GATE EC question library

The current $i_b$ through the base of a silicon $npn$ transistor is $1+0.1\cos(10000\pi t)\:mA$. At $300\:K$, the $r_\pi$ in the small signal model of the transistor is

The power spectral density of a real process $X(t)$ for positive frequencies is shown below. The values of $E[X^2(t)]$ and $\mid E[X(t)] \mid$, respectively, are

In a baseband communications link, frequencies upto $3500\:Hz$ are used for signaling. Using a raised cosine pulse with $75\%$ excess bandwidth and for no inter-symbol interference, the maximum possible signaling rate in the symbols per second is

A plane wave propagating in air with $\overrightarrow{E}=(8\hat{a}_x+6\hat{a}_y+5\hat{a}_z)e^{j(\omega t+3x-4y)}\:V/m$ is incident on a perfectly conducting slab positioned at $x\le0$ . The $\overrightarrow{E}$ field of the reflected wave is

The electric field of a uniform plane electromagnetic wave in free space, along the positive $x$ direction, is given by $\overrightarrow{E}=10(\hat{a}_y+j\hat{a}_z)e^{-j\:25x}$. The frequency and polarization of the wave, respectively, are

Consider the given circuit. In the circuit, the race around

The output $Y$ of a $2-\text{bit}$ comparator is logic $1$ whenever the $2-\text{bit}$ input $A$ is greater than the $2-\text{bit}$ input $B$. The number of combinations for which the output is logic $1$, is

The $i-v$ characteristics of the diode in the circuit given below are $$i = \begin{cases} \frac{v-0.07}{500}\:A, & v\ge0.7\:V \\ \:\:\:\:\:\:\:\:0\:A, & v \lt 0.7\:V \end{cases}$$ The current in the circuit is

In the following figure, $C_1$ and $C_2$ are ideal capacitors. $C_1$ has been charged to $12\:V$ before the ideal switch $S$ is closed at $t=0$. The current $i(t)$ for all $t$ is

The average power delivered to an impedance $(4-j3)\:\Omega$ by a current $5\cos(100\pi t+100)\:A$ is

The unilateral Laplace transform of $f(t)$ is $\frac{1}{s^2+s+1}$. The unilateral Laplace transform of $tf(t)$ is

With initial condition $x(1)=0.5$, the solution of the differential equation, $$t\frac{dx}{dt}+x=t$$ is

The diodes and capacitors in the circuit shown are ideal. The voltage $v(t)$ across the diode $D1$ is

In the circuit shown

A source alphabet consists of $N$ symbols with the probability of the first two symbols being the same. A source encoder increases the probability of the first symbol by a small amount $\varepsilon$ and decreases that of the second by $\varepsilon$. After encoding, the entropy of the source

A coaxial cable with an inner diameter of $1\:mm$ and outer diameter of $2.4\:mm$ is filled with a dielectric of relative permittivity $10.89$. Given $\mu_0=4\pi\times10^{-7}\:H/m$, $\varepsilon_0=\frac{10^{-9}}{36\pi}\:F/m$, the characteristic impedance of the cable is

The radiation pattern of an antenna in spherical co-ordinates is given by $$F(\theta)=\cos^4\theta\:\:\:;\:\:\:0\le \theta\le \frac{\pi}{2}$$ The directivity of the antenna is

If $x[n]=(\frac{1}{3})^{|n|}-(\frac{1}{2})^{|n|}u[n]$, then the region of convergence (ROC) of its Z-transform in the Z-plane will be

In the sum of products function $f(X,Y,Z)=\sum(2,3,4,5)$, the prime implicants are

A system with transfer function $$G(s)=\frac{(s^2+9)(s+2)}{(s+1)(s+3)(s+4)}$$ is excited by $\sin(\omega t)$. The steady-state output of the system is zero at