May/June 2025 Paper 33 Worked Answers (IGCSE Maths 0580 Core)

30 questions · 30 marks · 90 minutes

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1. Question 1a

   1 marksPrime Factorisation
   180 = 4 × 45 = 2² × 9 × 5 = 2² × 3² × 5. 4 × 9 × 5 uses non-primes. 2 × 3 × 30 has 30. 2³ × 3 × 5 has wrong powers.
   Examiner tips

   • Always check that all your factors are prime numbers (2, 3, 5, 7, 11, etc.).
   • Use a factor tree or repeated division to systematically find all prime factors.

2. Question 1b

   1 marksHCF
   48 = 2⁴ × 3, 60 = 2² × 3 × 5. HCF = 2² × 3 = 12. 4 is too small. 240 is LCM. 6 misses a factor.
   Examiner tips

   • HCF uses the LOWEST power of each COMMON prime factor.
   • Double-check by confirming both numbers are divisible by your answer.

3. Question 1c

   1 marksLCM
   12 = 2² × 3, 18 = 2 × 3². LCM = 2² × 3² = 36. 6 is HCF. 216 is product. 72 is 2 × LCM.
   Examiner tips

   • LCM uses the HIGHEST power of EACH prime factor from either number.
   • The LCM should be divisible by both original numbers.

4. Question 2a

   1 marksSimplifying Expressions
   7a − 3a = 4a, 2b + 5b = 7b. Answer: 4a + 7b. 10a + 7b adds a terms. 4a − 3b subtracts b terms. 11ab combines unlike terms.
   Examiner tips

   • Only combine terms that have exactly the same variable(s).
   • Write out the like terms grouped together before combining to avoid sign errors.

5. Question 2b

   1 marksExpanding and Simplifying
   15x − 10 − 8x − 2 = 7x − 12. 7x − 8 adds −10 and −2 wrong. 23x − 12 adds x terms. 7x + 12 has wrong sign.
   Examiner tips

   • When there's a negative sign before a bracket, it changes the sign of every term inside when expanding.
   • Expand both brackets first, then collect like terms.

6. Question 2c

   1 marksFactorisation
   HCF = 4ab. So 4ab(2a + 3b). 4(2a²b + 3ab²), 2ab(4a + 6b), and ab(8a + 12b) don't fully factorise or use wrong HCF.
   Examiner tips

   • 'Factorise completely' means find the largest possible common factor including both numbers and variables.
   • Check your answer by expanding - you should get back to the original expression.

7. Question 3a

   1 marksStandard Form
   3.27 × 10⁴ = 32700 (move decimal 4 places right). 327000 uses 10⁵. 3270 uses 10³. 0.000327 uses negative power.
   Examiner tips

   • Positive powers of 10 make numbers bigger - move decimal right.
   • Count carefully and add zeros as needed to fill the gaps.

8. Question 3b

   1 marksStandard Form
   0.00072 = 7.2 × 10⁻⁴ (4 places). 7.2 × 10⁻³ uses 3. 72 × 10⁻⁵ not standard form. 7.2 × 10⁴ positive power.
   Examiner tips

   • For small numbers (less than 1), the power of 10 is always negative.
   • The first number must be between 1 and 10 for valid standard form.

9. Question 3c

   1 marksStandard Form Calculations
   8.4 ÷ 4.2 = 2, 10⁶ ÷ 10² = 10⁴. Answer: 2 × 10⁴. 2 × 10⁸ adds powers. 2 × 10³ subtracts wrong. 4.2 × 10⁴ doesn't divide numbers.
   Examiner tips

   • When dividing powers of 10, subtract the indices: 10ᵃ ÷ 10ᵇ = 10⁽ᵃ⁻ᵇ⁾
   • Deal with the numbers and powers of 10 separately.

10. Question 4a

    1 marksSolving Linear Equations
    4x = 25 − 9 = 16, x = 4. x = 8.5 is (25−9)÷2. x = 16 forgets to divide. x = 3.5 uses wrong subtraction.
    Examiner tips

    • Always perform the same operation to both sides of the equation.
    • Check your answer by substituting back into the original equation.

11. Question 4b

    1 marksSolving Linear Equations
    12x − 8 = 2x + 10, 10x = 18, x = 18/10 = 9/5. x = 2 rounds. x = −1 has sign error. x = 18/10 not simplified.
    Examiner tips

    • Expand brackets carefully, especially with negative signs.
    • Always simplify fractions to their lowest terms in your final answer.

12. Question 4c

    1 marksSimultaneous Equations
    From eq2: x = 12 − 4y. Sub: 3(12−4y) + 2y = 16, 36 − 12y + 2y = 16, −10y = −20, y = 2, x = 4. Check: 3(4)+2(2)=16✓, 4+4(2)=12✓
    Examiner tips

    • Always check your answer by substituting both values into both original equations.
    • Write out each step clearly to avoid arithmetic errors.

13. Question 5a

    1 marksFrequency Polygons
    Frequency polygons use class midpoints. Midpoint of 20-30 is (20+30)/2 = 25. 20 and 30 are boundaries. 50 is sum.
    Examiner tips

    • Midpoint = (lower boundary + upper boundary) ÷ 2
    • This is different from histograms which use class widths.

14. Question 5b

    1 marksMean from Grouped Data
    Mean = (5×5 + 15×10 + 25×15)/(5+10+15) = (25+150+375)/30 = 550/30 ≈ 18.3. Closest is 17. 20, 15, and 10 are incorrect calculations.
    Examiner tips

    • The mean from grouped data is always an estimate because we don't know exact values.
    • Use midpoints to represent all values in each class.

15. Question 6a

    1 marksInequalities
    x ≥ −2 includes −2. x < 3 excludes 3. So: −2, −1, 0, 1, 2. −2, −1, 0, 1, 2, 3 includes 3. −1, 0, 1, 2 excludes −2. −2, −1, 0, 1 excludes 2.
    Examiner tips

    • ≤ means 'less than or equal to' - include the value. < means 'strictly less than' - exclude the value.
    • Don't forget to include 0 if it's in the range.

16. Question 6b

    1 marksSolving Inequalities
    5x − 3x < 10 − 2, 2x < 8, x < 4. x > 4 has wrong inequality sign. x < 6 uses wrong arithmetic. x < 1.5 divides wrong.
    Examiner tips

    • Solve inequalities the same way as equations, keeping the inequality sign pointing the same way.
    • Only reverse the sign when multiplying or dividing by a negative number.

17. Question 7a

    1 marksPythagoras' Theorem
    c² = 8² + 15² = 64 + 225 = 289, c = 17 cm. 23 cm adds. 289 cm forgets root. 12 cm subtracts.
    Examiner tips

    • The hypotenuse is always opposite the right angle and is the longest side.
    • Remember: Pythagoras only works for right-angled triangles.

18. Question 7b

    1 marksArea of Triangle
    Area = ½ × 6 × 10 = 30 cm². 60 cm² forgets ½. 16 cm² adds. 136 cm² finds c² + area.
    Examiner tips

    • In a right-angled triangle, the two shorter sides can be used as base and height.
    • Always remember the ½ in the triangle area formula.

19. Question 7c

    1 marksTrigonometry
    tan θ = 7/10 = 0.7, θ = tan⁻¹(0.7) = 35.0°. 55.0° is complement. 44.4° uses wrong ratio. 70.0° doubles.
    Examiner tips

    • SOHCAHTOA: Sin = O/H, Cos = A/H, Tan = O/A
    • Make sure your calculator is in degree mode, not radians.

20. Question 8a

    1 marksSubstitution
    y = 8/4 = 2. 32 multiplies. 4 uses same value. 0.5 inverts.
    Examiner tips

    • Substitute carefully - replace x with the given value.
    • Check whether to multiply or divide by looking at the formula.

21. Question 8b

    1 marksReciprocal Graphs
    6 = k/2, so k = 12. 3 is 6/2. 8 is 2+6. 4 is 6−2.
    Examiner tips

    • From y = k/x, rearranging gives k = xy.
    • For any point on the curve y = k/x, the product xy always equals k.

22. Question 8c

    1 marksSolving Equations Graphically
    10/x = 4, x = 10/4 = 2.5. x = 0.4 is 4/10. x = 6 is 10−4. x = 40 is 10×4.
    Examiner tips

    • On a graph, draw a horizontal line at y = 4 and read off the x-coordinate where it crosses the curve.
    • Algebraically: if a/x = b, then x = a/b.

23. Question 9a

    1 marksPercentage Decrease
    25% of 80 = 20. Sale price = 80 − 20 = $\$60$. $\$100$ adds. $\$55$ uses 31.25%. $\$20$ is just the discount.
    Examiner tips

    • For a decrease, multiply by (100 - percentage)/100 or find the percentage and subtract.
    • 25% = 1/4, so 25% of 80 = 80/4 = 20.

24. Question 9b

    1 marksPercentage Increase
    Increase = 12. Percentage = (12/40) × 100 = 30%. 23.1% uses 52 as base. 12% is just 12. 130% is ratio × 100.
    Examiner tips

    • Always divide by the ORIGINAL value when finding percentage change.
    • Formula: Percentage increase = (increase ÷ original) × 100.

25. Question 10a

    1 marksDistance-Time Graphs
    Horizontal line means distance isn't changing, so the object is stationary. The object is moving at constant speed would be a diagonal line. The object is accelerating and The object is decelerating would be curves.
    Examiner tips

    • Gradient of distance-time graph = speed. Horizontal = zero gradient = zero speed = stationary.
    • A steeper line means faster speed.

26. Question 10b

    1 marksSpeed
    Speed = distance/time = 90/1.5 = 60 km/h. 135 km/h multiplies. 91.5 km/h adds. 45 km/h uses 2 hours.
    Examiner tips

    • Speed = Distance ÷ Time. Remember the formula triangle: D = S × T.
    • Always include units in your answer.

27. Question 10c

    1 marksInterpreting Distance-Time Graphs
    Speed = gradient. Steepest gradient = fastest speed. The horizontal section is zero speed. The curved section shows changing speed. The lowest section is about position not speed.
    Examiner tips

    • On distance-time graphs: gradient = speed. Steeper = faster.
    • Look for the section where the line rises most quickly.

28. Question 11a

    1 marksArc Length
    Arc = (72/360) × 2π × 10 = (1/5) × 20π = 4π cm. 20π cm is full circumference. 2π cm uses wrong formula. 72π cm uses angle directly.
    Examiner tips

    • Arc length formula: (θ/360) × 2πr for angle θ in degrees.
    • Simplify the fraction (72/360 = 1/5) before multiplying.

29. Question 11b

    1 marksSector Area
    Area = (60/360) × π × 36 = (1/6) × 36π = 6π cm². 36π cm² is full circle. 12π cm² uses wrong fraction. 2π cm² uses arc formula.
    Examiner tips

    • Sector area formula: (θ/360) × πr² for angle θ in degrees.
    • 60° is 1/6 of 360°, so the sector is 1/6 of the full circle.

30. Question 11c

    1 marksPerimeter of Sector
    Perimeter = arc + 2r = 3π + 10 cm. 3π cm is just arc. 8π cm adds wrong. (3π + 5) cm only adds one radius.
    Examiner tips

    • Perimeter of sector = arc length + radius + radius = arc + 2r.
    • Don't forget both straight edges of the sector.