If $x = a\left(t - \dfrac{1}{t}\right)$, $y = b\left(t + \dfrac{1}{t}\right)$, then $\dfrac{dy}{dx} =$

If $y = t - \frac{1}{t}$ and $x = t + \frac{1}{t}$, then $\frac{dy}{dx}$ is equal to

Match the LIST-I with LIST-II

Choose the correct answer from the options given below:

If $x=4t^3$ and $y=6t^2$ then $\frac{d^2y}{dx^2}$ at $t=1$ is

If $x = 3\sin\theta - \sin 3\theta$, $y = 3\cos\theta - \cos 3\theta$, then $\frac{d^2y}{dx^2}$ at $\theta = \frac{\pi}{3}$ is

For the function $f(x) = \sqrt{x}$, which of the following statements are true?

(A) $f(x)$ is continuous for $x \geq 0$.

(B) $f(x)$ is continuous for $x \in R$ (Set of real numbers).

(C) $f(x)$ is differentiable for $x > 0$.

(D) $f(x)$ is differentiable for $x \in R$ (Set of real numbers).

Choose the correct answer from the options given below:

If $f(x) = \begin{cases} \frac{k\cos x}{\pi - 2x}, & x \neq \frac{\pi}{2} \\ 3, & x = \frac{\pi}{2} \end{cases}$ is continuous at $x = \frac{\pi}{2}$, then k is :