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WMA11/01 Jun 2023, Question 6: Surds

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 1 (WMA11) · Calculator · 7 marks

The question, in short

Show full working without calculator methods: expand and simplify a squared expression, rationalise a surd fraction into integer form, then combine results to prove a surd identity equals 2.

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

Part (a) · 2 marks

Expand and simplify (r − 1/r)², giving the result as a simplified expression.

  1. 1(r − 1/r)² = r² − r × (1/r) − (1/r) × r + 1/r²
  2. 2= r² − 2 + 1/r²

Answerr² − 2 + 1/r²

Part (b) · 2 marks

Express 1/(3 + 2√2) in the form p + q√2 with integers p and q.

  1. 1Multiply top and bottom by 3 − 2√2: (3 − 2√2)/((3 + 2√2)(3 − 2√2))
  2. 2Denominator = 9 − 8 = 1
  3. 31/(3 + 2√2) = 3 − 2√2

Answer3 − 2√2

Part (c) · 3 marks

Using earlier results, show that √(3 + 2√2) − 1/√(3 + 2√2) equals 2.

  1. 1Let r = √(3 + 2√2). By (a): (r − 1/r)² = r² + 1/r² − 2
  2. 2r² = 3 + 2√2 and 1/r² = 1/(3 + 2√2) = 3 − 2√2 by (b)
  3. 3(r − 1/r)² = 3 + 2√2 + 3 − 2√2 − 2 = 4
  4. 4r > 1, so r − 1/r > 0, hence r − 1/r = √4 = 2

AnswerShown: with r = √(3 + 2√2), (r − 1/r)² = 3 + 2√2 + (3 − 2√2) − 2 = 4, so the expression equals 2 (it is positive).

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Also searched as: WMA11 June 2023 · WMA11/01 · wma11 jun23 · IAS Maths Pure Mathematics 1 June 2023 · question 6.
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