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4MA1/2H Jan 2023, Question 17: Surds

Pearson Edexcel International GCSE Mathematics A 4MA1 · Paper 2H (Higher) · Calculator · 4 marks

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The question, in short

Simplify 8√m + √(49m) − √(9m) to find integer k, then express (5 − √18)/(1 − √2) in the form a + b√2 with integers a and b.

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

Part (a) · 1 mark

Work out the integer value of k in the equation involving m, a prime number.

  1. 18√m + √(49m) − √(9m) = k√m
  2. 2√(49m) = √49 × √m = 7√m
  3. 3√(9m) = √9 × √m = 3√m
  4. 48√m + 7√m − 3√m = 12√m
  5. 512√m = k√m, so k = 12

Answer12

Part (b) · 3 marks

Show, with each stage of working, that the fraction simplifies to a + b√2 for integers a and b.

  1. 1(5 − √18)/(1 − √2) × (1 + √2)/(1 + √2)
  2. 2Numerator: (5 − √18)(1 + √2) = 5 + 5√2 − √18 − √36
  3. 3Simplify: 5 + 5√2 − 3√2 − 6 = −1 + 2√2
  4. 4Denominator: (1 − √2)(1 + √2) = 1 − 2 = −1
  5. 5(−1 + 2√2)/(−1) = 1 − 2√2
  6. 6a + b√2 = 1 − 2√2

Answer1 − 2√2

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