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1

### IIT-JEE 2006

Subjective
Match the following

Column $$I$$

(A) $$\sum\limits_{i = 1}^\infty {{{\tan }^{ - 1}}\left( {{1 \over {2{i^2}}}} \right) = t,}$$ then tan $$t=$$

(B) Sides $$a, b, c$$ of a triangle $$ABC$$ are in $$AP$$ and
$$\cos {\theta _1} = {a \over {b + c}},\,\cos {\theta _2} = {b \over {a + c}},\cos {\theta _3} = {c \over {a + b}},$$
then $${\tan ^2}\left( {{{{\theta _1}} \over 2}} \right) + {\tan ^2}\left( {{{{\theta _3}} \over 2}} \right) =$$

(C) A line is perpendicular to $$x + 2y + 2z = 0$$ and
passes through $$(0, 1, 0)$$. The perpendicular distance of this line from the origin is

Column $$II$$

(p) $$1$$

(q) $${{\sqrt 5 } \over 3}$$

(r) $${2 \over 3}$$

$$\left( A \right) - \left( p \right),\left( B \right) - \left( r \right),\left( C \right) - \left( q \right)$$
2

### IIT-JEE 2002

Subjective
Prove that $$\cos \,ta{n^{ - 1}}\sin \,{\cot ^{ - 1}}x = \sqrt {{{{x^2} + 1} \over {{x^2} + 2}}}$$.

Solve it.
3

### IIT-JEE 1983

Subjective
Find all the solution of $$4$$ $${\cos ^2}x\sin x - 2{\sin ^2}x = 3\sin x$$

$$x = n\pi ,\,n\pi + {\left( { - 1} \right)^n}{\pi \over {10}},n\pi + {\left( { - 1} \right)^n}\left( {{{ - 3\pi } \over {10}}} \right)$$
4

### IIT-JEE 1981

Subjective
Find the value of : $$\cos \left( {2{{\cos }^{ - 1}}x + {{\sin }^{ - 1}}x} \right)$$ at $$x = {1 \over 5}$$, where
$$0 \le {\cos ^{ - 1}}x \le \pi$$ and $$- \pi /2 \le {\sin ^{ - 1}}x \le \pi /2$$.

$${{ - 2\sqrt 6 } \over 5}$$

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