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NDA 2019 Maths · Mathematics · Trigonometric Functions and Identities · medium

The equation sin θ=cos θ has no solution with θ in the first quadrant. 2.

Single correct option

Consider the following statements:
1. The equation sin⁡θ=cos⁡θ\sin\theta=\cos\theta has no solution with θ\theta in the first quadrant.
2. The maximum value of sin⁡xcos⁡x\sin x\cos x is 12\dfrac12.
Which of the statements given above is/are correct?
  1. A. 1 only
  2. B. 2 only
  3. C. Both 1 and 2
  4. D. Neither 1 nor 2
Show answer and solution

Answer: (B) 2 only

Statement 1 is false: θ=45∘\theta=45^\circ is in the first quadrant and gives sin⁡θ=cos⁡θ=12\sin\theta=\cos\theta=\dfrac1{\sqrt2}. Statement 2: sin⁡xcos⁡x=12sin⁡2x≤12\sin x\cos x=\dfrac12\sin2x\le\dfrac12, with equality at x=45∘x=45^\circ; it is true. Only statement 2 is correct. A maximum of 11 would be the maximum of sin⁡2x\sin2x, not of sin⁡xcos⁡x\sin x\cos x.
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Original question written in the NDA 2019 Maths pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake