If $f(x) = \begin{cases} e^x + a & \text{for } x < 0 \\ x - 3 & \text{for } x \geqslant 0 \end{cases}$ is differentiable at $x = 0$,then $a$ equals:

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

If $f(x) = \begin{cases} \frac{1}{|x|} & ; |x| \geq 1 \\ ax^2 + b & ; |x| < 1 \end{cases}$ is differentiable at every point of the domain,then the values of $a$ and $b$ are respectively

If $f(x) = \begin{cases} \frac{1}{|x|}, & |x| \geq 1 \\ ax^2 + b, & -1 < x < 1 \end{cases}$ is differentiable $\forall x \in \mathbb{R}$,then one of the values of $a$ and $b$ is-

If the function $f(x) = \begin{cases} \frac{(e^{kx}-1) \sin kx}{4 \tan x}, & x \neq 0 \\ P, & x=0 \end{cases}$ is differentiable at $x=0$,then

If $f(x) = \begin{cases} k \cos x - x \cos k, & x \in [0, \frac{\pi}{2}] \\ k \sin x + x \sin k, & x \in (\frac{\pi}{2}, \pi] \end{cases}$ is differentiable in $(0, \pi)$,then: