At $x = 2$, $f(x) = [x]$ is

[where [.] is greatest integer function]

Which of the following statements are correct ?

(A) If $f : R \to R$ then $f(x) = |x|$ is continuous everywhere.

(B) If $f : R \to R$ then $f(x) = |x|$ is continuous everywhere but not differentiable at $x = 0$.

(C) Let $f : R - \{0\} \to R$ then $f(x) = \frac{1}{x}$ is continuous everywhere.

(D) Let $f : R \to R$ then $f(x) = |x - 1| + |x - 2|$ is continuous everywhere but not differentiable at exactly 2 points.

(E) If $f : R \to R$ then $f(x) = \cot x$ is continuous everywhere.

Choose the correct answer from the options given below :

If $e^x + e^y = e^{x+y}$, then $\frac{dy}{dx}$ =

The function $f(x) = \begin{cases} \frac{(\sin 2x)}{x} + \cos x & , if \ x \neq 0 \\ K & , if \ x = 0 \end{cases}$ is continuous at $x = 0$, then the value of K is:

Match List-I with List-II

Choose the correct answer from the options given below:

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

If $x=2at^5$ and $y=at^{10}$, then $\dfrac{dy}{dx}=$