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

If $y=e^{\pi x}$ and $t=x^{\pi}$, then which of following statements is/are TRUE ?

(A) $\frac{d^2y}{dt^2}=\frac{y}{t^2}\left(\frac{\pi(x-1)+1}{\pi}\right)x$

(B) $\frac{d^2y}{dt^2}=y\left(\frac{\pi(x-1)+1}{\pi}\right)x$

(C) $\frac{d^2y}{dt^2}=1$ at $x=1$

(D) $\frac{d^2y}{dt^2}=\frac{e^{\pi}}{\pi}$ at $x=1$

Choose the correct answer from the options given below:

Let $f(x)=\begin{cases}\frac{1-\cos4x}{x^2}, & \text{if } x<0\\2\lambda, & \text{if } x=0\\\frac{\sqrt{x}}{\sqrt{16+\sqrt{x}}-4}, & \text{if } x>0\end{cases}$ such that $f$ continuous at $x=0$. Then, the value of $\lambda$ is equal to:

Match List - I with List - II.

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

If $x = a\sec^3 \theta$, $y = a \tan^3 \theta$, then $\frac{d^2y}{dx^2}$ equals.

If $e^y(x + 1) = 1$ and $\frac{d^2y}{dx^2} = k(\frac{dy}{dx})^2$, then k is equal to