PrepRung
Sign inSign up free

NDA 2017 Maths · Mathematics · Trigonometric Functions and Identities · hard

For real θ with sin θ≠0, the minimum value of 2sin ^2θ+3csc ^2θ is

Single correct option

For real θ\theta with sin⁡θ≠0\sin\theta\ne0, the minimum value of 2sin⁡2θ+3csc⁡2θ2\sin^2\theta+3\csc^2\theta is
  1. A. 262\sqrt6
  2. B. 66
  3. C. 33
  4. D. 55
Show answer and solution

Answer: (D) 55

AM–GM gives a bound 262\sqrt6 only if 2s2=3/s22s^2=3/s^2, i.e. s2=1.5>1s^2=\sqrt{1.5}>1, which is impossible. With t=s2∈(0,1]t=s^2\in(0,1] the function 2t+3/t2t+3/t is decreasing, so the minimum is at t=1t=1: 2+3=52+3=5.
Report a mistake

Original question written in the NDA 2017 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