The value of $$\left({}^{21}{C}_1-{}^{10}{C}_1\right)+\left({}^{21}{C}_2-{}^{10}{C}_2\right)+\left({}^{21}{C}_3-{}^{10}{C}_3\right)$$ $$\left({}^{21}{C}_4-{}^{10}{C}_4\right)$$$$+....+\left({}^{21}{C}_{10}-{}^{10}{C}_{10}\right)$$ is

If (27)999 is divided by 7, then the remainder is :

The coefficient of x−5 in the binomial expansion of {\left( {{{x + 1} \over {{x^{{2 \over 3}}} - {x^{{1 \over 3}}} + 1}} - {{x - 1} \over {x - {x^{{1 \over 2}}}}}} \right)^{10}}, where x $$\ne$$ 0, 1, is :

Let $$I_n=\int {\tan }^{n}xdx,\left(n>1\right).$$ If $$I_4+I_6$$ = $$a{\tan }^{5}x+b{x}^5+C$$, where C is a constant of integration, then the ordered pair $$\left(a,b\right)$$ is equal to

If f\left( {{{3x - 4} \over {3x + 4}}} \right) = x + 2, x $$\ne$$ $$-$$ {4 \over 3}, and $$\int$$f(x) dx = A log$$|$$1 $$-$$ x $$|$$ + Bx + C, then the ordered pair (A, B) is equal to : (where C is a constant of integration)

The integral $$\int \sqrt{1+2\cot x(\cos ecx+\cot x)}dx$$ \left( {0 < x < {\pi \over 2}} \right) is equal to : (where C is a constant of integration)

Let f(x) = 210.x + 1 and g(x)=310.x $$-$$ 1. If (fog) (x) = x, then x is equal to :

The function f:R \to \left[ { - {1 \over 2},{1 \over 2}} \right] defined as f\left( x \right) = {x \over {1 + {x^2}}}, is

Let $$a$$, b, c $$\in R$$. If $$f$$(x) = ax2 + bx + c is such that $$a$$ + b + c = 3 and $$f$$(x + y) = $$f$$(x) + $$f$$(y) + xy, $$\forall x,y\in R,$$ then $$\sum _{n=1}^{10}f(n)$$ is equal to

The function f : N $$\to$$ N defined by f (x) = x $$-$$ 5 \left[ {{x \over 5}} \right], Where N is the set of natural numbers and [x] denotes the greatest integer less than or equal to x, is :

If y = mx + c is the normal at a point on the parabola y2 = 8x whose focal distance is 8 units, then $$\left|c\right|$$ is equal to :

If the common tangents to the parabola, x2 = 4y and the circle, x2 + y2 = 4 intersect at the point P, then the distance of P from the origin, is :

The integral \int\limits_{{\pi \over 4}}^{{{3\pi } \over 4}} {{{dx} \over {1 + \cos x}}} is equal to

If \mathop {\lim }\limits_{n \to \infty } \,\,{{{1^a} + {2^a} + ...... + {n^a}} \over {{{(n + 1)}^{a - 1}}\left[ {\left( {na + 1} \right) + \left( {na + 2} \right) + ..... + \left( {na + n} \right)} \right]}} = {1 \over {60}} for some positive real number a, then a is equal to :

The integral \int_{{\pi \over {12}}}^{{\pi \over 4}} {\,\,{{8\cos 2x} \over {{{\left( {\tan x + \cot x} \right)}^3}}}} \,dx equals :

If \int\limits_1^2 {{{dx} \over {{{\left( {{x^2} - 2x + 4} \right)}^{{3 \over 2}}}}}} = {k \over {k + 5}}, then k is equal to :

Let z$$\in$$C, the set of complex numbers. Then the equation, 2|z + 3i| $$-$$ |z $$-$$ i| = 0 represents :

The equation Im \left( {{{iz - 2} \over {z - i}}} \right) + 1 = 0, z $$\in$$ C, z $$\ne$$ i represents a part of a circle having radius equal to :

Let $$\omega$$ be a complex number such that 2$$\omega$$ + 1 = z where z = $$\sqrt{-3}$$. If \left| {\matrix{ 1 & 1 & 1 \cr 1 & { - {\omega ^2} - 1} & {{\omega ^2}} \cr 1 & {{\omega ^2}} & {{\omega ^7}} \cr } } \right| = 3k, then k is equal to :

If a point P has co-ordinates (0, $$-$$2) and Q is any point on the circle, x2 + y2 $$-$$ 5x $$-$$ y + 5 = 0, then the maximum value of (PQ)2 is :