GATE EC question library

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

The Fourier transform of a signal $h(t)$ is $H(j\omega)=(2\cos\omega)(\sin2\omega)/\omega$. The value of $h(0)$ is

The state variable description of an LTI system is given by $$\begin{pmatrix} \dot{x_1}\\ \dot{x_2}\\ \dot{x_3} \end{pmatrix}=\begin{pmatrix} 0 & a_1 & 0\\ 0 & 0 & a_2\\a_3 & 0 & 0 \end{pmatrix} \begin{pmatrix} x_1\\ x_2\\x_3 \end{pmatrix}+\begin{pmatrix} 0\\ 0\\1 \end{pmatrix}u$$ …

Assuming both the voltage sources are in phase, the value of $R$ for which maximum power is transferred from circuit $A$ to circuit $B$ is

Consider the differential equation $\frac{d^2y(t)}{dt^2}+2\frac{dy(t)}{dt}+y(t)=\delta(t)$ with $y(t)\big|_{t=0^-}=-2$ and $\frac{dy}{dt}\big|_{t=0^-}=0$. The numerical value of $\frac{dy}{dt}\big|_{t=0^+}$ is

The direction of vector $A$ is radially outward from the origin, with $|A|=kr^n$ where $r^2=x^2+y^2+z^2$ and $k$ is a constant. The value of $n$ for which $\triangledown.A=0$ is

A fair coin is tossed till head appears for the first time. The probability that the number of required tosses is odd, is

In the CMOS circuit shown, electron and hole mobilities are equal, and $M1$ and $M2$ are equally sized. The device $M1$ is in the linear region if

A binary symmetric channel (BSC) has a transition probability of $\frac{1}{8}$. If the binary transmit symbol $X$ is such that $P(X=0)\:=\:\frac{9}{10}$, then the probability of error for an optimum receiver will be

The signal $m(t)$ as shown is applied both to a phase modulator (with $k_p$ as the phase constant) and a frequency modulator (with $k_f$ as the frequency constant) having the same carrier frequency. The ratio $\frac{k_p}{k_f}$ (in $rad/Hz$) for the same maximum phase deviation is

The magnetic field along the propagation direction inside a rectangular waveguide with the cross-section shown in the figure is $$H_Z=3\:\cos(2.094\times10^2x)\:\cos(2.618\times10^2y)\:\cos(6.283\times10^{10}t-\beta z)$$ The phase velocity $v_p$ of the wave inside the waveguide satisfies