The equation of a tangent to the curve $y \cot x = y^3 \tan x$ at the point where the abscissa is $\frac{\pi}{4}$ is:

Let $f:[0,3] \rightarrow R$ be defined by $f(x)=\min \{x-[x], 1+[x]-x\}$,where $[x]$ is the greatest integer less than or equal to $x$. Let $P$ denote the set containing all $x \in[0,3]$ where $f$ is discontinuous,and $Q$ denote the set containing all $x \in(0,3)$ where $f$ is not differentiable. Then the sum of number of elements in $P$ and $Q$ is equal to $......$

The number of real roots of the equation $e^{x-1} + \log x + x - 2 = 0$, where $x > 0$, is

If $f(x) = \cos x \cos 2x \cos 4x \cos 8x \cos 16x$,then $f'\left( \frac{\pi}{4} \right)$ is

If $y = \log(\tan(x/2)) + \sin^{-1}(\cos x)$,then $dy/dx$ is

If $F(x) = \left(f\left(\frac{x}{2}\right)\right)^2 + \left(g\left(\frac{x}{2}\right)\right)^2$,where $f^{\prime \prime}(x) = -f(x)$ and $g(x) = f^{\prime}(x)$,and given $F(5) = 5$,then $F(10)$ is equal to