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 $\operatorname{det}(M) \neq 0$ ?

Consider the two series, $S_A$ and $S_B$, where

$$ \begin{aligned} & S_A=\sum_{n=1}^{\infty} \frac{n^2}{2^n} \\ & S_A=1+\frac{1}{2}+\frac{1}{8}+\frac{1}{16}+\frac{1}{64}+\frac{1}{128}+\frac{1}{512}+\ldots \end{aligned} $$

Which of the following statements is correct for the two given series?