If $a+\alpha=1, b+\beta=2$ and $\operatorname{af}(x)+\alpha f\left(\frac{1}{x}\right)=b x+\frac{\beta}{x}, x \neq 0,$ then the value of expression $\frac{ f ( x )+ f \left(\frac{1}{ x }\right)}{ x +\frac{1}{ x }}$ is ..... .

Let  $a,b,c\; \in R.$ If $f\left( x \right) = a{x^2} + bx + c$ is such that $a + b + c = 3$ and $f\left( {x + y} \right) = f\left( x \right) + f\left( y \right) + xy,$ $\forall x,y \in R,$ then $\mathop \sum \limits_{n = 1}^{10} f\left( n \right)$ is equal to : 

Consider a function $f:\left[ { - 1,1} \right] \to R$ where $f(x) = {\alpha _1}{\sin ^{ - 1}}x + {\alpha _3}\left( {{{\sin }^{ - 1}}{x^3}} \right) + ..... + {\alpha _{(2n + 1)}}{({\sin ^{ - 1}}x)^{(2n + 1)}} - {\cot ^{ - 1}}x$ Where $\alpha _i\ 's$ are positive constants and $n \in N < 100$ , then $f(x)$ is

The value of $\sum \limits_{n=0}^{1947} \frac{1}{2^n+\sqrt{2^{1994}}}$ is equal to

The period of function

$f\left( x \right) = {\cos ^2}\left( {\sin x} \right) + {\sin ^2}\left( {\cos x} \right)$ is

If $f( x + y )=f( x ) f( y )$ and $\sum \limits_{ x =1}^{\infty} f( x )=2, x , y \in N$ where $N$ is the set of all natural numbers, then the value of $\frac{f(4)}{f(2)}$ is