Consider the system of linear equations: $A x=b$, where $A$ is an $\mathrm{n} \times \mathrm{n}$ matrix, and $x$ and $b$ are $n$-dimensional column vectors.

Suppose this system of equations has a unique solution. Which of the following statements is/are correct?

The magnitude of the contour integral

$$ \int_c \frac{(z+1)^2}{(z-i)(z-2)} d z $$

over the contour $C:|z-2-i|=\frac{3}{2}$ is $\_\_\_\_$ . [Round off to two decimal places]

Note : $z$ is a complex variable and $i=\sqrt{-1}$.

Let $X$ and $Y$ be two real-valued random variables with

$$ E(X)=1, E(Y)=2, E\left(X^2\right)=4, E\left(Y^2\right)=9, \text { and } E(X Y)=0.9, $$

where $E$ denotes the expectation operator.

The value of $\alpha$ that minimizes $\mathrm{E}\left((\mathrm{X}-\alpha \mathrm{Y})^2\right)$ is $\_\_\_\_$ .

(Round off to one decimal place)

The integral

$$ \frac{1}{\pi} \int\limits_0^{\infty} \frac{x^{2026}}{\left(1+x^{2026}\right)\left(1+x^2\right)} d x $$

evaluates to $\_\_\_\_$

(Round off to two decimal places)