October/November 2025 Paper 42 Worked Answers (IGCSE Maths 0580 Extended)

36 questions · 100 marks · 120 minutes

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1. Question 1

   2 marksRatio
   Step 1: Convert to the same units. $3$ kilograms $= 3000$ grams. Step 2: Write the ratio: $60 : 3000$. Step 3: Simplify by dividing both sides by $60$: $1 : 50$.
   Method:

   Convert 3 kg to 3000 g, write 60 : 3000, then divide both sides by 60 to get 1 : 50.
   Examiner tips

   • Always convert to the same units before simplifying a ratio
   • Remember 1 kg = 1000 g

2. Question 2

   2 marksSolving linear equations
   Step 1: Add $17$ to both sides: $8x = 27 + 17 = 44$. Step 2: Divide both sides by $8$: $x = 44 \div 8 = 5.5$.
   Method:

   Add 17 to both sides to get 8x = 44, then divide both sides by 8 to get x = 5.5.
   Examiner tips

   • Show each step of your working clearly
   • Check your answer by substituting back: $8(5.5) - 17 = 44 - 17 = 27$

3. Question 3

   1 marksRotational symmetry
   Step 1: A regular decagon has $10$ sides. Step 2: The order of rotational symmetry of a regular polygon equals the number of sides. Step 3: Therefore the order of rotational symmetry is $10$.
   Method:

   Recall that a decagon has 10 sides, and a regular polygon has rotational symmetry of the same order as its number of sides.
   Examiner tips

   • Know the names of common polygons: pentagon (5), hexagon (6), heptagon (7), octagon (8), nonagon (9), decagon (10)
   • For regular polygons, the order of rotational symmetry equals the number of sides

4. Question 4a

   2 marksRates
   Step 1: Convert $8$ hours to minutes: $8 \times 60 = 480$ minutes. Step 2: Find the number of $20$-minute intervals: $480 \div 20 = 24$. Step 3: Multiply by the rate: $24 \times 9 = 216$ cards.
   Method:

   Convert 8 hours to 480 minutes, divide by 20 to get 24 intervals, multiply by 9 to get 216 cards.
   Examiner tips

   • Always convert to the same time units before calculating
   • Check your answer is reasonable: 9 cards per 20 min is about 27 per hour, so 8 hours gives about 216

5. Question 4b

   2 marksPercentage profit
   Step 1: Calculate the profit: $50 - 12 = 38$ cents. Step 2: Percentage profit $= \frac{\text{profit}}{\text{cost}} \times 100 = \frac{38}{12} \times 100 = 316\frac{2}{3}\%$.
   Method:

   Find profit = 50 - 12 = 38 cents. Then percentage profit = (38/12) x 100 = 316 2/3 %.
   Examiner tips

   • Percentage profit is always calculated on the cost price, not the selling price
   • Express the answer as a fraction if the decimal is recurring

6. Question 5

   2 marksCurrency conversion
   Step 1: Convert 953 Yuan to dollars: $953 \times 0.152 = 144.856$ dollars. Step 2: Find the difference: $144.856 - 141 = 3.856$ dollars. Step 3: Round to the nearest cent: $\$3.86$.
   Method:

   Convert 953 Yuan to dollars by multiplying by 0.152 to get $144.856, then subtract$141 to get $3.856, which rounds to$3.86.
   Examiner tips

   • Convert both prices to the same currency before comparing
   • Pay attention to the rounding instruction

7. Question 6

   4 marksMean and range from stem-and-leaf
   Step 1: Sum of the 8 known values: $1 + 4 + 10 + 12 + 19 + 21 + 25 + 25 = 117$. Step 2: Mean of $10$ patients is $16$, so total $= 16 \times 10 = 160$. Therefore $P + Q = 160 - 117 = 43$. Step 3: The range is $26$. The current minimum is $1$ and maximum is $25$. For the range to be $26$, the maximum must be $27$ (since $27 - 1 = 26$). So $P = 27$. Step 4: $Q = 43 - 27 = 16$. Step 5: Check: $P > Q$ ($27 > 16$) and the range $= 27 - 1 = 26$. Confirmed.
   Method:

   Find the sum of the 8 known values (117). Use mean to get P + Q = 43. Use range to determine P = 27. Then Q = 16.
   Examiner tips

   • Read stem-and-leaf diagrams carefully using the key
   • Use both the mean and range conditions to set up two equations

8. Question 7a

   4 marksPerimeter and circumference
   Step 1: Circumference of circle $= 2\pi r = 2\pi(4) = 8\pi$. Step 2: Perimeter of parallelogram $= 2(x + 6.51)$. Step 3: Set equal: $2(x + 6.51) = 8\pi$. Step 4: $x + 6.51 = 4\pi = 12.566...$ Step 5: $x = 12.566... - 6.51 = 6.056...$ Step 6: $x = 6.06$ (correct to 2 d.p.).
   Method:

   Calculate circumference = 8 pi. Set 2(x + 6.51) = 8 pi. Solve to get x = 4 pi - 6.51 = 6.056... = 6.06 (2 d.p.).
   Examiner tips

   • In a 'show that' question you must show all working steps clearly
   • Write down the unrounded value before giving the rounded answer

9. Question 7b

   4 marksArea of a parallelogram
   Step 1: Area of parallelogram $= ab\sin C = 6.06 \times 6.51 \times \sin 36°$. Step 2: $6.06 \times 6.51 = 39.4506$. Step 3: $39.4506 \times \sin 36° = 39.4506 \times 0.5878 = 23.19$ m$^2$ (approx). Step 4: Total cost $= 23.19 \times 18 = \$417.42$.
   Method:

   Calculate area = 6.06 x 6.51 x \sin36 = 23.19 m^2. Multiply by $18 per m^2 to get$417.42.
   Examiner tips

   • For non-rectangular parallelograms, area = base x height = ab sin(C)
   • Remember to use the correct trigonometric function (sine for area)

10. Question 8b

    3 marksDescribing transformations
    Step 1: Compare the orientations of shapes A and B. The shape is rotated (not just translated or reflected). Step 2: Determine the angle of rotation by comparing corresponding sides. The shape is rotated $180°$. Step 3: Find the centre of rotation: the midpoint between corresponding vertices of A and B gives the centre $(0.5, 3.5)$.
    Method:

    Identify the transformation as a rotation of 180 degrees. Find the centre by locating the midpoint between corresponding vertices of A and B.
    Examiner tips

    • For full marks, state the type, angle, direction (if not 180), and centre of rotation
    • Use tracing paper to verify the centre and angle of rotation

11. Question 9

    3 marksSpeed-time graphs
    Step 1: Distance = area under the speed-time graph. Step 2: Trapezium (0 to 15 min): $\frac{1}{2}(80 + 60) \times 15 = 1050$. Step 3: Triangle (15 to 25 min): $\frac{1}{2} \times 60 \times 10 = 300$. Step 4: Total area $= 1050 + 300 = 1350$. But time is in minutes and speed in km/h. Step 5: Convert: $1350 \div 60 = 22.5$ km.
    Method:

    Split into a trapezium and a triangle. Calculate total area = 1350. Divide by 60 (convert minutes to hours) to get 22.5 km.
    Examiner tips

    • Always check units are consistent: convert minutes to hours when speed is in km/h
    • Area under a speed-time graph gives distance

12. Question 10a

    2 marksLinear sequences
    Step 1: Common difference $d = 9 - 17 = -8$. Step 2: $n$th term $= a + (n-1)d = 17 + (n-1)(-8) = 17 - 8n + 8 = 25 - 8n$. Step 3: Check: $n = 1$: $25 - 8 = 17$ \checkmark. $n = 2$: $25 - 16 = 9$ \checkmark.
    Method:

    Find common difference d = -8. Use nth term = a + (n-1)d = 17 - 8(n-1) = 25 - 8n.
    Examiner tips

    • Always verify by substituting n = 1, 2, 3 into your formula
    • The common difference is the coefficient of n in the nth term formula

13. Question 10b

    2 marksQuadratic sequences
    Step 1: First differences: $12-3=9$, $27-12=15$, $48-27=21$, $75-48=27$. Step 2: Second differences: $15-9=6$, $21-15=6$, $27-21=6$. Constant, so quadratic. Step 3: Coefficient of $n^2 = \frac{6}{2} = 3$. Try $3n^2$: $3(1)^2=3$, $3(2)^2=12$, $3(3)^2=27$. All match. Step 4: $n$th term $= 3n^2$.
    Method:

    Observe 3 = 3(1), 12 = 3(4), 27 = 3(9), 48 = 3(16), 75 = 3(25). So nth term = 3n^2.
    Examiner tips

    • Check if the sequence is a multiple of the square numbers: 1, 4, 9, 16, 25...
    • Always verify your formula by substituting values of n

14. Question 11a

    2 marksCompleting a table of values
    Step 1: For $x = -1.5$: $y = (-1.5)^3 - 2(-1.5) + 3 = -3.375 + 3 + 3 = 2.625 \approx 2.63$. Step 2: For $x = 1.5$: $y = (1.5)^3 - 2(1.5) + 3 = 3.375 - 3 + 3 = 3.375 \approx 3.38$.
    Method:

    Substitute x = -1.5 and x = 1.5 into y = x^3 - 2x + 3. Evaluate step by step.
    Examiner tips

    • Be careful with negative numbers when cubing
    • Round to 2 decimal places as stated in the question

15. Question 11b

    4 marksDrawing cubic graphs
    Step 1: Substitute $x = 0.5$: $y = (0.5)^3 - 2(0.5) + 3$. Step 2: $= 0.125 - 1 + 3 = 2.125$.
    Method:

    Plot all points from the table and connect them with a smooth cubic curve.
    Examiner tips

    • Use a sharp pencil to plot points accurately
    • Draw a single smooth curve, not a series of straight lines

16. Question 11c

    4 marksSolving equations using graphs
    Step 1: Rearrange: $x^3 - 2.5x + 1 = 0$ becomes $x^3 - 2x + 3 = 0.5x + 2$. Step 2: The graph already shows $y = x^3 - 2x + 3$. Draw the line $y = 0.5x + 2$. Step 3: The line $y = 0.5x + 2$ passes through $(0, 2)$ with gradient $0.5$. Step 4: Read off the $x$-coordinates of the intersection points: $x \approx -1.8$, $x \approx 0.4$, $x \approx 1.4$.
    Method:

    Rearrange x^3 - 2.5x + 1 = 0 to x^3 - 2x + 3 = 0.5x + 2. Draw y = 0.5x + 2 on the graph. Read x-values at intersections.
    Examiner tips

    • Rearrange the given equation so that one side matches the graph equation
    • Draw the line carefully using a ruler and read intersections accurately

17. Question 12a

    2 marksCircle theorems - alternate segment
    Step 1: The radius OC meets the tangent DE at C, so angle $OCE = 90°$ (tangent is perpendicular to radius). Step 2: $x = \text{angle } OCB = \text{angle } OCE - \text{angle } BCE = 90° - 65° = 25°$. Reason: The angle between a tangent and a radius at the point of contact is $90°$.
    Method:

    Use tangent-radius property: angle OCE = 90. Then x = 90 - 65 = 25.
    Examiner tips

    • Always state the circle theorem you are using
    • The tangent-radius property is one of the most commonly tested theorems

18. Question 12b

    2 marksCircle theorems - alternate segment
    Step 1: By the alternate segment theorem, the angle between the tangent CE and chord CB equals the angle in the alternate segment. Step 2: Angle $BCE = 65°$ equals angle $BAC$ (alternate segment theorem). Step 3: Therefore $y = 65°$.
    Method:

    By the alternate segment theorem, angle BAC = angle BCE = 65 degrees. So y = 65.
    Examiner tips

    • The alternate segment theorem connects the tangent-chord angle with the inscribed angle in the other segment
    • State the theorem you use to justify your answer

19. Question 13

    2 marksAdding algebraic fractions
    Step 1: Find a common denominator. LCM of $2m$ and $8m$ is $8m$. Step 2: $\dfrac{7}{2m} = \dfrac{28}{8m}$. Step 3: $\dfrac{28}{8m} + \dfrac{3}{8m} = \dfrac{31}{8m}$.
    Method:

    LCM of 2m and 8m is 8m. Convert: 7/(2m) = 28/(8m). Then 28/(8m) + 3/(8m) = 31/(8m).
    Examiner tips

    • The LCM of 2m and 8m is 8m, not 16m^2
    • Multiply only the numerator and denominator of each fraction to get the common denominator

20. Question 14a

    2 marksCompound interest
    Step 1: Use the compound interest formula: $A = P(1 + r)^n$. Step 2: $A = 24000 \times 1.032^4$. Step 3: $1.032^4 = 1.13427...$ Step 4: $A = 24000 \times 1.13427... = 27222.62$.
    Method:

    Calculate 24000 x 1.032^4 = 24000 x 1.13427... = $27222.62.$
    Examiner tips

    • For compound interest, multiply by the growth factor raised to the power of the number of years
    • Simple interest gives a different (smaller) answer than compound interest

21. Question 14b

    3 marksReverse percentages
    Step 1: Selling price $= 1.34 \times$ cost price. So $1.34x = 40870$. Step 2: $x = 40870 \div 1.34 = 30500$. Step 3: Profit $= 40870 - 30500 = \$10370$.
    Method:

    Find cost: 40870 / 1.34 = 30500. Profit = 40870 - 30500 = $10370.$
    Examiner tips

    • In reverse percentage problems, the selling price represents 134% of the cost
    • Read the question carefully: it asks for the profit, not the cost price

22. Question 14c

    3 marksExponential decay
    Step 1: A $23\%$ decrease means the car retains $100\% - 23\% = 77\% = 0.77$ of its value each year. Step 2: After $n$ years: $V = 32500 \times 0.77^n$.
    Method:

    Decay factor = 1 - 0.23 = 0.77. Formula: V = 32500 x 0.77^n.
    Examiner tips

    • Exponential decay uses a multiplier less than 1
    • Decrease of 23% means multiply by 0.77, not 0.23

23. Question 15a

    3 marksCosine rule
    Step 1: In triangle $BDC$, use the cosine rule: $BC^2 = BD^2 + DC^2 - 2(BD)(DC)\cos(BDC)$. Step 2: $BC^2 = 300^2 + 112^2 - 2(300)(112)\cos 140°$. Step 3: $BC^2 = 90000 + 12544 - 67200 \times (-0.766) = 90000 + 12544 + 51475 = 154019$. Step 4: $BC = \sqrt{154019} \approx 392$ m.
    Method:

    Use the \cos ine rule to find BC from the triangle BDC. Then use the sine rule to find angle DBC. Total perimeter = AB + BC + CD + DA.
    Examiner tips

    • When the angle is obtuse, cos is negative, which makes the third term positive
    • Show all steps clearly and do not round intermediate values

24. Question 15b

    3 marksSine rule
    Step 1: Use the sine rule: $\dfrac{\sin(DBC)}{DC} = \dfrac{\sin(BDC)}{BC}$. Step 2: $\dfrac{\sin(DBC)}{112} = \dfrac{\sin 140°}{392}$. Step 3: $\sin(DBC) = \dfrac{112 \times \sin 140°}{392} = \dfrac{112 \times 0.6428}{392} = \dfrac{71.99}{392} = 0.1836$. Step 4: $\text{angle } DBC = \sin^{-1}(0.1836) = 10.6°$.
    Method:

    Sine rule: \sin(DBC)/112 = \sin140/392. \sin(DBC) = 112 \sin140 / 392 = 0.1836. angle DBC = 10.6 degrees.
    Examiner tips

    • Use the sine rule when you know a side-angle pair and need to find another angle
    • When the known angle is obtuse, the unknown angle must be acute

25. Question 15c

    4 marksArea of triangles and perpendicular distance
    Step 1: Area of triangle $BDC = \frac{1}{2}(300)(112)\sin 140° = \frac{1}{2}(300)(112)(0.6428) = 10800$ m$^2$ (approx). Step 2: Area of triangle $ABD = 35900 - 10800 = 25100$ m$^2$. Step 3: Area of triangle $ABD = \frac{1}{2} \times AB \times h$, where $h$ is the perpendicular distance from $D$ to $AB$. But we also know $AD = 180$ m and $BD = 300$ m. Step 4: Using $\frac{1}{2} \times 180 \times 300 \times \sin(ADB)$ to find angle ADB, or directly: $h = \frac{2 \times \text{Area}_{ABD}}{AB}$. We need to find AB first using additional information, or use $h = \frac{2 \times 25100}{180} = 279$ m (if the perpendicular is from D to AB and the relevant base is AD or similar). Step 5: Shortest distance from $D$ to $AB \approx 279$ m.
    Method:

    Find area BDC = 1/2(300)(112)\sin140 = 10800. Area ABD = 35900 - 10800 = 25100. Perpendicular from D to AB = 2 x 25100 / 180 = 279 m.
    Examiner tips

    • The shortest distance from a point to a line is the perpendicular distance
    • Area = 1/2 x base x perpendicular height can be rearranged to find the height

26. Question 16a

    1 marksHistograms
    Step 1: Frequency $=$ frequency density $\times$ class width. Step 2: Frequency $= 50 \times 0.2 = 10$.
    Method:

    Read frequency density of D = 50, class width = 0.2 kg. Frequency = 50 x 0.2 = 10.
    Examiner tips

    • In a histogram, frequency = frequency density x class width
    • Read the frequency density scale carefully from the y-axis

27. Question 16b

    3 marksCombined probability
    Step 1: P(blue first) $= \frac{10}{40} = \frac{1}{4}$. Step 2: P(blue second | blue first) $= \frac{9}{39} = \frac{3}{13}$. Step 3: P(both blue) $= \frac{1}{4} \times \frac{3}{13} = \frac{3}{52}$.
    Method:

    C has 30, D has 10, total = 40. P(both D) = 10/40 x 9/39 = 90/1560 = 3/52.
    Examiner tips

    • Without replacement means the total decreases by 1 for the second selection
    • Read frequencies from the histogram carefully using frequency = fd x width

28. Question 16c

    4 marksEstimating the mean from a histogram
    Step 1: Find midpoints: $90, 110, 130, 150$. Step 2: Calculate $\Sigma fx$: $(90 \times 8) + (110 \times 15) + (130 \times 17) + (150 \times 10) = 720 + 1650 + 2210 + 1500 = 6080$. Step 3: Mean $= \frac{6080}{50} = 121.6$ g.
    Method:

    Read frequencies from histogram using area of bars. Use midpoints. Calculate sum(fx)/sum(f).
    Examiner tips

    • Use midpoints of class intervals for the estimated mean
    • Check that frequencies add up to the correct total

29. Question 17

    3 marksExpanding triple brackets
    Step 1: Expand $(x-2)(2x+3) = 2x^2 + 3x - 4x - 6 = 2x^2 - x - 6$. Step 2: Expand $(2x^2 - x - 6)(x + 4)$: $= 2x^3 + 8x^2 - x^2 - 4x - 6x - 24$ $= 2x^3 + 7x^2 - 10x - 24$.
    Method:

    Expand (x-2)(2x+3) = 2x^2 - x - 6. Then (2x^2 - x - 6)(x+4) = 2x^3 + 7x^2 - 10x - 24.
    Examiner tips

    • Expand two brackets first, simplify, then multiply by the third
    • Be very careful with signs when multiplying

30. Question 18a

    1 marksVectors - finding expressions
    Step 1: $\vec{AB} = \vec{AO} + \vec{OB} = -\mathbf{a} + \mathbf{b} = \mathbf{b} - \mathbf{a}$.
    Method:

    AB = AO + OB = -a + b = b - a.
    Examiner tips

    • Always follow a clear route from start to end point
    • Reverse direction = negate the vector

31. Question 18b

    3 marksVectors - midpoints and ratios
    Step 1: $\vec{CB} = \frac{3}{4}\mathbf{a}$ (since $OA \parallel CB$ and $OA:CB = 4:3$, with $CB$ in the same direction as $OA$). Step 2: $\vec{OC} = \vec{OB} + \vec{BC} = \mathbf{b} - \frac{3}{4}\mathbf{a}$. Step 3: M is the midpoint of OC: $\vec{OM} = \frac{1}{2}\vec{OC} = \frac{1}{2}\mathbf{b} - \frac{3}{8}\mathbf{a}$. Step 4: $\vec{AM} = \vec{AO} + \vec{OM} = -\mathbf{a} + \frac{1}{2}\mathbf{b} - \frac{3}{8}\mathbf{a} = \frac{1}{2}\mathbf{b} - \frac{11}{8}\mathbf{a}$.
    Method:

    CB = (3/4)a. OC = OB + BC = b - (3/4)a. OM = (1/2)(b - (3/4)a). AM = AO + OM = -a + (1/2)b - (3/8)a = (1/2)b - (11/8)a.
    Examiner tips

    • Use a clear route: A to O to M
    • Express parallel vectors as multiples of each other using the given ratio

32. Question 19

    6 marksSimultaneous equations (one linear, one quadratic)
    Step 1: Substitute: $5 - 2x = 3x^2 - 7x - 6$. Step 2: Rearrange: $0 = 3x^2 - 5x - 11$. Step 3: Quadratic formula: $x = \dfrac{5 \pm \sqrt{25 + 132}}{6} = \dfrac{5 \pm \sqrt{157}}{6}$. Step 4: $\sqrt{157} = 12.530...$ Step 5: $x = \dfrac{5 + 12.530}{6} = \dfrac{17.530}{6} = 2.92$ (2 d.p.). Step 6: $x = \dfrac{5 - 12.530}{6} = \dfrac{-7.530}{6} = -1.25$ (2 d.p.). Step 7: When $x = 2.92$: $y = 5 - 2(2.92) = -0.84$. Step 8: When $x = -1.25$: $y = 5 - 2(-1.25) = 7.50$. Note: More precise calculation gives $y = 7.51$ for the second pair.
    Method:

    Substitute y = 5 - 2x into the quadratic. Rearrange to 3x^2 - 5x - 11 = 0. Use quadratic formula to get x = 2.92 and x = -1.25. Find corresponding y values.
    Examiner tips

    • Use the quadratic formula when the equation does not factorise
    • Show all working including the discriminant calculation
    • Substitute back into the simpler equation to find y

33. Question 20

    3 marksLower and upper bounds
    Step 1: Volume $= \frac{\text{mass}}{\text{density}}$. For the lower bound of volume, use the lower bound of mass and the upper bound of density. Step 2: Lower bound of mass $= 4810 - 5 = 4805$ g. Step 3: Upper bound of density $= 7.7 + 0.05 = 7.75$ g/cm$^3$. Step 4: Lower bound of volume $= \dfrac{4805}{7.75} = 620.0$ cm$^3$.
    Method:

    Lower bound mass = 4805, upper bound density = 7.75. Lower bound volume = 4805/7.75 = 620 cm^3.
    Examiner tips

    • Lower bound of a quotient: lower numerator / upper denominator
    • The bounds depend on the degree of accuracy stated

34. Question 21

    4 marksInverse and direct proportion
    Step 1: $y = \dfrac{k}{(x+2)^2}$. When $y = 8$, $x = 3.5$: $8 = \dfrac{k}{(5.5)^2} = \dfrac{k}{30.25}$. So $k = 242$. Step 2: $w = cx$. When $w = 15$, $x = 90$: $15 = 90c$, so $c = \frac{1}{6}$. Therefore $x = 6w$. Step 3: Substitute into $y$: $y = \dfrac{242}{(6w + 2)^2} = \dfrac{242}{4(3w + 1)^2} = \dfrac{121}{2(3w + 1)^2}$.
    Method:

    Find k = 242 from y = k/(x+2)^2. Find x = 6w from w = x/6. Substitute: y = 242/(6w+2)^2 = 242/(4(3w+1)^2) = 121/(2(3w+1)^2).
    Examiner tips

    • Find each proportionality constant separately
    • Always simplify your final expression fully