Relations and F...

Let $$f\left( x \right)=\dfrac { \sin { x } \left( { 2 }^{ x }+{ 2 }^{ -x } \right) \sqrt { \tan ^{ -1 }{ \left( { x }^{ 2 }-x+1 \right)  }  }  }{ { \left( 7{ x }^{ 2 }+3x+1 \right)  }^{ 3 } } $$ then $$f'(0)$$ is equal to

Let $$X$$ be a family of sets and $$R$$ be a relation on $$X$$ defined by $$'A$$ is disjoint from $$B ^ { \prime }$$ . Then $$R$$ is _________.

A function $$ f : R \rightarrow R$$ satisfies x cos y (f (2x+2y)-f(2x-2y))=cos x sin y (f(2x+2y)+f(2x-2y)). If $$f'(0)=\dfrac{1}{2}$$, then 

If $$f(x) = \left\{ {\begin{array}{lllllllllllllll}{\left[ x \right]\quad \quad \quad ,if\quad  - 3 < x \le  - 1}\\{\left| x \right|\quad \quad \quad ,if\quad \; - 1 < x < 1}\\{\left| {\left[ { - x} \right]} \right|\quad \quad ,if\quad \;\;1 \le x \le 3}\end{array}} \right.$$, then $$\left\{x: f(x)\ge 0\right\}$$=

If $$f(x)=\left| \begin{matrix} sin{ x } & sin{ a } & sin{ b } \\ \cos { x }  & \cos { a }  & \cos { b }  \\ \tan { x }  & \tan { a }  & \tan { b }  \end{matrix} \right| $$, where $$0<a<b<\dfrac{\pi}{2}$$, then the equation $$f(x)=0$$ has in the interval $$(a,b)$$

The sum of infinite terms of the series $$1+2 \left(1-\dfrac{1}{n}\right)+3\left(1-\dfrac{1}{n}\right)^{2}+4\left(1-\dfrac{1}{n}\right)^{3}+........$$ is given by

The shortest distance between the line $$\dfrac{x-1}{2}=\dfrac{y-2}{3}=\dfrac{z-3}{4}$$ and $$\dfrac{x-2}{3}=\dfrac{y-2}{4}=\dfrac{z-5}{5}$$ is 

The complete set of values of $$x$$ for which the function $$f(x)=2\tan^{-1}x+\sin^{-1} \dfrac{2x}{1+x^{2}}$$ behaves like a constant function with positive output is equal to

If $$p$$ and $$q$$ are positive real numbers such that $$p^{2}+q^{2}=1$$, then the maximum value of $$(p+q)$$ is-

Let $$y$$ be an imlict function of $$x$$ defined by $$x^{2x}-2x^{x}\cot y-1=0$$.Then $$y'(1)$$ equals