The value of {\sin ^{ - 1}}\left( {{{12} \over {13}}} \right) - {\sin ^{ - 1}}\left( {{3 \over 5}} \right) is equal to :

If {\cos ^{ - 1}}x - {\cos ^{ - 1}}{y \over 2} = \alpha,where –1 $$\le$$ x $$\le$$ 1, – 2 $$\le$$ y $$\le$$ 2, x $$\le$$ {y \over 2} , then for all x, y, 4x2 – 4xy cos $$\alpha$$ + y2 is equal to :

Considering only the principal values of inverse functions, the set A = { x $$\ge$$ 0: tan$$-$$1(2x) + tan$$-$$1(3x) = {\pi \over 4}}

All x satisfying the inequality (cot–1 x)2– 7(cot–1 x) + 10 > 0, lie in the interval :

The value of $$\cot \left(\sum _{n=1}^{19}{\cot }^{-1}\left(1+\sum _{p=1}^{n}2p\right)\right)$$ is :

If x = sin$$-$$1(sin10) and y = cos$$-$$1(cos10), then y $$-$$ x is equal to :

If {\cos ^{ - 1}}\left( {{2 \over {3x}}} \right) + {\cos ^{ - 1}}\left( {{3 \over {4x}}} \right) = {\pi \over 2} (x > 3 \over 4), then x is equal to :

If 0 $$\le$$ x < {\pi \over 2}, then the number of values of x for which sin x $$-$$ sin 2x + sin 3x = 0, is :

The sum of all values of $$\theta$$ $$\in$$\left( {0,{\pi \over 2}} \right) satisfying sin2 2$$\theta$$ + cos4 2$$\theta$$ = {3 \over 4} is -

Let S = {$$\theta$$ $$\in$$ [–2$$\pi$$, 2$$\pi$$] : 2cos2$$\theta$$ + 3sin$$\theta$$ = 0}. Then the sum of the elements of S is

The number of solutions of the equation 1 + sin4 x = cos23x, x \in \left[ { - {{5\pi } \over 2},{{5\pi } \over 2}} \right] is :

If [x] denotes the greatest integer $$\le$$ x, then the system of linear equations [sin $$\theta$$]x + [–cos$$\theta$$]y = 0, [cot$$\theta$$]x + y = 0

Let S be the set of all $$\alpha$$ $$\in$$ R such that the equation, cos2x + $$\alpha$$sinx = 2$$\alpha$$– 7 has a solution. Then S is equal to :

The number of four-digit numbers strictly greater than 4321 that can be formed using the digits 0,1,2,3,4,5 (repetition of digits is allowed) is :

A group of students comprises of 5 boys and n girls. If the number of ways, in which a team of 3 students can randomly be selected from this group such that there is at least one boy and at least one girl in each team, is 1750, then n is equal to :

The number of ways of choosing 10 objects out of 31 objects of which 10 are identical and the remaining 21 are distinct, is :

Suppose that 20 pillars of the same height have been erected along the boundary of a circular stadium. If the top of each pillar has been connected by beams with the top of all its non-adjacent pillars, then the total number of beams is :

The number of 6 digit numbers that can be formed using the digits 0, 1, 2, 5, 7 and 9 which are divisible by 11 and no digit is repeated is :

A committee of 11 members is to be formed from 8 males and 5 females. If m is the number of ways the committee is formed with at least 6 males and n is the number of ways the committee is formed with at least 3 females, then :

All possible numbers are formed using the digits 1, 1, 2, 2, 2, 2, 3, 4, 4 taken all at a time. The number of such numbers in which the odd digits occupy even places is :