If the area of a circle increases at the rate of $\frac{1}{\sqrt{\pi}}$ sq. units/sec, then the rate (in units/sec) at which the perimeter of the circle changes, when perimeter is $\sqrt{\pi}$ units, is

Let $a$ be a fixed positive real number and $n$ be an arbitrary constant. For the curve $y=\frac{x^n}{a^{n-1}}$, if the length of the subnormal at any point $(\alpha, \beta)$ is proportional to $a^2$, then $n=$

In each of the choices given below, a function and an interval are given. The correct choice having a function and the associated interval for which the Lagrange's mean value theorem is not valid is

Let $P(x)$ be a polynomial of degree 3 having extreme value at $x=1$. If $\mathop {\lim }\limits_{x \to 0}\left(\frac{P(x)+4}{x^2}+2\right)=6$, then $\left(\frac{d P}{d x}\right)_{x=\frac{1}{2}}=$