Lemmara logoLemmaraWorked solution

4PM1/02R Jun 2020, Question 7: Roots α and β of a Quadratic

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2R · Calculator · 12 marks

اقرأ هذا الحل مشروحاً بالعربي ←

The question, in short

For kx² − 4x + 2k = 7, find the k values giving real roots, then use roots α and β to build a new quadratic with roots (α + 1)/α and (β + 1)/β, coefficients in k.

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

Part (a) · 4 marks

Find the range of values of k for which kx² − 4x + 2k = 7 has real roots.

  1. 1kx² − 4x + (2k − 7) = 0; real roots ⇒ b² − 4ac ≥ 0
  2. 216 − 4k(2k − 7) ≥ 0
  3. 32k² − 7k − 4 ≤ 0
  4. 4(2k + 1)(k − 4) ≤ 0
  5. 5−1/2 ≤ k ≤ 4

Answer−1/2 ≤ k ≤ 4

Part (b) · 8 marks

Given roots α and β of kx² − 4x + 2k = 7, form a quadratic whose roots are (α + 1)/α and (β + 1)/β, giving coefficients in terms of k.

  1. 1α + β = 4/k, αβ = (2k − 7)/k
  2. 2Sum = (α + 1)/α + (β + 1)/β = (2αβ + α + β)/(αβ) = 2(2k − 5)/(2k − 7)
  3. 3Product = (αβ + α + β + 1)/(αβ) = 3(k − 1)/(2k − 7)
  4. 4x² − [2(2k − 5)/(2k − 7)]x + 3(k − 1)/(2k − 7) = 0
  5. 5(2k − 7)x² − 2(2k − 5)x + 3(k − 1) = 0

Answer(2k − 7)x² − 2(2k − 5)x + 3(k − 1) = 0

Stuck on a step? Read the free lesson: Roots α and β of a Quadratic

Practise more roots α and β of a quadratic questions

Real 4PM1 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 roots α and β of a quadratic

Also searched as: 4PM1 2R June 2020 · 4PM1/02R · 4pm1 jun20 2r · question 7.
Lemmara is an independent study tool, not affiliated with or endorsed by Pearson Edexcel. The worked solutions are Lemmara’s own.