May/June 2025 Paper 42 Worked Answers (IGCSE Maths 0580 Extended)
41 questions · 100 marks · 120 minutes
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Worked answers for 31 questions
- Step 1: A rhombus has exactly two lines of symmetry, both along its diagonals (passing through vertices). Step 2: A rhombus has rotational symmetry of order 2. Step 3: No other quadrilateral has exactly these properties.Method:Identify the quadrilateral whose only lines of symmetry are its diagonals and that has rotational symmetry of order 2. This is a rhombus.Examiner tips
- Know the symmetry properties of all standard quadrilaterals
- Diagonals as lines of symmetry is a key property of a rhombus
- Step 1: The difference in ratio parts is $8 - 3 = 5$ parts. Step 2: One part $= 3.5 \div 5 = 0.7$ kg. Step 3: $P = 3 \times 0.7 = 2.1$ kg, $Q = 8 \times 0.7 = 5.6$ kg.Method:Find the difference in ratio parts (8 - 3 = 5). Divide 3.5 by 5 to get 0.7 per part. Multiply each ratio number by 0.7.Examiner tips
- The difference in mass corresponds to the difference in ratio parts, not the total
- Always check that your answers are in the correct ratio and have the correct difference
- Step 1: The exchange rate is £$1 = \$1.60$, meaning each pound costs $\$1.60$. Step 2: Divide the dollars by the rate: $480 \div 1.60 = 300$ pounds.Method:Divide 480 by 1.60 to convert dollars to pounds: $480 \div 1.60 = 300$.Examiner tips
- Check whether you should multiply or divide by the exchange rate
- The answer should make sense: converting to a stronger currency gives a smaller number
- Step 1: Calculate the volume: $V = \frac{1}{3} \times \pi \times 5^2 \times 9 = \frac{1}{3} \times \pi \times 225 = 75\pi \approx 235.6$ cm$^3$. Step 2: Mass $=$ density $\times$ volume $= 2.4 \times 235.6 \approx 565$ g.Method:Calculate volume using cone formula, then multiply by density to find mass.Examiner tips
- Remember to use the cone formula with the $\frac{1}{3}$ factor
- Mass = density $\times$ volume, not volume $\div$ density
- Step 1: Subtract $u$ from both sides: $v - u = at$. Step 2: Divide both sides by $t$: $a = \frac{v - u}{t}$.Method:Subtract $u$ from both sides to get $v - u = at$, then divide by $t$ to get $a = \frac{v - u}{t}$.Examiner tips
- Perform inverse operations in the correct order
- Whatever you do to one side, do to the other
- Step 1: Calculate $3.2^2 = 10.24$. Step 2: Subtract: $10.24 - 2.4 = 7.84$. Step 3: Divide: $7.84 \div 0.8 = 9.8$.Method:Square 3.2, subtract 2.4, then divide by 0.8: $(10.24 - 2.4) \div 0.8 = 9.8$.Examiner tips
- Evaluate powers first, then subtraction, then division
- Use a calculator carefully, entering the full expression
- Step 1: Multiply the second equation by 2: $6w + 2y = 26$. Step 2: Add to the first equation: $5w - 2y + 6w + 2y = 16 + 26$, giving $11w = 42$. Step 3: $w = \frac{42}{11}$. Step 4: Substitute into the second equation: $3 \times \frac{42}{11} + y = 13$, so $y = 13 - \frac{126}{11} = \frac{143 - 126}{11} = \frac{17}{11}$.Method:Multiply the second equation by 2, add to the first to eliminate $y$, solve for $w$, then substitute back.Examiner tips
- Check your solution by substituting both values back into the original equations
- Be careful with signs when adding or subtracting equations
- Step 1: Form the equation: $8n + 5(n - 6) = 192$. Step 2: Expand: $8n + 5n - 30 = 192$. Step 3: Simplify: $13n = 222$, so $n = \frac{222}{13} \approx 17.08$.Method:Form the equation $8n + 5(n-6) = 192$, expand and simplify to $13n = 222$, then solve for $n$.Examiner tips
- Set up the equation carefully before solving
- Check that your answer makes sense in context
- Step 1: A 20% reduction means the sale price is 80% of the original. Step 2: $36 = 0.80 \times \text{original}$. Step 3: Original $= 36 \div 0.80 = 45$.Method:The sale price is 80% of the original. Divide $36 by 0.80 to get the original price of $45.Examiner tips
- In reverse percentage questions, always divide by the percentage multiplier
- Never add the percentage of the reduced price back on
- Step 1: Lower bound of length $= 19.5$ cm. Lower bound of width $= 10.5$ cm. Step 2: Lower bound of perimeter $= 2(19.5 + 10.5) = 2 \times 30 = 60$ cm.Method:Find lower bounds of each dimension (19.5 and 10.5), then calculate perimeter: $2(19.5 + 10.5) = 60$.Examiner tips
- When rounded to the nearest cm, the lower bound is 0.5 less than the given value
- For the lower bound of a sum, use lower bounds of all values
- Step 1: Volume scale factor $= \left(\frac{5}{3}\right)^3 = \frac{125}{27}$. Step 2: Volume of larger $= 540 \times \frac{125}{27} = 2500$ cm$^3$.Method:Cube the linear scale factor to get the volume scale factor, then multiply the smaller volume.Examiner tips
- Volume scale factor = (linear scale factor)$^3$
- Make sure you identify which prism is larger and scale in the correct direction
- Step 1: Group the terms: $(3x - 12) + (bx - 4b)$. Step 2: Factor each group: $3(x - 4) + b(x - 4)$. Step 3: Factor out the common bracket: $(x - 4)(3 + b)$.Method:Group as $(3x - 12) + (bx - 4b)$, factor to $3(x-4) + b(x-4)$, then $(x-4)(3+b)$.Examiner tips
- Check your factorisation by expanding back to the original expression
- Be careful with signs when grouping
- Step 1: Exterior angle $= 180° - 156° = 24°$. Step 2: Number of sides $= \frac{360°}{24°} = 15$.Method:Find the exterior angle: $180 - 156 = 24$. Then $360 \div 24 = 15$ sides.Examiner tips
- Exterior angle = 180 - interior angle for regular polygons
- Sum of exterior angles of any polygon = 360 degrees
- Step 1: P(not rain) $= 1 -$ P(rain) $= 1 - 0.35 = 0.65$.Method:Use the complement rule: $1 - 0.35 = 0.65$.Examiner tips
- P(not A) = 1 - P(A)
- Probabilities must be between 0 and 1
- Step 1: P(warm and cycles) $= 0.6 \times 0.8 = 0.48$. Step 2: P(not warm and cycles) $= 0.4 \times 0.3 = 0.12$. Step 3: P(cycles) $= 0.48 + 0.12 = 0.6$.Method:P(cycles) = P(warm) $\times$ P(cycles|warm) + P(not warm) $\times$ P(cycles|not warm) = $0.6 \times 0.8 + 0.4 \times 0.3 = 0.6$.Examiner tips
- Draw a tree diagram to organise the information
- Remember: multiply along branches, add between branches
- Step 1: Interest per year $= \frac{3.5}{100} \times 600 = \$21$. Step 2: Total interest for 4 years $= 21 \times 4 = \$84$. Step 3: Total amount $= 600 + 84 = \$684$.Method:Calculate yearly interest ($600 \times 0.035 = 21$), multiply by 4 years ($84$), add to principal ($684$).Examiner tips
- Simple interest is the same amount each year
- Remember to add the interest to the principal for the total amount
- Step 1: January: $x$. February: $1.2x$. March: $1.2 \times 1.2x = 1.44x$. Step 2: Total spent $= x + 1.2x + 1.44x = 3.64x$. Step 3: $3.64x = 200 - 54.80 = 145.20$. Step 4: $x = 145.20 \div 3.64 = 40$.Method:Write each month as $x$, $1.2x$, $1.44x$. Total spent = $3.64x = 145.20$. Solve for $x = 40$.Examiner tips
- Each month's spending is a percentage increase on the previous month, not on the original
- Total spent + remainder = starting amount
- Step 1: $800 \times r^{12} = 1100.50$, so $r^{12} = \frac{1100.50}{800} = 1.375625$. Step 2: $r = 1.375625^{1/12} \approx 1.02688$. Step 3: Value after 15 years $= 800 \times r^{15} = 800 \times 1.02688^{15} \approx 1204$. Alternatively: $1100.50 \times r^3 = 1100.50 \times 1.02688^3 \approx 1204$.Method:Find $r$ from $800r^{12} = 1100.50$, then calculate $800r^{15}$. Or calculate $1100.50 \times r^3$.Examiner tips
- You can find the value after 15 years by multiplying the 12-year value by $r^3$
- Use the compound interest formula: $A = P \times r^n$
- Step 1: Substitute $x = -7$ into $f(x) = x + 4$. Step 2: $f(-7) = -7 + 4 = -3$.Method:Substitute $x = -7$: $f(-7) = -7 + 4 = -3$.Examiner tips
- Be careful with negative numbers when substituting
- Function notation f(a) means substitute x = a
- Step 1: $g(-2) = 7 - 3(-2) = 7 + 6 = 13$. Step 2: $g(1) = 7 - 3(1) = 4$. Step 3: $g(4) = 7 - 3(4) = 7 - 12 = -5$. Step 4: Range $= \{13, 4, -5\}$.Method:Substitute each domain value: $g(-2) = 13$, $g(1) = 4$, $g(4) = -5$. Range $= \{13, 4, -5\}$.Examiner tips
- Substitute each domain value carefully, especially negatives
- The range is the set of output values
- Step 1: $3^x = \frac{1}{81}$. Step 2: $81 = 3^4$, so $\frac{1}{81} = 3^{-4}$. Step 3: Therefore $x = -4$.Method:Recognise $\frac{1}{81} = 3^{-4}$, so $x = -4$.Examiner tips
- Write fractions as negative powers of the base
- $\frac{1}{a^n} = a^{-n}$
- Step 1: $h^{-1}(x) = 2$ means $h(2) = x$. Step 2: $h(2) = 5^2 = 25$. Step 3: So $x = 25$.Method:Since $h^{-1}(x) = 2$, we have $h(2) = x = 5^2 = 25$.Examiner tips
- If $h^{-1}(x) = a$ then $h(a) = x$
- The inverse function reverses the input and output
- Step 1: Multiply the coefficients: $4 \times 6 = 24$. Step 2: Add the indices: $m^3 \times m^4 = m^{3+4} = m^7$. Step 3: Answer: $24m^7$.Method:Multiply coefficients ($4 \times 6 = 24$) and add indices ($3 + 4 = 7$): $24m^7$.Examiner tips
- When multiplying: multiply coefficients, add indices
- When dividing: divide coefficients, subtract indices
- Step 1: $(27)^{2/3} = (\sqrt[3]{27})^2 = 3^2 = 9$. Step 2: $(p^{12})^{2/3} = p^{12 \times 2/3} = p^8$. Step 3: Answer: $9p^8$.Method:Apply the index to each part: $27^{2/3} = 9$ and $p^{12 \times 2/3} = p^8$. Answer: $9p^8$.Examiner tips
- $(a^m)^n = a^{mn}$
- For fractional indices: $a^{m/n} = (\sqrt[n]{a})^m$
- Step 1: P(French then not French) $= \frac{12}{20} \times \frac{8}{19} = \frac{96}{380}$. Step 2: P(not French then French) $= \frac{8}{20} \times \frac{12}{19} = \frac{96}{380}$. Step 3: P(one of each) $= \frac{96}{380} + \frac{96}{380} = \frac{192}{380} = \frac{96}{190}$.Method:P(one French, one not) = $\frac{12}{20} \times \frac{8}{19} + \frac{8}{20} \times \frac{12}{19} = \frac{96}{190}$.Examiner tips
- Without replacement means the denominator decreases by 1 for the second pick
- Always consider both orderings for 'one of each'
- Step 1: Factorise the numerator: $k^2 + 5k = k(k + 5)$. Step 2: Factorise the denominator: $k^2 - 25 = (k + 5)(k - 5)$. Step 3: Cancel the common factor $(k + 5)$: $\frac{k(k + 5)}{(k + 5)(k - 5)} = \frac{k}{k - 5}$.Method:Factorise: $\frac{k(k+5)}{(k+5)(k-5)}$. Cancel $(k+5)$: $\frac{k}{k-5}$.Examiner tips
- Always factorise before cancelling
- You can only cancel common factors, not individual terms
- Step 1: Time $=$ distance $\div$ speed. Time for first part $= \frac{4}{x}$. Time for second part $= \frac{5}{x+2}$. Step 2: $\frac{4}{x} + \frac{5}{x+2} = \frac{3}{2}$. Step 3: Multiply through by $2x(x+2)$: $8(x+2) + 10x = 3x(x+2)$. Step 4: Expand: $8x + 16 + 10x = 3x^2 + 6x$. Step 5: Simplify: $18x + 16 = 3x^2 + 6x$, so $3x^2 - 12x - 16 = 0$.Method:Write time = distance/speed for each part, set up the equation, multiply by the common denominator, expand and simplify to reach the required quadratic.Examiner tips
- In 'show that' questions, every algebraic step must be clearly shown
- Time = distance / speed
- Step 1: $a = 3$, $b = -12$, $c = -16$. Step 2: $x = \frac{-(-12) \pm \sqrt{(-12)^2 - 4(3)(-16)}}{2(3)} = \frac{12 \pm \sqrt{144 + 192}}{6} = \frac{12 \pm \sqrt{336}}{6}$. Step 3: $x = \frac{12 + 18.33}{6} \approx 5.06$ or $x = \frac{12 - 18.33}{6} \approx -1.06$. Step 4: Since $x$ represents a speed, $x = 5.08$ (positive root).Method:Substitute $a = 3$, $b = -12$, $c = -16$ into the quadratic formula. Evaluate and round to 2 d.p. Accept only the positive root.Examiner tips
- Show your substitution into the formula clearly
- Reject negative solutions when the variable represents a physical quantity like speed
- Step 1: Midpoint of AB $= \left(\frac{2+10}{2}, \frac{1+5}{2}\right) = (6, 3)$. Step 2: Gradient of AB $= \frac{5-1}{10-2} = \frac{4}{8} = \frac{1}{2}$. Step 3: Perpendicular gradient $= -2$ (negative reciprocal). Step 4: Equation: $y - 3 = -2(x - 6)$, so $y = -2x + 12 + 3 = -2x + 15$.Method:Find midpoint (6, 3), gradient of AB = 1/2, perpendicular gradient = -2, then $y - 3 = -2(x - 6)$ gives $y = -2x + 15$.Examiner tips
- The perpendicular bisector passes through the midpoint
- The perpendicular gradient is the negative reciprocal of the original gradient
- Step 1: Differentiating $ax^7$ gives $7ax^6 = 21x^6$, so $7a = 21$ and $a = 3$. Step 2: Differentiating $5x^b$ gives $5bx^{b-1} = 30x^c$, so $5b = 30$ giving $b = 6$, and $c = b - 1 = 5$.Method:From $7a = 21$, $a = 3$. From $5b = 30$, $b = 6$. Since $c = b - 1$, $c = 5$.Examiner tips
- Power rule: $\frac{d}{dx}(kx^n) = knx^{n-1}$
- Match coefficients and powers separately
- Step 1: The base diagonal $= \sqrt{6^2 + 6^2} = 6\sqrt{2}$ cm. Step 2: The space diagonal MN goes from one corner of the base to the opposite corner of the top face. The vertical height is 6 cm and the horizontal distance is $6\sqrt{2}$ cm. Step 3: $\tan \theta = \frac{6}{6\sqrt{2}} = \frac{1}{\sqrt{2}}$. Step 4: $\theta = \arctan\left(\frac{1}{\sqrt{2}}\right) \approx 35.3°$.Method:Find face diagonal = $6\sqrt{2}$. The angle with the base satisfies $\tan \theta = \frac{6}{6\sqrt{2}} = \frac{1}{\sqrt{2}}$, so $\theta \approx 35.3°$.Examiner tips
- Draw the right-angled triangle formed by the space diagonal, the base diagonal, and the vertical edge
- The angle with the base is at the base corner of this triangle
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