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Recent questions and answers in Vector Analysis
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GATE ECE 2016 Set 3  Question: 27
If the vectors $e_1=(1,0,2)$, $e_2=(0,1,0)$ and $e_3=(2,0,1)$ form an orthogonal basis of the threedimensional real space $\mathbb{R}^3$, then the vector $\textbf{u}=(4,3,3)\in \mathbb{R}^3 $ can be expressed as $\textbf{u}=$ ... \frac{2}{5}$e_1+3e_2+$\large\frac{11}{5}$e_3 \\$ $\textbf{u}=$\large\frac{2}{5}$e_1+3e_2$\large\frac{11}{5}$e_3$
answered
Jun 20
in
Vector Analysis
by
Divyanshu Shukla
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140
points)

55
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gate2016ec3
vectoranalysis
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2
GATE ECE 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 $\frac{1}{2}$ $\frac{1}{4}$ $\frac{1}{6}$ $\frac{1}{3}$
asked
Feb 20
in
Vector Analysis
by
Arjun
(
4.5k
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67
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gateec2021
vectoranalysis
vectorinplanes
0
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GATE ECE 2021  Question: 16
If the vectors $(1.0,\:1.0,\:2.0)$, $(7.0,\:3.0,\:x)$ and $(2.0,\:3.0,\:1.0)$ in $\mathbb{R}^{3}$ are linearly dependent, the value of $x$ is __________
asked
Feb 20
in
Vector Analysis
by
Arjun
(
4.5k
points)

45
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gateec2021
numericalanswers
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GATE ECE 2021  Question: 17
Consider the vector field $F\:=\:a_{x}\left ( 4yc_{1}z \right )+a_y\left ( 4x + 2z\right )+a_{z}\left ( 2y +z\right )$ in a rectangular coordinate system $(x,y,z)$ with unit vectors $a_{x},\:a_{y}$ and $a_{z}$. If the field $F$ is irrotational (conservative), then the constant $c_{1}$ (in integer) is _________________
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Feb 20
in
Vector Analysis
by
Arjun
(
4.5k
points)

61
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gateec2021
numericalanswers
vectoranalysis
vectorinplanes
0
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0
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GATE ECE 2021  Question: 37
For a vector field $D=\rho\cos^{2}\:\varphi \:a_{\rho }+z^{2}\sin^{2}\:\varphi \:a_{\varphi }$ in a cylindrical coordinate system $\left ( \rho ,\varphi ,z \right )$ with unit vectors $a_{\rho },a_{\varphi }$ and $a_{z}$, the ... $\left ( \rho =3, 0\leq z\leq 2 \right )$ (rounded off to two decimal places) is ________________
asked
Feb 20
in
Vector Analysis
by
Arjun
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4.5k
points)

25
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gateec2021
numericalanswers
vectoranalysis
0
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0
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GATE ECE 2020  Question: 1
If $v_{1},v_{2}, \dots ,v_{6}$ are six vectors in $\mathbb{R}^{4}$ , which one of the following statements is $\text{FALSE}$? It is not necessary that these vectors span $\mathbb{R}^{4}$. These vectors are not linearly independent. Any four of these vectors form a basis ... $\mathbb{R}^{4}$ , then it forms a basis for $\mathbb{R}^{4}$.
asked
Feb 13, 2020
in
Vector Analysis
by
jothee
(
1.9k
points)

156
views
gate2020ec
vectoranalysis
+1
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0
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GATE ECE 2020  Question: 2
For a vector field $\overrightarrow{A}$, which one of the following is $\text{FALSE}$? $\overrightarrow{A}$ is solenoidal if $\triangledown \cdot \overrightarrow{A}=0.$ $\triangledown \times \overrightarrow{A}$ ...
asked
Feb 13, 2020
in
Vector Analysis
by
jothee
(
1.9k
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170
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gate2020ec
vectoranalysis
0
votes
0
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8
GATE ECE 2020  Question: 24
The random variable $Y=\int_{\infty }^{\infty }W\left ( t \right )\phi \left ( t \right )dt, \text{ where } \phi \left ( t \right )=\begin{cases} 1; & 5\leq t\leq 7 &\\ 0; & \text{otherwise} \end{cases}$ and $W(t)$ is ... noise process with twosided power spectral density $S_{W}\left ( f \right )=3 W/Hz$, for all $f$. The variance of $Y$ is ________.
asked
Feb 13, 2020
in
Vector Analysis
by
jothee
(
1.9k
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55
views
gate2020ec
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vectoranalysis
gaussstheorem
0
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0
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GATE ECE 2016 Set 3  Question: 28
A triangle in the $xy$plane is bounded by the straight lines $2x=3y, \: y=0$ and $x=3$. The volume above the triangle and under the plane $x+y+z=6$ is _______
asked
Mar 28, 2018
in
Vector Analysis
by
Milicevic3306
(
15.8k
points)

42
views
gate2016ec3
numericalanswers
vectoranalysis
vectorinplanes
0
votes
0
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GATE ECE 2016 Set 3  Question: 50
A voicegrade AWGN (additive white Gaussian noise) telephone channel has a bandwidth of $4.0\:kHz$ and twosided noise power spectral density ${\large\frac{\eta}{2}}=2.5\times10^{5}Watt\:per\:Hz$. If information at the rate ... transmitted over this channel with arbitrarily small bit error rate, then the minimum bitenergy $E_b$ (in mJ/bit) necessary is _______
asked
Mar 28, 2018
in
Vector Analysis
by
Milicevic3306
(
15.8k
points)

30
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gate2016ec3
numericalanswers
vectoranalysis
gauss'stheorem
0
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GATE ECE 2016 Set 2  Question: 5
Consider the timevarying vector $\textbf{I}=\hat{x}15\cos(\omega t)+\hat{y}5\sin(\omega t)$ in Cartesian coordinates, where $\omega> 0$ is a constant. When the vector magnitude $\mid \textbf{I} \mid$ is at its minimum value, the angle $\theta$ that $\textbf{I}$ makes with the $x$ axis (in degrees, such that $ 0\leq \theta \leq 180)$ ________
asked
Mar 28, 2018
in
Vector Analysis
by
Milicevic3306
(
15.8k
points)

94
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gate2016ec2
numericalanswers
vectoranalysis
0
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0
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GATE ECE 2016 Set 2  Question: 55
A positive charge $q$ is placed at $x=0$ between two infinte metal plates placed at $x=d$ and at $x=+d$ respectively. The metal plates lie in the $yz$ plane. The charge is at rest at $t=0$, when a voltage $+V$ is applied to the plate at $d$ and ... that the charge $q$ takes to reach the right plate is proportional to $d/V$ $\sqrt{d}/V$ $d/\sqrt{V}$ $\sqrt{d/V}$
asked
Mar 28, 2018
in
Vector Analysis
by
Milicevic3306
(
15.8k
points)

21
views
gate2016ec2
vectoranalysis
0
votes
0
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13
GATE ECE 2016 Set 1  Question: 29
The region specified by $\{ (\rho,\varphi,z):3 \leq\rho\leq 5,\frac{\pi}{8}\leq\varphi\leq\frac{\pi}{4}, \: 3\leq z\leq4.5\}$ in cylindrical coordinates has volume of _________
asked
Mar 28, 2018
in
Vector Analysis
by
Milicevic3306
(
15.8k
points)

37
views
gate2016ec1
numericalanswers
vectoranalysis
0
votes
0
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14
GATE ECE 2016 Set 1  Question: 50
An analog pulse $s(t)$ is transmitted over an additive white Gaussian noise (AWGN) channel. The received signal is $r(t) = s(t) + n(t)$, where $n(t)$ is additive white Gaussian noise with power spectral density $\frac{N_0}{2}$. The received signal is passed ... $E_s > E_h$ ; $SNR_{max}>\frac{2E_s}{N_0} \\ $ $E_s < E_h$ ; $SNR_{max}=\frac{2E_h}{N_0}$
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Mar 28, 2018
in
Vector Analysis
by
Milicevic3306
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15.8k
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27
views
gate2016ec1
vectoranalysis
gauss'stheorem
0
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0
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GATE ECE 2015 Set 3  Question: 4
Consider the function $g(t) = e^{t}\sin(2\pi t)u(t)$ where $u(t)$ is the unit step function. The area under $g(t)$ is ______.
asked
Mar 28, 2018
in
Vector Analysis
by
Milicevic3306
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15.8k
points)

58
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gate2015ec3
numericalanswers
vectoranalysis
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GATE ECE 2015 Set 3  Question: 29
A vector field $\textbf{D} = 2\rho^{2}\:\textbf{a}_{\rho} + z\: \textbf{a}_{z}$ exists inside a cylindrical region enclosed by the surfaces $\rho =1,z = 0$ and $z = 5.$ Let $S$ be the surface bounding this cylindrical region. The surface integral of this field on $S(∯_{S} \textbf{D.ds})$ is _______.
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Mar 28, 2018
in
Vector Analysis
by
Milicevic3306
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15.8k
points)

42
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gate2015ec3
numericalanswers
vectoranalysis
0
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0
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GATE ECE 2015 Set 2  Question: 49
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 $\dfrac{3}{2}A^{2}N_{0} \\$ $\dfrac{3}{4}A^{2}N_{0} \\$ $A^{2}N_{0} \\$ $\dfrac{1}{2}A^{2}N_{0}$
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Mar 28, 2018
in
Vector Analysis
by
Milicevic3306
(
15.8k
points)

26
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gate2015ec2
vectoranalysis
gauss'stheorem
0
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0
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18
GATE ECE 2015 Set 1  Question: 25
A vector $\overrightarrow{P}$ is given by $\overrightarrow{P} = x^3y\overrightarrow{a}_x  x^2y^2\overrightarrow{a}_y  x^2 yz \overrightarrow{a}_z$. Which one of the statements is TRUE? $\overrightarrow{P}$ is ... irrotational, but not solenoidal $\overrightarrow{P}$ is neither solenoidal, nor irrotational $\overrightarrow{P}$ is both solenoidal and irrotational
asked
Mar 28, 2018
in
Vector Analysis
by
Milicevic3306
(
15.8k
points)

37
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gate2015ec1
vectoranalysis
0
votes
0
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19
GATE ECE 2015 Set 1  Question: 54
The electric field intensity of a plane wave traveling in free space is given by the following expression $\textbf{E}(x,t)=\textbf{a}_y \: 24 \: \pi \: \: \cos(\omega t  k_0 x) \: \: (V/m)$ ... $x+y=1$. The total timeaveraged power (in mW) passing through the square area is _____________.
asked
Mar 28, 2018
in
Vector Analysis
by
Milicevic3306
(
15.8k
points)

27
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gate2015ec1
numericalanswers
vectoranalysis
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GATE ECE 2015 Set 1  Question: 55
Consider a uniform plane wave with amplitude $(E_0)$ of $10 \: V/m$ and $1.1$ GHz frequency travelling in air, and incident normally on a dielectric medium with complex relative permittivity $(\varepsilon _r)$ ... electric field component (in V/m) after it has travelled a distance of $10$ cm inside the dielectric region is ____________.
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Mar 28, 2018
in
Vector Analysis
by
Milicevic3306
(
15.8k
points)

26
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gate2015ec1
numericalanswers
vectoranalysis
0
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0
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21
GATE ECE 2014 Set 4  Question: 2
The magnitude of the gradient for the function $f(x,y,z) = x^2 +3y^2 +z^3$ at the point $(1,1,1)$ is ___________.
asked
Mar 26, 2018
in
Vector Analysis
by
Milicevic3306
(
15.8k
points)

24
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gate2014ec4
numericalanswers
vectoranalysis
gradient
0
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0
answers
22
GATE ECE 2014 Set 4  Question: 3
Let $X$ be a zero mean unit variance Gaussian random variable. $E[ \mid X \mid ]$ is equal to __________
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Mar 26, 2018
in
Vector Analysis
by
Milicevic3306
(
15.8k
points)

28
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gate2014ec4
numericalanswers
vectoranalysis
gaussstheorem
randomvariable
0
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GATE ECE 2014 Set 4  Question: 5
The directional derivative of $f(x,y)= \frac{xy}{\sqrt{2}} (x+y)$ at $(1,1)$ in the direction of the unit vector at an angle of $\frac{\pi}{4}$ with $y$axis, is given by _________.
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Mar 26, 2018
in
Vector Analysis
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Milicevic3306
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15.8k
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23
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gate2014ec4
vectoranalysis
numericalanswers
0
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0
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GATE ECE 2014 Set 4  Question: 22
If calls arrive at a telephone exchange such that the time of arrival of any call is independent of the time of arrival of earlier or future calls, the probability distribution function of the total number of calls in a fixed time interval will be Poisson Gaussian Exponential Gamma
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Mar 26, 2018
in
Vector Analysis
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Milicevic3306
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15.8k
points)

39
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gate2014ec4
vectoranalysis
gauss'stheorem
0
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0
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25
GATE ECE 2014 Set 4  Question: 49
Consider a communication scheme where the binary valued signal $X$ satisfies $P\{X=+1\}=0.75$ and $P\{X=1 \}=0.25$. The received signal $Y=X+Z$, where $Z$ is a Gaussian random variable with zero mean and variance ... $\sigma ^2$
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Mar 26, 2018
in
Vector Analysis
by
Milicevic3306
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15.8k
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gate2014ec4
vectoranalysis
gauss'stheorem
0
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26
GATE ECE 2014 Set 4  Question: 52
Consider a discretetime channel $Y=X +Z$, where the additive noise $Z$ is signaldependent. In particular, given the transmitted symbol $ X \in \{a , +a\}$ at any instant, the noise sample $Z$ is chosen independently from a Gaussian distribution with mean $\beta X$ and unit ... $\beta = 0.3$, the BER is closest to $10^{7}$ $10^{6}$ $10^{4}$ $10^{2}$
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Mar 26, 2018
in
Vector Analysis
by
Milicevic3306
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15.8k
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23
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gate2014ec4
vectoranalysis
gaussstheorem
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27
GATE ECE 2014 Set 4  Question: 54
Gven $\overrightarrow{F} = z \hat{a}_x + x \hat{a}_y + y \hat{a}_z$. If $S$ represents the portion of the sphere $x^2 +y^2+z^2=1$ for $z \geq 0$, then $\int _s \nabla \times \overrightarrow{F} \cdot \overrightarrow{ds}$ is __________.
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Mar 26, 2018
in
Vector Analysis
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Milicevic3306
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15.8k
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30
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gate2014ec4
numericalanswers
vectoranalysis
0
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28
GATE ECE 2014 Set 3  Question: 53
Given the vector $\textbf{A}= ( \cos x ) ( \sin y )\hat{a_{x}}+( \sin x )( \cos y )\hat{a_{y}},$ where $\hat{a_{x}},$ $\hat{a_{y}}$ denote unit vectors along $x$, $y$ directions, respectively. The magnitude of curl of $\textbf{A}$ is __________
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Mar 26, 2018
in
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Milicevic3306
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gate2014ec3
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0
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29
GATE ECE 2014 Set 2  Question: 22
The capacity of a bandlimited additive white Gaussian noise (AWGN) channel is given by $C = W \log_{2}\left ( 1+\frac{p} {\sigma ^{2}w} \right )$ bits per second (bps), where $W$ is the channel bandwidth, $P$ is the average power received ... channel capacity (in kbps) with infinite bandwidth $(W\rightarrow \infty )$ is approximately $1.44$ $1.08$ $0.72$ $0.36$
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Mar 26, 2018
in
Vector Analysis
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Milicevic3306
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51
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gate2014ec2
vectoranalysis
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GATE ECE 2014 Set 2  Question: 29
If $\overrightarrow {r}= x\hat{a_{x}}+y\hat{a_{y}}+z\hat{a_{z}}$ and $\mid \overrightarrow{r} \mid= r$ , then $\text{div} ( r^{2} \nabla ( \text{ln }r ) )$ = _______ .
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Mar 26, 2018
in
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Milicevic3306
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gate2014ec2
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31
GATE ECE 2014 Set 1  Question: 28
The volume under the surface $z(x,y) = x + y$ and above the triangle in the $x – y$ plane defined by $\{0 \leq y \leq x \: \text{and} \: 0 \leq x \leq 12\}$ is _______.
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Mar 26, 2018
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Vector Analysis
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GATE ECE 2014 Set 1  Question: 46
Consider the state space model of a system, as given below ... The system is controllable and observable uncontrollable and observable uncontrollable and unobservable controllable and unobservable
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Mar 26, 2018
in
Vector Analysis
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Milicevic3306
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gate2014ec1
vectoranalysis
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33
GATE ECE 2014 Set 1  Question: 53
In spherical coordinates, let $\hat{a_{\theta}},\hat{a_{\phi}}$ denote unit vectors along the $\theta,\phi$ directions. $\textbf{E} = \dfrac{100}{r}\sin\theta \cos (\omega t  \beta r)\hat{a_{\theta}}\: V/m$ ... free space. The average power $(W)$ crossing the hemispherical shell located at $r = 1\:km,0\leq \theta \leq \pi/2$ is ______.
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Mar 26, 2018
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GATE ECE 2013  Question: 53
A monochromatic plane wave of wavelength $\lambda = 600 \mu m$ is propagating in the direction as shown in the figure below. $\vec{E_{i}},\vec{E_{r}},$ and $\vec{E_{t}}$ ...
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Mar 26, 2018
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vectoranalysis
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GATE ECE 2013  Question: 52
A monochromatic plane wave of wavelength $\lambda = 600 \mu m$ is propagating in the direction as shown in the figure below. $\vec{E_{i}},\vec{E_{r}},$ and $\vec{E_{t}}$ ...
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Mar 26, 2018
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Milicevic3306
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vectoranalysis
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GATE ECE 2013  Question: 39
The $\text{DFT}$ of a vector $\begin{bmatrix} a & b & c & d \end{bmatrix}$ is the vector $\begin{bmatrix} \alpha & \beta & \gamma & \delta \end{bmatrix}.$ ... $\begin{bmatrix} \alpha & \beta & \gamma & \delta \end{bmatrix}$
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37
GATE ECE 2013  Question: 7
The divergence of the vector field $\overrightarrow{A} = x\hat{a}_{x} + y\hat{a}_{y} + z\hat{a}_{z}$ is $0$ $1/3$ $1$ $3$
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38
GATE ECE 2013  Question: 2
Consider a vector field $\vec{A}(\vec{r}).$ The closed loop line integral $\displaystyle {} \int \vec{A}\bullet\vec{dl}$ can be expressed as $\displaystyle {} \iint (\triangledown \times \vec{A}) \bullet\vec{ds}$ over the closed surface ... by the loop $\displaystyle {} \iiint (\triangledown \times \vec{A}) \bullet\vec{ds}$ over the open surface bounded by the loop
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vectoranalysis
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39
GATE ECE 2012  Question: 35
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 $2$ $2$ $1$ $0$
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Mar 25, 2018
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Vector Analysis
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gate2012ec
vectoranalysis
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40
GATE ECE 2012  Question: 23
Given $f(z)=\frac{1}{z+1}\frac{2}{z+3}$. If $C$ is a counterclockwise path in the zplane such that $\mid z+1 \mid=1$, the value of $\frac{1}{2\pi j}\oint_cf(z)dz$ is $2$ $1$ $1$ $2$
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Vector Analysis
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