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TMUA 2023 · Paper 1 · Question 15 of 20

TMUA 2023 Paper 1 Question 15

Exponentials and logarithms — Exponential functions · range and reciprocals. Try it first; the answer and a full worked solution are below.

TMUA 2023 · Paper 1Exponentials and logarithmsExponential functions · range and reciprocals6 optionshard
The difference between the maximum and minimum values of the function f(x) = acos x, where a > 0 and x is real, is 3.

Find the sum of the possible values of a.

  1. A0
  2. B32
  3. C52
  4. D3
  5. E√10
  6. F√13
Show the answer and worked solution
answer · F
  1. A0
  2. B32
  3. C52
  4. D3
  5. E√10
  6. F√13
As x varies, cos x takes every value in [−1,  1], so acos x runs between a−1 and a. The difference between the extremes is |a − 1a| = 3. If a > 1 then a − 1a = 3, so a2 − 3a − 1 = 0 and a = 3+√132. If a < 1 then 1a − a = 3, so a2 + 3a − 1 = 0 and a = −3+√132. Their sum is √13.