GATE EC question library

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

The circuit shown is a

Let $y[n]$ denote the convolution of $h[n]$ and $g[n]$, where $h[n]=(\frac{1}{2})^nu[n]$ and $g[n]$ is a casual sequence. If $y[0]=1$ and $y[1]=\frac{1}{2}$, then $g[1]$ equals

The state transition diagram for the logic circuit shown is

The voltage gain $A_v$ of the circuit shown below is

If $V_A-V_B=6\:V$, then $V_C-V_D$ is

The maximum value of $f(x)=x^3-9x^2+24x+5$ in the interval $[1,6]$ is

Given that $A=\begin{bmatrix} -5 &-3 \\ 2 &0\end{bmatrix}$ and $I=\begin{bmatrix} 1 & 0 \\ 0 &1\end{bmatrix}$, the value of $A^3$ is

With $10\:V$ dc connected at port A in the linear nonreciprocal two-port network shown below, the following were observed: $1\: \Omega$ connected at port B draws a current of $3\:A$ $2.5\: \Omega$ connected at port B draws a current of $2\:A$ With $10\: V$ dc connected at port A, the current drawn by $7\:\Omega$ connected at port B is

With $10\:V$ dc connected at port A in the linear nonreciprocal two-port network shown below, the following were observed: $1\: \Omega$ connected at port B draws a current of $3\:A$ $2.5\: \Omega$ connected at port B draws a current of $2\:A$ For the same network, with $6\: V$ dc connected at port A, $1\:\Omega$ connected at port B draws …

In the three dimensional view of a silicon n-channel MOS transistor shown below, $\delta=20\:nm$. The transistor is of width $1\: \mu m$. The depletion width formed at every p-n junction is $10\:nm$. The relative permittivities of $Si$ and $SiO_2$, respectively, are $11.7$ and $3.9$, and $\varepsilon_0=8.9 \times 10^{-12} \: F/m$. The …