GATE EC 2025 Question Paper (15-Feb-2025) (Shift 2)

Let $G(s)=\frac{1}{10s^2}$ be the transfer function of a second-order system. A controller $M(s)$ is connected to the system $G(s)$ in the configuration shown below. Consider the following statements.
(i) There exists no controller of the form $M(s)=\frac{K_I}{s}$, where $K_I$ is a positive real number, such that the closed loop system is stable.
(ii) There exists at least one controller of the form $M(s)=K_P+s{K}_D$, where $K_P$ and $K_D$ are positive real numbers, such that the closed loop system is stable.
Which one of the following options is correct?

Consider the polynomial $p(s)=s^5+7s^4+3s^3\text{–}33s^2+2s\text{–}40$. Let (L, I, R) be defined as follows.
L is the number of roots of $p(s)$ with negative real parts.
I is the number of roots of $p(s)$ that are purely imaginary.
R is the number of roots of $p(s)$ with positive real parts.
Which one of the following options is correct?

Consider a continuous-time finite-energy signal $f(t)$ whose Fourier transform vanishes outside the frequency interval $[-{\omega }_c,{\omega }_c]$, where ${\omega }_c$ is in rad/sec.
The signal $f(t)$ is uniformly sampled to obtain $y(t)=f(t)p(t)$. Here
$p(t)={\sum }_{n=-\infty }^{\infty }\delta (t-\tau -n{T}_s)$,
with $\delta (t)$ being the Dirac impulse, $T_s>0$, and $\tau >0$. The sampled signal $y(t)$ is passed through an ideal lowpass filter $h(t)={\omega }_c{T}_s\frac{\sin ({\omega }_{c}t)}{\pi {\omega }_{c}t}$ with cutoff frequency ${\omega }_c$ and passband gain $T_s$.
The output of the filter is given by ________.

In the circuit below, $M_1$ is an ideal AC voltmeter and $M_2$ is an ideal AC ammeter. The source voltage (in Volts) is $v_s(t)=100\cos (200t)$.
What should be the value of the variable capacitor C such that the RMS readings on $M_1$ and $M_2$ are 25 V and 5 A, respectively?

The Z-parameter matrix of a two port network relates the port voltages and port currents as follows:

$\left[\begin{array}{c}{V}_1\\ V_2\end{array}\right]=Z\left[\begin{array}{c}{I}_1\\ I_2\end{array}\right]$

The Z-parameter matrix (with each entry in Ohms) of the network shown below is ________.