$f(x) = [x]$, where $[\,]$ represents greatest integer function.

(A) For $2 \leq x < 3$, $[x] = 3$

(B) For $2 \leq x < 3$, $[x] = 2$

(C) Right hand derivative of f at $x = 2$ is not defined

(D) Left hand derivative of f at $x = 2$ is zero

(E) $f(x)$ is not differentiable at $x = 2$

Choose the correct answer from the options given below :

If $x = t^3$, $y = t^2$ then $\frac{d^2y}{dx^2}$ is equal to:

For the function $y(x)=\sqrt{x+1}-\sqrt{x-1}$, which of the following are true?

(A) $\left(\frac{dy}{dx}\right)_{at\,x=2}=\frac{1-\sqrt{3}}{2\sqrt{3}}$

(B) $(x^2-1)\frac{d^2y}{dx^2}-\frac{dy}{dx}+4y=0$

(C) $(x^2-1)\left(\frac{dy}{dx}\right)^2-4y^2=0$

(D) If $P(x)\frac{d^2y}{dx^2}+Q(x)\frac{dy}{dx}+\alpha y=0$, then $P(x)-xQ(x)-4\alpha=0$

Choose the correct answer from the options given below:

If $y = x^x$, $\frac{dy}{dx}$ will be:

The value of k for which the function, defined by, $f(x) = \begin{cases} \frac{3x + 4 \tan x}{x} & : x \neq 0 \\ k & : x = 0 \end{cases}$ is continuous at $x = 0$, is

Match List-I with List-II. Here [x] denotes the greatest integer function

$\begin{array}{|l|l|} \hline \rule{0pt}{2.8ex}\text{List-I} & \text{List-II} \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(A) } f(x) = [x] & \text{(I) is continuous everywhere but not differentiable at } x=-1 \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(B) } f(x) = |x-1| & \text{(II) is continuous everywhere except at all integral values} \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(C) } f(x) = e^{|x|} & \text{(III) is continuous everywhere but not differentiable at } x=1 \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(D) } f(x) = |x+1| & \text{(IV) is continuous everywhere but not differentiable at } x=0 \\[1.2ex] \hline \end{array}$

Choose the correct answer from the options given below:

For $x > y > 0$, if $x^5 y^6 = (x + y)^{11}$, then $\frac{d^2y}{dx^2}$ is