$A$ rotating body has angular momentum $L$. If its frequency of rotation is halved and its rotational kinetic energy is doubled,what will be its new angular momentum?

Let $f: R \rightarrow R$ be a function such that $f(x)=x^3+x^2 f^{\prime}(1)+x f^{\prime \prime}(2)+6, x \in R$,then $f(2)$ is

If $f(x)$ is a polynomial of degree $n$ with rational coefficients and $1+2i, 2-\sqrt{3},$ and $5$ are three roots of $f(x)=0$,then the least value of $n$ is

$A$ parallel plate capacitor of capacitance $C$ is connected to a battery and is charged to a potential difference $V$. Another capacitor of capacitance $2C$ is connected to another battery and is charged to potential difference $2V$. The charging batteries are now disconnected and the capacitors are connected in parallel to each other in such a way that the positive terminal of one is connected to the negative terminal of the other. The final energy of the configuration is

If $A_i$ $(i=1, 2, 3, \ldots, n)$ are $n$ independent events with $P(A_i) = \frac{1}{1+i}$ for each $i$, then the probability that none of $A_i$ occurs is

The maximum value of $x^4 e^{-x^2}$ is