Recent questions and answers in Vector Analysis

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1

GATE EC 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 (4.4k points) | 18 views

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GATE EC 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 (4.4k points) | 13 views

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GATE EC 2021 | Question: 17
Consider the vector field $F\:=\:a_{x}\left ( 4y-c_{1}z \right )+a_y\left ( 4x + 2z\right )+a_{z}\left ( 2y +z\right )$ 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 _________________

asked Feb 20 in Vector Analysis by Arjun (4.4k points) | 12 views

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GATE EC 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 (4.4k points) | 13 views

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5

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 (1.8k points) | 112 views

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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 (1.8k points) | 118 views

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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 two-sided 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 (1.8k points) | 33 views

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8

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 three-dimensional 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}$e_3$

asked Mar 28, 2018 in Vector Analysis by Milicevic3306 (15.8k points) | 22 views

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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) | 18 views

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GATE ECE 2016 Set 3 | Question: 50
A voice-grade AWGN (additive white Gaussian noise) telephone channel has a bandwidth of $4.0\:kHz$ and two-sided 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 bit-energy $E_b$ (in mJ/bit) necessary is _______

asked Mar 28, 2018 in Vector Analysis by Milicevic3306 (15.8k points) | 17 views

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11

GATE ECE 2016 Set 2 | Question: 5
Consider the time-varying 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) | 46 views

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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) | 11 views

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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) | 23 views

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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} \\ $ $E_s < E_h$ ; $SNR_{max}=\frac{2E_h}{N_0}$

asked Mar 28, 2018 in Vector Analysis by Milicevic3306 (15.8k points) | 16 views

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15

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 (15.8k points) | 37 views

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16

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 _______.

asked Mar 28, 2018 in Vector Analysis by Milicevic3306 (15.8k points) | 25 views

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17

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 $\dfrac{3}{2}A^{2}N_{0} \\$ $\dfrac{3}{4}A^{2}N_{0} \\$ $A^{2}N_{0} \\$ $\dfrac{1}{2}A^{2}N_{0}$

asked Mar 28, 2018 in Vector Analysis by Milicevic3306 (15.8k points) | 15 views

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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) | 21 views

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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 time-averaged power (in mW) passing through the square area is _____________.

asked Mar 28, 2018 in Vector Analysis by Milicevic3306 (15.8k points) | 11 views

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20

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 ____________.

asked Mar 28, 2018 in Vector Analysis by Milicevic3306 (15.8k points) | 15 views

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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) | 11 views

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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 __________

asked Mar 26, 2018 in Vector Analysis by Milicevic3306 (15.8k points) | 15 views

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23

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 _________.

asked Mar 26, 2018 in Vector Analysis by Milicevic3306 (15.8k points) | 12 views

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24

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

asked Mar 26, 2018 in Vector Analysis by Milicevic3306 (15.8k points) | 19 views

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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 ... $\sigma ^2$

asked Mar 26, 2018 in Vector Analysis by Milicevic3306 (15.8k points) | 9 views

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26

GATE ECE 2014 Set 4 | Question: 52
Consider a discrete-time channel $Y=X +Z$, where the additive noise $Z$ is signal-dependent. 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 $10^{-7}$ $10^{-6}$ $10^{-4}$ $10^{-2}$

asked Mar 26, 2018 in Vector Analysis by Milicevic3306 (15.8k points) | 11 views

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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 __________.

asked Mar 26, 2018 in Vector Analysis by Milicevic3306 (15.8k points) | 13 views

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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 __________

asked Mar 26, 2018 in Vector Analysis by Milicevic3306 (15.8k points) | 22 views

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29

GATE ECE 2014 Set 2 | Question: 22
The capacity of a band-limited 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 $1.44$ $1.08$ $0.72$ $0.36$

asked Mar 26, 2018 in Vector Analysis by Milicevic3306 (15.8k points) | 40 views

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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 $\text{div} ( r^{2} \nabla ( \text{ln }r ) )$ = _______ .

asked Mar 26, 2018 in Vector Analysis by Milicevic3306 (15.8k points) | 20 views

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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 _______.

asked Mar 26, 2018 in Vector Analysis by Milicevic3306 (15.8k points) | 24 views

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32

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

asked Mar 26, 2018 in Vector Analysis by Milicevic3306 (15.8k points) | 17 views

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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 ______.

asked Mar 26, 2018 in Vector Analysis by Milicevic3306 (15.8k points) | 34 views

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34

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 $\vec{E_{t}}$ ...

asked Mar 26, 2018 in Vector Analysis by Milicevic3306 (15.8k points) | 21 views

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35

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 $\vec{E_{t}}$ ...

asked Mar 26, 2018 in Vector Analysis by Milicevic3306 (15.8k points) | 21 views

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36

GATE ECE 2013 | Question: 39
The $\text{DFT}$ of a vector $\begin{bmatrix} a & b & c & d \end{bmatrix}$ is the vector $\begin{bmatrix} \alpha & \beta & \gamma & \delta \end{bmatrix}.$ ... $\begin{bmatrix} \alpha & \beta & \gamma & \delta \end{bmatrix}$

asked Mar 26, 2018 in Vector Analysis by Milicevic3306 (15.8k points) | 14 views

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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 $0$ $1/3$ $1$ $3$

asked Mar 26, 2018 in Vector Analysis by Milicevic3306 (15.8k points) | 14 views

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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

asked Mar 26, 2018 in Vector Analysis by Milicevic3306 (15.8k points) | 14 views

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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 $-2$ $2$ $1$ $0$

asked Mar 25, 2018 in Vector Analysis by Milicevic3306 (15.8k points) | 33 views

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GATE ECE 2012 | Question: 23
Given $f(z)=\frac{1}{z+1}-\frac{2}{z+3}$. If $C$ is a counterclockwise path in the z-plane such that $\mid z+1 \mid=1$, the value of $\frac{1}{2\pi j}\oint_cf(z)dz$ is $-2$ $-1$ $1$ $2$

asked Mar 25, 2018 in Vector Analysis by Milicevic3306 (15.8k points) | 17 views