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GATE2020EC: 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 $\mathbb{R}^{4}$.
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vectoranalysis
engineeringmathematics
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2
GATE2020EC: 2
For a vector field $\overrightarrow{A}$, which one of the following is $\text{FALSE}$? $\overrightarrow{A}$ is solenoidal if $\triangledown \cdot \overrightarrow{A}=0.$ $\triangledown \times \overrightarrow{A}$ ...
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gate2020ec
engineeringmathematics
vectors
overrightarrow
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3
GATE2020EC: 3
The partial derivative of the function $f(x, y, z) = e^{1x\cos y} + xze^{1/(1+y^{2})}$ with respect to $x$ at the point $(1,0,e)$ is $1$ $0$ $1 \\$ $\dfrac{1}{e}$
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gate2020ec
partialdifferentialequations
engineeringmathematics
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4
GATE2020EC: 4
The general solution of $\dfrac{\mathrm{d^{2}} y}{\mathrm{d} x^{2}}6\dfrac{\mathrm{d} y}{\mathrm{d} x}+9y=0$ is $y=C_{1}e^{3x}+C_{2}e^{3x}$ $y=(C_{1}+C_{2}x)e^{3x}$ $y=(C_{1}+C_{2}x)e^{3x}$ $y=C_{1}e^{3x}$
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gate2020ec
engineeringmathematics
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5
GATE2020EC: 22
A binary random variable $X$ takes the value $+2$ or $2$. The probability $P(X=+2)=\alpha $. The value of $\alpha$ (rounded off to one decimal place), for which the entropy of $X$ is maximum, is __________.
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numericalanswers
probability
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6
GATE2020EC: 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 twosided power spectral density $S_{W}\left ( f \right )=3 W/Hz$, for all $f$. The variance of $Y$ is ________.
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gate2020ec
numericalanswers
vectoranalysis
engineeringmathematics
gaussstheorem
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7
GATE2020EC: 25
The two sides of a fair coin are labelled as $0$ to $1$. The coin is tossed two times independently. Let $M$ and $N$ denote the labels corresponding to the outcomes of those tosses. For a random variable $X$, defined as $X = \text{min}(M, N)$, the expected value $E(X)$ (rounded off to two decimal places) is ___________.
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numericalanswers
engineeringmathematics
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8
GATE2020EC: 26
Consider the following system of linear equations. $\begin{array}{llll} x_{1}+2x_{2}=b_{1} ; & 2x_{1}+4x_{2}=b_{2}; & 3x_{1}+7x_{2}=b_{3} ; & 3x_{1}+9x_{2}=b_{4} \end{array}$ ... $b_{2}=2b_{1}$ and $3b_{1}6b_{3}+b_{4}=0$ $b_{3}=2b_{1}$ and $3b_{1}6b_{3}+b_{4}=0$
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gate2020ec
engineeringmathematics
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9
GATE2020EC: 27
Which one of the following options contains two solutions of the differential equation $\dfrac{\mathrm{d} y}{\mathrm{d} x}=\left ( y1 \right )x?$ $\ln\mid y1 \mid=0.5x^{2}+C$ and $y=1$ $\ln\mid y1 \mid=2x^{2}+C$ and $y=1$ $\ln\mid y1 \mid=0.5x^{2}+C$ and $y=1$ $\ln\mid y1 \mid=2x^{2}+C$ and $y=1$
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gate2020ec
differentialequations
engineeringmathematics
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10
GATE2020EC: 51
For the solid $S$ shown below, the value of $\underset{S}{\iiint} xdxdydz$ (rounded off to two decimal places) is _______________.
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Feb 13
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gate2020ec
numericalanswers
engineeringmathematics
+2
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0
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11
GATE2019 EC: 1
Which one of the following functions is analytic over the entire complex plane? $\ln(z)$ $e^{1/z}$ $\frac{1}{1z}$ $\cos(z)$
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Feb 12, 2019
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gate2019ec
complexanalysis
engineeringmathematics
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0
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12
GATE2019 EC: 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 Parabolas and Circles Circles and Hyperbolas Hyperbolas and Circles Hyperbolas and Parabolas
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gate2019ec
differentialequations
engineeringmathematics
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13
GATE2019 EC: 14
In the circuit shown, what are the values of $F$ for $EN=0$ and $EN=1,$ respectively? $\text{0 and D}$ $\text{HiZ and D}$ $\text{0 and 1}$ $\text{HiZ and}$ $ \overline{D}$
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gate2019ec
engineeringmathematics
overline
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0
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14
GATE2019 EC: 16
The value of the contour integral $\frac{1}{2\pi j} \oint\left(z+\frac{1}{z}\right)^{2}dz$ evaluated over the unit circle $\mid z \mid=1$ is_______.
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Feb 12, 2019
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gate2019ec
numericalanswers
calculus
integrals
engineeringmathematics
0
votes
1
answer
15
GATE2019 EC: 17
The number of distinct eigenvalues of the matrix $A=\begin{bmatrix} 2&2&3&3\\0&1&1&1\\0&0&3&3\\0&0&0&2 \end{bmatrix}$ is equal to ____________.
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Feb 12, 2019
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gate2019ec
numericalanswers
matrixalgebra
linearalgebra
engineeringmathematics
0
votes
0
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16
GATE2019 EC: 18
If $X$ and $Y$ are random variables such that $E\left[2X+Y\right]=0$ and $E\left[X+2Y\right]=33$, then $E\left[X\right]+E\left[Y\right]=$___________.
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gate2019ec
numericalanswers
engineeringmathematics
0
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0
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17
GATE2019 EC: 19
The value of the integral $\int_{0}^{\pi} \int_{y}^{\pi}\dfrac{\sin x}{x}dx\: dy ,$ is equal to __________.
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gate2019ec
numericalanswers
integrals
calculus
engineeringmathematics
differentialequation
0
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0
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18
GATE2019 EC: 20
Let $Z$ be an exponential random variable with mean $1$. That is, the cumulative distribution function of $Z$ is given by $F_{Z}(x)= \left\{\begin{matrix} 1e^{x}& \text{if}\: x \geq 0 \\ 0& \text{if}\: x< 0 \end{matrix}\right.$ Then $Pr\left(Z>2 \mid Z>1\right),$ rounded off to two decimal places, is equal to ___________.
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gate2019ec
numericalanswers
engineeringmathematics
matrix
linearalgebra
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0
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19
GATE2019 EC: 26
Consider a differentiable function $f(x)$ on the set of real numbers, such that $f(1)=0$ and $ \mid f’(x) \mid \leq 2.$ Given these conditions, which one of the following inequalities is necessarily true for all $x \in[2,2]?$ $f(x)\leq \frac{1}{2} \mid x+1 \mid$ $f(x)\leq 2 \mid x+1 \mid $ $f(x)\leq \frac{1}{2} \mid x \mid$ $f(x)\leq 2 \mid x \mid$
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gate2019ec
differentiablefunction
engineeringmathematics
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0
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20
GATE2019 EC: 27
Consider the line integral $\int_{c} (xdyydx)$ the integral being taken in a counterclockwise direction over the closed curve $C$ that forms the boundary of the region $R$ shown in the figure below. The region $R$ is the area enclosed by the union of a $2 \times 3$ rectangle and ... circle of radius $1$. The line integral evaluates to $6+ \dfrac{\pi}{2}$ $8+\pi$ $12+\pi$ $16+2\pi$
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gate2019ec
integrals
calculus
engineeringmathematics
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0
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21
GATE2019 EC: 29
It is desired to find a threetap casual filter which gives zero signal as an output to an input of the form $x[n]= c_{1}exp\left(\dfrac{j\pi n}{2}\right)+c_{2}\left(\dfrac{j\pi n}{2}\right),$ where $c_{1}$ and $c_{2}$ ... $n$, when $x[n]$ is as given above ? $a=1,b=1$ $a=0,b=1$ $a=1,b=1$ $a=0,b=1$
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gate2019ec
diffrentialequation
engineeringmathematics
0
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0
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22
GATE2019 EC: 43
Consider the homogenous ordinary differential equation $x^{2}\frac{d^{2}y}{dx^{2}}3x\frac{dy}{dx}+3y=0, \quad x>0$ with $y(x)$ as a general solution. Given that $y(1)=1 \quad \text{and} \quad y(2)=14$ the value of $y(1.5),$ rounded off to two decimal places, is________.
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gate2019ec
numericalanswers
differentialequations
engineeringmathematics
0
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0
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23
GATE2019 EC: 47
A random variable $X$ takes values $1$ and $+1$ with probabilities $0.2$ and $0.8$, respectively. It is transmitted across a channel which adds noise $N,$ so that the random variable at the channel output is $Y=X+N$. The noise $N$ is independent of ... the probability of error $Pr[ \hat{X} \neq X].$ The minimum probability of error, rounded off to $1$ decimal place, is _________.
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gate2019ec
numericalanswers
probability
engineeringmathematics
0
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0
answers
24
GATE2019 EC: 48
A Germanium sample of dimensions $1\: cm \times 1\: cm$ is illuminated with a $20\:mW,$ $600\: nm$ laser light source as shown in the figure. The illuminated sample surface has a $100\: nm$ of lossless Silicon dioxide layer that reflects onefourth of the ... the bandgap is $0.66\: eV,$ the thickness of the Germanium layer, rounded off to $3$ decimal places, is ________ $\mu m.$
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gate2019ec
numericalanswers
engineeringmathematics
0
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0
answers
25
GATE201631
Consider a $2\times2$ sqaure matrix $\textbf{A}= \begin{bmatrix} \sigma &x\\ \omega &\sigma \end{bmatrix},$ where $x$ is unknown. If the eigen values of the matrix $\textbf{A}$ are $(\sigma + j\omega)$ and $(\sigma  j\omega)$, then $x$ is equal to $+j\omega$ $j\omega$ $+\omega$ $\omega$
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Mar 28, 2018
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gate2016ec3
matrixalgebra
linearalgebra
engineeringmathematics
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0
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26
GATE201632
For $f(z)= \large\frac{\sin(z)}{z^2}$, the residue of the pole at $z = 0$ is _______
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Mar 28, 2018
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Milicevic3306
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gate2016ec3
numericalanswers
engineeringmathematics
0
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0
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27
GATE201633
The probability of getting a “head” in a single toss of a biased coin is $0.3$. The coin is tossed repeatedly till a head is obtained. If the tosses are independent, then the probability of getting “head” for the first time in the fifth toss is _______
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gate2016ec3
numericalanswers
probability
engineeringmathematics
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28
GATE201634
The integral $\int\limits_{0}^{1}\large\frac{dx}{\sqrt{(1x)}}$ is equal to _______
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gate2016ec3
numericalanswers
integrals
calculus
engineeringmathematics
0
votes
0
answers
29
GATE201635
Consider the first order initial value problem $y’= y+2xx^2 ,\ y(0)=1,\ (0 \leq x < \infty)$ with exact solution $y(x) = x^2 + e^x$. For $x = 0.1$, the percentage difference between the exact solution and the solution obtained using a single iteration of the secondorder RungaKutta method with stepsize $h=0.1$ is _______
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gate2016ec3
numericalanswers
engineeringmathematics
0
votes
0
answers
30
GATE201637
If the signal $x(t) = \large \frac{\sin(t)}{\pi t}$*$\large \frac{\sin(t)}{\pi t}$ with $*$ denoting the convolution operation, then $x(t)$ is equal to $\large\frac{\sin(t)}{\pi t}$ $\large\frac{\sin(2t)}{2\pi t}$ $\large\frac{2\sin(t)}{\pi t}$ $\bigg(\frac{\sin(t)}{\pi t}\bigg)^2$
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gate2016ec3
engineeringmathematics
0
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0
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31
GATE2016326
The particular solution of the initial value problem given below is $\frac{d^2y}{dx^2}+12\frac{dy}{dx}+36y=0\hspace{0.3cm} \text{ with } \hspace{0.3cm}y(0)=3\hspace{0.3cm} \text{ and }\hspace{0.3cm} \frac{dy}{dx} \bigg _{x=0} =36$ $(318x)e^{6x}$ $(3+25x)e^{6x}$ $(3+20x)e^{6x}$ $(312x)e^{6x}$
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gate2016ec3
engineeringmathematics
0
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0
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32
GATE2016327
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 $\textbf{u}=$\large\frac{2}{5}$e_13e_2$\ ... $e_1+3e_2+$\large\frac{11}{5}$e_3 \\$ $\textbf{u}=$\large\frac{2}{5}$e_1+3e_2$\large\frac{11}{5}$e_3$
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gate2016ec3
vectoranalysis
engineeringmathematics
0
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0
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33
GATE2016328
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 _______
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gate2016ec3
numericalanswers
vectorinplans
engineeringmathematics
0
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0
answers
34
GATE2016329
The values of the integral $\large\frac{1}{2\pi j}\oint_c\frac{e^z}{(z2)} \small dz$ along a closed contour $c$ in anticlockwise direction for the point $z_0=2$ inside the contour $c$, and the point $z_0=2$ outside the contour $c$, respectively,are $(i)2.72, \: (ii) 0$ $(i)7.39, \: (ii) 0$ $(i)0, \: (ii) 2.72$ $(i)0, \: (ii) 7.39$
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gate2016ec3
integrals
calculus
engineeringmathematics
0
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0
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35
GATE2016331
The ROC (region of convergence) of the $z$transform of a discretetime signal is represented by the shaded region in the $z$plane. If the signal $x[n]=(2.0)^{\mid n\mid},\infty<n<+\infty$, then the ROC of its $z$transform is represented by
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gate2016ec3
numericalmethods
engineeringmathematics
convergence
0
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0
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36
GATE2016334
The $z$parameter matrix $\begin{bmatrix} z_{11} &z_{12}\\ z_{21} &z_{22} \end{bmatrix}$ for the twoport network shown is $\begin{bmatrix} 2 &2\\2 &2 \end{bmatrix} \\$ $\begin{bmatrix} 2 &2\\2 &2 \end{bmatrix} \\$ $\begin{bmatrix} 9 &3\\6 &9 \end{bmatrix} \\$ $\begin{bmatrix} 9 &3\\6 &9 \end{bmatrix}$
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gate2016ec3
matrixalgebra
linearalgebra
engineeringmathematics
0
votes
0
answers
37
GATE2016350
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 of $52\:kbps$ is to be transmitted over this channel with arbitrarily small bit error rate, then the minimum bitenergy $E_b$ (in mJ/bit) necessary is _______
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gate2016ec3
numericalanswers
vectoranalysis
gauss'stheorem
engineeringmathematics
0
votes
0
answers
38
GATE2016351
The bit error probability of a memoryless binary symmetric channel is $10^{5}$. If $10^5$ bits are sent over this channel, then the probability that not more than one bit will be in error is _______
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gate2016ec3
numericalanswers
probability
engineeringmathematics
0
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0
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39
GATE201621
The value of $x$ for which the matrix $A= \begin{bmatrix} 3& 2 &4 \\ 9& 7 & 13\\ 6&4 &9+x \end{bmatrix}$ has zero as an eigenvalue is ________
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gate2016ec2
numericalanswers
matrixalgebra
linearalgebra
engineeringmathematics
0
votes
0
answers
40
GATE201622
Consider the complex valued function $f(z)=2z^{3}+b\mid z \mid^{3}$ where $z$ is a complex variable. The value of $b$ for which function $f(z)$ is analytic is _________
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gate2016ec2
numericalanswers
complexanalysis
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