If $\frac{d}{d x}\left(\frac{1+x^2+x^4}{1+x+x^2}\right)=a x+b$,then $(a, b)=$

If $y=a \sin x+b \cos x$,then $y^2+\left(\frac{d y}{d x}\right)^2$ is a

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

If $f(x) = \sum_{p=1}^7 p^2 \sin^{-1}\left(\frac{4}{5} \sin(px) - \frac{3}{5} \cos(px)\right)$, then the value of $\frac{df}{dx}$ at $x = 1$ is (Given that $\sin^{-1}(\sin x) = x$)

If $[x]$ represents the greatest integer function and $f(x) = x - [x] - \cos x$,then $f^{\prime}\left(\frac{\pi}{2}\right) = $

Which one of the following is false?