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1

IIT-JEE 2005

Subjective
Find the range of values of $$\,t$$ for which $$$2\,\sin \,t = {{1 - 2x + 5{x^2}} \over {3{x^2} - 2x - 1}},\,\,\,\,\,t\, \in \,\left[ { - {\pi \over 2},\,{\pi \over 2}} \right].$$$

Answer

$$\,\left[ {{{ - \pi } \over 2},\,{{ - \pi } \over {10}}} \right]\, \cup \,\left[ {{{3\pi } \over {10}},\,{\pi \over 2}} \right]$$
2

IIT-JEE 2000

Subjective
In any triangle $$ABC,$$ prove that $$$\cot {A \over 2} + \cot {B \over 2} + \cot {C \over 2} = \cot {A \over 2}\cot {B \over 2}\cot {C \over 2}.$$$

Answer

Solve it.
3

IIT-JEE 1998

Subjective
Prove that $$\tan \,\alpha + 2\tan 2\alpha + 4\tan 4\alpha + 8\cot 8\alpha = \cot \alpha $$

Answer

Solve it.
4

IIT-JEE 1997

Subjective
Prove that $$\sum\limits_{k = 1}^{n - 1} {\left( {n - k} \right)\,\cos \,{{2k\pi } \over n} = - {n \over 2},} $$ where $$n \ge 3$$ is an integer.

Answer

Solve it.

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