Q204 / 270 · NDA 2023 (Phase 1) · Mathematics

Paragraph: Given that $m(\theta)=\cot ^2 \theta+n^2 \tan ^2 \theta+2 n$, where $n$ is a fixed positive real number.
Question: Under what condition does $m$ attain the least value?

  • A
    $n=\tan ^2 \theta$
  • Correct answer
    B
    $n=\cot ^2 \theta$
  • C
    $n=\sin ^2 \theta$
  • D
    $n=\cos ^2 \theta$

Solution

For minimum value, $\cot ^2 \theta+n^2 \tan ^2 \theta+2 n=4 n$ $(\cot \theta)^2+(n \tan \theta)^2-2(\cot \theta \cdot n \tan \theta)=0$ $(\cot \theta-n \tan \theta)^2=0$ $\cot \theta=n \tan \theta$ $\therefore \quad \cot ^2 \theta=n$

Hence, the correct answer is option: 'B'.

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