For three vectors $u, v, w$,which of the following expressions is not equal to the other three?

Match the appropriate pairs from List-$I$ and List-$II$ :

List-$I$	List-$II$
$1$. Nitrogen molecules	$(A)$ Continuous spectrum
$2$. Incandescent solids	$(B)$ Absorption spectrum
$3$. Fraunhoffer lines	$(C)$ Band spectrum
$4$. Electric arc between iron rods	$(D)$ Emission spectrum

$A$ uniform cylinder of length $L$ and mass $M$ having cross-sectional area $A$ is suspended,with its length vertical,from a fixed point by a massless spring such that it is half-submerged in a liquid of density $\sigma$ at equilibrium position. The extension $x_0$ of the spring when it is in equilibrium is:

$\left|\begin{array}{ccc} \log e & \log e^2 & \log e^3 \\ \log e^2 & \log e^3 & \log e^4 \\ \log e^3 & \log e^4 & \log e^5 \end{array}\right|$ is equal to :

The $P-V$ diagram of $2\,g$ of helium gas for a certain process $A \to B$ is shown in the figure. What is the heat given to the gas during the process $A \to B$ (in $P_0V_0$)?

$\left(\frac{1+i}{1-i}\right)^4+\left(\frac{1-i}{1+i}\right)^4$ is equal to