Consider that the concentration of electrons in a semiconductor bar varies linearly from $2 \times 10^{17} \mathrm{~cm}^{-3}$ at $x=1 \mu \mathrm{~m}$ to $1 \times 10^{16} \mathrm{~cm}^{-3}$ at $x=4 \mu \mathrm{~m}$ along the $x$-direction. Assume that the concentration of electrons is not varying along other directions (that is along $y$ and $z$-directions).

[Given: the mobility of electron is $1400 \mathrm{~cm}^2 \mathrm{~V}^{-1} \mathrm{~s}^{-1}$, thermal voltage is 25 mV and electronic charge is $1.6 \times 10^{-19}$ Coulomb.]

The density of electron diffusion current (in $\mathrm{A} / \mathrm{mm}^2$ ) is $\_\_\_\_$ .

(rounded off to two decimal places)

Consider the differential equation $\dot{\vec{w}}=A \vec{w}$, with $\vec{w}(t=0)=\left[\begin{array}{l}1 \\ 1\end{array}\right]$.

If $\vec{w}(t)=e^t \vec{u}_x+e^{-2 t} \vec{u}_y$ be the solution to the equation where $\vec{u}_x$ and $\vec{u}_y$ are unit vectors along the positive x and y axes respectively, then which of the following options is the correct matrix representing $A$ ?

A surface is given by $z^2=2 x^2-y^2$ and $\vec{n}$ and $-\vec{n}$ are unit normal vectors to the surface at the point $\vec{P}=\hat{i}+\sqrt{2} \hat{k}$.

Which of the following vectors can be $\vec{n}$, where $\hat{i}, \hat{j}$ and $\hat{k}$ and are the unit vectors along $x, y$ and $z$ axes, respectively?

Consider the matrix $M=\left[\begin{array}{ccc}2 & 1 & 1 \\ 1 & 3 & 0 \\ -1 & a & b\end{array}\right]$.

Which of the following options is/ are TRUE if $\operatorname{det}(M) \neq 0$ ?