JEE MAIN - Mathematics (2020 - 7th January Morning Slot - No. 14)
If $$y\left( \alpha \right) = \sqrt {2\left( {{{\tan \alpha + \cot \alpha } \over {1 + {{\tan }^2}\alpha }}} \right) + {1 \over {{{\sin }^2}\alpha }}} ,\alpha \in \left( {{{3\pi } \over 4},\pi } \right)$$
$${{dy} \over {d\alpha }}\,\,at\,\alpha = {{5\pi } \over 6}is$$ :
$${{dy} \over {d\alpha }}\,\,at\,\alpha = {{5\pi } \over 6}is$$ :
4
-4
$${4 \over 3}$$
-$${1 \over 4}$$
Explanation
$$y\left( \alpha \right) = \sqrt {2\left( {{{\tan \alpha + \cot \alpha } \over {1 + {{\tan }^2}\alpha }}} \right) + {1 \over {{{\sin }^2}\alpha }}}$$
= $$\sqrt {2\left( {{{1 + {{\tan }^2}\alpha } \over {\tan \alpha \left( {1 + {{\tan }^2}\alpha } \right)}}} \right) + {1 \over {{{\sin }^2}\alpha }}} $$
= $$\sqrt {2\left( {{{\cos \alpha } \over {\sin \alpha }}} \right) + {1 \over {{{\sin }^2}\alpha }}} $$
= $$\sqrt {{{\sin 2\alpha + 1} \over {{{\sin }^2}\alpha }}} $$
= $${{\left| {\sin \alpha + \cos \alpha } \right|} \over {\left| {\sin \alpha } \right|}}$$
At $$\alpha $$ = $${{5\pi } \over 6}$$
$${\left| {\sin \alpha + \cos \alpha } \right|}$$ = -(sin$$\alpha $$ + cos$$\alpha $$)
and |sin$$\alpha $$| = sin$$\alpha $$
$$ \therefore $$ y($$\alpha $$) = $${{ - \left( {\sin \alpha + \cos \alpha } \right)} \over {\sin \alpha }}$$
= -1 - cot$$\alpha $$
$$ \therefore $$ $${{dy} \over {d\alpha }}$$ = cosec2$$\alpha $$
So $${{{dy} \over {d\alpha }}}$$ at $$\alpha $$ = $${{5\pi } \over 6}$$,
= cosec2$${{5\pi } \over 6}$$ = 4
= $$\sqrt {2\left( {{{1 + {{\tan }^2}\alpha } \over {\tan \alpha \left( {1 + {{\tan }^2}\alpha } \right)}}} \right) + {1 \over {{{\sin }^2}\alpha }}} $$
= $$\sqrt {2\left( {{{\cos \alpha } \over {\sin \alpha }}} \right) + {1 \over {{{\sin }^2}\alpha }}} $$
= $$\sqrt {{{\sin 2\alpha + 1} \over {{{\sin }^2}\alpha }}} $$
= $${{\left| {\sin \alpha + \cos \alpha } \right|} \over {\left| {\sin \alpha } \right|}}$$
At $$\alpha $$ = $${{5\pi } \over 6}$$
$${\left| {\sin \alpha + \cos \alpha } \right|}$$ = -(sin$$\alpha $$ + cos$$\alpha $$)
and |sin$$\alpha $$| = sin$$\alpha $$
$$ \therefore $$ y($$\alpha $$) = $${{ - \left( {\sin \alpha + \cos \alpha } \right)} \over {\sin \alpha }}$$
= -1 - cot$$\alpha $$
$$ \therefore $$ $${{dy} \over {d\alpha }}$$ = cosec2$$\alpha $$
So $${{{dy} \over {d\alpha }}}$$ at $$\alpha $$ = $${{5\pi } \over 6}$$,
= cosec2$${{5\pi } \over 6}$$ = 4
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