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201
TIFR ECE 2011 | Question: 13
If $a_k$ is an increasing function of $k$, i.e. $a_1<a_2<\ldots<a_k \ldots$. Then which of the following is $\text{TRUE.}$ $\lim _{n \rightarrow \infty} \sum_{k=1}^{n} \frac{1}{a_{k}}=\infty$ ... . Either $(a)$ or $(b)$. $\lim _{n \rightarrow \infty} \sum_{k=1}^{n} \frac{1}{a_{k}}=0$. None of the above.
If $a_k$ is an increasing function of $k$, i.e. $a_1<a_2<\ldots<a_k \ldots$. Then which of the following is $\text{TRUE.}$$\lim _{n \rightarrow \infty} \sum_{k=1}^{n} \fr...
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205
GATE ECE 2014 Set 2 | Question: 5
If the characteristic equation of the differential equation $\frac{\mathrm{d}^2 y}{\mathrm{dx}^2}+2\alpha \frac{\mathrm{d}y}{\mathrm{d} x}+y= 0$ has two equal roots, then the value of $\alpha$ are $\pm 1$ $0,0$ $\pm j$ $\pm 1/2$
If the characteristic equation of the differential equation $$\frac{\mathrm{d}^2 y}{\mathrm{dx}^2}+2\alpha \frac{\mathrm{d}y}{\mathrm{d} x}+y= 0$$ has two equal roots, th...
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206
GATE ECE 2012 | Question: 23
Given $f(z)=\frac{1}{z+1}-\frac{2}{z+3}$. If $C$ is a counterclockwise path in the z-plane such that $\mid z+1 \mid=1$, the value of $\frac{1}{2\pi j}\oint_cf(z)dz$ is $-2$ $-1$ $1$ $2$
Given $f(z)=\frac{1}{z+1}-\frac{2}{z+3}$. If $C$ is a counterclockwise path in the z-plane such that $\mid z+1 \mid=1$, the value of $\frac{1}{2\pi j}\oint_cf(z)dz$ is$-2...
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207
GATE ECE 2012 | Question: 18
If $x[n]=(\frac{1}{3})^{|n|}-(\frac{1}{2})^{|n|}u[n]$, then the region of convergence (ROC) of its Z-transform in the Z-plane will be $\frac{1}{3}\lt |z|\lt 3$ $\frac{1}{3}\lt |z|\lt \frac{1}{2}$ $\frac{1}{2}\lt |z|\lt 3$ $\frac{1}{3}\lt |z|$
If $x[n]=(\frac{1}{3})^{|n|}-(\frac{1}{2})^{|n|}u[n]$, then the region of convergence (ROC) of its Z-transform in the Z-plane will be$\frac{1}{3}\lt |z|\lt 3$$\frac{1}{3}...
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208
TIFR ECE 2014 | Question: 6
Let $g:[0, \pi] \rightarrow \mathbb{R}$ be continuous and satisfy \[ \int_{0}^{\pi} g(x) \sin (n x) d x=0 \] for all integers $n \geq 2$. Then which of the following can you say about $g?$ $g$ must be identically zero. $g(\pi / 2)=1$. $g$ need not be identically zero. $g(\pi)=0$. None of the above.
Let $g:[0, \pi] \rightarrow \mathbb{R}$ be continuous and satisfy\[\int_{0}^{\pi} g(x) \sin (n x) d x=0\]for all integers $n \geq 2$. Then which of the following can you ...
1 votes
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210
TIFR ECE 2022 | Question: 13
Calculate the minimum value attained by the function \[\sin (\pi x)-\sqrt{2} \pi x^{2}\] for values of $x$ which lie in the interval $[0,1]$. $\frac{1}{\sqrt{2}}\left(1-\frac{\pi}{8}\right)$ $0$ $1-\frac{\pi}{2 \sqrt{2}}$ $-\frac{1}{\sqrt{2}}\left(1+\frac{9 \pi}{2}\right)$ $-\sqrt{2} \pi$
Calculate the minimum value attained by the function\[\sin (\pi x)-\sqrt{2} \pi x^{2}\]for values of $x$ which lie in the interval $[0,1]$.$\frac{1}{\sqrt{2}}\left(1-\fra...
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211
GATE ECE 2015 Set 3 | Question: 1
For $A = \begin{bmatrix} 1 &\tan x \\ -\tan x &1 \end{bmatrix},$ the determinant of $A^{T}A^{-1}$ is $\sec^{2}x$ $\cos 4x$ $1$ $0$
For $A = \begin{bmatrix} 1 &\tan x \\ -\tan x &1 \end{bmatrix},$ the determinant of $A^{T}A^{-1}$ is$\sec^{2}x$$\cos 4x$$1$$0$
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212
GATE ECE 2015 Set 3 | Question: 5
The value of $\displaystyle{}\sum_{n=0}^{\infty} n \left(\dfrac{1}{2}\right)^{n}$ is ________.
The value of $\displaystyle{}\sum_{n=0}^{\infty} n \left(\dfrac{1}{2}\right)^{n}$ is ________.
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213
GATE ECE 2015 Set 2 | Question: 46
The state variable representation of a system is given as $\dot{x} = \begin{bmatrix} 0 &1 \\ 0 &-1 \end{bmatrix}\: ; x(0)=\begin{bmatrix} 1\\0 \end{bmatrix}$ $y=\begin{bmatrix} 0 &1 \end{bmatrix} x$ The response $y(t)$ is $\sin(t)$ $1-e^{t}$ $1-\cos(t)$ $0$
The state variable representation of a system is given as$\dot{x} = \begin{bmatrix} 0 &1 \\ 0 &-1 \end{bmatrix}\: ; x(0)=\begin{bmatrix} 1\\0 \end{bmatrix}$$y=\begin{bm...
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214
GATE ECE 2014 Set 2 | Question: 2
Let $X$ be a random variable which is uniformly chosen from the set of positive odd numbers less than $100$. The expectation, $E[X]$, is ________.
Let $X$ be a random variable which is uniformly chosen from the set of positive odd numbers less than $100$. The expectation, $E[X]$, is ________.
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215
GATE ECE 2014 Set 1 | Question: 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 $-4\pi(1+j2)$ $4\pi(3-j2)$ $-4\pi(3+j2)$ $4\pi(1-j2)$
$C$ 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...
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218
GATE ECE 2015 Set 3 | Question: 26
The Newton-Raphson method is used to solve the equation $f(x) = x^{3} – 5x^{2} + 6x – 8 = 0.$ Taking the initial guess as $x = 5,$ the solution obtained at the end of the first iteration is __________.
The Newton-Raphson method is used to solve the equation $f(x) = x^{3} – 5x^{2} + 6x – 8 = 0.$ Taking the initial guess as $x = 5,$ the solution obtained at the end of...
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219
GATE ECE 2015 Set 2 | Question: 29
Let the random variable $X$ represent the number of times a fair coin needs to be tossed till two consecutive heads appear for the first time. The expectation of $X$ is _______.
Let the random variable $X$ represent the number of times a fair coin needs to be tossed till two consecutive heads appear for the first time. The expectation of $X$ is _...
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220
GATE ECE 2014 Set 3 | Question: 26
The maximum value of $f(x)$= $2x^{3}$ – $9x^{2}$ + $12x – 3$ in the interval $0\leq x\leq 3$ is _______.
The maximum value of $f(x)$= $2x^{3}$ – $9x^{2}$ + $12x – 3$ in the interval $0\leq x\leq 3$ is _______.
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221
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 $\text{div} ( r^{2} \nabla ( \text{ln }r ) )$ = _______ .
If $\overrightarrow {r}= x\hat{a_{x}}+y\hat{a_{y}}+z\hat{a_{z}}$ and $\mid \overrightarrow{r} \mid= r$ , then $\text{div} ( r^{2} \nabla ( \text{ln }r ) )$ = _______ .
1 votes
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222
TIFR ECE 2016 | Question: 7
Suppose $X$ and $Y$ are independent Gaussian random variables, whose pdfs are represented below. Which of the following describes the pdf of the $X+Y?$
Suppose $X$ and $Y$ are independent Gaussian random variables, whose pdfs are represented below. Which of the following describes the pdf of the $X+Y?$
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223
GATE ECE 2015 Set 1 | Question: 29
The maximum area (in square units) of a rectangle whose vertices lie on the eclipse $x^2+4y^2=1$ is __________.
The maximum area (in square units) of a rectangle whose vertices lie on the eclipse $x^2+4y^2=1$ is __________.
1 votes
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225
TIFR ECE 2015 | Question: 9
Consider a random variable $X$ that takes integer values $1$ through $10$ each with equal probability. Now consider random variable \[ Y=\min (7, \max (X, 4)). \] What is the variance of $Y?$ $121 / 4$ $37 / 20 $ $9 / 5$ $99 / 12$ None of the above
Consider a random variable $X$ that takes integer values $1$ through $10$ each with equal probability. Now consider random variable\[Y=\min (7, \max (X, 4)).\]What is the...
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226
GATE ECE 2016 Set 2 | Question: 19
The response of the system $G(s)=\frac{s-2}{(s+1)(s+3)}$ to the unit step input $u(t)$ is $y(t)$. The value of $\frac{dy}{dt}$ at $t=0^{+}$ is _________
The response of the system $G(s)=\frac{s-2}{(s+1)(s+3)}$ to the unit step input $u(t)$ is $y(t)$. The value of $\frac{dy}{dt}$ at $t=0^{+}$ is _________
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227
GATE ECE 2012 | Question: 24
Two independent random variables $X$ and $Y$ are uniformly distributed in the interval $[-1,1]$. The probability that max$[X,Y]$ is less than $\frac{1}{2}$ is $\frac{3}{4}$ $\frac{9}{16}$ $\frac{1}{4}$ $\frac{2}{3}$
Two independent random variables $X$ and $Y$ are uniformly distributed in the interval $[-1,1]$. The probability that max$[X,Y]$ is less than $\frac{1}{2}$ is$\frac{3}{4}...
1 votes
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228
TIFR ECE 2015 | Question: 8
Let $X$ and $Y$ be two independent and identically distributed random variables. Let $Z=\max (X, Y)$ and $W=\min (X, Y)$. Which of the following is true? $Z$ and $W$ are independent $E(X Z)=E(Y W)$ $E(X Y)=E(Z W)$ $(a), (b)$, and $(c)$ $(a)$ and $(b)$ only
Let $X$ and $Y$ be two independent and identically distributed random variables. Let $Z=\max (X, Y)$ and $W=\min (X, Y)$. Which of the following is true?$Z$ and $W$ are i...
1 votes
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229
TIFR ECE 2011 | Question: 10
Let $f(x)=|x|$, for $x \in(-\infty, \infty)$. Then $f(x)$ is not continuous but differentiable. $f(x)$ is continuous and differentiable. $f(x)$ is continuous but not differentiable. $f(x)$ is neither continuous nor differentiable. None of the above.
Let $f(x)=|x|$, for $x \in(-\infty, \infty)$. Then$f(x)$ is not continuous but differentiable.$f(x)$ is continuous and differentiable.$f(x)$ is continuous but not differe...
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232
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 ___________.
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 ___________.
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233
GATE ECE 2014 Set 2 | Question: 4
The value of $\lim_{x\rightarrow \infty }(1 +\tfrac{1}{x})^{x}$ is $\text{ln }2$ $1.0$ $e$ $\infty$
The value of $$\lim_{x\rightarrow \infty }(1 +\tfrac{1}{x})^{x}$$ is$\text{ln }2$$1.0$$e$$\infty$
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238
GATE ECE 2015 Set 1 | Question: 5
The value of $p$ such that the vector $\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$ is an eigenvector of the matrix $\begin{bmatrix} 4 & 1 & 2 \\ p & 2 & 1 \\ 14 & -4 & 10 \end{bmatrix}$ is _________.
The value of $p$ such that the vector $\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$ is an eigenvector of the matrix $\begin{bmatrix} 4 & 1 & 2 \\ p & 2 & 1 \\ 14 & -4 & 10 ...
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