GATE EC question library

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

The impedance looking into nodes $1$ and $2$ in the given circuit is

In the circuit shown below, the current through the inductor is

Given $f(z)=\frac{1}{z+1}-\frac{2}{z+3}$. If $C$ is a counterclockwise path in the z-plane such that $\mid z+1 \mid=1$, the value of $\frac{1}{2\pi j}\oint_cf(z)dz$ is

Two independent random variables $X$ and $Y$ are uniformly distributed in the interval $[-1,1]$. The probability that max$[X,Y]$ is less than $\frac{1}{2}$ is

If $x=\sqrt{-1}$, then the value of $x^x$ is

The source of a silicon ($n_i=10^{10}\:per\:cm^3$) n-channel MOS transistor has an area of $1\:sq\:\mu m$ and a depth of $1\:\mu m$. If the dopant density in the source is $10^{19}/cm^3$, the number of holes in the source region with the above volume is approximately

A BPSK scheme operating over an AWGN channel with noise power spectral density of $\frac{N_o}{2}$, uses equiprobable signals $s_1(t)=\sqrt{\frac{2E}{T}}\sin(\omega_ct)$ and $s_2(t)=-\sqrt{\frac{2E}{T}}\sin(\omega_ct)$ over the symbol interval $(0,T)$. If the local oscillator in a coherent receiver is ahead in phase by $45^\circ$ with respect to the …

A trasmission line with a characteristic impedance of $100\:\Omega$ is used to match a $50\:\Omega$ section to a $200\:\Omega$ section. If the matching is to be done both at $429\:MHz$ and $1\:GHz$, the length of the transmission line can be approximately

The input $x(t)$ and output $y(t)$ of a system are related as $y(t)=\underset{-\infty}{\int}x(\tau)\cos(3\tau)d\tau$. The system is

The feedback system shown below oscillates at $2\:rad/s$ when