October/November 2025 Paper 32 Worked Answers (IGCSE Maths 0580 Core)
27 questions · 27 marks · 90 minutes
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Worked answers for 27 questions
- 200 = 8 × 25 = 2³ × 5². 4 × 50 and 8 × 25 use non-primes. 2² × 5² has wrong power of 2.Examiner tips
- Use a factor tree to systematically break down the number
- Check your answer by multiplying the prime factors back together
- Remember: 2, 3, 5, 7, 11, 13... are prime numbers
- 84 = 2² × 3 × 7, 120 = 2³ × 3 × 5. HCF = 2² × 3 = 12. 4 is too small. 840 is LCM. 24 is wrong.Examiner tips
- HCF = Highest Common Factor - look for the largest number that divides into both
- Use prime factorisation and take the LOWEST power of common primes
- The HCF is always smaller than or equal to both numbers
- 18 = 2 × 3², 24 = 2³ × 3. LCM = 2³ × 3² = 72. 6 is HCF. 432 is product. 48 is wrong.Examiner tips
- LCM = Lowest Common Multiple - the smallest number both divide into exactly
- Use the HIGHEST power of each prime factor from either number
- The LCM is always greater than or equal to the larger number
- 1⅔ = 5/3, 2¼ = 9/4. (5/3) × (9/4) = 45/12 = 15/4 = 3¾. 3 11/12, 2¼, and 4⅙ miscalculate.Examiner tips
- Always convert mixed numbers to improper fractions before multiplying
- Simplify before multiplying if possible to make calculations easier
- Convert back to a mixed number and ensure it's in simplest form
- 3½ = 7/2, 1¾ = 7/4. (7/2) ÷ (7/4) = (7/2) × (4/7) = 4/2 = 2. 6⅛ multiplies. 1¾ and 2½ wrong.Examiner tips
- KFC: Keep the first fraction, Flip the second, Change to multiplication
- Convert mixed numbers to improper fractions first
- Cancel common factors before multiplying for easier calculation
- 4a + 3a = 7a, −2b + 5b = 3b. Answer: 7a + 3b. a + 3b, 7a − 7b, and 10ab have calculation errors.Examiner tips
- Collect like terms systematically - deal with one variable at a time
- Be very careful with negative signs when adding
- Unlike terms (different letters) cannot be combined
- 12x − 6 − 2x − 10 = 10x − 16. 10x + 4 adds wrong. 14x − 16 doesn't subtract 2x. 10x − 6 miscalculates constant.Examiner tips
- Expand brackets one at a time before simplifying
- Be especially careful with negative signs in front of brackets
- Check by substituting a simple value like x = 1
- HCF = 3a. So 3a(2a + 3b). 3(2a² + 3ab), a(6a + 9b), and 3a(2a + 3) don't fully factorise.Examiner tips
- 'Completely' means extract ALL common factors - both numerical and algebraic
- Check your answer by expanding back - you should get the original expression
- Find the HCF of coefficients and the lowest power of each common variable
- 5x − 10 = 3x + 12, 2x = 22, x = 11. x = 1, x = 7, and x = −11 have calculation errors.Examiner tips
- Expand brackets on both sides first
- Collect all x terms on one side, all numbers on the other
- Always check your answer by substituting back into the original equation
- 8000 × 1.04² = 8000 × 1.0816 = . uses simple interest. uses 1 year. uses wrong rate.Examiner tips
- For compound interest, use the multiplier method: Amount = P(1 + r/100)^n
- 4% increase means multiply by 1.04 each year
- Do not confuse with simple interest which is P(1 + rn/100)
- 25000 × 0.85² = 25000 × 0.7225 = . uses simple. is 1 year. wrong.Examiner tips
- Depreciation is like reverse compound interest - use (1 - r/100)^n
- 15% depreciation means 85% remains, so multiply by 0.85
- After 2 years, multiply by 0.85²
- 3.45 × 10⁵ = 345000 (5 places right). 34500 uses 10⁴. 3450000 uses 10⁶. 0.0000345 uses negative power.Examiner tips
- Positive power = move decimal right = bigger number
- The power tells you exactly how many places to move
- Fill in zeros as needed when moving the decimal
- 0.00084 = 8.4 × 10⁻⁴ (4 places). 8.4 × 10⁻³ counts 3. 84 × 10⁻⁵ not standard form. 8.4 × 10⁴ positive power.Examiner tips
- Standard form: A × 10ⁿ where 1 ≤ A < 10
- Negative powers for numbers less than 1
- Count decimal places moved to the RIGHT to get the negative power
- 6.3 ÷ 9 = 0.7. 10⁴ ÷ 10⁻³ = 10⁷. 0.7 × 10⁷ = 7 × 10⁶. 7 × 10¹ subtracts powers. 7 × 10⁷ doesn't adjust. 0.7 × 10⁷ not standard form.Examiner tips
- Deal with numbers and powers of 10 separately
- When dividing powers: subtract the exponents (a ÷ 10⁻ⁿ = a × 10ⁿ)
- Adjust final answer to proper standard form if needed
- y = 16 − 8 − 3 = 5. −3 is just constant. 13 adds. 9 miscalculates.Examiner tips
- Replace every x with the given value in brackets
- Calculate powers before multiplication, then subtraction
- Double-check arithmetic, especially with negative signs
- Minimum at x = −b/(2a) = 6/2 = 3. Or roots at 1 and 5, midpoint is 3. −3 wrong sign. 6 and 5 are other values.Examiner tips
- Use x = -b/(2a) to find the x-coordinate of the turning point
- The turning point lies on the line of symmetry of the parabola
- If roots are known, the minimum is exactly halfway between them
- x² − 5x + 6 = 0, (x−2)(x−3) = 0, x = 2 or x = 3. x = −2 and x = −3, x = 1 and x = 6, and x = 5 and x = 6 have wrong values.Examiner tips
- X-intercepts occur where y = 0, so solve the quadratic = 0
- Factorise or use the quadratic formula
- Check by substituting your answers back into the equation
- Arc = (60/360) × 2π × 15 = (1/6) × 30π = 5π cm. 30π cm is full circumference. 2.5π cm uses radius/2. 15π cm uses π × radius.Examiner tips
- Arc length = (θ/360) × 2πr or (θ/360) × πd
- 60° is 1/6 of a full circle (60/360 = 1/6)
- Leave answer in terms of π when asked
- Area = (90/360) × π × 100 = (1/4) × 100π = 25π cm². 100π cm² is full circle. 5π cm² is arc. 50π cm² uses wrong fraction.Examiner tips
- Sector area = (θ/360) × πr²
- 90° is 1/4 of a full circle
- Remember: area involves r² (squared), arc length involves r (not squared)
- c² = 81 + 144 = 225. c = 15 cm. 21 cm adds. 225 cm forgets root. 10.5 cm halves.Examiner tips
- Pythagoras: a² + b² = c² (c is the hypotenuse)
- The hypotenuse is always the longest side
- Don't forget to take the square root at the end
- tan θ = 8/6 = 1.333. θ = tan⁻¹(1.333) = 53.1°. 36.9° is complement. 48.6° and 75° wrong calculations.Examiner tips
- SOHCAHTOA: Sin = O/H, Cos = A/H, Tan = O/A
- Use inverse function (tan⁻¹) to find the angle from a ratio
- Make sure calculator is in degrees mode
- Total = 12. P(blue) = 4/12 = 1/3. 4/12 not simplified. 1/2 is P(red). 2/3 is P(not green).Examiner tips
- P(event) = favourable outcomes / total outcomes
- Always simplify your fraction
- Check that all probabilities are between 0 and 1
- P(BB) = (2/5) × (1/4) = 2/20 = 1/10. 4/25 uses replacement. 2/20 not simplified. 1/5 wrong.Examiner tips
- Without replacement: total decreases by 1, and specific colour decreases if that colour was picked
- Multiply along branches for 'and' probability
- First pick: 2/5 blue, Second pick: 1/4 blue (one fewer blue, one fewer total)
- P(same) = 1/6 + 1/10 + 1/30 = 5/30 + 3/30 + 1/30 = 9/30 = 3/10. 1/1800 multiplies. 1/6 and 7/15 wrong.Examiner tips
- OR means ADD probabilities (for mutually exclusive events)
- AND means MULTIPLY probabilities
- Find common denominator when adding fractions
- Vector from centre: (8−4, 6−2) = (4, 4). Multiply by 1/2: (2, 2). Add to centre: (6, 4). (4, 3), (2, 2), and (12, 10) wrong calculations.Examiner tips
- For enlargement: Image = Centre + SF × (Original - Centre)
- SF < 1 means the image is smaller (reduction)
- The centre of enlargement stays in the same place
- 90° anticlockwise: (x, y) → (−y, x). (3, 5) → (−5, 3). (5, −3) is clockwise. (−3, −5) is 180°. (5, 3) swaps.Examiner tips
- 90° anticlockwise: (x, y) → (-y, x)
- 90° clockwise: (x, y) → (y, -x)
- 180°: (x, y) → (-x, -y)
- Reflection in y = x: swap coordinates. (4, −2) → (−2, 4). (−4, 2), (2, −4), and (4, 2) apply wrong transformations.Examiner tips
- Reflection in y = x: swap x and y coordinates
- Reflection in y = -x: swap and change both signs
- Reflection in x-axis: (x, y) → (x, -y)
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