Given the function $f(x) = \frac{a^x + a^{-x}}{2}$,where $a > 2$. Then $f(x + y) + f(x - y) = $

Let $f(x+y)=f(x) \cdot f(y)$ for all $x, y \in R$. Suppose that $f(3)=3$ and $f^{\prime}(0)=11$,then $f^{\prime}(3)$ is given by

If $f(x) = \frac{x}{x - 1}$,then $\frac{f(a)}{f(a + 1)} = $

If $f : \mathbb{Z} \rightarrow \mathbb{Z}$ is defined by $f(x) = x^{9} - 11 x^{8} - 2 x^{7} + 22 x^{6} + x^{4} - 12 x^{3} + 11 x^{2} + x - 3, \forall x \in \mathbb{Z}$,then $f(11) = $

Let $R$ be the set of all real numbers and let $f$ be a function from $R$ to $R$ such that $f(x) + (x + \frac{1}{2}) f(1 - x) = 1$,for all $x \in R$. Then $2 f(0) + 3 f(1)$ is equal to

If $f(x) = \frac{x}{x - 1} = \frac{1}{y}$,then $f(y) = $