The sum of all those terms which are rational numbers in the expansion of (21/3 + 31/4)12 is :

If the greatest value of the term independent of 'x' in the expansion of {\left( {x\sin \alpha + a{{\cos \alpha } \over x}} \right)^{10}} is {{10!} \over {{{(5!)}^2}}}, then the value of 'a' is equal to :

The lowest integer which is greater than {\left( {1 + {1 \over {{{10}^{100}}}}} \right)^{{{10}^{100}}}} is ______________.

If the coefficients of x7 in {\left( {{x^2} + {1 \over {bx}}} \right)^{11}} and x$$-$$7 in {\left( {{x} - {1 \over {bx^2}}} \right)^{11}}, b $$\ne$$ 0, are equal, then the value of b is equal to :

If $${}^{20}{C}_r$$ is the co-efficient of xr in the expansion of (1 + x)20, then the value of $$\sum _{r=0}^{20}{r}^2.{}^{20}{C}_r$$ is equal to :

$$\sum _{k=0}^{20}{\left({}^{20}{C}_k\right)}^2$$ is equal to :

If \int {{{\cos x - \sin x} \over {\sqrt {8 - \sin 2x} }}} dx = a{\sin ^{ - 1}}\left( {{{\sin x + \cos x} \over b}} \right) + c, where c is a constant of integration, then the ordered pair (a, b) is equal to :

The value of the integral \int {{{\sin \theta .\sin 2\theta ({{\sin }^6}\theta + {{\sin }^4}\theta + {{\sin }^2}\theta )\sqrt {2{{\sin }^4}\theta + 3{{\sin }^2}\theta + 6} } \over {1 - \cos 2\theta }}} \,d\theta is :

The integral \int {{{{e^{3{{\log }_e}2x}} + 5{e^{2{{\log }_e}2x}}} \over {{e^{4{{\log }_e}x}} + 5{e^{3{{\log }_e}x}} - 7{e^{2{{\log }_e}x}}}}} dx, x > 0, is equal to : (where c is a constant of integration)

The integral \int {{{(2x - 1)\cos \sqrt {{{(2x - 1)}^2} + 5} } \over {\sqrt {4{x^2} - 4x + 6} }}} dx is equal to (where c is a constant of integration)

The integral \int {{1 \over {\root 4 \of {{{(x - 1)}^3}{{(x + 2)}^5}} }}} \,dx is equal to : (where C is a constant of integration)

Let f : R → R be defined as f (x) = 2x – 1 and g : R - {1} → R be defined as g(x) = {{x - {1 \over 2}} \over {x - 1}}. Then the composition function f(g(x)) is :

Let f, g : N $$\to$$ N such that f(n + 1) = f(n) + f(1) $$\forall$$ n$$\in$$N and g be any arbitrary function. Which of the following statements is NOT true?

A function f(x) is given by f(x) = {{{5^x}} \over {{5^x} + 5}}, then the sum of the series f\left( {{1 \over {20}}} \right) + f\left( {{2 \over {20}}} \right) + f\left( {{3 \over {20}}} \right) + ....... + f\left( {{{39} \over {20}}} \right) is equal to :

Let x denote the total number of one-one functions from a set A with 3 elements to a set B with 5 elements and y denote the total number of one-one functions form the set A to the set A $$\times$$ B. Then :

Let $$A=\{1,2,3,....,10\}$$ and $$f:A\to A$$ be defined as f(k) = \left\{ {\matrix{ {k + 1} & {if\,k\,is\,odd} \cr k & {if\,k\,is\,even} \cr } } \right. Then the number of possible functions $$g:A\to A$$ such that $$gof=f$$ is :

The range of a$$\in$$R for which the function f(x) = (4a $$-$$ 3)(x + loge 5) + 2(a $$-$$ 7) cot\left( {{x \over 2}} \right) sin2\left( {{x \over 2}} \right), x $$\ne$$ 2n$$\pi$$, n$$\in$$N has critical points, is :

The inverse of $$y=5^{\log x}$$ is :

The real valued function f(x) = {{\cos e{c^{ - 1}}x} \over {\sqrt {x - [x]} }}, where [x] denotes the greatest integer less than or equal to x, is defined for all x belonging to :

If the functions are defined as $$f(x)=\sqrt{x}$$ and $$g(x)=\sqrt{1-x}$$, then what is the common domain of the following functions : f + g, f $$-$$ g, f/g, g/f, g $$-$$ f where (f \pm g)(x) = f(x) \pm g(x),(f/g)x = {{f(x)} \over {g(x)}}