If cot$$-$$1($$\alpha$$) = cot$$-$$1 2 + cot$$-$$1 8 + cot$$-$$1 18 + cot$$-$$1 32 + ...... upto 100 terms, then $$\alpha$$ is :

The number of solutions of the equation {\sin ^{ - 1}}\left[ {{x^2} + {1 \over 3}} \right] + {\cos ^{ - 1}}\left[ {{x^2} - {2 \over 3}} \right] = {x^2}, for x$$\in$$[$$-$$1, 1], and [x] denotes the greatest integer less than or equal to x, is :

The number of real roots of the equation {\tan ^{ - 1}}\sqrt {x(x + 1)} + {\sin ^{ - 1}}\sqrt {{x^2} + x + 1} = {\pi \over 4} is :

The value of \tan \left( {2{{\tan }^{ - 1}}\left( {{3 \over 5}} \right) + {{\sin }^{ - 1}}\left( {{5 \over {13}}} \right)} \right) is equal to :

If the domain of the function f(x) = {{{{\cos }^{ - 1}}\sqrt {{x^2} - x + 1} } \over {\sqrt {{{\sin }^{ - 1}}\left( {{{2x - 1} \over 2}} \right)} }} is the interval ($$\alpha$$, $$\beta$$], then $$\alpha$$ + $$\beta$$ is equal to :

The domain of the function {{\mathop{\rm cosec}\nolimits} ^{ - 1}}\left( {{{1 + x} \over x}} \right) is :

If \sum\limits_{r = 1}^{50} {{{\tan }^{ - 1}}{1 \over {2{r^2}}} = p}, then the value of tan p is :

If $$({\sin }^{-1}x{)}^2-({\cos }^{-1}x{)}^2=a$$; 0 < x < 1, a $$\ne$$ 0, then the value of 2x2 $$-$$ 1 is :

Let M and m respectively be the maximum and minimum values of the function f(x) = tan$$-$$1 (sin x + cos x) in \left[ {0,{\pi \over 2}} \right], then the value of tan(M $$-$$ m) is equal to :

The domain of the function f(x) = {\sin ^{ - 1}}\left( {{{3{x^2} + x - 1} \over {{{(x - 1)}^2}}}} \right) + {\cos ^{ - 1}}\left( {{{x - 1} \over {x + 1}}} \right) is :

$${\cos }^{-1}(\cos (-5))+{\sin }^{-1}(\sin (6))-{\tan }^{-1}(\tan (12))$$ is equal to : (The inverse trigonometric functions take the principal values)

All possible values of $$\theta$$ $$\in$$ [0, 2$$\pi$$] for which sin 2$$\theta$$ + tan 2$$\theta$$ > 0 lie in :

The number of roots of the equation, (81)sin2x + (81)cos2x = 30 in the interval [ 0, $$\pi$$ ] is equal to :

The number of solutions of the equation x + 2tanx = {\pi \over 2} in the interval [0, 2$$\pi$$] is :

The number of solutions of sin7x + cos7x = 1, x$$\in$$ [0, 4$$\pi$$] is equal to

The sum of all values of x in [0, 2$$\pi$$], for which sin x + sin 2x + sin 3x + sin 4x = 0, is equal to :

The sum of solutions of the equation {{\cos x} \over {1 + \sin x}} = \left| {\tan 2x} \right|, x \in \left( { - {\pi \over 2},{\pi \over 2}} \right) - \left\{ {{\pi \over 4}, - {\pi \over 4}} \right\} is :

The number of solutions of the equation {32^{{{\tan }^2}x}} + {32^{{{\sec }^2}x}} = 81,\,0 \le x \le {\pi \over 4} is :

If n is the number of solutions of the equation 2\cos x\left( {4\sin \left( {{\pi \over 4} + x} \right)\sin \left( {{\pi \over 4} - x} \right) - 1} \right) = 1,x \in [0,\pi ] and S is the sum of all these solutions, then the ordered pair (n, S) is :

A scientific committee is to be formed from 6 Indians and 8 foreigners, which includes at least 2 Indians and double the number of foreigners as Indians. Then the number of ways, the committee can be formed, is :