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0580/42 Oct/Nov 2022, Question 6: Algebraic Fractions

Cambridge IGCSE Mathematics 0580 · Paper 42 (Extended) · Calculator · 21 marks

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The question, in short

Solve, factorise, simplify, and work with proportional relationships, expansions, and a derivative across eight parts.

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

Part (a) · 2 marks

Solve the linear equation 4x + 15 = 9 for x.

  1. 14x = 9 − 15 = −6
  2. 2x = −6/4 = −1.5

Answer−1.5 or −1 1/2 or −3/2

Part (b) · 1 mark

Factorise the difference of two squares a² − 9.

  1. 1a² − 9 = a² − 3² = (a − 3)(a + 3)

Answer(a − 3)(a + 3)

Part (c) · 3 marks

Divide (4a)/5 by (3ad)/(10c) and give the result as one simplified fraction.

  1. 1(4a)/5 × (10c)/(3ad)
  2. 2= (40ac)/(15ad)
  3. 3= (8c)/(3d)

Answer(8c)/(3d)

Part (d) · 2 marks

Five copies of 5ⁿ equal 5ᵐ; express m in terms of n.

  1. 15ⁿ + 5ⁿ + 5ⁿ + 5ⁿ + 5ⁿ = 5 × 5ⁿ = 5ⁿ⁺¹
  2. 2m = n + 1

Answern + 1

Part (e) · 3 marks

Solve 4x² + 8x − 5 = 0 using factorisation.

  1. 14x² + 8x − 5 = 0
  2. 2(2x − 1)(2x + 5) = 0
  3. 3x = 1/2 or x = −5/2

Answer1/2 or 0.5 and −2.5 or −2 1/2 or −5/2

Part (fi) · 3 marks

Given y directly proportional to (x + 3)³ with x = 2 giving y = 13.5, find x when y = 108.

  1. 1y = k(x + 3)³
  2. 213.5 = k × 5³, so k = 0.108
  3. 3108 = 0.108(x + 3)³, so (x + 3)³ = 1000
  4. 4x + 3 = 10, x = 7

Answer7

Part (fii) · 2 marks

For g inversely proportional to d², find the factor n multiplying g when d is halved.

  1. 1g = k/d²
  2. 2d → d/2: g = k/(d/2)² = 4k/d²
  3. 3n = 4

Answer4

Part (g) · 3 marks

Expand and simplify the product (2x + 3)(x − 1)(x + 3).

  1. 1(2x + 3)(x − 1) = 2x² + x − 3
  2. 2(2x² + x − 3)(x + 3) = 2x³ + 6x² + x² + 3x − 3x − 9
  3. 3= 2x³ + 7x² − 9

Answer2x³ + 7x² − 9

Part (h) · 2 marks

Differentiate y = 3x² + 4x − 1 to find dy/dx.

  1. 1Differentiate each term: axⁿ becomes naxⁿ⁻¹.
  2. 23x² → 2 × 3x = 6x
  3. 34x → 4, and the constant −1 → 0
  4. 4dy/dx = 6x + 4

Answer6x + 4

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Also searched as: 0580/42/O/N/22 · 0580_w22_qp_42 · 0580 w22 42 · question 6.
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