Consider a non-negative function $f(x)$ which is continuous and bounded over the interval $[2,8]$. Let $M$ and $m$ denote, respectively, the maximum and the minimum values of $f(x)$ over the interval.

Among the combinations of $\alpha$ and $\beta$ given below, choose the one(s) for which the inequality

$$ \beta \leq \int_2^8 f(x) d x \leq \alpha $$

is guaranteed to hold.

Which of the following statements involving contour integrals (evaluated counter-clockwise) on the unit circle $C$ in the complex plane is/are TRUE?

Consider the vectors

$$ a=\left[\begin{array}{l} 1 \\ 1 \end{array}\right], b=\left[\begin{array}{c} 0 \\ 3 \sqrt{2} \end{array}\right] $$

For real-valued scalar variable $x$, the value of

$$ \min _x\|a x-b\|_2 $$

is___________(rounded off to two decimal places).

$\|\cdot\|_2$ denotes the Euclidean norm, i.e., for $y=\left[\begin{array}{l}y_1 \\ y_2\end{array}\right],\|y\|_2=\sqrt{y_1^2+y_2^2}$.