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)

The figure shows a straight-line approximation for the forward characteristics of a power diode. A continuous on-state current of 15 A is flowing through the diode.

What is the power loss in the diode?

Consider the circuit shown in Figure (a). A gate pulse $v_g$ is applied between time instants $t_0$ and $t_1$. After $t_1$, during the MOSFET turn OFF process, it experiences a voltage overshoot.

Based on the $v_{d s}$ waveforms shown in Figure (b), which one of the following options is correct?