GATE EC question library

Consider the integral $$\oint _{c}\frac{sin\left ( x \right )}{x^{2}\left ( x^{2}+4 \right )}dx$$ where $C$ is a counter-clockwise oriented circle defined as $\left | x-i \right |=2$. The value of the integral is

A box contains the following three coins. A fair coin with head on one face and tail on the other face. A coin with heads on both the faces. A coin with tails on both the faces. A coin is picked randomly from the box and tossed. Out of the two remaining coins in the box, one coin is then picked randomly and tossed. If the first toss results in a head, the …

The switch in the circuit in the figure is in position $P$ for a long time and then moved to position $Q$ at time $t=0$. The value of $\dfrac{dv\left ( t \right )}{dt}$ at $t=0^{+}$ is

Consider the two-port network shown in the figure. The admittance parameters, in siemens, are

For an $n$-channel silicon $\text{MOSFET}$ with $10\:nm$ gate oxide thickness, the substrate sensitivity $\left ( \partial V_{T}/\partial \left | V_{BS} \right | \right )$ is found to be $50\:mV/V$ at a substrate voltage $\left | V_{BS} \right |=2V$, where $V_{T}$ is the threshold voltage of the $\text{MOSFET}$. Assume that, …

​​​​​​​The propagation delays of the $\text{XOR}$ gate, $\text{AND}$ gate and multiplexer $\text{(MUX)}$ in the circuit shown in the figure are $4\:ns$, $2\:ns$ and $1\:ns$, respectively. If all the inputs $\text{P, Q, R, S and T}$ are applied simultaneously and held constant, the maximum propagation delay of the circuit is

The content of the registers are $R_{1} = 25H, R_{2} = 30H$ and $R_{3} = 40H$. The following machine instructions are executed. $\text{PUSH } \{R_{1}\}$ $\text{PUSH } \{R_{2}\}$ $\text{PUSH } \{R_{3}\}$ $\text{POP } \{R_{1}\}$ $\text{POP } \{R_{2}\}$ $\text{POP } \{R_{3}\}$ After execution, the content of registers $R_{1},\:R_{2},\:R_{3}$ …

The electrical system shown in the figure converts input source current $i_{s}(t)$ to output voltage $v_{o}(t)$ Current i $i_{L}(t)$ in the inductor and voltage $v_{c}(t)$ across the capacitor are taken as the state variables, both assumed to be initially equal to zero, i.e., $i_{L}(0) =0$ and $v_{c}(0)=0$. The system is

A digital transmission system uses a ($7,4$) systematic linear Hamming code for transmitting data over a noisy channel. If three of the message-codeword pairs in this code ($\text{m}_{i}$ ; $\text{c}_{i}$), where $\text{c}_{i}$ is the codeword corresponding to the $i^{th}$ message $\text{m}_{i}$, are known to be $(1100; 0101 100)$, …

The impedance matching network shown n the figure is to match a lossless line having characteristic impedance $Z_{0}= 50 \:\Omega$ with a load impedance $Z_{L}$. A quarter-wave line having a characteristic impedance $Z_1=75\:\Omega$ is connected to $Z_{L}$. Two stubs having characteristic impedance of $75\:\Omega$ each are connected to this …

A real $2\times2$ non-singular matrix $A$ with repeated eigenvalue is given as $$A=\begin{bmatrix} x & -3.0\\ 3.0 & 4.0 \end{bmatrix}$$ where $x$ is a real positive number. The value of $x$ (rounded off to one decimal place) is ________________

For a vector field $D=\rho\cos^{2}\:\varphi \:a_{\rho }+z^{2}\sin^{2}\:\varphi \:a_{\varphi }$ in a cylindrical coordinate system $\left ( \rho ,\varphi ,z \right )$ with unit vectors $a_{\rho },a_{\varphi }$ and $a_{z}$, the net flux of $D$ leaving the closed surface of the cylinder $\left ( \rho =3, 0\leq z\leq 2 \right )$ (rounded off to two …

In the circuit shown in the figure, the switch is closed at time $t=0$, while the capacitor is initially charged to $-5\:V$ (i.e., $v_{c}(0)=-5V$). The time after which the voltage across the capacitor becomes zero (rounded off to three decimal places) is ________________ $\text{ms}$.

The exponential Fourier series representation of a continuous-time periodic signal $x(t)$ is defined as $$x\left ( t \right )=\sum_{k=-\infty }^{\infty }a_{k}e^{jk\omega _{0}t}$$ where $\omega _0$ is the fundamental angular frequency of $x(t)$ and the coefficients of the series are $a_{k}$. The following information is given about $x(t)$ and …

For a unit step input $u[n]$, a discrete-time $\text{LTI}$ system produces an output signal $\left ( 2\delta \left [ n+1 \right ] +\delta \left [ n \right ]+\delta \left [ n-1 \right ]\right )$. Let $y[n]$ be the output of the system for an input $\left ( \left ( \dfrac{1}{2} \right )^n u\left [ n \right ]\right )$. The value of $y[0]$ is …

Consider the signals $x\left [ n \right ]=2^{n-1}u\left [ -n+2 \right ]$ and $y\left [ n \right ]=2^{-n+2}u\left [ n+1 \right ]$, where $u[n]$ is the unit step sequence. Let $X(e^{jw})$ and $Y(e^{jw})$ be the discrete-time Fourier transform of $x[n]$ and $y[n]$, respectively. The value of the integral …

A silicon $P-N$ junction is shown in the figure. The doping in the $P$ region is $5\times10^{16}\:cm^{-3}$ and doping in the $N$ region is $10\times10^{16}\:cm^{-3}$. The parameters given are Built-in voltage $\left ( \Phi _{\text{bi}} \right )$ = $0.8\:V$ Electron charge $(q)$ = $1.6\times10^{-19} C\:$ Vacuum permittivity …

An asymmetrical periodic pulse train $v_{in}$ of $10\:V$ amplitude with on-time $T_{\text{ON}}=1\:ms$ and off-time $T_{\text{OFF}}=1\:\mu s$ is applied to the circuit shown in the figure. The diode $D_{1}$ is ideal. The difference between the maximum voltage and minimum voltage of the output waveform $v_{o}$ (in integer) is ______________ $V$.

For the transistor $M_{1}$ in the circuit shown in the figure, $\mu_{n} C_{\text{ox}} = 100\:\mu A/V^{2}$ and $(W/L)=10$, where $\mu_{n}$ is the mobility of electron, $C_{\text{ox}}$ is the oxide capacitance per unit area , $W$ is the width and $L$ is the length. The channel length modulation coefficient is ignored. If the gate-to-source …

A circuit with an ideal $\text{OPAMP}$ is shown in the figure. A pulse $V_{\text{IN}}$ of $20\:ms$ duration is applied to the input. The capacitors are initially uncharged. The output voltage $V_{\text{OUT}}$ of this circuit at $t=0^{+}$ (in integer) is ____________________ $V$.