GATE EC question library

The rank of the matrix $\left[\begin{array}{ccc}1 & 1 & 1 \\ 1 & -1 & 0 \\ 1 & 1 & 1\end{array}\right]$ is

$\nabla \times \nabla \times \mathrm{P},$ where $\mathrm{P}$ is a vector, is equal to

$\displaystyle{}\iint(\nabla \times \mathrm{P}) \bullet \mathrm{ds},$ where $\mathrm{P}$ is a vector, is equal to

A probability density function is of the form $\qquad p(x)=\mathrm{Ke}^{-\alpha|x|}, x \in(-\infty, \infty)$ The value of $\text{K}$ is

A solution for the differential equation $\qquad \dot{x}(t)+2 x(t)=\delta(t)$ with initial condition $x(0-)=0$ is

A low-pass filter having a frequency response $\mathrm{H}(\mathrm{j} \omega)=\mathrm{A}(\omega) \;e^{j{\phi(\omega)}}$ does not produce any phase distortion, if

The values of voltage $\left(\mathrm{V}_{\mathrm{D}}\right)$ across a tunnel-diode corresponding to peak and valley currents are $\text{V}_{\text{P}}$ and $\text{V}_{\text{V}}$ respectively. The range of tunnel-diode voltage $\text{V}_{\text{D}}$ for which the slope of its $\text{I}-\text{V}_{\text{D}}$ characteristics is negative would be

The concentration of minority carriers in an extrinsic semiconductor under equilibrium is

Under low level injection assumption, the injected minority carrier current for an extrinsic semiconductor is essentially the

The phenomenon known as "Early Effect" in a bipolar transistor refers to a reduction of the effective base-width caused by

The input impedance $\left(\text{Z}_i\right)$ and the output impedance $\left(\text{Z}_o\right)$ of an ideal transconductance (voltage controlled current source) amplifier are

An $n$-channel depletion MOSFET has following two points on its $\mathrm{I}_\text{D}-\mathrm{V}_{\text{GS}}$ curve $\text{V}_{\text{Gs}}=0$ at $\text{I}_\text{D}=12 \mathrm{~mA}$ and $\mathrm{V}_{\mathrm{GS}}=-6$ Volts at $\mathrm{I}_{\mathrm{D}}=0$ Which of the following $\text{Q}$-points will give the highest trans-conductance gain for small …

The number of product terms in the minimized sum-of-product expression obtained through the following $\text{K}$-map is (where, " $d$ " denotes don't care states) $$\begin{array}{|c|c|c|c|} \hline 1 & 0 & 0 & 1 \\ \hline 0 & d & 0 & 0 \\ \hline 0 & 0 & d & 1 \\ \hline 1 & 0 & 0 & 1 \\ \hline \end{array}$$

Let $x(t) \longleftrightarrow \mathrm{X}(j \omega)$ be Fourier Transform pair. The Fourier Transform of the signal $x(5 t-3)$ in terms of $X(j \omega)$ is given as

The Dirac delta function $\delta(t)$ is defined as

If the region of convergence of $x_1[n]+x_2[n]$ is $\frac{1}{3}<|z|<\frac{2}{3}$, then the region of convergence of $x_1[n]-x_2[n]$ includes

The open-loop transfer function of a unity-gain feedback control system is given by $$\text{G}(s)=\frac{\text{K}}{(s+1)(s+2)}$$ The gain margin of the system in $\text{dB}$ is given by

In the system shown below, $x(t)=(\sin t) u(t)$. In steady-steady-state, the response $y(t)$ will be

The electric field of an electomagnetic wave propagating in the positive $z$-direction is given by $$\left.\text{E}=\hat{a_x} \sin (\omega t-\beta z\right)+\hat{a_y} \sin \left(\omega t-\beta z+\frac{\pi}{2}\right)$$ The wave is

A transmission line is feeding $1 \mathrm{Watt}$ of power to a horn antenna having a gain of $10 \mathrm{~dB}$. The antenna is matched to the transmission line. The total power radiated by the horn antenna into the free-space is