## GATE Electronics and Communication 2023

 Question 1
Let $v_{1}=\left[\begin{array}{l}1 \\ 2 \\ 0\end{array}\right]$ and $v_{2}=\left[\begin{array}{l}2 \\ 1 \\ 3\end{array}\right]$ be two vectors. The value of the coefficient $\alpha$ in the expression $v_{1}=\alpha v_{2}+e$, which minimizes the length of the error vector $e$, is
 A $\frac{7}{2}$ B $-\frac{2}{7}$ C $\frac{2}{7}$ D $-\frac{7}{2}$
Engineering Mathematics   Linear Algebra
Question 1 Explanation:
\begin{aligned} e & =V_{1}-\alpha V_{2} \\ e & =(i+2 k+0 k)-\alpha(2 i+j+3 k) \\ \hat{e} & =(1-2 \alpha) \hat{i}+(2-\alpha) \hat{j}+(0-3 \alpha) \hat{k} \\ |\hat{e}| & =\sqrt{(1-2 \alpha)^{2}+(2-\alpha)^{2}+(-3 \alpha)^{2}} \\ |\hat{e}|^{2} & =5+14 \alpha^{2}-8 \alpha \text { to be minimum at } \frac{\partial e^{2}}{\partial \alpha}=28 \alpha-8=0 \\ \alpha & =\frac{2}{7} \text { stationary point } \end{aligned}
 Question 2
The rate of increase, of a scalar field $f(x, y, z)=x y z$ in the direction $v=(2,1,2)$ at a point $(0,2,1)$ is
 A $\frac{2}{3}$ B $\frac{4}{3}$ C $2$ D $4$
Engineering Mathematics   Linear Algebra
Question 2 Explanation:
\begin{aligned} f(x, y, z) & =x y z \\ \overline{\nabla f} & =\hat{i} f_{x}+\hat{j} f_{y}+\hat{k} f_{z} \\ & =\hat{i}(y z)+\hat{j}(x z)+\hat{k}(x y) \\ \overline{\nabla f}_{(0,2,1)} & =\hat{i}(2)+0 \hat{j}+0 \hat{k} \end{aligned}

Directional derivative,
\begin{aligned} D \cdot D & =\overline{\nabla f} \cdot \frac{\bar{a}}{|\bar{a}|} \\ & =(2 \hat{i}+0 \hat{j}+0 \hat{k}) \cdot \frac{(2 \hat{i}+\hat{j}+2 \hat{k})}{\sqrt{2^{2}+1^{2}+2^{2}}}=\frac{4}{\sqrt{9}}=\frac{4}{3} \end{aligned}

 Question 3
Let $w^{4}=16 j$. Which of the following cannot be a value of $w$ ?
 A $2 e^{\frac{j 2 \pi}{8}}$ B $2 e^{\frac{j \pi}{8}}$ C $2 e^{\frac{j 5 \pi}{8}}$ D $2 e^{\frac{j 9 \pi}{8}}$
Engineering Mathematics   Complex Variables
Question 3 Explanation:
$w=(2) j^{1 / 4}$
$w=2(0+j)^{1 / 4}$
$w=2\left[e^{j(2 n+1) \pi / 2}\right]^{1 / 4}$ $=2\left[e^{j(2 n+1) \pi / 8}\right]$

For $n=0$, $w=e^{j \pi / 8}$
For $n=2$, $w=2 e^{5 \pi j / 8}$
For $n=4$, $w=2 e^{9 \pi j / 8}$
 Question 4
The value of the contour integral, $\oint_{c}\left(\frac{z+2}{z^{2}+2 z+2}\right) d z$, where the contour $C$ is $\left\{z:\left|z+1-\frac{3}{2} j\right|=1\right\}$, taken in the counter clockwise direction, is
 A $-\pi(1+j)$ B $\pi(1+j)$ C $\pi(1-j)$ D $-\pi(1-j)$
Engineering Mathematics   Calculus
Question 4 Explanation:
$I=\oint_{c} \frac{z+2}{z^{2}+2 z+2} d z ; \quad c=\left|z+1-\frac{3}{2} i\right|=1$

Poles are given $(z+1)^{2}+1=0$
\begin{aligned} & z+1= \pm \sqrt{-1} \\ & z=-1+j,-1-j \end{aligned}
where $-1-i$ lies outside ' $c$ '
$z=(-1,1) \text { lies inside } 'c'$.

by $\mathrm{CRT}$
\begin{aligned} \oint_{c} f(z) d z & =2 \pi i \operatorname{Res}(f(z), z=-1+j) \\ & =2 \pi i\left(\frac{z+2}{2(z+1)}\right)_{z=-1+i}\\ & =2 \pi i\left(\frac{-1+j+2}{2(-1+j+1)}\right) \\ & =\pi(1+j) \end{aligned}
 Question 5
Let the sets of eigenvalues and eigenvectors of a matrix $B$ be $\left\{\lambda_{k} \mid 1 \leq k \leq n\right\}$ and $\left\{v_{k} \mid 1 \leq k \leq n\right\}$, respectively. For any invertible matrix $P$, the sets of eigenvalues and eigenvectors of the matrix $A$, where $B=P^{-1} A B$, respectively, are
 A $\left\{\lambda_{k} \operatorname{det}\mid 1 \leq k \leq n\right\}$ and $\left\{P v_{k} \mid 1 \leq k \leq n\right\}$ B $\left\{\lambda_{k} \mid 1 \leq k \leq n\right\}$ and $\left\{v_{k} \mid 1 \leq k \leq n\right\}$ C $\left\{\lambda_{k} \mid 1 \leq k \leq n\right\}$ and $\left\{P v_{k} \mid 1 \leq k \leq n\right\}$ D $\left\{\lambda_{k} \mid 1 \leq k \leq n\right\}$ and $\left\{P^{-1} v_{k} \mid 1 \leq k \leq n\right\}$
Engineering Mathematics   Linear Algebra
Question 5 Explanation:
\begin{aligned} & B & =P^{-1} A P \\ A & =P B P^{-1}\end{aligned}

$\Rightarrow A, B$ are called matrices similar.
$\Rightarrow$ Both $A, B$ have same set 7 eigen values
But eigen vectors of $A, B$ are different.

Let $B X=\lambda X$
$\Rightarrow \quad\left(P^{-1} A P\right) X=\lambda X$
$\Rightarrow \quad A(P X)=\lambda(P X)$

$\therefore$ Eigen vectors of $A$ are $P X$.

There are 5 questions to complete.

## GATE Electronics and Communication 2022

 Question 1
Consider the two-dimensional vector field $\vec{F}(x,y)=x\vec{i}+y\vec{j}$, where $\vec{i}$ and $\vec{j}$ denote the unit vectors along the x-axis and the y-axis, respectively. A contour C in the x-y plane, as shown in the figure, is composed of two horizontal lines connected at the two ends by two semicircular arcs of unit radius. The contour is traversed in the counter-clockwise sense. The value of the closed path integral
$\oint _c \vec{F}(x,y)\cdot (dx\vec{i}+dy\vec{j})$

 A 0 B 1 C $8+2 \pi$ D -1
Engineering Mathematics   Calculus
Question 1 Explanation:
$\oint \vec{F} (x,y)\cdot [dx\vec{i}+dy\vec{j}]$
Given $\vec{F} (x,y)=x\vec{i}+y\vec{j}$
$\therefore \int_{c}xdx+ydy=0$
Because here vector is conservative.
If the integral function is the total derivative over the closed contoure then it will be zero
 Question 2
Consider a system of linear equations $Ax=b$, where
$A=\begin{bmatrix} 1 & -\sqrt{2} & 3\\ -1& \sqrt{2}& -3 \end{bmatrix},b=\begin{bmatrix} 1\\ 3 \end{bmatrix}$
This system of equations admits ______.
 A a unique solution for x B infinitely many solutions for x C no solutions for x D exactly two solutions for x
Engineering Mathematics   Linear Algebra
Question 2 Explanation:
Here equation will be
$x-\sqrt{2}y+3z=1$
$-x+\sqrt{2}y-3z=3$
therefore inconsistant solution i.e. there will not be any solution.

 Question 3
The current $I$ in the circuit shown is ________

 A $1.25 \times 10^{-3}A$ B $0.75 \times 10^{-3}A$ C $-0.5 \times 10^{-3}A$ D $1.16 \times 10^{-3}A$
Network Theory   Basics of Network Analysis
Question 3 Explanation:

Applying Nodal equation at Node-A
\begin{aligned} \frac{V_A}{2k}+\frac{V_A-5}{2k}&=10^{-3}\\ \Rightarrow 2V_A-5&=2k \times 10^{-3}\\ V_A&=3.5V\\ Again,&\\ I&=\frac{5-V_A}{2k}\\ &=\frac{5-3.5}{2k}\\ &=0.75 \times 10^{-3}A \end{aligned}
 Question 4
Consider the circuit shown in the figure. The current $I$ flowing through the $10\Omega$ resistor is _________.

 A 1A B 0A C 0.1A D -0.1A
Network Theory   Basics of Network Analysis
Question 4 Explanation:
Here, there is no any return closed path for Current (I) . Hence I=0
Current always flow in loop.
 Question 5
The Fourier transform $X(j\omega )$ of the signal $x(t)=\frac{t}{(1+t^2)^2}$ is _________.
 A $\frac{\pi}{2j}\omega e^{-|\omega|}$ B $\frac{\pi}{2}\omega e^{-|\omega|}$ C $\frac{\pi}{2j} e^{-|\omega|}$ D $\frac{\pi}{2} e^{-|\omega|}$
Signals and Systems   DTFS, DTFT and DFT
Question 5 Explanation:
$x(t)=\frac{t}{(1+t^2)^2}$
As we know that FT of $te^{-|t|} \; \underleftrightarrow{FT} \;\frac{-j4\omega }{(1+\omega ^2)^2}$
Duality $\frac{-j4\omega }{(1+t ^2)^2} \leftrightarrow 2 \pi(-\omega )e^{-|-\omega |}$
$\Rightarrow \frac{t}{(1+t^2)^2} \underrightarrow{FT} \frac{-2\pi}{-j4}\omega e^{-|\omega |}$
$\Rightarrow \;\;\;\rightarrow\frac{\pi}{j2} \omega e^{-|\omega |}$

There are 5 questions to complete.

## GATE Electronics and Communication 2021

 Question 1
The vector function $F\left ( r \right )=-x\hat{i}+y\hat{j}$ is defined over a circular arc C shown in the figure.

The line integral of $\int _{C} F\left ( r \right ).dr$ is
 A $\frac{1}{2}$ B $\frac{1}{4}$ C $\frac{1}{6}$ D $\frac{1}{3}$
Engineering Mathematics   Calculus
Question 1 Explanation:
\begin{aligned} \bar{F} &=-x i+y j \\ \int \vec{F} \cdot \overrightarrow{d r} &=\int_{c}-x d x+y d y \\ &=\int_{\theta=0}^{45^{\circ}}(-\cos \theta(-\sin \theta)+\sin \theta \cos \theta) d \theta \\ \int_{\theta=0}^{\pi / 4} \sin 2 \theta d \theta &\left.=-\frac{\cos 2 \theta}{2}\right]_{0}^{\pi / 4} \\ &=-\frac{1}{2}[0-1]=\frac{1}{2} \end{aligned}

 Question 2
Consider the differential equation given below.
$\frac{dy}{dx}+\frac{x}{1-x^{2}}y=x\sqrt{y}$
The integrating factor of the differential equation is
 A $\left ( 1-x^{2} \right )^{-3/4}$ B $\left ( 1-x^{2} \right )^{-1/4}$ C $\left ( 1-x^{2} \right )^{-3/2}$ D $\left ( 1-x^{2} \right )^{-1/2}$
Engineering Mathematics   Differential Equations
Question 2 Explanation:
\begin{aligned} \frac{d y}{d x}+\frac{x}{1-x^{2}} y&=x \sqrt{y}, \quad \text { IF }=?\\ \text{Divided by }\sqrt{y}\\ \frac{1}{\sqrt{y}} \frac{d y}{d x}+\frac{x}{1-x^{2}} \sqrt{y}&=x \\ 2 \frac{d u}{d x}+\frac{x}{1-x^{2}} u&=x\\ \text{Let }\qquad x \sqrt{y}&=u\\ \frac{1}{2 \sqrt{v}} \frac{d y}{d x}&=\frac{d u}{d x}\\ \Rightarrow \qquad \frac{d u}{d x}+\frac{x}{2\left(1-x^{2}\right)} u&=\frac{x}{2} \rightarrow \text{ lines diff. equ.} \\ \text { I. } F&=e^{\int \frac{x}{2\left(1-x^{2}\right)} d x}=e^{-\frac{1}{4} \log \left(1-x^{2}\right)}&=e^{\log \left(1-x^{2}\right) \frac{-1}{4}} \\ \text { I.F }&=\frac{1}{\left(1-x^{2}\right)^{\frac{1}{4}}} \end{aligned}

 Question 3
Two continuous random variables X and Y are related as
$Y=2X+3$
Let $\sigma ^{2}_{X}$ and $\sigma ^{2}_{Y}$ denote the variances of X and Y, respectively. The variances are related as
 A $\sigma ^{2}_{Y}=2 \sigma ^{2}_{X}$ B $\sigma ^{2}_{Y}=4 \sigma ^{2}_{X}$ C $\sigma ^{2}_{Y}=5 \sigma ^{2}_{X}$ D $\sigma ^{2}_{Y}=25 \sigma ^{2}_{X}$
Communication Systems   Random Signals and Noise
Question 3 Explanation:
\begin{aligned} Y &=2 X+3 \\ \operatorname{Var}[Y] &=E\left[(Y-\bar{Y})^{2}\right] \\ E[Y] &=\bar{Y}=2 \bar{X}+3 \\ \operatorname{Var}[Y] &=E\left[(2 X+3-2 \bar{X}-3)^{2}\right] \\ &=E\left[4(X-\bar{X})^{2}\right] \\ &=4 \cdot E\left[(X-\bar{X})^{2}\right] \\ \sigma_{Y}^{2} &=4 \cdot \sigma_{X}^{2} \end{aligned}
 Question 4
Consider a real-valued base-band signal x(t), band limited to $\text{10 kHz}$. The Nyquist rate for the signal $y\left ( t \right )=x\left ( t \right )x\left ( 1+\dfrac{t}{2} \right )$ is
 A $\text{15 kHz}$ B $\text{30 kHz}$ C $\text{60 kHz}$ D $\text{20 kHz}$
Signals and Systems   Sampling
Question 4 Explanation:

$\mathrm{NR}=2 \times f_{\mathrm{max}}=2 \times 15=30 \mathrm{kHz}$
 Question 5
Consider two 16-point sequences x[n] and h[n]. Let the linear convolution of x[n] and h[n] be denoted by y[n], while z[n] denotes the 16-point inverse discrete Fourier transform (IDFT) of the product of the 16-point DFTs of x[n] and h[n]. The value(s) of k for which z[k]=y[k] is/are
 A k=0,1,2,,15 B k=0 C k=15 D k=0 and k=15
Signals and Systems   DTFS, DTFT and DFT
Question 5 Explanation:
If two' N' point signals x(n) and h(n) are convolving with each other linearly and circularly
then
$y(k)=z(k)$ at $k=N-1$
where, y(n)= Linear convolution of x(n) and h(n)
z(n)= Circular convolution of x(n) and h(n)
Since, $N=16$ (Given)
Therefore, $\quad y(k)=z(k)$ at $k=N-1=15$

There are 5 questions to complete.

## GATE Notes – Electronics and Communications (EC)

GATE Electronics and Communications notes for all subjects as per syllabus of GATE 2024 Electronics and Communications.

• GATE Notes – Communication Systems

NOTE : These GATE Electronics and Communications notes are not design by us and we encourage to read standard books for GATE Electronics and Communications preparation. All notes are available on internet. We have just organize it properly to helps students who are interested to refer the GATE Electronics and Communications notes. All GATE Electronics and Communications aspirants are suggested to use these notes as just reference for quick revision but not entirely depends on it.

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## GATE Electronics and Communication-Topic wise Previous Year Questions

Prepare for GATE 2024 with practice of GATE Electronics previous year questions and solution

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Prepare for GATE 2024 with practice of GATE Electronics previous year questions and solution

Topic-wise practice of GATE Electronics and Communications Engineering previous year questions is an effective approach for candidates preparing for the GATE 2024 Electronics and Communications Engineering examination. This approach involves practicing previous year question papers topic-wise to develop a strong understanding of the fundamental concepts and their application.

Candidates should attempt as many previous year question papers topic-wise as possible to increase their Rank in the GATE 2024 Electronics and Communications Engineering examination and securing admission to their dream postgraduate program.

## GATE 2024 Electronics and Communications Syllabus

Revised syllabus of GATE 2024 Electronics and Communications by IIT.

Practice GATE Electronics and Communications previous year questions for GATE 2024

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## GATE EC 2018

 Question 1
Two identical nMOS transistors $M_{1}$ and $M_{2}$ are connected as shown below. The circuit is used as an amplifier with the input connected between G and S terminals and the output taken between D and S terminals. $V_{bias}$ and $V_{D}$ are so adjusted that both transistors are in saturation. The transconductance of this combination is defined as $g_{m}=\frac{\partial i_{D}}{\partial V_{GS}}$ while the output resistance is $r_{0}=\frac{\partial V_{GS}}{\partial i_{D}}$ , where $i_{D}$ is the current flowing into the drain of $M_{2}$. Let $g_{m1}$ , $g_{m2}$ be the transconductances and $r_{01}$ , $r_{02}$ be the output resistances of transistors $M_{1}$ and $M_{2}$ , respectively.

Which of the following statements about estimates for $g_{m}$ and $r_{0}$ is correct?
 A $g_{m}\approx g_{m1}\cdot g_{m2}\cdot r_{02} \;and \; r_0 \approx r_{01}+r_{02}.$ B $g_{m}\approx g_{m1}\ + g_{m2} \; and \; r_{0} \approx r_{01}+r_{02}.$ C $g_{m}\approx g_{m1} \; and \; r_{0}\approx r_{01} \cdot g_{m2}\cdot r_{02}.$ D $g_{m}\approx g_{m1} \; and \; r_{0}\approx r_{02}$.
Analog Circuits   FET and MOSFET Analysis
Question 1 Explanation:

$g_{m}=\frac{\Delta I_{D}}{\Delta V_{\text {in }}}=\frac{i_{D}}{v_{g s}}=\frac{i_{D 1}}{v_{g s}}=g_{m 1}$
To calculate $r_{o}$ :

\begin{aligned} v_{\pi 2} &=-I_{x} r_{01} \\ I_{x} &=g_{m 2} v_{\pi 2}+\frac{\left(V_{x}-I_{x} r_{01}\right)}{r_{02}} \\ I_{x} &=-g_{m 2} r_{01} I_{x}+\frac{V_{x}}{r_{02}}-I_{x} \frac{r_{01}}{r_{02}} \\ V_{x} &=r_{02}\left[1+r_{01} g_{m 2}+\frac{r_{01}}{r_{02}}\right] I_{x} \\ r_{0} &=\frac{V_{x}}{I_{x}}=r_{01}+r_{02}+r_{01} r_{02} g_{m 2} \\ & \approx r_{01} r_{02} g_{m 2} \end{aligned}
 Question 2
In the circuit shown below, the op-amp is ideal and Zener voltage of the diode is 2.5 volts. At the input, unit step voltage is applied, i.e. $v_{IN}(t)= u(t)$ volts. Also, at t= 0, the voltage across each of the capacitors is zero.
The time t, in milliseconds, at which the output voltage $v_{OUT}$ crosses -10 V is
 A 2.5 B 5 C 7.5 D 10
Analog Circuits   Operational Amplifiers
Question 2 Explanation:
$\text{For} \quad t \gt 0,$

$I=\frac{1 V}{1 \mathrm{k} \Omega}=1 \mathrm{mA}$
Till $t=2.5 \mathrm{msec}$, both $V_{1}$ and $V_{2}$ will increase and after $t=2.5 \mathrm{msec}$, $V_{2}=2.5 \mathrm{V}$ and $V_{1}$ increases with time.
\begin{aligned} \text { when } v_{\text {out }}(t) &=-10 \mathrm{V} \\ & V_{1}=7.5 \mathrm{V}\\ \text{So,}\\ \frac{1}{1 \mu F} \int_{0}^{t}(1 \mathrm{m} \mathrm{A}) d t &=7.5 \mathrm{V} \\ 10^{3} t &=7.5 \\ t &=7.5 \mathrm{msec} \end{aligned}

 Question 3
A good transimpedance amplifier has
 A low input impedance and high output impedance. B high input impedance and high output impedance. C high input impedance and low output impedance. D low input impedance and low output impedance.
Analog Circuits   Feedback Amplifiers
Question 3 Explanation:
A good transimpedance amplifier should have low input impedance and low output impedance
 Question 4
Let the input be u and the output be y of a system, and the other parameters are real constants. Identify which among the following systems is not a linear system:
 A $\frac{d^{3}y}{dt^{3}} + a_{1} \frac{d^{2}y}{dt^{2}} + a_{2}\frac{dy}{dt} + a_{3}y =$ $b_{3}u+b_{2}\frac{du}{dt}+b_{1}\frac{d^{2}u}{dt^{2}}$ (with initial rest conditions) B $y(t)=\int_{0}^{t}e^{a(t-r)}\beta u(\tau)d \tau$ C $y= au +b$, $b \neq 0$ D $y=au$
Signals and Systems   Basics of Signals and Systems
Question 4 Explanation:
$y=a u+b, b \neq 0$ is a non-linear system.
 Question 5
The Nyquist stability criterion and the Routh criterion both are powerful analysis tools for determining the stability of feedback controllers. Identify which of the following statements is FALSE:
 A Both the criteria provide information relative to the stable gain range of the system. B The general shape of the Nyquist plot is readily obtained from the Bode magnitude plot for all minimum-phase systems. C The Routh criterion is not applicable in the condition of transport lag, which can be readily handled by the Nyquist criterion. D The closed-loop frequency response for a unity feedback system cannot be obtained from the Nyquist plot.
Control Systems   Frequency Response Analysis

There are 5 questions to complete.

## GATE Electronics and Communication 2019

 Question 1
Which one of the following functions is analytic over the entire complex plane?
 A $ln(z)$ B $e^{1/z}$ C $\frac{1}{1-z}$ D $cos(z)$
Engineering Mathematics   Complex Analysis
Question 1 Explanation:
$f(z) = \cos z$ is analytic every where.
 Question 2
The families of curves represented by the solution of the equation

$\frac{dy}{dx}=-\left (\frac{x}{y} \right )^n$

for $n = -1$ and $n = +1$, respectively, are
 A Parabolas and Circles B Circles and Hyperbolas C Hyperbolas and Circles D Hyperbolas and Parabolas
Engineering Mathematics   Differential Equations
Question 2 Explanation:
\begin{aligned} \frac{d y}{d x} &=-\left(\frac{x}{y}\right)^{n} \\ n=-1\quad\quad \frac{d y}{d x} &=-\frac{y^{\prime}}{x} \\ \frac{d y}{y} &=-\frac{d x}{x} \\ \int \frac{1}{y} d y &=-\int \frac{1}{x} d x \\ \ln y &=-\ln x+\ln c \\ \ln (y x) &=\ln c \end{aligned}
$x y=c \quad$ (Represents rectangular hyporbola)
\begin{aligned} n=1, \quad \frac{d y}{d x}&=-\frac{x}{y} \\ y d y &=-x d x \\ y d y &=-\int x d x \\ \frac{y^{2}}{2} &=-\frac{x^{2}}{2}+c \end{aligned}
$x^{2}+y^{2}=2 c \quad$ (Represents family of circles)

 Question 3
Let H(z) be the z-transform of a real-valued discrete-time signal h[n]. If $P(z)=H(z)H\left (\frac{1}{z} \right )$ has a zero at $z=\frac{1}{2}+\frac{1}{2}j$, and P(z) has a total of four zeros, which one of the following plots represents all the zeros correctly?
 A A B B C C D D
Signals and Systems   Z-Transform
Question 3 Explanation:
$P(Z)=H(Z)H\left ( \frac{1}{Z} \right )$
(i) $h(n)$ is real. Som $p(n)$ will be also real
(ii) $P(z)=P(z^{-1})$
From (i) : if $z_1$ is a zero of $P(z)$, then $z_1^*$ will be also a zero of $P(z)$.
From (ii): If $z_1$ is a zero of $P(z)$, then $\frac{1}{z_1}$ will be also a zero of $P(z)$.
So, the 4 zeros are,
\begin{aligned} z_1&= \frac{1}{2}+\frac{1}{2}j\\ z_2&= z_1^*=\frac{1}{2}-\frac{1}{2}j\\ z_3&=\frac{1}{z_1}=\frac{1}{\frac{1}{2}-\frac{1}{2}j}=1-j \\ z_4&=\left ( \frac{1}{z_1} \right )^*=z_3^*=1+j \end{aligned}
 Question 4
Consider the two-port resistive network shown in the figure. When an excitation of 5 V is applied across Port 1, and Port 2 is shorted, the current through the short circuit at Port 2 is measured to be 1 A (see (a) in the figure).
Now, if an excitation of 5 V is applied across Port 2, and Port 1 is shorted (see(b) in the figure), what is the current through the short circuit at Port 1?
 A 0.5 A B 1.0 A C 2.0 A D 2.5 A
Network Theory   Network Theorems
Question 4 Explanation:
According to reciprocity theorem,
In a linear bilateral single source network the ratio of response to excitation remains the same even after their positions get interchanged.
$\therefore \quad \frac{I}{5}=\frac{1}{5} \Rightarrow I=1 \mathrm{A}$
 Question 5
Let Y(s) be the unit-step response of a causal system having a transfer function
$G(s)=\frac{3-s}{(s+1)(s+3)}$

that is, $Y(s)=\frac{G(s)}{s}$. The forced response of the system is
 A $u(t)-2e^{-t}u(t)+e^{-3t}u(t)$ B $2u(t)-2e^{-t}u(t)+e^{-3t}u(t)$ C $2u(t)$ D $u(t)$
Signals and Systems   Laplace Transform
Question 5 Explanation:
Given, $\quad G(s)=\frac{3-s}{(s+1)(s+3)}$
$\therefore \quad Y(s)=\frac{G(s)}{s}=\frac{3-s}{s(s+1)(s+3)}$
Using partial fractions, we get,
\begin{aligned} Y(s)&=\frac{A}{s}+\frac{B}{(s+1)}+\frac{C}{(s+3)} \\ A\left(s^{2}+4 s+3\right)&+B\left(s^{2}+3 s\right)+C\left(s^{2}+s\right)=3-s \\ A+B+C&=0\\ 4 A+3 B+C&=-1 \\ \text{and }3 A&=3 \\ \text{Therefore, }&\text{we get,}\\ A=1, B&=-2 \text { and } C=1\\ \text{So, }\quad Y(s)&=\frac{1}{s}-\frac{2}{(s+1)}+\frac{1}{(s+3)} \\ \text{and}\quad \mathrm{y}(t)&=u(t)-2 e^{-t} u(t)+e^{-3 t} u(t) \\ \end{aligned}
Forced response,
$y_{t}(t)=u(t) \Rightarrow \text { option }(D)$

There are 5 questions to complete.

## GATE Electronics and Communication 2020

 Question 1
If $v_1, v_2,..., v_6$ are six vectors in $\mathbb{R}^4$, which one of the following statements is False?
 A It is not necessary that these vectors span $\mathbb{R}^4$. B These vectors are not linearly independent. C Any four of these vectors form a basis for $\mathbb{R}^4$. D If {$v_1, v_3,v_5, v_6$} spans $\mathbb{R}^4$, then it forms a basis for $\mathbb{R}^4$.
Engineering Mathematics   Calculus
Question 1 Explanation:
$v_1, v_2,..., v_6$ are six vectors in $\mathbb{R}^4$.
For a 4-dimensional vector space,
(i) any four linearly independent vectors form a basis (or)
(ii) Any set of four vectors in $\mathbb{R}^4$ spans $\mathbb{R}^4$, then it forms a basis.
Therefore, clearly options (A), (B), (D) are true.
Option (C) is FALSE
 Question 2
For a vector field $\vec{A}$, which one of the following is False?
 A $\vec{A}$ is solenoidal if $\bigtriangledown \cdot \vec{A}=0$ B $\bigtriangledown \times \vec{A}$ is another vector field. C $\vec{A}$ is irrotational if $\bigtriangledown ^2 \vec{A}=0$. D $\bigtriangledown \times(\bigtriangledown \times \vec{A})=\bigtriangledown (\bigtriangledown \cdot \vec{A})-\bigtriangledown ^2 \vec{A}$
Engineering Mathematics   Calculus
Question 2 Explanation:
Divergence and curl operator is performed on a vector field $\vec{A}$
Curl operation provides a vector orthogonal to the given vector field $\vec{A}$
$\bigtriangledown \times(\bigtriangledown \times \vec{A})=\bigtriangledown (\bigtriangledown \cdot \vec{A})-\bigtriangledown ^2 \vec{A}$
If a vector field is irrortational then $\bigtriangledown \times \vec{A}=0$
If a vector field is solenoidal then $\bigtriangledown \cdot \vec{A}=0$
If a field is scalar A, then $\bigtriangledown ^2 \vec{A}=0$, is a laplacian equation.
Hence option (C) is incorrect

 Question 3
The partial derivative of the function

$f(x,y,z)=e^{1-x \cos y}+xze^{-1/(1+y^2)}$

with respect to x at the point (1,0,e) is
 A -1 B 0 C 1 D $\frac{1}{e}$
Engineering Mathematics   Calculus
Question 3 Explanation:
\begin{aligned} \text{Given, } f(x,y,z)&=e^{1-x\cos y}+xze^{-1/(1+y^{2})} \\ \frac{\partial f}{\partial x}&=e^{1-x\cos y}(0-\cos y)+ze^{-1/1+y^{2}} \\ \left ( \frac{\partial f }{\partial x} \right )_{(1,0,e)}&=e^{0}(0-1)+e\cdot e^{-1/(1+0)} \\ &=-1+1=0 \end{aligned}
 Question 4
The general solution of $\frac{d^2y}{dx^2}-6\frac{dy}{dx}+9y=0$ is
 A $y=C_1e^{3x}+C_2e^{-3x}$ B $y=(C_1+C_2x)e^{-3x}$ C $y=(C_1+C_2x)e^{3x}$ D $y=C_1e^{3x}$
Engineering Mathematics   Differential Equations
Question 4 Explanation:
Taking $\frac{\mathrm{d} }{\mathrm{d} x}=D$
Given, $D^{2}-6D+9=0$
$(D-3)^2=0$
$D=3,3$
So, Solution of the given Differential equation
$y=(C_{1}+C_{2}x)e^{3x}$
 Question 5
The output y[n] of a discrete-time system for an input x[n] is

$y[n]=\begin{matrix} max\\ -\infty \leq k\leq n \end{matrix}\; |x[k]|$.

The unit impulse response of the system is
 A 0 for all n B 1 for all n C unit step signal u[n]. D unit impulse signal $\delta$[n].
Signals and Systems   LTI Systems Continuous and Discrete

There are 5 questions to complete.

## GATE EC 2016 SET-1

 Question 1
Let $M^{4}=I$, (where $I$ denotes the identity matrix) and $M \neq I$, $M^{2} \neq I$ and $M^{3} \neq I$. Then, for any natural number $k,M^{-1}$ equals
 A $M^{4k+1}$ B $M^{4k+2}$ C $M^{4k+3}$ D $M^{4k}$
Engineering Mathematics   Linear Algebra
Question 1 Explanation:
Given that $M^{4}=I$ or $M^{4 k}=I$ or $M^{4(k+1)}=I$
\begin{aligned} \therefore \quad M^{-1} \times I & =M^{4(k+1)} \times M^{-1} \\ \therefore \quad M^{-1} & =M^{4 k+3}\end{aligned}
 Question 2
The second moment of a Poisson-distributed random variable is 2. The mean of the random variable is _______
 A 0.5 B 1 C 2 D 3
Engineering Mathematics   Probability and Statistics
Question 2 Explanation:
In Poisson distribution,
Mean = First moment $=\lambda$
secondmoment $=\lambda^{2}+\lambda$
Given, that second moment is 2
$\begin{array}{r} \lambda^{2}+\lambda=2 \\ \lambda^{2}+\lambda-2=0 \\ (\lambda+2)(\lambda-1)=0 \\ \lambda=1 \end{array}$

 Question 3
Given the following statements about a function $f:\mathbb{R}\rightarrow \mathbb{R}$, select the right option:
P: If f(x) is continuous at x=$x_{0}$, then it is also differentiable at x =$x_{0}$
Q: If f(x) is continuous at x = $x_{0}$, then it may not be differentiable at x= $x_{0}$.
R: If f(x) is differentiable at x= $x_{0}$, then it is also continuous at x= $x_{0}$.
 A P is true, Q is false, R is false B P is false, Q is true, R is true C P is false, Q is true, R is false D P is true, Q is false, R is true
Engineering Mathematics   Calculus
Question 3 Explanation:
P: If f(x) is continuous at $x=x_{0},$ then it is also differentiable at $x=x_{0}$
Q: If f(x) is continuous at $x=x_{0},$ then it may or may not be derivable at $x=x_{0}$
R: If f(x) is differentiable at $x=x_{0}$, then it is also continuous at $x=x_{0}$
P is false
Q is true
R is true
Option (B) is correct
 Question 4
Which one of the following is a property of the solutions to the Laplace equation: $\bigtriangledown ^{2} f= 0$?
 A The solutions have neither maxima nor minima anywhere except at the boundaries. B The solutions are not separable in the coordinates. C The solutions are not continuous D The solutions are not dependent on the boundary conditions
Signals and Systems   Laplace Transform
 Question 5
Consider the plot of $f(x)$ versus $x$ as shown below.

Suppose F(x)=$\int_{-5}^{x}f(y)dy$ . Which one of the following is a graph of F(x) ?
 A A B B C C D D
Engineering Mathematics   Calculus
Question 5 Explanation:
$F^{\prime}(x)=f(x)$ which is density function
$F^{\prime}(x)=f(x) \lt 0$ when $x \lt 0$
$\therefore \quad F(x)$ is decreasing for $x \lt 0$
$F^{\prime}(x)=f(x) \gt 0$
when $x\gt 0$
$\therefore \quad F(x)$ is increasing for $x\gt 0$.

There are 5 questions to complete.