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0580/42 May/Jun 2024, Question 5: Equations That Become Quadratics

Cambridge IGCSE Mathematics 0580 · Paper 42 (Extended) · Calculator · 18 marks

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The question, in short

Work through five algebra and geometry items: simplify a power expression, find a sequence rule, expand a product of three brackets, reduce an equation to a quadratic and solve it, and relate cylinder and hemisphere surface areas.

Summarised in our own words. Use the official paper for the exact wording and any diagrams.

Part (a) · 2 marks

Simplify the expression (25x⁶)^(3/2).

  1. 125^(3/2) = (√25)³ = 125
  2. 2(x⁶)^(3/2) = x⁹
  3. 3= 125x⁹

Answer125x⁹

Part (b) · 2 marks

For the terms 1/6, 1, 6, 36, 216, give the formula for the nth term.

  1. 1Each term is 6 × the one before, so terms are powers of 6.
  2. 21/6 = 6⁻¹, 1 = 6⁰, 6 = 6¹, 36 = 6²
  3. 3Term n has power n − 2.
  4. 4nth term = 6ⁿ⁻²

Answer6ⁿ⁻²

Part (c) · 3 marks

Multiply out and collect terms in (x + 4)(x − 3)(3x − 1).

  1. 1(x + 4)(x − 3) = x² + x − 12
  2. 2(x² + x − 12)(3x − 1) = 3x³ − x² + 3x² − x − 36x + 12
  3. 3= 3x³ + 2x² − 37x + 12

Answer3x³ + 2x² − 37x + 12

Part (di) · 4 marks

Starting from (3x + 5) + 7/(x − 2) = x, derive the quadratic 2x² + x − 3 = 0.

  1. 1Multiply by (x − 2): (3x + 5)(x − 2) + 7 = x(x − 2)
  2. 23x² − 6x + 5x − 10 + 7 = x² − 2x
  3. 33x² − x − 3 = x² − 2x
  4. 42x² + x − 3 = 0

Answer(3x + 5)(x − 2) + 7 = x(x − 2) 3x² + 5x − 6x − 10 + 7 = x² − 2x 2x² + x − 3 = 0

Part (dii) · 3 marks

Find the roots of 2x² + x − 3 = 0 using factorisation.

  1. 1(2x + 3)(x − 1) = 0
  2. 2x = −1.5 and x = 1

Answer−1.5 and 1

Part (e) · 4 marks

A cylinder with base radius x and height 3x has the same total surface area as a hemisphere of radius 5y; prove x² = 75y²/8, using A = 4πr² for a sphere.

  1. 1Cylinder: 2πx² + 2πx × 3x = 8πx²
  2. 2Hemisphere: 1/2 × 4π(5y)² + π(5y)² = 50πy² + 25πy² = 75πy²
  3. 38πx² = 75πy²
  4. 4x² = 75y²/8

Answerx² = 75y²/8

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Also searched as: 0580/42/M/J/24 · 0580_s24_qp_42 · 0580 s24 42 · question 5.
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