If $f(x) = \begin{cases} \frac{x^2 - 9}{x - 3}, & x \neq 3 \\ 5, & x = 3 \end{cases}$ then $f(x)$ :

If $e^y = \log x$, then which of the following is true?

If $y = \left(\log\left(x + \sqrt{x^2+a^2}\right)\right)^2$ and $x \neq \frac{1-a^2}{2}$, then $(x^2+a^2)\frac{d^2y}{dx^2} + x\frac{dy}{dx}$ is equal to:

Consider the following statements for the curve $f(x) = x|x|$ and find that which of the following(s) are/is correct :

(A) $f(x) = x|x|$ is differentiable at $x = 0$.

(B) $f(x) = x|x|$ is continuous at $x = 0$ but not differentiable at $x = 0$.

(C) $f(x) = x|x|$ has point of infection at $x = 0$.

(D) $f(x) = x|x|$ is symmetrical about y-axis.

Choose the correct answer from the options given below :

If $x = 4t$ and $y = \frac{4}{t}$, then $\frac{d^2y}{dx^2}$ is

If the function defined by $f(x) = \begin{cases} \ kx^2 + 1, & \text{if } x \le 1 \\ 2 , & \text{if } x > 1 \end{cases}$ is continuous at $x = 1$, then k is equal to

If $f(x) = \begin{cases} \frac{\tan(\frac{\pi}{4} - x)}{\cot 2x} & , x ≠ \frac{\pi}{4} \\ 2K + 1 & , x = \frac{\pi}{4} \end{cases}$ is continuous at $x = \frac{\pi}{4}$, then the value of K is equal to