If $f(x) = \sin \log x$, then the value of $f(xy) + f\left( {\frac{x}{y}} \right) - 2f(x).\cos \log y$ is equal to

Let $R$ be the set of all real numbers and $f(x)=\sin ^{10} x\left(\cos ^8 x+\cos ^4 x+\cos ^2 x+1\right)$ $x \in R$. Let  $S=\{\lambda \in R$ there exists a point $c \in(0,2 \pi)$ with $\left.f^{\prime}(c)=\lambda f(c)\right\}$ Then,

If ${e^x} = y + \sqrt {1 + {y^2}} $, then $y =$

Let $f(x) = cos(\sqrt P \,x),$ where $P = [\lambda], ([.]$ is $G.I.F.)$ If the period of $f(x)$ is $\pi$. then

The number of points, where the curve $f(x)=e^{8 x}-e^{6 x}-3 e^{4 x}-e^{2 x}+1, x \in R$ cuts $x$-axis, is equal to

If $f(x) = \frac{{\alpha \,x}}{{x + 1}},\;x \ne - 1$. Then, for what value of $\alpha $ is $f(f(x)) = x$