6π + 8) cm doesn't simplify. (16π + 4) cm uses wrong straight length.Examiner tips
- Arc of semicircle = πr (half of 2πr).
- Don't forget the straight edges connecting the semicircles.
- Leave answer in terms of π as requested.
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