October/November 2025 Paper 13 Worked Answers (IGCSE Maths 0580 Core)
29 questions · 29 marks · 90 minutes
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Worked answers for 29 questions
- Eight million = 8000000, forty-two thousand = 42000, nine = 9. Combined: 8042009. 842009 drops a zero. 8420009 misplaces digits. 80042009 adds extra zeros.Examiner tips
- Write each part with its full value first (8000000 + 42000 + 9), then combine by aligning place values
- Check your answer has the correct number of digits for a number in the millions (7 digits)
- 0.03 = 0.03, 3% = 0.03, 0.3 = 0.3, 3/10 = 0.3. Order: 0.03 = 3% < 0.3 = 3/10. Answer: 0.03, 3%, 0.3, 3/10. 3%, 0.03, 0.3, 3/10, C, D have wrong orders.Examiner tips
- Always convert to the same form (decimals are usually easiest) before comparing
- Remember that 3% = 3/100 = 0.03 and 3/10 = 0.3
- 9² = 9 × 9 = 81. 18 is 9 × 2. 99 concatenates. 27 is 9 × 3.Examiner tips
- Remember: n² means n × n (the number multiplied by itself)
- Memorize common square numbers up to 15² to save time in exams
- ³√8 = 2 because 2 × 2 × 2 = 8. 4 is 8/2. 2.67 is 8/3. 512 is 8³.Examiner tips
- Cube root is the inverse of cubing: if a³ = b, then ³√b = a
- Learn the first few cube numbers: 1, 8, 27, 64, 125
- Negative × negative = positive. −8 × (−4) = 32. −32 ignores sign rule. −12 adds. 12 adds with wrong sign.Examiner tips
- Remember the sign rules: (+)(+) = +, (−)(−) = +, (+)(−) = −, (−)(+) = −
- Calculate the magnitude first (8 × 4 = 32), then determine the sign
- 90 = 2 × 45 = 2 × 9 × 5 = 2 × 3² × 5. 9 × 10 uses non-primes. 2 × 3 × 15 has 15. 2² × 3 × 5 has wrong power.Examiner tips
- Use a factor tree to systematically break down the number
- Check your answer by multiplying back: 2 × 9 × 5 = 90
- 09:25 to 12:25 = 3 hours. 12:25 to 12:08 = −17 min. 3h − 17min = 2h 43min. 3 hours 17 minutes adds wrong. 2 hours 17 minutes subtracts wrong. 3 hours 43 minutes adds both.Examiner tips
- Use a step method: 09:25 → 10:00 (35 min) → 12:00 (2 hours) → 12:08 (8 min) = 2h 43min
- Remember there are 60 minutes in an hour when borrowing
- In y = mx + c, gradient m = 4. 7 is y-intercept. 4/7 divides them. 11 adds them.Examiner tips
- In y = mx + c form, m is always the gradient (slope) and c is always the y-intercept
- The gradient tells you how much y increases for every 1 unit increase in x
- Y-axis crossing at x = 0: y = 0 + 9 = 9. Point (0, 9). (9, 0) is x-intercept format. (0, 2) uses gradient. (−4.5, 0) is x-intercept.Examiner tips
- The y-intercept is where x = 0, so the answer is always (0, c) for y = mx + c
- Don't confuse with x-intercept which is where y = 0
- 8x + 3x = 11x, −2y − 5y = −7y. Answer: 11x − 7y. 5x − 7y subtracts x terms. 11x + 3y adds y terms. 4xy combines illegally.Examiner tips
- Only like terms (same variable and power) can be combined
- Be careful with negative signs - write each term with its sign
- 6x − 3x = 17 − 5, 3x = 12, x = 4. x = 2.44 divides by 9. x = 12 is 3x. x = 7.33 divides by 3 wrong.Examiner tips
- Always do the same operation to both sides of the equation
- Check your answer by substituting back: 6(4) + 5 = 29 and 3(4) + 17 = 29
- Back bearing = 72° + 180° = 252°. 072° is original. 288° is 360 − 72. 108° is 180 − 72.Examiner tips
- Back bearing formula: add 180° if original < 180°; subtract 180° if original > 180°
- Bearings are always measured clockwise from North and written with 3 digits
- 0.00027 = 2.7 × 10⁻⁴ (4 places). 2.7 × 10⁻³ counts 3. 27 × 10⁻⁵ not standard form. 2.7 × 10⁴ positive power.Examiner tips
- For numbers less than 1, the power of 10 is negative
- Standard form requires a number between 1 and 10 multiplied by a power of 10
- A 5-sided polygon is a pentagon. Hexagon has 6, quadrilateral has 4, octagon has 8.Examiner tips
- Learn the polygon names: triangle (3), quadrilateral (4), pentagon (5), hexagon (6), heptagon (7), octagon (8), nonagon (9), decagon (10)
- The prefixes come from Greek numbers
- Exterior angle = 360° ÷ 10 = 36°. 144° is interior angle. 1440° is sum of interior. 40° uses 9.Examiner tips
- Formula: Exterior angle = 360° ÷ n (where n is number of sides)
- Remember: exterior + interior = 180°, and sum of all exterior angles = 360°
- P(black) = 1 − 0.3 − 0.45 = 0.25. 0.75 adds red and yellow. 0.15 subtracts wrong. 1.75 adds all.Examiner tips
- Probabilities of all mutually exclusive outcomes must sum to 1
- Always check your answer is between 0 and 1
- HCF = 5ab. So 5ab(2a − 3b). 5(2a²b − 3ab²), ab(10a − 15b), and 5ab(2a − 3) don't fully factorise.Examiner tips
- Find HCF of coefficients (10 and 15) = 5
- Find HCF of variables: both have 'a' and 'b', so HCF includes 'ab'
- 4⅓ = 13/3, 2⅚ = 17/6. 13/3 − 17/6 = 26/6 − 17/6 = 9/6 = 3/2 = 1½. 2½ adds. 1⅙ and 2⅙ calculate wrong.Examiner tips
- Convert to improper fractions first to avoid borrowing complications
- Always simplify your final answer
- 5n − 3 = 47, 5n = 50, n = 10. So 10th term. 9th term and 11th term off by one. 44th term uses 47−3.Examiner tips
- To find which term has a given value, set the nth term = value and solve for n
- Check: 5(10) - 3 = 50 - 3 = 47
- Actual = 12 × 40 = 480 cm = 4.8 m. 480 m doesn't convert to m. 0.3 m divides by 40. 48 m wrong conversion.Examiner tips
- Scale 1 : 40 means actual = 40 × model
- Remember to convert units: 100 cm = 1 m
- From eq1: x = 10 − 2y. Sub: 3(10−2y) − y = 9, 30 − 7y = 9, y = 3, x = 4. Check: 4+6=10✓, 12−3=9✓Examiner tips
- Always check your answer in BOTH original equations
- Use elimination or substitution - choose the method that avoids fractions
- C = 2πr = 2π × 5 = 10π cm. 25π cm is area formula. 5π cm is πr. 20π cm is 4πr.Examiner tips
- C = 2πr or C = πd - know both forms
- 'In terms of π' means leave π in your answer, don't calculate a decimal
- Radius = 6. A = πr² = π × 36 = 36π cm². 144π cm² uses diameter. 12π cm² is circumference. 6π cm² is πr.Examiner tips
- Always use radius (not diameter) in the area formula A = πr²
- If given diameter, divide by 2 to get radius first
- c² = 5² + 12² = 25 + 144 = 169. c = 13 cm. 17 cm adds. 169 cm forgets root. 7 cm subtracts.Examiner tips
- Pythagoras: c² = a² + b² for finding hypotenuse
- 5, 12, 13 is a Pythagorean triple - worth memorizing!
- Mean = (5×5 + 15×12 + 25×8)/(25) = (25+180+200)/25 = 405/25 = 16.2. Closest is A. 15, 20, and 13.33 miscalculate.Examiner tips
- Midpoint = (lower boundary + upper boundary) ÷ 2
- This gives an estimate because we don't know exact values within each group
- (x + 7)(x − 4) = x² − 4x + 7x − 28 = x² + 3x − 28. x² − 3x − 28 has wrong middle sign. x² + 11x − 28 adds coefficients. x² + 3x + 28 has wrong constant sign.Examiner tips
- FOIL: First (x×x), Outside (x×-4), Inside (7×x), Last (7×-4)
- Be careful with signs, especially when multiplying by negative numbers
- Reflection in x = 0 (y-axis): x becomes −x, y stays same. (2,3)→(−2,3), etc. (2, −3), (4, −3), (3, −5) reflects in y=0. (−2, −3), (−4, −3), (−3, −5) reflects in both. (3, 2), (3, 4), (5, 3) swaps coordinates.Examiner tips
- x = 0 is the y-axis, so reflection changes x to -x
- y = 0 is the x-axis, so reflection would change y to -y
- Line of symmetry is midway between roots: (1 + 5)/2 = 3. So x = 3. y = 3 is wrong axis. x = 5 and x = 1 are the roots.Examiner tips
- Line of symmetry is x = (root1 + root2)/2 or x = -b/(2a) from ax² + bx + c
- Parabola symmetry line is always vertical (x = constant)
- V = πr²h = π × 16 × 9 = 144π cm³. 36π cm³ is πr²h/4. 72π cm³ is πr²h/2. 288π cm³ doubles.Examiner tips
- V = πr²h - remember to square the radius
- Don't confuse with surface area formulas
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