October/November 2025 Paper 33 Worked Answers (IGCSE Maths 0580 Core)
27 questions · 27 marks · 90 minutes
Question papers and mark schemes are copyright Cambridge International. We do not reproduce them: the worked answers here are written by The Practice Book. Have the paper open alongside. Get the official paper from Cambridge
Worked answers for 27 questions
- 300 = 4 × 75 = 2² × 3 × 25 = 2² × 3 × 5². 4 × 75 and 6 × 50 use non-primes. 2³ × 3 × 5 has wrong powers.Examiner tips
- Use a factor tree or systematic division to find all prime factors
- Always check your answer by multiplying the factors back together
- Remember that 1 is not a prime number
- 90 = 2 × 3² × 5, 150 = 2 × 3 × 5². HCF = 2 × 3 × 5 = 30. 10 is too small. 450 is LCM. 15 misses 2.Examiner tips
- Always find prime factorisations first when dealing with HCF
- For HCF, take the LOWEST power of each COMMON prime factor
- Check your answer divides exactly into both original numbers
- 14 = 2 × 7, 21 = 3 × 7. LCM = 2 × 3 × 7 = 42. 7 is HCF. 294 is product. 35 is wrong.Examiner tips
- For LCM, take the HIGHEST power of each prime factor that appears in either number
- The LCM must be divisible by both original numbers
- LCM × HCF = product of the two numbers - use this to check your answer
- 2⅔ = 8/3, 1⅘ = 9/5. (8/3) × (9/5) = 72/15 = 24/5 = 4⅘. 3 7/15, 2⅖, and 5⅓ miscalculate.Examiner tips
- Always convert mixed numbers to improper fractions before multiplying
- Cancel common factors before multiplying to make calculations easier
- Remember to convert back to mixed number if required
- 3¾ = 15/4, 1½ = 3/2. (15/4) ÷ (3/2) = (15/4) × (2/3) = 30/12 = 5/2 = 2½. 5⅝ multiplies. 2¼ and 3 wrong.Examiner tips
- Remember KCF: Keep the first fraction, Change to multiplication, Flip the second fraction
- Convert mixed numbers to improper fractions first
- Cancel common factors before multiplying when possible
- 7a + 2a = 9a, −4b + 6b = 2b. Answer: 9a + 2b. 5a + 2b, 9a − 10b, and 11ab have calculation errors.Examiner tips
- Underline or circle like terms in different colours to avoid mistakes
- Always check signs carefully when combining terms
- Terms with different letters cannot be combined
- 6x − 8 + 4x + 12 = 10x + 4. 10x − 20 adds wrong. 6x + 4 misses 4x. 10x + 20 wrong constant.Examiner tips
- Expand each bracket separately first, writing out all terms
- Pay special attention to negative signs when multiplying
- Collect like terms as a final step
- HCF = 6a. So 6a(2a − 3b). 6(2a² − 3ab), 2a(6a − 9b), and a(12a − 18b) don't fully factorise.Examiner tips
- The question says 'completely' - this means you must extract ALL common factors
- Find the HCF of the coefficients AND the common variable factors
- Check by expanding your answer - you should get the original expression
- 6x + 8 = 5x − 5, x = −13. x = 13 wrong sign. x = 3 and x = −3 wrong calculations.Examiner tips
- Expand all brackets before collecting terms
- Show clear steps when rearranging the equation
- Check your answer by substituting back into the original equation
- 6000 × 1.05³ = 6000 × 1.157625 = . uses simple interest. uses 1 year. uses wrong rate.Examiner tips
- Use the multiplier method: multiply by (1 + r)^n where r is the decimal rate
- For 5% increase, multiply by 1.05 not 0.05
- Show your working clearly, especially the power calculation
- 600 × 0.75² = 600 × 0.5625 = . uses simple. uses 1 year. uses wrong multiplier.Examiner tips
- Depreciation uses the same formula as compound interest but with a multiplier less than 1
- 25% depreciation means multiply by 0.75 (100% - 25% = 75%)
- Check your answer is less than the original value
- 7.2 × 10⁶ = 7200000 (6 places right). 720000 uses 10⁵. 72000000 uses 10⁷. 0.0000072 uses negative power.Examiner tips
- Positive power = big number (move decimal right)
- The power tells you exactly how many places to move
- Fill in zeros as placeholders when needed
- 0.00095 = 9.5 × 10⁻⁴ (4 places). 9.5 × 10⁻³ counts 3. 95 × 10⁻⁵ not standard form. 9.5 × 10⁴ positive power.Examiner tips
- The first part must be between 1 and 10
- Negative power = small number (less than 1)
- Count decimal places carefully from the original position to the new position
- 2.5 × 4 = 10 = 1 × 10. 10³ × 10⁻⁵ = 10⁻². Total: 1 × 10⁻¹. 1 × 10⁻² doesn't adjust. 10 × 10⁻² not standard. 1 × 10⁸ adds powers.Examiner tips
- Multiply the number parts separately from the power parts
- When multiplying powers of 10, add the indices
- Always check your final answer is in proper standard form (1 ≤ a < 10)
- y = 1 + 3 + 2 = 6. 0 is when x=1 or x=2. 2 is constant only. −2 miscalculates.Examiner tips
- Be careful with negative numbers, especially when squaring
- (-1)² = 1, not -1
- Remember that negative × negative = positive
- Vertex at x = −b/(2a) = 4/2 = 2. −2 wrong sign. 4 is −b/a. 1 is constant.Examiner tips
- Use the formula x = -b/(2a) to find the x-coordinate of the vertex
- The vertex is the minimum point for a positive quadratic (U-shape)
- Check by substituting back to find the y-coordinate if needed
- x² = 2, x = ±√2 ≈ ±1.41. x = ±2 solves x² = 4. x = 2 only forgets negative. x = 4 is 2².Examiner tips
- When a line crosses a curve, set the equations equal to each other
- Always consider both positive and negative square roots
- Use the graph to verify your algebraic solution
- Area = ½ × 8 × 3 = 12 cm². Volume = 12 × 15 = 180 cm³. 360 cm³ forgets ½. 24 cm³ is just area. 12 cm³ is ½ area.Examiner tips
- For any prism: Volume = Area of cross-section × length
- Remember to halve for triangle area: ½ × b × h
- Volume units are always cubic (cm³)
- TSA = 2(20) + 30 + 40 + 50 = 40 + 120 = 160 cm². 120 cm² forgets triangles. 140 cm² forgets one rectangle. 200 cm² doubles.Examiner tips
- A triangular prism has 5 faces: 2 triangles and 3 rectangles
- Remember to count BOTH triangular end faces
- Surface area units are always squared (cm²)
- c² = 400 + 441 = 841. c = 29 cm. 41 cm adds. 841 cm forgets root. 20.5 cm averages.Examiner tips
- Pythagoras: a² + b² = c² where c is the longest side (hypotenuse)
- Remember to square root at the end to find the length
- 20-21-29 is a Pythagorean triple
- tan θ = 15/8 = 1.875. θ = tan⁻¹(1.875) = 61.9°. 28.1° is complement. 58.0° and 45° wrong calculations.Examiner tips
- Remember SOH CAH TOA: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent
- Use the correct trig function based on what sides you're given
- Make sure your calculator is in degree mode
- Total = 8. P(green) = 5/8. 3/8 is P(yellow). 5/3 and 8/5 are ratios not probabilities.Examiner tips
- Probability = Number of favourable outcomes ÷ Total number of outcomes
- Probability is always between 0 and 1 (or 0% and 100%)
- Simplify fractions where possible
- P(R then B) = (4/7) × (3/6) = 12/42 = 2/7. 12/49 uses replacement. 12/42 not simplified. 1/7 wrong.Examiner tips
- Without replacement: total decreases by 1 after each pick
- Multiply along branches for 'and' (consecutive events)
- Always simplify your final answer
- P(same) = 2/7 + 1/7 = 3/7. 2/49 multiplies. 1/7 is just P(BB). 4/7 is P(different).Examiner tips
- For 'or' with mutually exclusive events, ADD the probabilities
- Same colour = (both red) OR (both blue)
- Check: P(same) + P(different) should equal 1
- 180° rotation: (x, y) → (−x, −y). (2, 5) → (−2, −5). (5, 2) swaps. (−5, −2) swaps and negates. (2, −5) reflects in x-axis.Examiner tips
- 180° rotation about origin: (x, y) → (−x, −y)
- Think of it as a half-turn - the point ends up directly opposite
- 90° clockwise: (x, y) → (y, −x), 90° anticlockwise: (x, y) → (−y, x)
- Reflection in y = x swaps coordinates: (1, 4) → (4, 1). (−1, −4) and (−4, −1) negate. (1, −4) reflects in x-axis.Examiner tips
- Reflection in y = x: (x, y) → (y, x) - just swap the coordinates
- Reflection in y = −x: (x, y) → (−y, −x)
- The line y = x is diagonal through the origin at 45°
- Vector from centre: (3−1, 2−1) = (2, 1). Multiply by 3: (6, 3). Add to centre: (7, 4). (9, 6), (10, 7), and (4, 3) wrong calculations.Examiner tips
- Vector method: Image = Centre + SF × (Original − Centre)
- The distance from centre to image is SF times the distance from centre to original
- For SF > 1, image is larger and further from centre
Sit this paper in the app
Timed mock papers, instant marking and worked solutions for every question, free.