question_answer

A rod AB of length 16 cm. rests between the wall AD and a smooth peg, 1 cm from the wall and makes an angle \[\theta \] with the horizontal. The value of \[\theta \] for which the height of G, the mid-point of the rod above the peg is minimum, is

A) \[15{}^\circ \]

B) \[30{}^\circ \]

C) \[60{}^\circ \]

D) \[75{}^\circ \]

Correct Answer: C

Solution :

[c] We have \[AC=\sec \theta ,AG=8\] \[\therefore CG=8-\sec \theta \] (C being the peg).             But \[u=CG\sin \theta =(8-sec\theta )sin\theta \]             \[u=8\sin \theta -\tan \theta \]             \[\frac{du}{d\theta }=8\cos \theta -{{\sec }^{2}}\theta ,\]             \[\frac{{{d}^{2}}u}{d{{\theta }^{2}}}=-8\sin \theta -2{{\sec }^{2}}\theta \tan \theta \]             \[\frac{du}{d\theta }=0,\] when \[{{\cos }^{3}}\theta =\frac{1}{8},\cos \theta =\frac{1}{2},\]             \[\frac{{{d}^{2}}u}{d{{\theta }^{2}}}>0(at\theta =60{}^\circ ),\therefore \theta =60{}^\circ \]