Lemmara logoLemmaraWorked solution

0580/43 Oct/Nov 2023, Question 6: Algebraic Fractions

Cambridge IGCSE Mathematics 0580 · Paper 43 (Extended) · Calculator · 12 marks

اقرأ هذا الحل مشروحاً بالعربي ←

The question, in short

Given f(x) = 5x − 3, g(x) = 64ˣ and h(x) = 2/(x + 1) with x ≠ −1, evaluate values, invert h, solve an equation and combine fractions.

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

Part (ai) · 1 mark

Work out the output of f when the input is 2.

  1. 1f(2) = 5 × 2 − 3
  2. 2= 7

Answer7

Part (aii) · 2 marks

Work out g applied to the result of f at 0.5.

  1. 1f(0.5) = 5 × 0.5 − 3 = −0.5
  2. 2gf(0.5) = g(−0.5) = 64^(−0.5)
  3. 3= 1/√64 = 1/8

Answer1/8

Part (b) · 3 marks

Obtain an expression for the inverse function of h.

  1. 1y = 2/(x + 1); swap: x = 2/(y + 1)
  2. 2x(y + 1) = 2
  3. 3xy = 2 − x
  4. 4y = (2 − x)/x
  5. 5h⁻¹(x) = (2 − x)/x

Answer(2 − x)/x or 2/x − 1

Part (c) · 2 marks

Determine x such that g(x) equals 1/2⁵.

  1. 164ˣ = (2⁶)ˣ = 2⁶ˣ
  2. 21/2⁵ = 2⁻⁵
  3. 36x = −5
  4. 4x = −5/6

Answer−5/6 or −0.833

Part (d) · 4 marks

Express 1/f(x) − h(x) as one fraction in its simplest form.

  1. 11/f(x) − h(x) = 1/(5x − 3) − 2/(x + 1)
  2. 2= (x + 1 − 2(5x − 3))/((5x − 3)(x + 1))
  3. 3= (x + 1 − 10x + 6)/((5x − 3)(x + 1))
  4. 4= (7 − 9x)/((5x − 3)(x + 1))

Answer(7 − 9x)/((5x − 3)(x + 1)) or (7 − 9x)/(5x² + 2x − 3)

Stuck on a step? Read the free lesson: Algebraic Fractions

Practise more algebraic fractions questions

Real 0580 past-paper questions, marked on your working like the exam, with a worked solution for every part.

Practise free on Lemmara

More past-paper questions on algebraic fractions

Also searched as: 0580/43/O/N/23 · 0580_w23_qp_43 · 0580 w23 43 · question 6.
Lemmara is an independent study tool, not affiliated with or endorsed by Cambridge International Education. The worked solutions are Lemmara’s own.