If $\tan^{-1}(1) + \tan^{-1}(3) + \tan^{-1}(5) + \tan^{-1}\left(\dfrac{1}{4}\right) = \pi + \tan^{-1}\left(\dfrac{\alpha}{2}\right)$, then the value of $\alpha$ is...

The domain of the function $f(x) = \sqrt{x - 1} + \sqrt{3 - x}$ is ...

Let the function $f(x)$ be defined as:
$f(x) = \begin{cases} \left[\tan\left(\dfrac{\pi}{4} + x\right)\right]^{\dfrac{1}{x}}, & x \neq 0 \\ k, & x = 0 \end{cases}$
If $f(x)$ is continuous at $x = 0$, then the value of $k$ is...

If $y = f\left(\dfrac{3 + 2x}{3 - 2x}\right)$, where $f(x) = \tan(\log x)$ and $\dfrac{dy}{dx} = \left(\dfrac{A}{B + C x^2}\right) \cdot \sec^2\left(\log\left(\dfrac{3 + 2x}{3 - 2x}\right)\right)$, then the respective values of A, B and C are ......