GATE EC 2024 Question Paper (11-Feb-2024) (Shift 1)

In the circuit shown below, the transistors $M_1$ and $M_2$ are biased in saturation. Their small signal transconductances are $g_{m1}$ and $g_{m2}$ respectively. Neglect body effect, channel length modulation and intrinsic device capacitances.

Assuming that capacitor $C_1$ is a short circuit for AC analysis, the magnitude of small signal voltage gain $|\frac{v_{out}}{v_{in}}|$ is _________

Which of the following statements is/are true for a BJT with respect to its DC current gain $\beta$?

Consider a system $S$ represented in state space as
$\frac{dx}{dt}=\left[\begin{array}{cc}0& -2\\ 1& -3\end{array}\right]x+\left[\begin{array}{c}1\\ 0\end{array}\right]r,y=[2-5]x$.
Which of the state space representations given below has/have the same transfer function as that of $S$?

Let $F_1,F_2$, and $F_3$ be functions of $(x,y,z)$. Suppose that for every given pair of points $A$ and $B$ in space, the line integral ${\int }_C(F_{1}dx+F_{2}dy+F_{3}dz)$ evaluates to the same value along any path $C$ that starts at $A$ and ends at $B$. Then which of the following is/are true?

Consider the matrix $\left[\begin{array}{cc}1& k\\ 2& 1\end{array}\right]$, where $k$ is a positive real number. Which of the following vectors is/are eigenvector(s) of this matrix?