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TMUA 2018 · Paper 1 · Question 6 of 20

TMUA 2018 Paper 1 Question 6

Trigonometry — Counting solutions · line meeting a tangent curve. Try it first; the answer and a full worked solution are below.

TMUA 2018 · Paper 1TrigonometryCounting solutions · line meeting a tangent curve5 optionshard
Find the number of solutions of the equation xsin 2x = cos 2x with 0 ≤ x ≤ 2π.
  1. A0
  2. B1
  3. C2
  4. D3
  5. E4
Show the answer and worked solution
answer · E
  1. A0
  2. B1
  3. C2
  4. D3
  5. E4
Where cos 2x = 0 the left side would have to be zero too, but there sin 2x = ±1 and x ≠ 0, so no solution is lost by dividing: the equation is tan 2x = 1x, and x = 0 fails as well. Sketch y = tan 2x, which has asymptotes at x = π4,  3π4,  5π4,  7π4, against the decreasing positive curve y = 1x. On (0,  π4) the tangent curve climbs from 0 to +∞ while 1x falls from +∞, so they cross once; on each of the three full branches (π4, 3π4), (3π4, 5π4) and (5π4, 7π4) the tangent curve runs from −∞ to +∞ and crosses once more. On the last piece (7π4,  2π] it only reaches 0 at x = 2π, still below 1x > 0, so there is no crossing there. That is 4 solutions.