The coefficient of x101 in the expression $$(5+x{)}^{500}+x(5+x{)}^{499}+x^2(5+x{)}^{498}+.....+x^{500}$$, x > 0, is

If {1 \over {2\,.\,{3^{10}}}} + {1 \over {{2^2}\,.\,{3^9}}} + \,\,.....\,\, + \,\,{1 \over {{2^{10}}\,.\,3}} = {K \over {{2^{10}}\,.\,{3^{10}}}}, then the remainder when K is divided by 6 is :

The remainder when 32022 is divided by 5 is :

For two positive real numbers a and b such that {1 \over {{a^2}}} + {1 \over {{b^3}}} = 4, then minimum value of the constant term in the expansion of {\left( {a{x^{{1 \over 8}}} + b{x^{ - {1 \over {12}}}}} \right)^{10}} is :

The remainder when $$(11{)}^{1011}+(1011{)}^{11}$$ is divided by 9 is

$$\sum _{\text{matrix}i,j=0\text{cr}i\ne j\text{cr}}^n{}^n{C}_i{}^n{C}_j$$ is equal to

The remainder when $$(2021{)}^{2022}+(2022{)}^{2021}$$ is divided by 7 is

The remainder when $$7^{2022}+3^{2022}$$ is divided by 5 is :

$$\sum _{r=1}^{20}\left(r^2+1\right)(r!)$$ is equal to

If \int {{{({x^2} + 1){e^x}} \over {{{(x + 1)}^2}}}dx = f(x){e^x} + C}, where C is a constant, then {{{d^3}f} \over {d{x^3}}} at x = 1 is equal to :

If \int {{1 \over x}\sqrt {{{1 - x} \over {1 + x}}} dx = g(x) + c}, $$g(1)=0$$, then g\left( {{1 \over 2}} \right) is equal to :

For $$I(x)=\int \frac{{\sec }^{2}x-2022}{{\sin }^{2022}x}dx$$, if $$I\left(\frac{\pi }{4}\right)=2^{1011}$$, then

Let a function f : N $$\to$$ N be defined by f(n) = \left[ {\matrix{ {2n,} & {n = 2,4,6,8,......} \cr {n - 1,} & {n = 3,7,11,15,......} \cr {{{n + 1} \over 2},} & {n = 1,5,9,13,......} \cr } } \right. then, f is

Let f(x) = {{x - 1} \over {x + 1}},\,x \in R - \{ 0, - 1,1\}. If $$f^{n+1}(x)=f(f^n(x))$$ for all n $$\in$$ N, then $$f^6(6)+f^7(7)$$ is equal to :

Let f : R $$\to$$ R be defined as f (x) = x $$-$$ 1 and g : R $$-$$ {1, $$-$$1} $$\to$$ R be defined as g(x) = {{{x^2}} \over {{x^2} - 1}}. Then the function fog is :

Let f : N $$\to$$ R be a function such that $$f(x+y)=2f(x)f(y)$$ for natural numbers x and y. If f(1) = 2, then the value of $$\alpha$$ for which \sum\limits_{k = 1}^{10} {f(\alpha + k) = {{512} \over 3}({2^{20}} - 1)} holds, is :

Let $$f:R\to R$$ and $$g:R\to R$$ be two functions defined by $$f(x)={\log }_e(x^2+1)-e^{-x}+1$$ and g(x) = {{1 - 2{e^{2x}}} \over {{e^x}}}. Then, for which of the following range of $$\alpha$$, the inequality f\left( {g\left( {{{{{(\alpha - 1)}^2}} \over 3}} \right)} \right) > f\left( {g\left( {\alpha -{5 \over 3}} \right)} \right) holds ?

The total number of functions, $$f:\{1,2,3,4\}\to \{1,2,3,4,5,6\}$$ such that $$f(1)+f(2)=f(3)$$, is equal to :

The number of bijective functions $$f:\{1,3,5,7,\dots ,99\}\to \{2,4,6,8,\dots .100\}$$, such that $$f(3)\ge f(9)\ge f(15)\ge f(21)\ge \dots ..f(99)$$, is ____________.