Q$2)(50$ pts$.)$ Consider the MOS differential pair given in the figure below. Assume that $M_1$ and $M_2$ are perfectly matched.

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In part $a,b$ and c $,$ you can assume ${}_{1-8}=0$

a$)$ Determine the value of $R_B$ to set $I_{\text{R}\text{E}\text{F}}=1mA.$

b$)$ Find the drain currents of $M_1$ and $M_2$ transistors.

c$)$ Derive the expressions and calculate the input common$-$mode range $(V_{CM,\text{m}\text{i}\text{n}},V_{CM,\text{m}\text{a}\text{x}}).$

d$)$ Derive the expressions and calculate the output resistance of the current source $R_{\text{o}\text{u}\text{t},Is}.$

e$)$ Find the differential mode gain $A_d=\frac{v_{\text{o}\text{u}\text{t}}}{v_l-v_2}.$

f$)$ Find the common mode gain $A_{cm}=\frac{v_{\text{o}\text{u}\text{t}}}{v_{cm}}.$