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TMUA 2023 · Paper 2 · Question 5 of 20

TMUA 2023 Paper 2 Question 5

Logic and arguments — Necessary and sufficient conditions. Try it first; the answer and a full worked solution are below.

TMUA 2023 · Paper 2Logic and argumentsNecessary and sufficient conditions4 options
Consider the two statements

R:   k is an integer multiple of π

S:   ∫0k sin 2x dx = 0

Which of the following statements is true?

  1. AR is necessary and sufficient for S.
  2. BR is necessary but not sufficient for S.
  3. CR is sufficient but not necessary for S.
  4. DR is not necessary and not sufficient for S.
Show the answer and worked solution
answer · A
  1. AR is necessary and sufficient for S.
  2. BR is necessary but not sufficient for S.
  3. CR is sufficient but not necessary for S.
  4. DR is not necessary and not sufficient for S.
Integrating, ∫0k sin 2x dx = [−cos 2x2]0k = 1 − cos 2k2. This is zero exactly when cos 2k = 1, that is when 2k is a multiple of 2π, which is exactly when k is an integer multiple of π. So R and S are equivalent, and R is both necessary and sufficient.