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NDA 2023
Exam
NDA
Year
2023
Subject
Mathematics
Chapter
Application of Derivatives

NDA 2023 · Mathematics · Application of Derivatives

Paragraph: Let x=( ^2 A+ A+1)/( A) where 0 Question: What is the...

Paragraph: Let x=sin⁡2A+sin⁡A+1sin⁡Ax=\frac{\sin ^2 A+\sin A+1}{\sin A} where 0<A≤π20 < A \leq \frac{\pi}{2}
Question: What is the minimum value of xx ?

Options

  1. A11
  2. B22
  3. C33
  4. D44

Correct answer

C. 33

Step-by-step solution

x=sin⁡2A+sin⁡A+1sin⁡A=sin⁡A+1sin⁡A+1x=\frac{\sin ^2 A+\sin A+1}{\sin A}=\sin A+\frac{1}{\sin A}+1 Using AM≥GMsin⁡A+1sin⁡A2≥(sin⁡A)⋅(1sin⁡A)sin⁡A+1sin⁡A≥2\mathrm{AM} \geq \mathrm{GM}\frac{\sin A+\frac{1}{\sin A}}{2} \geq \sqrt{(\sin A) \cdot\left(\frac{1}{\sin A}\right)}\sin A+\frac{1}{\sin A} \geq 2 Minimum value of sin⁡A+1sin⁡A=2xminimum⁡=2+1=3\sin A+\frac{1}{\sin A}=2x_{\operatorname{minimum}}=2+1=3
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