Cambridge IGCSE · 0580

Mathematics

Number and algebra — Topics 1-2

Name: ____________________Date: October 10, 2026

1.
A laptop is reduced by 15% to a sale price of $612. Calculate its price before the reduction.
[3 marks]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. A 15% reduction leaves 100% - 15% = 85% of the original price. If that price is p, the sale price is 0.85p.
2. Solve 0.85p = 612 by division: p = 612/0.85 = 720 dollars. Adding 15% to 612 would use the wrong base.
3. Check forward: 15% of 720 is 108, and 720 - 108 = 612. The original price is $720.
Marking points
- Recognises that $612 represents 85% of the original price.
- Uses original price = 612 ÷ 0.85.
- Obtains $720.
Examiner tip: A percentage reduction is calculated from the original price, not the sale price.
2.
Solve simultaneously: 2x + y = 11 and x − y = 1. Show your elimination or substitution method.
[4 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
2. Work through this mathematical step: Adds the equations or substitutes correctly to eliminate one variable. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Obtains 3x = 12. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: Finds x = 4. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Work through this mathematical step: Substitutes to find y = 3. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Check both values in both original equations.
Marking points
- Adds the equations or substitutes correctly to eliminate one variable.
- Obtains 3x = 12.
- Finds x = 4.
- Substitutes to find y = 3.
Examiner tip: Check both values in both original equations.
3.
Simplify fully: 3(2x − 5) − 2(x − 4).
[3 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Keep each expression equivalent to the previous one. Expand brackets with their signs intact, combine only like terms, or take out a common factor as requested. Check an algebraic result by expanding it back or substituting a permitted value.
2. Work through this mathematical step: Expands both brackets: 6x − 15 − 2x + 8. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Collects x terms to obtain 4x. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: States the simplified expression 4x − 7. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Take care with the sign when expanding a bracket preceded by a minus sign.
Marking points
- Expands both brackets: 6x − 15 − 2x + 8.
- Collects x terms to obtain 4x.
- States the simplified expression 4x − 7.
Examiner tip: Take care with the sign when expanding a bracket preceded by a minus sign.
4.
$5000 is invested at 4% per year compound interest. Calculate the value of the investment after 3 years.
[3 marks]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
2. Work through this mathematical step: Uses the multiplier 1.04 for each year. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Calculates 5000 × 1.04³. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: Obtains $5624.32 (to the nearest cent). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Raising the multiplier to the power of the number of years is faster and more accurate than calculating year by year.
Marking points
- Uses the multiplier 1.04 for each year.
- Calculates 5000 × 1.04³.
- Obtains $5624.32 (to the nearest cent).
Examiner tip: Raising the multiplier to the power of the number of years is faster and more accurate than calculating year by year.
5.
Factorise fully: 6x² − 15x.
[2 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Keep each expression equivalent to the previous one. Expand brackets with their signs intact, combine only like terms, or take out a common factor as requested. Check an algebraic result by expanding it back or substituting a permitted value.
2. Work through this mathematical step: Identifies the highest common factor 3x. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: States the factorised form 3x(2x − 5). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Check by expanding your answer back out — it should match the original expression exactly.
Marking points
- Identifies the highest common factor 3x.
- States the factorised form 3x(2x − 5).
Examiner tip: Check by expanding your answer back out — it should match the original expression exactly.
6.
Express 0.(45) — the recurring decimal 0.454545... — as a fraction in its simplest form.
[4 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Break the command into its requested parts. For each part, connect a relevant fact or observation to the conclusion it supports. Describing what happens and explaining why it happens are different tasks.
2. Work through this mathematical step: Sets x = 0.454545... and 100x = 45.4545.... Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Subtracts to obtain 99x = 45. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: States x = 45/99. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Work through this mathematical step: Simplifies to 5/11. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Multiply by 100 (not 10) when a two-digit block repeats, so that subtraction cancels the recurring part exactly.
Marking points
- Sets x = 0.454545... and 100x = 45.4545....
- Subtracts to obtain 99x = 45.
- States x = 45/99.
- Simplifies to 5/11.
Examiner tip: Multiply by 100 (not 10) when a two-digit block repeats, so that subtraction cancels the recurring part exactly.
7.
Solve the inequality 3x − 7 ≤ 11, and represent the solution on a number line.
[3 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
2. Work through this mathematical step: Adds 7 to both sides: 3x ≤ 18. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Divides by 3: x ≤ 6. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: Represents x ≤ 6 correctly on a number line, with a filled circle at 6 and shading to the left. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: A filled circle indicates the boundary value is included (≤ or ≥); an open circle indicates it is not (< or >).
Marking points
- Adds 7 to both sides: 3x ≤ 18.
- Divides by 3: x ≤ 6.
- Represents x ≤ 6 correctly on a number line, with a filled circle at 6 and shading to the left.
Examiner tip: A filled circle indicates the boundary value is included (≤ or ≥); an open circle indicates it is not (< or >).
8.
Write 384 as a product of its prime factors, giving your answer in index form.
[3 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Break the command into its requested parts. For each part, connect a relevant fact or observation to the conclusion it supports. Describing what happens and explaining why it happens are different tasks.
2. Work through this mathematical step: Divides repeatedly by prime factors, e.g. 384 = 2 × 192 = 2 × 2 × 96 = ... Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Continues to full factorisation: 2⁷ × 3. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: Confirms the factorisation multiplies back to 384. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Use a factor tree or repeated division by the smallest prime each time to avoid missing a factor.
Marking points
- Divides repeatedly by prime factors, e.g. 384 = 2 × 192 = 2 × 2 × 96 = ...
- Continues to full factorisation: 2⁷ × 3.
- Confirms the factorisation multiplies back to 384.
Examiner tip: Use a factor tree or repeated division by the smallest prime each time to avoid missing a factor.
9.
Make r the subject of the formula V = (4/3)πr³.
[3 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Break the command into its requested parts. For each part, connect a relevant fact or observation to the conclusion it supports. Describing what happens and explaining why it happens are different tasks.
2. Work through this mathematical step: Multiplies both sides by 3/(4π) to isolate r³: r³ = 3V/(4π). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Takes the cube root of both sides. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: States r = ∛(3V/(4π)). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Undo operations in reverse order: since r is cubed and multiplied by constants, isolate r³ first, then take the cube root last.
Marking points
- Multiplies both sides by 3/(4π) to isolate r³: r³ = 3V/(4π).
- Takes the cube root of both sides.
- States r = ∛(3V/(4π)).
Examiner tip: Undo operations in reverse order: since r is cubed and multiplied by constants, isolate r³ first, then take the cube root last.
10.
A number increased by 20% is then decreased by 20%. Determine whether the final result is greater than, less than, or equal to the original number, showing your reasoning with an example.
[4 marks]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
2. Work through this mathematical step: Chooses a starting value, e.g. 100. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Increases by 20% to obtain 120. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: Decreases the new value by 20% to obtain 120 × 0.8 = 96. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Work through this mathematical step: Concludes that the final result (96) is less than the original number (100), since the second percentage is taken from a larger base. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: A percentage increase followed by the same percentage decrease never returns to the original value, because the decrease is calculated from a larger (or smaller) base than the increase was.
Marking points
- Chooses a starting value, e.g. 100.
- Increases by 20% to obtain 120.
- Decreases the new value by 20% to obtain 120 × 0.8 = 96.
- Concludes that the final result (96) is less than the original number (100), since the second percentage is taken from a larger base.
Examiner tip: A percentage increase followed by the same percentage decrease never returns to the original value, because the decrease is calculated from a larger (or smaller) base than the increase was.
11.
A jacket costs $250. It is first reduced by 20% in a sale, then 8% sales tax is added to the reduced price. Calculate the final price.
[3 marks]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
2. Work through this mathematical step: Calculates the reduced price: 250 × 0.8 = 200. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Applies the tax multiplier: 200 × 1.08. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: Obtains a final price of $216. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Apply each percentage change in sequence to the running total, not both to the original price.
Marking points
- Calculates the reduced price: 250 × 0.8 = 200.
- Applies the tax multiplier: 200 × 1.08.
- Obtains a final price of $216.
Examiner tip: Apply each percentage change in sequence to the running total, not both to the original price.
12.
Write 0.000473 in standard form.
[2 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Break the command into its requested parts. For each part, connect a relevant fact or observation to the conclusion it supports. Describing what happens and explaining why it happens are different tasks.
2. Work through this mathematical step: Identifies the significant digits as 4.73. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: States the answer as 4.73 × 10⁻⁴. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Count how many places the decimal point moves to reach a number between 1 and 10 — that count, with the correct sign, is the power of 10.
Marking points
- Identifies the significant digits as 4.73.
- States the answer as 4.73 × 10⁻⁴.
Examiner tip: Count how many places the decimal point moves to reach a number between 1 and 10 — that count, with the correct sign, is the power of 10.
13.
Calculate (3.2 × 10⁵) × (4 × 10³), giving your answer in standard form.
[3 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
2. Work through this mathematical step: Multiplies the decimal parts: 3.2 × 4 = 12.8. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Adds the powers of 10: 10⁵ × 10³ = 10⁸. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: Rewrites 12.8 × 10⁸ in correct standard form as 1.28 × 10⁹. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: If multiplying the decimal parts gives a result of 10 or more, you must adjust it back into the range 1 to 10 and add one to the power of 10.
Marking points
- Multiplies the decimal parts: 3.2 × 4 = 12.8.
- Adds the powers of 10: 10⁵ × 10³ = 10⁸.
- Rewrites 12.8 × 10⁸ in correct standard form as 1.28 × 10⁹.
Examiner tip: If multiplying the decimal parts gives a result of 10 or more, you must adjust it back into the range 1 to 10 and add one to the power of 10.
14.
Simplify fully: (2x³)⁴ ÷ (4x⁵).
[3 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Keep each expression equivalent to the previous one. Expand brackets with their signs intact, combine only like terms, or take out a common factor as requested. Check an algebraic result by expanding it back or substituting a permitted value.
2. Work through this mathematical step: Expands the bracket: (2x³)⁴ = 16x¹². Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Divides the coefficients: 16 ÷ 4 = 4. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: Subtracts the powers of x: x¹² ÷ x⁵ = x⁷, giving 4x⁷. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Raise both the coefficient and the power inside a bracket to the outer index before combining with anything else.
Marking points
- Expands the bracket: (2x³)⁴ = 16x¹².
- Divides the coefficients: 16 ÷ 4 = 4.
- Subtracts the powers of x: x¹² ÷ x⁵ = x⁷, giving 4x⁷.
Examiner tip: Raise both the coefficient and the power inside a bracket to the outer index before combining with anything else.
15.
The nth term of a sequence is given by Uₙ = 5n − 3. Find U₁ and U₁₀, and determine whether 152 is a term of this sequence.
[4 marks]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
2. Work through this mathematical step: Substitutes n = 1 to obtain U₁ = 2. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Substitutes n = 10 to obtain U₁₀ = 47. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: Sets 5n − 3 = 152 and solves to obtain n = 31. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Work through this mathematical step: Concludes that since n = 31 is a positive integer, 152 is a term of the sequence (the 31st term). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: A value belongs to a sequence only if solving for n gives a positive whole number — a fractional or negative n means it is not a term.
Marking points
- Substitutes n = 1 to obtain U₁ = 2.
- Substitutes n = 10 to obtain U₁₀ = 47.
- Sets 5n − 3 = 152 and solves to obtain n = 31.
- Concludes that since n = 31 is a positive integer, 152 is a term of the sequence (the 31st term).
Examiner tip: A value belongs to a sequence only if solving for n gives a positive whole number — a fractional or negative n means it is not a term.
16.
A geometric sequence has first term 3 and common ratio 2. Find the 6th term and the sum of the first 6 terms.
[4 marks]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
2. Work through this mathematical step: Uses the nth term formula Uₙ = ar⁽ⁿ⁻¹⁾ with a = 3, r = 2. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Calculates the 6th term U₆ = 3 × 2⁵ = 96. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: Uses the sum formula Sₙ = a(rⁿ − 1)/(r − 1). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Work through this mathematical step: Calculates S₆ = 3(2⁶ − 1)/(2 − 1) = 189. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: The exponent in the nth term formula is (n − 1), not n — a very common off-by-one error.
Marking points
- Uses the nth term formula Uₙ = ar⁽ⁿ⁻¹⁾ with a = 3, r = 2.
- Calculates the 6th term U₆ = 3 × 2⁵ = 96.
- Uses the sum formula Sₙ = a(rⁿ − 1)/(r − 1).
- Calculates S₆ = 3(2⁶ − 1)/(2 − 1) = 189.
Examiner tip: The exponent in the nth term formula is (n − 1), not n — a very common off-by-one error.
17.
Solve x² − 5x + 3 = 0 using the quadratic formula, giving your answers correct to 2 decimal places.
[4 marks]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
2. Work through this mathematical step: States the quadratic formula x = (−b ± √(b² − 4ac))/(2a) with a = 1, b = −5, c = 3. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Calculates the discriminant b² − 4ac = 25 − 12 = 13. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: Substitutes to obtain x = (5 ± √13)/2. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Work through this mathematical step: States both solutions x = 4.30 and x = 0.70 (2 d.p.). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Identify a, b and c carefully from the equation in the form ax² + bx + c = 0 before substituting into the formula.
Marking points
- States the quadratic formula x = (−b ± √(b² − 4ac))/(2a) with a = 1, b = −5, c = 3.
- Calculates the discriminant b² − 4ac = 25 − 12 = 13.
- Substitutes to obtain x = (5 ± √13)/2.
- States both solutions x = 4.30 and x = 0.70 (2 d.p.).
Examiner tip: Identify a, b and c carefully from the equation in the form ax² + bx + c = 0 before substituting into the formula.
18.
Solve simultaneously: y = x + 1 and y = x² − 5.
[5 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
2. Work through this mathematical step: Sets the two expressions for y equal: x + 1 = x² − 5. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Rearranges into a quadratic equal to zero: x² − x − 6 = 0. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: Factorises to (x − 3)(x + 2) = 0. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Work through this mathematical step: Obtains x = 3 and x = −2. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
6. Work through this mathematical step: Substitutes back to find the corresponding y-values: (3, 4) and (−2, −1). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
7. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Always substitute each x-value back into the simpler (linear) equation to find its matching y-value.
Marking points
- Sets the two expressions for y equal: x + 1 = x² − 5.
- Rearranges into a quadratic equal to zero: x² − x − 6 = 0.
- Factorises to (x − 3)(x + 2) = 0.
- Obtains x = 3 and x = −2.
- Substitutes back to find the corresponding y-values: (3, 4) and (−2, −1).
Examiner tip: Always substitute each x-value back into the simpler (linear) equation to find its matching y-value.
19.
y is directly proportional to x². When x = 3, y = 45. Find y when x = 5.
[4 marks]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
2. Work through this mathematical step: States the relationship y = kx². Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Substitutes x = 3, y = 45 to obtain 45 = 9k. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: Solves to find k = 5. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Work through this mathematical step: Substitutes x = 5 into y = 5x² to obtain y = 125. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Always find the constant of proportionality k first using the given pair of values, before using the formula for any other value.
Marking points
- States the relationship y = kx².
- Substitutes x = 3, y = 45 to obtain 45 = 9k.
- Solves to find k = 5.
- Substitutes x = 5 into y = 5x² to obtain y = 125.
Examiner tip: Always find the constant of proportionality k first using the given pair of values, before using the formula for any other value.
20.
p is inversely proportional to q. When q = 4, p = 15. Find p when q = 10.
[4 marks]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
2. Work through this mathematical step: States the relationship p = k/q. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Substitutes q = 4, p = 15 to obtain 15 = k/4. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: Solves to find k = 60. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Work through this mathematical step: Substitutes q = 10 into p = 60/q to obtain p = 6. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: As q increases in an inverse proportion, p must decrease — use this to sanity-check the direction of your final answer.
Marking points
- States the relationship p = k/q.
- Substitutes q = 4, p = 15 to obtain 15 = k/4.
- Solves to find k = 60.
- Substitutes q = 10 into p = 60/q to obtain p = 6.
Examiner tip: As q increases in an inverse proportion, p must decrease — use this to sanity-check the direction of your final answer.
21.
$720 is shared between A, B and C in the ratio 2 : 3 : 4. Calculate B's share.
[3 marks]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
2. Work through this mathematical step: Finds the total number of parts: 2 + 3 + 4 = 9. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Calculates the value of one part: 720 ÷ 9 = 80. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: States B's share as 3 × 80 = $240. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Always find the value of a single part first by dividing by the total number of parts, then multiply by the number of parts required.
Marking points
- Finds the total number of parts: 2 + 3 + 4 = 9.
- Calculates the value of one part: 720 ÷ 9 = 80.
- States B's share as 3 × 80 = $240.
Examiner tip: Always find the value of a single part first by dividing by the total number of parts, then multiply by the number of parts required.
22.
A length is measured as 15 cm, correct to the nearest centimetre. Write down the upper and lower bounds of this length.
[2 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Break the command into its requested parts. For each part, connect a relevant fact or observation to the conclusion it supports. Describing what happens and explaining why it happens are different tasks.
2. Work through this mathematical step: States the lower bound as 14.5 cm. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: States the upper bound as 15.5 cm. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: For a measurement rounded to the nearest unit, the bounds are half a unit below and half a unit above the given value.
Marking points
- States the lower bound as 14.5 cm.
- States the upper bound as 15.5 cm.
Examiner tip: For a measurement rounded to the nearest unit, the bounds are half a unit below and half a unit above the given value.
23.
Factorise fully: 2x² + 7x + 3.
[3 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Keep each expression equivalent to the previous one. Expand brackets with their signs intact, combine only like terms, or take out a common factor as requested. Check an algebraic result by expanding it back or substituting a permitted value.
2. Work through this mathematical step: Finds two numbers that multiply to give 2 × 3 = 6 and add to give 7: these are 6 and 1. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Splits the middle term: 2x² + 6x + x + 3. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: Factorises by grouping to obtain (2x + 1)(x + 3). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Check the factorisation by expanding it back out; the middle and constant terms must match the original expression exactly.
Marking points
- Finds two numbers that multiply to give 2 × 3 = 6 and add to give 7: these are 6 and 1.
- Splits the middle term: 2x² + 6x + x + 3.
- Factorises by grouping to obtain (2x + 1)(x + 3).
Examiner tip: Check the factorisation by expanding it back out; the middle and constant terms must match the original expression exactly.
24.
Simplify fully: (x² − 9)/(x² + x − 6).
[4 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Keep each expression equivalent to the previous one. Expand brackets with their signs intact, combine only like terms, or take out a common factor as requested. Check an algebraic result by expanding it back or substituting a permitted value.
2. Work through this mathematical step: Factorises the numerator using the difference of two squares: x² − 9 = (x − 3)(x + 3). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Factorises the denominator: x² + x − 6 = (x + 3)(x − 2). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: Cancels the common factor (x + 3). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Work through this mathematical step: States the simplified fraction as (x − 3)/(x − 2). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Always factorise the numerator and denominator fully before attempting to cancel any common factor.
Marking points
- Factorises the numerator using the difference of two squares: x² − 9 = (x − 3)(x + 3).
- Factorises the denominator: x² + x − 6 = (x + 3)(x − 2).
- Cancels the common factor (x + 3).
- States the simplified fraction as (x − 3)/(x − 2).
Examiner tip: Always factorise the numerator and denominator fully before attempting to cancel any common factor.
25.
Make x the subject of the formula y = (3x + 2)/(x − 1).
[5 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Break the command into its requested parts. For each part, connect a relevant fact or observation to the conclusion it supports. Describing what happens and explaining why it happens are different tasks.
2. Work through this mathematical step: Multiplies both sides by (x − 1): y(x − 1) = 3x + 2. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Expands the left side: yx − y = 3x + 2. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: Collects all x terms on one side: yx − 3x = y + 2. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Work through this mathematical step: Factorises out x: x(y − 3) = y + 2. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
6. Work through this mathematical step: States x = (y + 2)/(y − 3). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
7. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: When the subject appears more than once, collect every term containing it on one side before factorising it out as a common factor.
Marking points
- Multiplies both sides by (x − 1): y(x − 1) = 3x + 2.
- Expands the left side: yx − y = 3x + 2.
- Collects all x terms on one side: yx − 3x = y + 2.
- Factorises out x: x(y − 3) = y + 2.
- States x = (y + 2)/(y − 3).
Examiner tip: When the subject appears more than once, collect every term containing it on one side before factorising it out as a common factor.
26.
A jacket costs 68 after a 15% discount. Calculate its original price. All prices use the same currency.
[3 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Let the original price be p. Removing 15% leaves 0.85p, so 0.85p = 68.
2. Divide by the remaining proportion rather than adding 15% to the reduced price: p = 80.
Marking points
- The sale price is 85% of the original price.
- Original price = 68 / 0.85.
- The original price is 80.
Examiner tip: The discount is a percentage of the original price, not of 68.
27.
Adult tickets cost 12 and child tickets cost 7. Nine tickets cost 93 in total. Form simultaneous equations and find how many of each ticket were bought.
[4 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. The first equation counts people, while the second adds their costs; the two constraints must hold together.
2. Eliminating c gives six adults. Check: 6 + 3 = 9 and 6(12) + 3(7) = 93.
Marking points
- Let a and c be adult and child counts; a + c = 9.
- 12a + 7c = 93.
- Subtract 7(a + c) = 63 to get 5a = 30, so a = 6.
- c = 3: six adult tickets and three child tickets.
Examiner tip: Check both the total number and total cost, not only one equation.
28.
Write 0.272727... as a fraction in its simplest form, showing an algebraic method.
[3 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. A repeating block has two digits, so multiplication by 100 aligns the recurring decimal tails.
2. Subtracting removes the infinite tail exactly. Reduce 27/99 by dividing numerator and denominator by nine.
Marking points
- Let x = 0.272727...; then 100x = 27.272727... .
- Subtract to obtain 99x = 27.
- x = 27/99 = 3/11.
Examiner tip: Use an exact recurring decimal, not a finite rounded decimal such as 0.27.
29.
Solve x^2 - x - 6 < 0. Explain why the endpoints are not included.
[4 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Split the number line at the two roots. At x = 0 the expression is -6, whereas outside the roots both factors have the same sign.
2. The required interval is open because the question asks for a value strictly below zero.
Marking points
- Factor as (x - 3)(x + 2).
- The zeros are -2 and 3.
- The upward-opening quadratic is negative between its zeros: -2 < x < 3.
- At either endpoint the expression is zero, which does not satisfy a strict inequality.
Examiner tip: Roots alone are not the solution to an inequality; identify the correct interval.
30.
A rectangle has sides x cm and (x + 3) cm and area 40 cm^2. Form an equation, find its dimensions and calculate its perimeter.
[5 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Multiply the stated side lengths to model the area, then bring all terms to one side to solve the quadratic.
2. Only the positive root describes a physical rectangle. Its area checks as 5(8) = 40, and its perimeter adds all four sides.
Marking points
- x(x + 3) = 40.
- x^2 + 3x - 40 = (x + 8)(x - 5) = 0.
- Reject x = -8 because a length must be positive; x = 5.
- The sides are 5 cm and 8 cm.
- Perimeter = 2(5 + 8) = 26 cm.
Examiner tip: Do not carry a negative algebraic root into a physical length.
31.
An investment grows by 8% in each of two years, then falls by 10%. Calculate the overall percentage change to two decimal places and explain why 8 + 8 - 10 is not correct.
[5 marks]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Using an initial balance of 100 gives 108, then 116.64, then 104.976.
2. The final balance exceeds the initial balance by 4.976 out of 100. Multipliers capture compounding and the changed base for the fall.
Marking points
- Each growth year has multiplier 1.08.
- The fall has multiplier 0.90.
- Combined multiplier = 1.08^2(0.90) = 1.04976.
- Overall change is a 4.98% increase, rounded from 4.976%.
- The changes act on successive different balances, not the same original amount.
Examiner tip: Keep the full multiplier until the final percentage is rounded.