Recent questions and answers in Vector Analysis - GO Electronics

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GATE2020-EC: 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 $\mathbb{R}^{4}$ ... $\mathbb{R}^{4}$ , then it forms a basis for $\mathbb{R}^{4}$.

asked Feb 13 in Vector Analysis by jothee (1.4k points)

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2

GATE2020-EC: 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 in Vector Analysis by jothee (1.4k points)

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3

GATE2020-EC: 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 a real white Gaussian 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 in Vector Analysis by jothee (1.4k points)

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4

GATE2016-3-27
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 $\textbf{u}=-$\large\frac{2}{5}$e_1-3e_2-$\ ... $e_1+3e_2+$\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)

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5

GATE2016-3-28
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)

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6

GATE2016-3-50
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 of $52\:kbps$ is to be 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)

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7

GATE2016-2-5
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)

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8

GATE2016-2-55
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 voltage $-V$ is ... time 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)

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9

GATE2016-1-50
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 through a filter with ... $E_s > E_h$ ; $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)

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10

GATE2015-3-29
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)

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11

GATE2015-2-49
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)

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12

GATE2015-1-25
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 solenoidal, but ... $\overrightarrow{P}$ is neither solenoidal, nor irrotational $\overrightarrow{P}$ is both solenoidal and irrotational

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

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13

GATE2015-1-54
The electric field intensity of a plane wave traveling in free space is given by the following expression $\textbf{E}(x,t)=\textbf{a}_y \: 24 \: \pi \: \: \cos(\omega t – k_0 x) \: \: (V/m)$ In this field, consider a square area $10 \: \text{cm} \times 10 \: \text{cm}$ on a plane $x+y=1$. 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)

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14

GATE2015-1-55
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 $(\varepsilon _r)$ and permeability $(\mu _r)$ as ... transmitted 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)

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15

GATE2014-4-2
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)

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16

GATE2014-4-3
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)

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17

GATE2014-4-22
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)

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18

GATE2014-4-49
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 $Z$ ... $\tau$ should be strictly positive zero strictly negative strictly positive, zero, or strictly negative depending on the nonzero value of $\sigma ^2$

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

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19

GATE2014-4-52
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 variance. Assume a ... $\beta = -0.3$, 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)

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20

GATE2014-3-53
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)

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21

GATE2014-2-22
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, $P$ ... $(W\rightarrow \infty )$ is approximately $1.44$ $1.08$ $0.72$ $0.36$

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

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22

GATE2014-1-3
$C$ is a closed path in the $z$-plane given by $\mid z \mid = 3.$ The value of the integral $\displaystyle{}\oint_{C}\bigg(\dfrac{z^{2}-z+4j}{z+2j}\bigg)dz$ is $-4\pi(1+j2)$ $4\pi(3-j2)$ $-4\pi(3+j2)$ $4\pi(1-j2)$

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

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23

GATE2014-1-28
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)

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24

GATE2014-1-46
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)

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25

GATE2014-1-53
In spherical coordinates, let $\hat{a_{\theta}},\hat{a_{\phi}}$ denote unit vectors along the $\theta,\phi$ directions. $\textbf{E} = \dfrac{100}{r}\sin\theta \cos (\omega t - \beta r)\hat{a_{\theta}}\: V/m$ ... antenna, in 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)

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26

GATE2013-52
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 $\vec{E_{t}}$ denote incident, reflected, and transmitted electric field vectors associated with the wave. The angle of incidence $\theta_{i}$ and ...

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

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27

GATE2013-53
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 $\vec{E_{t}}$ denote incident, reflected, and transmitted electric field vectors associated with the wave. The expression for $\vec{E_{r}}$ ...

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

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28

GATE2013-39
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)

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29

GATE2013-7
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)

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30

GATE2013-2
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 bounded by the ... bounded 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)

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31

GATE2012-23
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)

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32

GATE2017 EC-2: 28
If the vector function $\overrightarrow{F}=\widehat{a_x}(3y-k_1z)+\widehat{a_y}(k_2x-2z)-\widehat{a_z}(k_3y+z)$ is irrotational, then the values of the constants $k_1$,$k_2$ and $k_3$, respectively, are $0.3, -2.5, 0.5$ $0.0, 3.0, 2.0$ $0.3, 0.33, 0.5$ $4.0, 3.0, 2.0$

asked Nov 23, 2017 in Vector Analysis by admin (2.8k points)

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33

GATE2017 EC-2: 3
The smaller angle (in degrees) between the planes $x+y+z=1$ and $2x-y+2z=0$ is ________.

asked Nov 23, 2017 in Vector Analysis by admin (2.8k points)

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34

GATE2017 EC-1: 28
Let $I=\int_{c}\left ( 2zdx+2ydy+2xdx \right )$ where $x,y,z$ are real, and let $C$ be the straight line segment from point $A:(0,2,1)$ to point $B:(4,1,-1)$.The value of $I$ is ____________.

asked Nov 17, 2017 in Vector Analysis by admin (2.8k points)

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