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WMA11/01 Jun 2021, Question 3: The Quadratic Formula and Completing the Square

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 1 (WMA11) · Calculator · 9 marks

The question, in short

Use triangle geometry with sides p, q, 2√2 and 60° angle to derive a relation, solve for p and q, then find the area.

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

Part (a) · 2 marks

Show that applying the cosine rule to triangle ABC with the given sides and 60° angle gives p² + q² − pq = 8.

  1. 1Cosine rule: BC² = p² + q² − 2pq cos60°
  2. 2(2√2)² = 8 and cos60° = 1/2
  3. 38 = p² + q² − pq

AnswerShown: (2√2)² = p² + q² − 2pq cos60° ⇒ 8 = p² + q² − pq

Part (b) · 5 marks

Given q = p + 2, solve algebraically for the exact values of p and q.

  1. 1q = p + 2, so p² + (p + 2)² − p(p + 2) = 8
  2. 2p² + p² + 4p + 4 − p² − 2p = 8, so p² + 2p − 4 = 0
  3. 3p = (−2 ± √(4 + 16))/2 = −1 ± √5
  4. 4p must be positive, so p = −1 + √5
  5. 5q = p + 2 = 1 + √5

Answerp = −1 + √5, q = 1 + √5

Part (c) · 2 marks

Calculate the exact area of the triangle using the values of p and q found.

  1. 1Area = ½pq sin60°
  2. 2pq = (√5 − 1)(√5 + 1) = 5 − 1 = 4
  3. 3Area = ½ × 4 × (√3/2) = √3 m²

AnswerArea = √3 m²

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