GATE EC question library

Let $G(s)=\frac{1}{10 s^{2}}$ 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. 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 …

Consider the polynomial $p(s)=s^{5}+7 s^{4}+3 s^{3}-33 s^{2}+2 s-40$. 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 …

Consider a continuous-time finite-energy signal $f(t)$ whose Fourier transform vanishes outside the frequency interval $\left[-\omega_{c}, \omega_{c}\right]$, where $\omega_{c}$ is in $\mathrm{rad}$ / $\mathrm{sec}$. The signal $f(t)$ is uniformly sampled to obtain $y(t)=f(t) \: p(t)$. Here, …

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 $v_{s}(t)=100 \cos (200 t)$. What should be the value of the variable capacitor $C$ such that the RMS readings on $M_{1}$ and $M_{2}$ are $25 \: \mathrm{V}$ and $5 \: \mathrm{A}$, respectively?

The $Z$-parameter matrix of a two port network relates the port voltages and port currents as follows: $$\left[\begin{array}{l} V_{1} \\ V_{2} \end{array}\right]=Z\left[\begin{array}{l} I_{1} \\ I_{2} \end{array}\right]$$ The $Z$-parameter matrix (with each entry in Ohms) of the network shown below is $\_\_\_\_\_\_\_$

A source transmits symbol $S$ that takes values uniformly at random from the set $\{-2,0,2\}$. The receiver obtains $Y=S+N$, where $N$ is a zero-mean Gaussian random variable independent of $S$. The receiver uses the maximum likelihood decoder to estimate the transmitted symbol $S$. Suppose the probability of symbol estimation error $P_{e}$ is …

Consider a real-valued random process $$f(t)=\sum_{n=1}^{N} a_{n} \: p(t-n T),$$ where $T>0$ and $N$ is a positive integer. Here, $p(t)=1$ for $t \in[0,0.5 T]$ and $0$ otherwise. The coefficients $a_{n}$ are pairwise independent, zero-mean unit-variance random variables. Read the following statements about the random process and choose the …

The identical MOSFETs $M_{1}$ and $M_{2}$ in the circuit given below are ideal and biased in the saturation region. $\mathrm{M}_{1}$ and $\mathrm{M}_{2}$ have a transconductance $g_{m}$ of $5$ mS. The input signals (in Volts) are: $$\begin{array}{l} V_{1}=2.5+0.01 \sin \omega t \\ V_{2}=2.5-0.01 \sin \omega t \end{array}$$ The output signal …

A $10$-bit analog-to-digital converter (ADC) has a sampling frequency of $1\: \mathrm{MHz}$ and a full scale voltage of $3.3 \: \mathrm{V}$. For an input sinusoidal signal with frequency $500 \: \mathrm{kHz}$ , the maximum SNR (in $\mathrm{dB}$ , rounded off to two decimal places) and the data rate (in $\mathrm{Mbps}$) at the output of the ADC …

A positive-edge-triggered sequential circuit is shown below. There are no timing violations in the circuit. Input $P 0$ is set to logic '$0$ ' and $P 1$ is set to logic '$1$' at all times. The timing diagram of the inputs $S E L$ and $S$ are also shown below. The sequence of output $Y$ from time $T_{0}$ to $T_{3}$ is _______.

The intrinsic carrier concentration of a semiconductor is $2.5 \times 10^{16} / \mathrm{m}^{3}$ at $300 \: \mathrm{K}$. If the electron and hole mobilities are $0.15 \mathrm{~m}^{2} / \mathrm{Vs}$ and $0.05 \mathrm{~m}^{2} / \mathrm{Vs}$, respectively, then the intrinsic resistivity of the semiconductor (in $\mathrm{k} \Omega . \mathrm{m}$ ) at …

In the circuit shown, the identical transistors $\text{Q1}$ and $\text{Q2}$ are biased in the active region with $\beta=120$. The Zener diode is in the breakdown region with $\mathrm{V}_{\mathrm{Z}}=5 \mathrm{~V}$ and $\mathrm{I}_{\mathrm{Z}}=25 \mathrm{~mA}$. If $\mathrm{I}_{\mathrm{L}}=12 \mathrm{~mA}$ and …

The electron mobility $\mu_{\mathrm{n}}$ in a non-degenerate germanium semiconductor at $300 \: \mathrm{K}$ is $0.38 \mathrm{~m}^{2} / \mathrm{Vs}$. The electron diffusivity $D_{n}$ at $300 \: \mathrm{K}$ (in $\mathrm{cm}^{2} / \mathrm{s}$, rounded off to the nearest integer) is _________. (Consider the Boltzmann constant …

A square metal sheet of $4 \mathrm{~m} \times 4 \mathrm{~m}$ is placed on the $x-y$ plane as shown in the figure below. If the surface charge density (in $\mu \mathrm{C} / \mathrm{m}^{2}$ ) on the sheet is $\rho_{s}(x, y)=4\mid y \mid$, then the total charge (in $\mu \mathrm{C}$, rounded off to the nearest integer) on the sheet is $\_\_\_\_\_\_$

An electric field of $0.01 \mathrm{~V} / \mathrm{m}$ is applied along the length of a copper wire of circular cross-section with diameter $1 \: \mathrm{mm}$. Copper has a conductivity of $5.8 \times 10^{7} \mathrm{~S} / \mathrm{m}$. The current (in Amperes, rounded off to two decimal places) flowing through the wire is $\_\_\_\_\_\_$.

Consider a non-negative function $f(x)$ which is continuous and bounded over the interval $[2,8]$. Let $M$ and $m$ denote, respectively, the maximum and the minimum values of $f(x)$ over the interval. Among the combinations of $\alpha$ and $\beta$ given below, choose the one(s) for which the inequality …

Which of the following statements involving contour integrals (evaluated counter-clockwise) on the unit circle $C$ in the complex plane is/are TRUE?

Consider a system where $x_{1}(t), x_{2}(t)$, and $x_{3}(t)$ are three internal state signals and $u(t)$ is the input signal. The differential equations governing the system are given by …

The random variable $X$ takes values in $\{-1,0,1\}$ with probabilities $P(X=-1)=P(X=1)=\alpha$ and $P(X=0)=1-2 \alpha$, where $0<\alpha<1 / 2$. Let $g(\alpha)$ denote the entropy of $X$ (in bits), parameterized by $\alpha$. Which of the following statements is/are TRUE?

Let $f(t)$ be a periodic signal with fundamental period $T_{0}>0$. Consider the signal $y(t)=f(\alpha t)$, where $\alpha>1$. The Fourier series expansions of $f(t)$ and $y(t)$ are given by …