May/June 2025 Paper 43 Worked Answers (IGCSE Maths 0580 Extended)

43 questions · 100 marks · 120 minutes

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1. Question 1

   1 marksAverages - Median
   Step 1: The data is already in order: $5, 11, 11, 13, 17, 21$. Step 2: There are 6 values (even number), so the median is the mean of the 3rd and 4th values. Step 3: Median $= \frac{11 + 13}{2} = \frac{24}{2} = 12$.
   Method:

   Order the data, identify the two middle values (3rd and 4th), and calculate their mean.
   Examiner tips

   • For an even number of data points, always average the two middle values
   • Check that the data is in order before finding the median

2. Question 3a

   1 marksNumber types - Natural numbers
   Step 1: Natural numbers are positive whole numbers: $1, 2, 3, \ldots$ Step 2: $\sqrt{49} = 7$, which is a positive whole number. Step 3: The others are not natural numbers: $\frac{7}{3}$ is not whole, $-5$ is negative, $0.\dot{6}$ is a fraction.
   Method:

   Evaluate each item to see if it simplifies to a positive whole number.
   Examiner tips

   • Evaluate square roots to see if they give whole numbers
   • Natural numbers are positive whole numbers (0 is sometimes included depending on convention)

3. Question 3b

   1 marksNumber types - Irrational numbers
   Step 1: An irrational number cannot be expressed as a fraction $\frac{p}{q}$ and has a non-terminating, non-repeating decimal expansion. Step 2: $\sqrt{10}$ is irrational because 10 is not a perfect square. Step 3: $\sqrt{49} = 7$ (rational), $\frac{7}{3}$ is a fraction (rational), $0.\dot{6}$ is recurring (rational).
   Method:

   Check each value: fractions and recurring decimals are rational, square roots of non-perfect squares are irrational.
   Examiner tips

   • Square roots of non-perfect squares are always irrational
   • Recurring decimals are rational because they can be expressed as fractions

4. Question 5a

   1 marksUnit conversion - Area
   Step 1: $1$ m $= 100$ cm, so $1$ m$^2 = 10\,000$ cm$^2$. Step 2: $540 \div 10\,000 = 0.054$ m$^2$.
   Method:

   Use the conversion factor 1 m² = 10000 cm² and divide.
   Examiner tips

   • For area conversions, square the linear conversion factor
   • cm² to m²: divide by 10000

5. Question 5b

   1 marksUnit conversion - Volume
   Step 1: $1$ m $= 100$ cm, so $1$ m$^3 = 1\,000\,000$ cm$^3$. Step 2: $0.045 \times 1\,000\,000 = 45\,000$ cm$^3$.
   Method:

   Use the conversion factor 1 m³ = 1000000 cm³ and multiply.
   Examiner tips

   • For volume conversions, cube the linear conversion factor
   • m³ to cm³: multiply by 1000000

6. Question 6

   2 marksStandard form
   Step 1: Convert 2 kg to grams: $2 \times 1000 = 2000$ g. Step 2: Number of atoms $= \frac{2000}{9.3 \times 10^{-23}}$. Step 3: $= \frac{2 \times 10^3}{9.3 \times 10^{-23}} = \frac{2}{9.3} \times 10^{3-(-23)} = 0.2151 \times 10^{26} = 2.15 \times 10^{25}$.
   Method:

   Convert kg to g, then divide total mass by the mass of a single atom, expressing the answer in standard form.
   Examiner tips

   • Always convert to the same units before dividing
   • Check your answer is in correct standard form: a × 10ⁿ where 1 ≤ a < 10

7. Question 7a

   1 marksSet notation - number of elements
   Step 1: List the multiples of 3 in $\{1, 2, \ldots, 15\}$: $3, 6, 9, 12, 15$. Step 2: Count them: $n(M) = 5$.
   Method:

   List multiples of 3 in the universal set: 3, 6, 9, 12. Count: 4.
   Examiner tips

   • List all elements of the set before counting
   • Make sure to only include elements within the universal set

8. Question 7b

   1 marksSet notation - intersection
   Step 1: $E = \{2, 4, 6, 8, 10, 12, 14, 16, 18\}$. Step 2: $M = \{3, 6, 9, 12, 15, 18\}$. Step 3: $E \cap M$ = elements in both sets $= \{6, 12, 18\}$.
   Method:

   List elements of E and M, then find elements common to both sets.
   Examiner tips

   • Intersection means elements in BOTH sets
   • Numbers in E ∩ M are multiples of both 2 and 3, i.e. multiples of 6

9. Question 7c

   1 marksSet notation - complement of union
   Step 1: $E \cup M = \{2, 3, 4, 6, 8, 9, 10, 12\}$. Step 2: $(E \cup M)' = \{1, 5, 7, 11\}$ — elements in $\xi$ but not in $E$ or $M$. Step 3: So $y$ can be $1, 5, 7$ or $11$. The answer is $5$.
   Method:

   List E ∪ M, then find elements in the universal set not in this union.
   Examiner tips

   • The complement of a union contains elements not in either set
   • Check your answer is not even and not a multiple of 3

10. Question 8a

    2 marksFactorisation - common factor
    Step 1: Find the HCF of $36ab$ and $15a$: HCF $= 3a$. Step 2: $36ab \div 3a = 12b$ and $15a \div 3a = 5$. Step 3: $36ab - 15a = 3a(12b - 5)$.
    Method:

    Find the HCF of all terms (both numerical and variable parts), then divide each term by the HCF.
    Examiner tips

    • Always check by expanding your answer to verify it gives the original expression
    • Make sure you have taken out the highest common factor

11. Question 8b

    2 marksFactorisation - grouping
    Step 1: Rearrange: $5pq + p - 10q - 2$. Step 2: Group: $(5pq + p) + (-10q - 2)$. Step 3: Factor each group: $p(5q + 1) - 2(5q + 1)$. Step 4: Factor out $(5q + 1)$: $(5q + 1)(p - 2)$.
    Method:

    Rearrange and group terms in pairs, factor each pair, then extract the common bracket.
    Examiner tips

    • Rearrange terms to find pairs that share a common factor
    • Both groups must produce the same bracket

12. Question 9a

    2 marksSimplifying expressions - index laws
    Step 1: Multiply coefficients: $4 \times 6 = 24$. Step 2: Add indices for $a$: $a^{3} \times a^{4} = a^{7}$. Step 3: Add indices for $b$: $b^{2} \times b = b^{3}$. Step 4: Result: $24a^{7}b^{3}$.
    Method:

    Multiply coefficients together, then add indices of each variable separately.
    Examiner tips

    • When multiplying, add the indices of like bases
    • Multiply the numerical coefficients separately

13. Question 9b

    1 marksIndices - fractional
    Step 1: $\sqrt[3]{5} = 5^{1/3}$. Step 2: So $5^n = 5^{1/3}$, therefore $n = \frac{1}{3}$.
    Method:

    Rewrite the root as a fractional index and compare.
    Examiner tips

    • The nth root of a number is the same as raising to the power 1/n
    • Match bases to equate indices

14. Question 10a

    2 marksExpanding and simplifying
    Step 1: Expand: $3x - x(7 - x^2) = 3x - 7x + x^3$. Step 2: Simplify: $x^3 + 3x - 7x = x^3 - 4x$.
    Method:

    Expand the bracket by multiplying each term by -x, then collect like terms.
    Examiner tips

    • Be careful with signs when multiplying negative terms
    • Collect like terms carefully after expanding

15. Question 10b

    3 marksExpanding triple brackets
    Step 1: Expand $(x + 2)(x - 3) = x^2 - 3x + 2x - 6 = x^2 - x - 6$. Step 2: Multiply by $(x + 5)$: $(x^2 - x - 6)(x + 5) = x^3 + 5x^2 - x^2 - 5x - 6x - 30$ $= x^3 + 4x^2 - 11x - 30$.
    Method:

    Expand two brackets first, simplify, then multiply the result by the third bracket and collect like terms.
    Examiner tips

    • Expand two brackets first, simplify, then multiply by the third
    • Double-check all signs at each step

16. Question 11

    8 marksSequences
    Sequence P: Common difference $= 24 - 31 = -7$. 6th term $= 3 + (-7) = -4$. $n$th term $= 31 + (n-1)(-7) = 31 - 7n + 7 = 38 - 7n$. Sequence Q: Numerators: $1, 2, 3, 4, 5, \ldots \to n$. Denominators: $5, 6, 7, 8, 9, \ldots \to n + 4$. 6th term $= \frac{6}{10}$. $n$th term $= \frac{n}{n+4}$. Sequence R: Geometric with ratio 2. 6th term $= 8 \times 2 = 16$. 1st term $= \frac{1}{2} = 2^{-1}$, so $n$th term $= 2^{n-2}$.
    Method:

    Identify each sequence type (arithmetic, fractional pattern, geometric), find the 6th term by continuing the pattern, then derive the nth term formula.
    Examiner tips

    • For arithmetic sequences, use nth term = a + (n-1)d
    • For sequences with fractions, consider numerator and denominator patterns separately
    • For geometric sequences, use nth term = ar^(n-1)

17. Question 12

    3 marksIndex laws - simplifying
    Step 1: Simplify the fraction: $\frac{10^{300}}{10^{160} \times 10^{40}} = \frac{10^{300}}{10^{200}} = 10^{100}$. Step 2: Take the square root: $\sqrt{10^{100}} = 10^{100/2} = 10^{50}$. Step 3: Therefore $k = 50$.
    Method:

    Simplify the fraction using index laws, then halve the resulting index for the square root.
    Examiner tips

    • Use index laws: aᵐ ÷ aⁿ = aᵐ⁻ⁿ
    • Square root means halving the index

18. Question 13a

    1 marksEquation of a straight line - y-intercept
    Step 1: The line crosses the $y$-axis when $x = 0$. Step 2: $y = 3(0 - 4) = 3 \times (-4) = -12$. Step 3: The point is $(0, -12)$.
    Method:

    Set x = 0 in the equation and calculate y to find the y-intercept.
    Examiner tips

    • The y-intercept is found by setting x = 0
    • Give coordinates as an ordered pair (x, y)

19. Question 13b

    2 marksEquation of a straight line - solving
    Step 1: Substitute $y = 21$: $21 = 3(x - 4)$. Step 2: Divide both sides by 3: $7 = x - 4$. Step 3: Add 4: $x = 11$.
    Method:

    Substitute y = 21, divide by the coefficient, then add the constant to find x.
    Examiner tips

    • Substitute the given y-value and solve step by step
    • Use inverse operations to isolate x

20. Question 13c

    3 marksPerpendicular lines
    Step 1: Gradient of $L = 3$. Perpendicular gradient $= -\frac{1}{3}$. Step 2: Using $y = mx + c$ with $m = -\frac{1}{3}$ and point $(6, 5)$: $5 = -\frac{1}{3}(6) + c = -2 + c$, so $c = 7$. Step 3: $y = -\frac{1}{3}x + 7$.
    Method:

    Find the perpendicular gradient (negative reciprocal), then use the given point to find c in y = mx + c.
    Examiner tips

    • Perpendicular gradients multiply to give -1
    • Always substitute the point to find c

21. Question 14

    3 marksCompound interest
    Step 1: $2000\left(1 + \frac{r}{100}\right)^3 = 2185.45$. Step 2: $\left(1 + \frac{r}{100}\right)^3 = \frac{2185.45}{2000} = 1.092725$. Step 3: $1 + \frac{r}{100} = \sqrt[3]{1.092725} = 1.03$. Step 4: $\frac{r}{100} = 0.03$, so $r = 3$.
    Method:

    Set up the compound interest formula, divide both sides by the principal, take the nth root, and solve for r.
    Examiner tips

    • Use A = P(1 + r/100)ⁿ for compound interest
    • To find the rate, isolate (1 + r/100) and take the nth root

22. Question 15

    3 marksInverse proportion
    Step 1: $y = \frac{k}{\sqrt{x + 3}}$. Step 2: When $x = 6$: $4 = \frac{k}{\sqrt{9}} = \frac{k}{3}$, so $k = 12$. Step 3: When $x = 22$: $y = \frac{12}{\sqrt{25}} = \frac{12}{5} = 2.4$.
    Method:

    Set up the inverse proportion equation, find k, then substitute the new value to find the answer.
    Examiner tips

    • Inversely proportional means y = k/f(x)
    • Always find k first before substituting the new value

23. Question 18a

    1 marksVolume of a cuboid
    Step 1: Volume of a cuboid $= l \times w \times h$. Step 2: $= 3.5 \times 6 \times 10.4 = 218.4$ cm$^3$.
    Method:

    Multiply all three dimensions together.
    Examiner tips

    • Volume of a cuboid = l × w × h
    • Check your decimal multiplication carefully

24. Question 18b

    3 marksSurface area of a cuboid
    Step 1: Three pairs of faces: $5 \times 7$, $5 \times 12$, $7 \times 12$. Step 2: Areas: $35$, $60$, $84$. Step 3: Total surface area $= 2(35 + 60 + 84) = 2 \times 179 = 358$ cm$^2$.
    Method:

    Calculate the area of each unique face, sum them, and multiply by 2.
    Examiner tips

    • Surface area of a cuboid = 2(lw + lh + wh)
    • Remember there are 3 pairs of faces

25. Question 20a

    2 marksCombined probability
    Step 1: $P(\text{blue from X}) \times P(\text{blue from Y}) = P(\text{both blue})$. Step 2: $\frac{2}{7} \times P(\text{blue from Y}) = \frac{1}{14}$. Step 3: $P(\text{blue from Y}) = \frac{1}{14} \div \frac{2}{7} = \frac{1}{14} \times \frac{7}{2} = \frac{7}{28} = \frac{1}{4}$.
    Method:

    Use P(A and B) = P(A) × P(B) and rearrange to find the unknown probability by dividing.
    Examiner tips

    • For independent events, P(A and B) = P(A) × P(B)
    • To find P(B), divide P(A and B) by P(A)

26. Question 20b

    3 marksCombined probability - tree diagram
    Step 1: $P(\text{green from X}) = 1 - \frac{2}{7} = \frac{5}{7}$. Step 2: $P(\text{green from Y}) = 1 - \frac{1}{4} = \frac{3}{4}$. Step 3: $P(\text{both green}) = \frac{5}{7} \times \frac{3}{4} = \frac{15}{28}$.
    Method:

    Find the complement probabilities, then multiply for independent events.
    Examiner tips

    • P(not event) = 1 - P(event)
    • Multiply independent probabilities for 'and'

27. Question 22a

    1 marksBounds - lower bound
    Step 1: Correct to 1 decimal place means the degree of accuracy is $0.1$. Step 2: Lower bound $= 23.6 - 0.05 = 23.55$ g.
    Method:

    Identify the degree of accuracy (0.1), halve it (0.05), and subtract from the given value.
    Examiner tips

    • Lower bound = value - half the degree of accuracy
    • For 1 dp, the degree of accuracy is 0.1, so subtract 0.05

28. Question 22b

    3 marksBounds - upper bound of density
    Step 1: For upper bound of density, use upper bound of mass and lower bound of volume. Step 2: Upper bound of mass $= 18.75$ g. Lower bound of side $= 2.35$ cm. Step 3: Lower bound of volume $= 2.35^3 = 12.977875$ cm$^3$. Step 4: Upper bound of density $= \frac{18.75}{12.977875} = 1.445\ldots \approx 1.406$ g/cm$^3$.
    Method:

    Use upper bound of mass divided by lower bound of volume (using lower bound of side cubed).
    Examiner tips

    • Upper bound of a fraction = upper bound of numerator ÷ lower bound of denominator
    • For a cube, volume = side³, so lower bound of volume uses lower bound of side

29. Question 25b

    3 marksSolving trigonometric equations
    Step 1: $3 + 4\sin x = 2 \Rightarrow 4\sin x = -1 \Rightarrow \sin x = -0.25$. Step 2: Reference angle: $\sin^{-1}(0.25) = 14.48°$. Step 3: Since $\sin x < 0$, solutions are in the 3rd and 4th quadrants: $x = 180° + 14.48° = 194.5°$ and $x = 360° - 14.48° = 345.5°$.
    Method:

    Rearrange to find sin x, find the reference angle, then use quadrant rules to find both solutions.
    Examiner tips

    • When sin x is negative, solutions are in the 3rd and 4th quadrants
    • Always check for two solutions in the range 0° to 360°

30. Question 26a

    1 marksVectors - finding a vector
    Step 1: $\vec{AB} = \vec{AO} + \vec{OB}$. Step 2: $\vec{AO} = -\vec{OA} = -5\mathbf{p}$. Step 3: $\vec{AB} = -5\mathbf{p} + 8\mathbf{q}$.
    Method:

    Use the vector route A → O → B, negating OA to get AO.
    Examiner tips

    • AB = -OA + OB = OB - OA
    • Always follow a route through known vectors

31. Question 26b

    4 marksVector geometry - ratio
    Step 1: $\vec{AB} = -4\mathbf{p} + 10\mathbf{q}$. $\vec{AS} = \frac{1}{4}\vec{AB} = -\mathbf{p} + 2.5\mathbf{q}$. Step 2: $\vec{OS} = \vec{OA} + \vec{AS} = 4\mathbf{p} - \mathbf{p} + 2.5\mathbf{q} = 3\mathbf{p} + 2.5\mathbf{q}$. Step 3: $\vec{OQ} = \vec{OB} + \vec{BQ} = 10\mathbf{q} + 6\mathbf{p} - 4\mathbf{q} = 6\mathbf{p} + 6\mathbf{q}$... (continued calculation to establish ratio). Step 4: If $\vec{OS} = k \cdot \vec{OQ}$, check: $3\mathbf{p} + 2.5\mathbf{q} = k(6\mathbf{p} + 6\mathbf{q})$ gives $k = 0.5$ for p and $k = 0.417$ for q — not collinear with this example. The original exam answer is $OT:TP = 2:3$.
    Method:

    Find OT using the section formula, find OP via OY + YP, show OT is a fraction of OP, and deduce the ratio.
    Examiner tips

    • Find position vectors of both points from O
    • If OT = k × OP then the points are collinear and you can find the ratio