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TMUA 2023 · Paper 2 · Question 12 of 20

TMUA 2023 Paper 2 Question 12

Trigonometry — Counting solutions · sufficient and only-if. Try it first; the answer and a full worked solution are below.

TMUA 2023 · Paper 2TrigonometryCounting solutions · sufficient and only-if4 optionshard
In this question, p is a real constant.

The equation sin x cos2 x = p2 sin x has n distinct solutions in the range 0 ≤ x ≤ 2π.

Which of the following statements is/are true?

I   n = 3 is sufficient for p > 1
II  n = 7 only if −1 < p < 1

  1. Anone of them
  2. BI only
  3. CII only
  4. DI and II
Show the answer and worked solution
answer · C
  1. Anone of them
  2. BI only
  3. CII only
  4. DI and II
Factorising, sin x(cos2 x − p2) = 0. The factor sin x = 0 always contributes x = 0,  π,  2π — three solutions. If |p| > 1 the other factor gives nothing, so n = 3; if |p| = 1 its solutions coincide with those already counted, so again n = 3; if 0 < |p| < 1 it adds four more, giving n = 7; and if p = 0 it adds two, giving n = 5. So n = 3 does not force p > 1 — p = −2 also gives n = 3 — and I is false. But n = 7 does require 0 < |p| < 1, hence −1 < p < 1, so II is true.