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0580/43 May/Jun 2021, Question 2: Algebraic Fractions

Cambridge IGCSE Mathematics 0580 · Paper 43 (Extended) · Calculator · 15 marks

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The question, in short

Algebra skills: substitute into y = px² + t and rearrange it, factorise and solve a quadratic, factorise a cubic fully, and simplify an algebraic fraction.

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

Part (ai) · 2 marks

Substitute p = 3, x = 2, t = −13 into y = px² + t.

  1. 1y = 3 × 2² + (−13)
  2. 2= 12 − 13 = −1

Answer−1

Part (aii) · 3 marks

Make x the subject of y = px² + t.

  1. 1y − t = px²
  2. 2x² = (y − t)/p
  3. 3x = ±√((y − t)/p)

Answer±√((y − t)/p)

Part (bi) · 2 marks

Factorise the quadratic 15x² − 2x − 8.

  1. 115x² − 2x − 8 = 15x² − 12x + 10x − 8
  2. 2= 3x(5x − 4) + 2(5x − 4)
  3. 3= (5x − 4)(3x + 2)

Answer(5x−4)(3x+2)

Part (bii) · 1 mark

Use the factors to solve 15x² − 2x − 8 = 0.

  1. 1(5x − 4)(3x + 2) = 0
  2. 2x = 4/5 or x = −2/3

Answerx = 4/5 or x = −2/3

Part (c) · 3 marks

Take out the common factor and use the difference of two squares on x³ − 16xy².

  1. 1Take out the common factor x: x(x² − 16y²)
  2. 2x² − 16y² is a difference of two squares: x² − (4y)²
  3. 3So x³ − 16xy² = x(x + 4y)(x − 4y)

Answerx(x + 4y)(x − 4y)

Part (d) · 4 marks

Factorise the top and bottom of (2x − 1 − 4ax + 2a)/(2x² − x) and cancel.

  1. 1Numerator: 2x − 1 − 2a(2x − 1) = (2x − 1)(1 − 2a)
  2. 2Denominator: 2x² − x = x(2x − 1)
  3. 3(2x − 1)(1 − 2a)/(x(2x − 1)) = (1 − 2a)/x

Answer(1−2a)/x

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Also searched as: 0580/43/M/J/21 · 0580_s21_qp_43 · 0580 s21 43 · question 2.
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