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TMUA 2022 · Paper 1 · Question 3 of 20

TMUA 2022 Paper 1 Question 3

Differentiation and integration — Integration with several conditions. Try it first; the answer and a full worked solution are below.

TMUA 2022 · Paper 1Differentiation and integrationIntegration with several conditions6 options
Given the following statements about a function f
  • f''(x) = a for all x
  • f(0) = 1, f(1) = 2
  • ∫01 f(x) dx = 1
find the value of a.
  1. A−6
  2. B−3
  3. C−2
  4. D2
  5. E3
  6. F6
Show the answer and worked solution
answer · F
  1. A−6
  2. B−3
  3. C−2
  4. D2
  5. E3
  6. F6
Since f''  is constant, f(x) = a2x2 + bx + c. From f(0) = 1 we get c = 1, and from f(1) = 2 we get a2 + b = 1. The integral gives a6 + b2 + 1 = 1, so b = −a3. Substituting, a2 − a3 = 1, so a6 = 1 and a = 6.