For the curve $\frac{x^n}{a^n}+\frac{y^n}{b^n}=2,(n \in N$ and $n>1)$ the line $\frac{x}{a}+\frac{y}{b}=2$ is

The height of a cone with semi-vertical angle $\frac{\pi}{3}$ is increasing at the rate of 2 units $/ \mathrm{min}$. The rate at which the radius of the cone is to be decreased so as to have a fixed volume always is

The function $f(x)=2 x^3-9 a x^2+12 a^2 x+1$ where $a>0$ attains its local maximum and local minimum at $p$ and $q$ respectively. If $p^2=q$, then $a=$

Consider all functions given in List I in the interval [1,3]. The list II has the value of ' $c$ ' obtained by applying Lagrange's mean value theorem on the function of List I . Match the function and values of ' c '

$$ \begin{array}{llll} \hline & \text { List I } & & \text { List II } \\ \hline \text { A } & |x-1| & \text { I } & 2 \log \left(e^3+e^2\right) \\ \hline \text { B } & \log x & \text { II } & 2 \\ \hline \text { C } & x^2+x+1 & \text { III } & \log _3 e^2 \\ \hline \text { D } & e^x & \text { IV } & \sqrt{2} \\ \hline & & \text { V } & \log \left(\frac{e^3-e}{2}\right) \\ \hline \end{array} $$