GATE EC question library

Consider the Friis' transmission equation $P_{R}=\left(P_{T} G_{T} G_{R} \lambda^{2}\right) /(4 \pi D)^{2}$, where $P_{R}$ and $P_{T}$ are the received and the transmitted powers, respectively. $G_{T}$ and $G_{R}$ are the gain of transmitting and receiving antennas, respectively, $D$ is the distance between the transmitting and receiving …

Consider a discrete memoryless source with an alphabet of four source symbols. $s(t)$ is a multi-level $(-1,0,+1,+2)$ signal representing a long sequence of random symbols from the above source which is generating $10^{4}$ symbols per second. Which of the following options is the correct value of equivalent Nyquist bandwidth of $s(t)$?

The relation between the input current ( $I$ ) and the output voltage ( $V$ ) of a circuit is governed by the equation: $C \dfrac{d V}{d t}=I(t)-m(t)$. The circuit is excited by $I(t)=q \delta(t)$, where $q$ is a real valued constant. $V$ at $t=0^{-}$is $V_{0}$. Which of the following is an equivalent representation of the above case?

The electric field of a monochromatic plane wave travelling in a lossless isotropic and homogenous medium is given by $$\vec{E}(z, t)=E_{0}[\hat{x} \cos (\omega t-k z)+\hat{y} \sin (\omega t-k z)]$$ in a right-handed orthogonal co-ordinate system. Which of the following is the correct polarization of the electromagnetic wave?

Consider a $\text{p-n}$ junction diode when it is forward biased with $2$ V. Which of the following is/are the correct magnitude(s) of the energy difference between quasi Fermi-levels, $E_{f n}$ in the $\text{n -side}$ and $E_{f p}$ in the $\text{p -side}?$

Consider the matrix $\mathrm{M}=\left[\begin{array}{ccc}2 & 1 & 1 \\ 1 & 3 & 0 \\ -1 & a & b\end{array}\right]$. Which of the following options is/ are TRUE if $\operatorname{det}(\mathrm{M}) \neq 0 ?$

A binary ripple counter is designed to count $(0)_{10}$ to $(64)_{10}$. Which of the following is/are the number of flip-flops required to design the counter?

Which option(s) represents/ represent the dielectric loss tangent of a substrate?

Figure shows the output characteristics of two different Bipolar Junction Transistors (BJT), BJT $1$ with magnitude of Early voltage $\left|V_{A 1}\right|$, and BJT $2$ with magnitude of Early voltage $\left|V_{A 2}\right|$. Which of the following options is/are correct regarding the Early voltages?

The output voltage $V_{o}$ (in Volt) for the network given in the Figure is $\_\_\_\_$ . (rounded off to two decimal places)

Consider the circuit shown in the Figure, where the input $v_{i}(t)$ is in Volt. The average power (in $\text{mW}$ ) dissipated in the load resistance of $1 ~\mathrm{k} \Omega$ at the resonant frequency is $\_\_\_\_$ . (rounded off to two decimal places)

A wireless digital transmission scheme is using $16$-QAM over an additive white Gaussian noise channel and a maximum-likelihood receiver. Consider the information bit rate from source to be $4 \times 10^{6}$ bits per second. The minimum transmission bandwidth (in MHz) of the modulated signal necessary for optimum recovery of information at the receiver …

The cutoff frequency ( $\text{in GHz}$ ) for the dominant $\mathrm{TE}_{10}$ mode of an air-filled rectangular waveguide of inner dimension $0.28$ inch $\times 0.14$ inch is $\_\_\_\_$ . (rounded off to two decimal places).

For a lossless passive two-port network, $\left|S_{11}\right|$ and $\left|S_{21}\right|$ intersect at $-3$ dB . For a lossy passive two-port network, $\left|S_{11}\right|$ and $\left|S_{21}\right|$ intersect at $-4$ dB . The percentage of power dissipated in the lossy network at the intersection frequency is $\_\_\_\_$. (rounded off to two …

The negative edge triggered JK flip-flop in the Figure has J and K inputs tied to Logic High and a square wave of $10$ cycles/second is applied to its clock (C) input. The frequency of the output Q (in cycles/second) is $\_\_\_\_$. (rounded off to two decimal places)

Consider the two series, $S_{A}$ and $S_{B}$, where $$\begin{array}{l} S_{A}=\sum_{n=1}^{\infty} \frac{n^{2}}{2^{n}} \\ S_{B}=1+\frac{1}{2}+\frac{1}{8}+\frac{1}{16}+\frac{1}{64}+\frac{1}{128}+\frac{1}{512}+\cdots \end{array}$$ Which of the following statements is correct for the two given series?

The continuous time signal $x(t)$ is real, periodic with period $T$ and satisfies the Dirichlet conditions. The Fourier series representation of $x(t)=\sum_{-\infty}^{\infty} a_{n} e^{j\left(\frac{2 \pi n t}{T}\right)}$ and $x(t)$ satisfies the following: $x\left(t-\frac{T}{2}\right)=-x(t)$. For any integer $m$, which of the following options is …

Let $X, N, Y$ and $Z$ be random variables. The variables $X$ and $N$ are independent of each other. $X$ is uniformly distributed between $-1$ and $1 ; N$ follows Normal distribution with zero mean and unity variance. $Y$ and $Z$ are defined as, $Y=X+N$ and $Z=X^{2}+N$. Which of the following pairs represents the values of correlation …

In the given circuit, $L=1 \mu \mathrm{H}$ and $C=1 \mu \mathrm{~F}$. The phasor diagram for $I_{C}$ and $I_{L}$ is also shown. Assume that the phase $\left(\theta_{1}+\theta_{2}\right)$ is $90^{\circ}$ at a frequency of $159.15 ~\text{kHz}.$ Among the following options, what is the nearest integer value of $R_{C} \times R_{L}?$

For the control system shown in the Figure, the transfer function of a plant, $G(s)=\frac{1}{(s+1)(s+2)}$ is connected in cascade with a compensator $C(s)=K(s+\alpha)$, where $K$ and $\alpha$ are positive real valued constants. Which of the following pairs $(K, \alpha)$ represent the correct values for the closed loop system to have poles at …