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CUET · MATHS · PYQ PAPER 2025

The semi vertical angle of a right circular cone of maximum volume of a given slant height is

  1. A \(\tan ^{-1} 2\)
  2. B \(\sin ^{-1}\left(\frac{1}{3}\right)\)
  3. C \(\cos ^{-1}\left(\frac{1}{\sqrt{3}}\right)\)
  4. D \(\sin ^{-1}\left(\frac{1}{\sqrt{3}}\right)\)
Verified Solution

Answer & Solution

Correct Answer

(C) \(\cos ^{-1}\left(\frac{1}{\sqrt{3}}\right)\)

Step-by-step Solution

Detailed explanation

\(V = \frac{1}{3}\pi r^2 h = \frac{1}{3}\pi (l\sin\theta)^2 (l\cos\theta) = \frac{1}{3}\pi l^3 \sin^2\theta \cos\theta\) \(\frac{dV}{d\theta} = \frac{1}{3}\pi l^3 (2\sin\theta\cos^2\theta - \sin^3\theta)\)…