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?

Let $X, N, Y$ and $Z$ be random variables. The variables $X$ and $N$ are independent of each other. $X$ is uniformly distributed between -1 and $1 ; N$ follows Normal distribution with zero mean and unity variance.

$Y$ and $Z$ are defined as, $Y=X+N$ and $Z=X^2+N$.

Which of the following pairs represents the values of correlation between $X$ and $Y$ and that between $X$ and $Z$ ?