Lemmara logoLemmaraWorked solution

4PM1/01R Jun 2025, Question 8: Roots α and β of a Quadratic

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1R · Calculator · 15 marks

اقرأ هذا الحل مشروحاً بالعربي ←

The question, in short

Given f(x) = 2x² − 4x + 9, curve C and tangent line l: y − 8x + 9 = 0, find p where perpendicular tangent k touches C at x = p, prove two identities, then form an integer quadratic with roots α³ − β and β³ − α.

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

Part (a) · 5 marks

Find the value of p where the tangent to C perpendicular to l touches C.

  1. 1dy/dx = 4x − 4
  2. 2Gradient of l = 8
  3. 3Gradient of k = −1/8
  4. 44p − 4 = −1/8
  5. 5p = 31/32

Answerp = 31/32

Part (b) · 2 marks

Show the two given algebraic identities involving α + β, αβ, α³ + β³ and α⁴ + β⁴.

  1. 1(i) (α + β)³ = α³ + 3α²β + 3αβ² + β³
  2. 2= α³ + β³ + 3αβ(α + β)
  3. 3(ii) α⁴ + β⁴ = (α² + β²)² − 2α²β²
  4. 4α² + β² = (α + β)² − 2αβ
  5. 5So α⁴ + β⁴ = ((α + β)² − 2αβ)² − 2(αβ)²

Answer(i) (α + β)³ = α³ + β³ + 3αβ(α + β) shown; (ii) α⁴ + β⁴ = ((α + β)² − 2αβ)² − 2(αβ)² shown

Part (c) · 8 marks

Using part (b) and roots α, β of f(x) = 0 without solving, form an integer quadratic with roots α³ − β and β³ − α.

  1. 1α + β = 2, αβ = 9/2
  2. 2Sum = α³ + β³ − (α + β) = (α + β)³ − 3αβ(α + β) − (α + β) = 8 − 27 − 2 = −21
  3. 3Product = (αβ)³ − (α⁴ + β⁴) + αβ
  4. 4α⁴ + β⁴ = (4 − 9)² − 2(81/4) = −31/2
  5. 5Product = 729/8 + 31/2 + 9/2 = 889/8
  6. 6x² + 21x + 889/8 = 0, so 8x² + 168x + 889 = 0

Answer8x² + 168x + 889 = 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 1R June 2025 · 4PM1/01R · 4pm1 jun25 1r · question 8.
Lemmara is an independent study tool, not affiliated with or endorsed by Pearson Edexcel. The worked solutions are Lemmara’s own.