GATE EC question library

A simplified small-signal equivalent circuit of a BJT-based amplifier is given below. The small-signal voltage gain $V_{o} / V_{S}$ (in $\mathrm{V} / \mathrm{V})$ is $\_\_\_\_\_\_$

The ideal BJT in the circuit given below is biased in the active region with a $\beta$ of $100$. If $I_{B}$ is $10 \: \mu \mathrm{A}$, then $V_{C E}$ (in Volts, rounded off to two decimal places) is $\_\_\_\_\_\_\_\_$.

A $3$-input majority logic gate has inputs $X, Y$, and $Z$. The output $F$ of the gate is logic '$1$' if two or more of the inputs are logic '$1$'. The output $F$ is logic '$0$' if two or more of the inputs are logic '$0$'. Which one of the following options is a Boolean expression of the output $F$ ?

A full adder and an XOR gate are used to design a digital circuit with inputs $X, Y$, and $Z$, and output $F$, as shown below. The input $Z$ is connected to the carry-in input of the full adder. If the input $Z$ is set to logic ' $l$ ', then the circuit functions as $\_\_\_\_\_\_\_$ with $X$ and $Y$ as inputs.

Consider the function $f: \mathbb{R} \rightarrow \mathbb{R}$, defined as $$f(x)=2 x^{3}-3 x^{2}-12 x+1$$ Which of the following statements is/are correct? (Here, $\mathbb{R}$ is the set of real numbers.)

Consider the unity-negative-feedback system shown in Figure $(i)$ below, where gain $K \geq 0$. The root locus of this system is shown in Figure $(ii)$ below. For what value(s) of $K$ will the system in Figure $(i)$ have a pole at $-1+j 1$ ?

Let $x[n]$ be a discrete-time signal whose $z$-transform is $X(z)$. Which of the following statements is/are TRUE?

Consider a message signal $m(t)$ which is bandlimited to $[-W, W]$, where $W$ is in $\mathrm{Hz}$ . Consider the following two modulation schemes for the message signal: Double sideband-suppressed carrier (DSB-SC): $$f_{D S B}(t)=A_{c} \: m(t) \cos \left(2 \pi f_{c} t\right)$$ Amplitude modulation (AM): …

Which of the following statements is/are TRUE with respect to an ideal opamp?

Which of the following statements is/are TRUE with respect to ideal MOSFET-based DC-coupled single-stage amplifiers having finite load resistors?

Which of the following can be used as an n-type dopant for silicon? Select the correct option(s).

The function $y(t)$ satisfies $$t^{2} y^{\prime \prime}(t)-2 t y^{\prime}(t)+2 y(t)=0,$$ where $y^{\prime}(t)$ and $y^{\prime \prime}(t)$ denote the first and second derivatives of $y(t)$, respectively. Given $y^{\prime}(0)=1$ and $y^{\prime}(1)=-1$, the maximum value of $y(t)$ over $[0,1]$ is $\_\_\_\_\_\_\_$ (rounded off to two decimal …

The generator matrix of a $(6,3)$ binary linear block code is given by $$G=\left[\begin{array}{llllll} 1 & 0 & 0 & 1 & 0 & 1 \\ 0 & 1 & 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 1 & 1 & 0 \end{array}\right]$$ The minimum Hamming distance $d_{\text{min }}$ between codewords equals _________ (answer in …

All the components in the bandpass filter given below are ideal. The lower $-3$ dB frequency of the filter is $1$ MHz. The upper $-3$ dB frequency (in MHz , rounded off to the nearest integer) is $\_\_\_\_\_\_\_$

A $4$-bit weighted-resistor DAC with inputs $b_{3}, b_{2}, b_{1}$, and $b_{0}$ (MSB to LSB) is designed using an ideal opamp, as shown below. The switches are closed when the corresponding input bits are logic ' $l$ ' and open otherwise. When the input $b_{3} b_{2} b_{1} b_{0}$ changes from $1110$ to $1101$, the magnitude of the change in the …

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 $\_\_\_\_\_\_\_$