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NDA 2021
Exam
NDA
Year
2021
Subject
Mathematics
Chapter
Trigonometric Ratios & Identities

NDA 2021 · Mathematics · Trigonometric Ratios & Identities

Let the equation x · cosec x=p have a solution, where p is a...

Let the equation sec⁡x⋅cosec⁡x=p\sec x \cdot \operatorname{cosec} x=p have a solution, where pp is a positive real number. What should be the smallest value of pp ?

Options

  1. A12\frac{1}{2}
  2. B1
  3. C2
  4. DMinimum does not exist

Correct answer

C. 2

Step-by-step solution

1sin⁡x⋅cos⁡x=p⇒2sin⁡x⋅cos⁡x=2P⇒sin⁡2x=2P since −1≤sin⁡≤1 and P>0∴2P≥1⇒P≤2\begin{aligned} & \frac{1}{\sin x \cdot \cos x}=p \Rightarrow 2 \sin x \cdot \cos x=\frac{2}{P} \\ & \Rightarrow \sin 2 x=\frac{2}{P} \\ & \text{ since }-1 \leq \sin \leq 1 \text{ and } \mathrm{P} > 0 \\ & \therefore \frac{2}{P} \geq 1 \Rightarrow \mathrm{P} \leq 2 \end{aligned} So, smallest value of PP is 2 .
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