GATE EC question library

A function of Boolean variables $X, Y$ and $Z$ is expressed in terms of the min-terms as $$F(X, Y, Z) = \Sigma (1, 2, 5, 6, 7)$$ Which one of the product of sums given below is equal to the function $F(X, Y, Z)?$

The figure shows a binary counter with synchronous clear input. With the decoding logic shown, the counter works as a

A $1$-to-$8$ demultiplexer with data input $D_{in}$ , address inputs $S_{0}, S_{1}, S_{2}$ (with $S_{0}$ as the LSB) and $\overline{Y_{0}}$ to $\overline{Y_{7}}$ as the eight demultiplexed outputs, is to be designed using two $2$-to-$4$ decoders (with enable input $\overline{E}$ and address inputs $A_{0}$ and $A_{1}$) as shown in the …

The diode in the circuit given below has $V_{ON} = 0.7\:V$ but is ideal otherwise. The current $(\text{in}\: mA)$ in the $4\: k\Omega$ resistor is _______.

Assuming that the opamp in the circuit shown below is ideal, the output voltage ܸ$V_{o}$ (in volts) is________.

For the voltage regulator circuit shown, the input voltage $(V_{in})$ is $20 V \pm 20\%$ and the regulated output voltage $(V_{out})$ is $10\: V.$ Assume the opamp to be ideal. For a load $R_L$ drawing $200\: mA,$ the maximum power dissipation in $Q_{1}\:\text{(in Watts)}$ is ________.

In the ac equivalent circuit shown, the two BJTs are biased in active region and have identical parameters with $\beta \gg 1$. The open circuit small signal voltage gain is approximately _______.

Input $x(t)$ and output $y(t)$ of an LTI system are related by the differential equation $y’’(t) – y’(t) – 6y(t) = x(t).$ If the system is neither causal nor stable, the impulse response $h(t)$ of the system is

Consider two real sequences with time-origin marked by the bold value, $$x_{1}[n] = \{\textbf{1},2,3,0\},\:\:x_{2}[n] = \{\textbf{1},3,2,1\}$$ Let ܺ$X_{1}(k)$ and ܺ$X_{2}(k)$ be $4$-point DFTs of $x_{1}[n]$ and $x_{2}[n]$, respectively. Another sequence $x_{3}[n]$ is derived by taking $4$-point inverse DFT of $X_{3}(k) = X_{1}(k)X_{2}(k).$ …

Let $x(t) = \alpha s(t) + s(-t)$ with $s(t) = \beta e^{-4t}u(t),$ where $u(t)$ is unit step function. If the bilateral Laplace transform of $x(t)$ is $$X(s) = \dfrac{16}{s^{2} – 16}\:\: -4 < Re\{s\}<4;$$ then the value of $\beta$ is ______.

The state variable representation of a system is given as $\dot{x} = \begin{bmatrix} 0 &1 \\ 0 &-1 \end{bmatrix}\: ; x(0)=\begin{bmatrix} 1\\0 \end{bmatrix}$ $y=\begin{bmatrix} 0 &1 \end{bmatrix} x$ The response $y(t)$ is

The output of a standard second-order system for a unit step input is given as $y(t) = 1-\dfrac{2}{\sqrt{3}}e^{-t}\cos \left(\sqrt{3t}-\dfrac{\pi}{6}\right)$. The transfer function of the system is

The transfer function of a mass-spring-damper system is given by $$G(S) = \dfrac{1}{Ms^{2}+Bs+K}$$ The frequency response data for the system are given in the following table. …

A zero mean white Gaussian noise having power spectral density $\dfrac{N_{0}}{2}$ is passed through an LTI filter whose impulse response $h(t)$ is shown in the figure. The variance of the filtered noise at $t = 4$ is

$\begin{Bmatrix} X_{n}\\ \end{Bmatrix}_{n=-\infty}^{n=\infty}$ is an independent and identically distributed (i.i.d.) random process with ܺ$X_{n}$ equally likely to be $+1$ or $−1.\:\:\begin{Bmatrix} Y_{n}\\ \end{Bmatrix}_{n=-\infty}^{n=\infty}$ is another random process obtained as ܻ$Y_{n}=X_{n} + 0.5X_{n-1}.$ The autocorrelation function of …

Consider a binary, digital communication system which uses pulses $g(t)$ and $-g(t)$ for transmitting bits over an AWGN channel. If the receiver uses a matched filter, which one of the following pulses will give the minimum probability of bit error?

Let $X\in \{0,1\}$ and $Y\in \{0,1\}$ be two independent binary random variables. If $P(X=0)=p$ and $P(Y=0)=q,$ then $P(X+Y\geq 1)$ is equal to

An air-filled rectangular waveguide of internal dimensions $a\: cm \times b\: cm\:(a>b)$ has a cutoff frequency of $6\:GHz$ for the dominant $TE_{10}$ mode. For the same waveguide, if the cutoff frequency of the $TM_{11}$mode is $15\: GHz,$ the cutoff frequency of the $TE_{01}$ mode in $GHz$ is __________.

Two half-wave dipole antennas placed as shown in the figure are excited with sinusoidally varying currents of frequency $3\: MHz$ and phase shift of $\frac{\pi}{2}$ between them (the element at the origin leads in phase). If the maximum radiated $E$-field at the point $P$ in the $x-y$ plane occurs at an azimuthal angle of $60^{\circ},$ the …

The electric field of a plane wave propagating in a lossless non-magnetic medium is given by the following expression $$\text{E}(z,t) = \textbf{a}_{x} 5 \cos(2\pi \times 10^{9}t + \beta z) + \textbf{a}_{y} 3 \cos\left(2\pi \times 10^{9} t + \beta z – \frac{\pi}{2}\right)$$ The type of the polarization is