GATE EC question library

In the circuit shown below, $D_1$ and $D_2$ are silicon diodes with cut-in voltage of $0.7 \mathrm{~V}$. $\mathrm{V}_{\mathrm{IN}}$ and $\mathrm{V}_{\mathrm{OUT}}$ are input and output voltages in volts. The transfer characteristic is

A closed loop system is shown in the figure where $\mathrm{k}>0$ and $\alpha>0$. The steady state error due to a ramp input $\left(\mathrm{R}(\mathrm{s})=\alpha / \mathrm{s}^2\right)$ is given by

In the following block diagram, $\mathrm{R}(\mathrm{s})$ and $\mathrm{D}(\mathrm{s})$ are two inputs. The output $\mathrm{Y}(\mathrm{s})$ is expressed as $Y(s)=G_1(s) R(s)+G_2(s) D(s)$. $\mathrm{G}_1(\mathrm{~s})$ and $\mathrm{G}_2(\mathrm{~s})$ are given by

The state equation of a second order system is $\dot{\boldsymbol{x}}(t)=\mathrm{A} \boldsymbol{x}(t), \; \boldsymbol{x}(0)$ is the initial condition. Suppose $\lambda_1$ and $\lambda_2$ are two distinct eigenvalues of $\mathrm{A}$ and $v_1$ and $v_2$ are the corresponding eigenvectors. For constants $\alpha_1$ and $\alpha_2$, the solution, …

The switch $\mathrm{S}_{1}$ was closed and $\mathrm{S}_{2}$ was open for a long time. At $t=0$, switch $\mathrm{S}_{1}$ is opened and $\mathrm{S}_{2}$ is closed, simultaneously. The value of $\mathrm{i}_{\mathrm{c}}\left(0^{+}\right)$, in amperes, is

Let a frequency modulated $\text{(FM)}$ signal $x(t)=A \cos \left(\omega_{c} t+k_{f} \int_{-\infty}^{t} m(\lambda) d \lambda\right)$, where $m(t)$ is a message signal of bandwidth $\text{W.}$ It is passed through a non-linear system with output $y(t)=2 x(t)+5(x(t))^{2}$. Let $B_{T}$ denote the $\text{FM}$ bandwidth. The minimum value of …

The $\text{h}$-parameters of a two port network are shown below. The condition for the maximum small signal voltage gain $\frac{\mathrm{V}_{\text{out }}}{\mathrm{V}_{\mathrm{s}}}$ is

Consider a discrete-time periodic signal with period $N=5$. Let the discrete-time Fourier series $\text{(DTFS}$) representation be $x[n]=\sum_{k=0}^4 a_k e^{\frac{j k 2 \pi n}{5}}$, where $a_0=1, a_1=$ $3 j, a_2=2 j, a_3=-2 j$ and $a_4=-3 j$. The value of the sum $\sum_{n=0}^4 x[n] \sin \frac{4 \pi n}{5}$ is

Let an input $x[n]$ having discrete-time Fourier transform $X\left(e^{j \Omega}\right)=1-e^{-j \Omega}+2 e^{-3 j \Omega}$ be passed through an LTI system. The frequency response of the LTI system is $H\left(e^{j \Omega}\right)=1-\frac{1}{2} e^{-j 2 \Omega}$. The output $y[n]$ of the system is

Let $x(t)=10 \cos (10.5 W t)$ be passed through an LTI system having impulse response $h(t)=\pi\left(\frac{\sin W t}{\pi t}\right)^{2} \cos 10 Wt$. The output of the system is

Let $\mathrm{x}_1(\mathrm{t})$ and $\mathrm{x}_2(\mathrm{t})$ be two band-limited signals having bandwidth $B=4 \pi \times 10^3 \; \mathrm{rad} / \mathrm{s}$ each. In the figure below, the Nyquist sampling frequency, in $\mathrm{rad} / \mathrm{s}$, required to sample $y(\mathrm{t})$, is

The $S$-parameters of a two-port network is given as $$[S]=\left[\begin{array}{ll} S_{11} & S_{12} \\ S_{21} & S_{22} \end{array}\right]$$ with reference to $Z_0$. Two lossless transmission line sections of electrical lengths $\theta_1=\beta l_1$ and $\theta_2=\beta l_2$ are added to the input and output ports for measurement purposes, …

The standing wave ratio on a $50 \; \Omega$ lossless transmission line terminated in an unknown load impedance is found to be $2.0$. The distance between successive voltage minima is $30 \mathrm{~cm}$ and the first minimum is located at $10 \mathrm{~cm}$ from the load. $Z_L$ can be replaced by an equivalent length $l_m$ and terminating resistance …

The electric field of a plane electromagnetic wave is $\boldsymbol{E}=\boldsymbol{a}_{x} C_{1 x} \cos (\omega t-\beta z)+\boldsymbol{a}_{y} C_{1 y} \cos (\omega t-\beta z+\theta) \quad \mathrm{V} / \mathrm{m}$. Which of the following combination(s) will give rise to a left handed elliptically polarized $\text{(LHEP)}$ wave?

The following circuit(s) representing a lumped element equivalent of an infinitesimal section of a transmission line is/are

The value of the integral $\iint_R \text{xy dx dy}$ over the region $R$, given in the figure, __________ is (rounded off to the nearest integer).

In an extrinsic semiconductor, the hole concentration is given to be $1.5 n_{i}$ where $n_{i}$ is the intrinsic carrier concentration of $1 \times 10^{10} \mathrm{~cm}^{-3}$. The ratio of electron to hole mobility for equal hole and electron drift current is given as ________ (rounded off to two decimal places).

The asymptotic magnitude Bode plot of a minimum phase system is shown in the figure. The transfer function of the system is $(s)=\frac{k(s+z)^{a}}{s^{b}(s+p)^{c}}$, where $k, z, p, a, b$ and $c$ are positive constants. The value of $(a+b+c)$ is ____________ (rounded off to the nearest integer).

Let $\mathrm{x}_1(\mathrm{t})=\mathrm{u}(\mathrm{t}+1.5)-\mathrm{u}(\mathrm{t}-1.5)$ and $\mathrm{x}_2(\mathrm{t})$ is shown in the figure below. For $\mathrm{y}(\mathrm{t})=\mathrm{x}_1(\mathrm{t}) * \mathrm{x}_2(\mathrm{t})$, the $\int_{-\infty}^{\infty} \mathrm{y}(\mathrm{t}) \mathrm{dt}$ is ______________ (rounded off to the nearest integer).

Let $X(t)$ be a white Gaussian noise with power spectral density $\frac{1}{2} \mathrm{~W} / \mathrm{Hz}$. If $X(t)$ is input to an LTI system with impulse response $e^{-t} u(t)$. The average power of the system output is _____________ $\mathrm{W}$ (rounded off to two decimal places).