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TIFR ECE 2012 | Question: 1
The minimum value of $f(x)=\ln \left(1+\exp \left(x^{2}-3 x+2\right)\right)$ for $x \geq 0$, where $\ln (\cdot)$ denotes the natural logarithm, is $\ln \left(1+e^{-1 / 4}\right)$ $\ln (5 / 3)$ $0$ $\ln \left(1+e^{2}\right)$ None of the above
The minimum value of $f(x)=\ln \left(1+\exp \left(x^{2}-3 x+2\right)\right)$ for $x \geq 0$, where $\ln (\cdot)$ denotes the natural logarithm, is$\ln \left(1+e^{-1 / 4}\...
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46.4k
points
114
views
admin
asked
Dec 8, 2022
Calculus
tifr2012
calculus
maxima-minima
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1
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0
answers
202
TIFR ECE 2012 | Question: 2
Let $\alpha_{1}, \alpha_{2}, \cdots, \alpha_{k}$ be complex numbers. Then \[ \lim _{n \rightarrow \infty}\left|\sum_{i=1}^{k} \alpha_{i}^{n}\right|^{1 / n} \] is $0$ $\infty$ $\alpha_{k}$ $\alpha_{1}$ $\max _{j}|\alpha_{j}|$
Let $\alpha_{1}, \alpha_{2}, \cdots, \alpha_{k}$ be complex numbers. Then\[\lim _{n \rightarrow \infty}\left|\sum_{i=1}^{k} \alpha_{i}^{n}\right|^{1 / n}\]is$0$$\infty$$\...
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46.4k
points
86
views
admin
asked
Dec 8, 2022
Calculus
tifr2012
calculus
limits
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1
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203
TIFR ECE 2012 | Question: 3
A sequence of numbers $\left(x_{n}: n=1,2,3, \ldots\right)$ is said to have a limit $x$, if given any number $\epsilon>0$, there exists an integer $n_{\epsilon}$ ... $6$ and has a limit that equals $6$ . None of the above statements are true.
A sequence of numbers $\left(x_{n}: n=1,2,3, \ldots\right)$ is said to have a limit $x$, if given any number $\epsilon>0$, there exists an integer $n_{\epsilon}$ such tha...
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46.4k
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Others
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204
TIFR ECE 2012 | Question: 4
The signal $x_{n}=0$ for $n<0$ and $x_{n}=a^{n} / n$ ! for $n \geq 0$. Its $z$-transform $X(z)=\sum_{n=-\infty}^{\infty} x_{n} z^{-n}$ is $1 /\left(z^{-1}-a\right)$, region of convergence $\text{(ROC)}$: $|z| \leq 1 / a$ ... $|z|>a$ Item $(a)$ if $a>1$, Item $(b)$ if $a<1$ $\exp \left(a z^{-1}\right)$, $\text{ROC}$: entire complex plane.
The signal $x_{n}=0$ for $n<0$ and $x_{n}=a^{n} / n$ ! for $n \geq 0$. Its $z$-transform $X(z)=\sum_{n=-\infty}^{\infty} x_{n} z^{-n}$ is$1 /\left(z^{-1}-a\right)$, regio...
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46.4k
points
70
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Dec 8, 2022
Others
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205
TIFR ECE 2012 | Question: 5
Consider a periodic square wave $f(t)$ with a period of $1$ second such that $f(t)=1$ for $t \in[0,1 / 2)$ and $f(t)=-1$ for $t \in[1 / 2,1)$. It is passed through an ideal low-pass filter with cutoff at $2 \mathrm{~Hz}$. Then the output is $\sin (2 \pi t)$ ... $\sin (2 \pi t)-\cos (2 \pi t)$ None of the above
Consider a periodic square wave $f(t)$ with a period of $1$ second such that $f(t)=1$ for $t \in[0,1 / 2)$ and $f(t)=-1$ for $t \in[1 / 2,1)$. It is passed through an ide...
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84
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Dec 8, 2022
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206
TIFR ECE 2012 | Question: 6
Let $u(t)$ be the unit step function that takes value $1$ for $t \geq 0$ and is zero otherwise. Let $f(t)=e^{-t} u(t)$ and $g(t)=u(t) u(1-t)$. Then the convolution of $f(t)$ and $g(t)$ is $(e-1) e^{-t} u(t)$ $1-e^{-t}$ for $0 \leq t \leq 1,(e-1) e^{-t}$ for $t \geq 1$ and zero otherwise $t e^{-t} u(t)$ The convolution integral is not well defined None of the above
Let $u(t)$ be the unit step function that takes value $1$ for $t \geq 0$ and is zero otherwise. Let $f(t)=e^{-t} u(t)$ and $g(t)=u(t) u(1-t)$. Then the convolution of $f(...
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46.4k
points
83
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admin
asked
Dec 8, 2022
Others
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207
TIFR ECE 2012 | Question: 7
A linear time-invariant system has a transfer function $H(s)=1 /(1+s)$. If the input to the system is $\cos (t)$, the output is $\left(e^{j t}+e^{-j t}\right) / 2$ where $j=\sqrt{-1}$ $\cos (t) / 2$ $(\cos (t)+\sin (t)) / 2 \sqrt{ }$ $\sin (t) / 2$. The system is unstable and the output is not well-defined.
A linear time-invariant system has a transfer function $H(s)=1 /(1+s)$. If the input to the system is $\cos (t)$, the output is$\left(e^{j t}+e^{-j t}\right) / 2$ where $...
admin
46.4k
points
76
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admin
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Dec 8, 2022
Others
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208
TIFR ECE 2012 | Question: 8
The input to a series $\text{RLC}$ circuit is a sinusoidal voltage source and the output is the current in the circuit. Which of the following is true about the magnitude frequency response of this system? Dependending on the values of $\text{R, L}$ ... $1 /(2 \pi \sqrt{\text{LC}})$.
The input to a series $\text{RLC}$ circuit is a sinusoidal voltage source and the output is the current in the circuit. Which of the following is true about the magnitude...
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46.4k
points
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Others
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209
TIFR ECE 2012 | Question: 9
$x(t)$ is a signal of bandwidth $4 \mathrm{~kHz}$. It was sampled at a rate of $16 \mathrm{~kHz}$. \[ x_{n}=x(n T), \quad n \text { integer, } \quad T=\frac{1}{16} \mathrm{~ms} . \] Due to a data handling error alternate samples were erased ... $y(t)$ over a low pass filter of bandwidth $4\text{ KHz}$ any of the above none of the above
$x(t)$ is a signal of bandwidth $4 \mathrm{~kHz}$. It was sampled at a rate of $16 \mathrm{~kHz}$.\[x_{n}=x(n T), \quad n \text { integer, } \quad T=\frac{1}{16} \mathrm{...
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46.4k
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Others
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TIFR ECE 2012 | Question: 10
Suppose three dice are rolled independently. Each dice can take values $1$ to $6$ with equal probability. Find the probability that the second highest outcome equals the average of the other two outcomes. Here, the ties may be resolved arbitrarily. $1 / 6$ $1 / 9$ $39 / 216$ $7 / 36$ $43 / 216$
Suppose three dice are rolled independently. Each dice can take values $1$ to $6$ with equal probability. Find the probability that the second highest outcome equals the ...
admin
46.4k
points
80
views
admin
asked
Dec 8, 2022
Probability and Statistics
tifr2012
probability-and-statistics
probability
independent-events
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211
TIFR ECE 2012 | Question: 11
A Poisson random variable $X$ is given by $\operatorname{Pr}\{X=k\}=\mathrm{e}^{-\lambda} \lambda^{k} / k !, k=0,1,2, \ldots$ for $\lambda>0$. The variance of $X$ scales as $\lambda$ $\lambda^{2}$ $\lambda^{3}$ $\sqrt{\lambda}$ None of the above
A Poisson random variable $X$ is given by $\operatorname{Pr}\{X=k\}=\mathrm{e}^{-\lambda} \lambda^{k} / k !, k=0,1,2, \ldots$ for $\lambda>0$. The variance of $X$ scales ...
admin
46.4k
points
70
views
admin
asked
Dec 8, 2022
Probability and Statistics
tifr2012
probability-and-statistics
probability
poisson-distribution
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0
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212
TIFR ECE 2012 | Question: 12
In modeling the number of health insurance claims filed by an individual during a three year period, an analyst makes a simplifying assumption that for all non-negative integer up to $5$. \[ p_{n+1}=\frac{1}{2} p_{n} \] where $p_{n}$ denotes the probability that a ... files more than two claims in this period? $7 / 31$ $29 / 125$ $1 / 3$ $13 / 125$ None of the above
In modeling the number of health insurance claims filed by an individual during a three year period, an analyst makes a simplifying assumption that for all non-negative i...
admin
46.4k
points
89
views
admin
asked
Dec 8, 2022
Probability and Statistics
tifr2012
probability-and-statistics
probability
conditional-probability
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213
TIFR ECE 2012 | Question: 13
Consider a single amoeba that at each time slot splits into two with probability $p$ or dies otherwise with probability $1-p$. This process is repeated independently infinitely at each time slot, i.e. if there are any amoebas left at time slot $t$, then they all split independently into ... $\min \left\{\frac{1 \pm \sqrt{1-4 p(1-p)}}{2(1-p)}\right\}$ None of the above
Consider a single amoeba that at each time slot splits into two with probability $p$ or dies otherwise with probability $1-p$. This process is repeated independently infi...
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46.4k
points
87
views
admin
asked
Dec 8, 2022
Probability and Statistics
tifr2012
probability-and-statistics
probability
independent-events
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214
TIFR ECE 2012 | Question: 14
Let $X$ and $Y$ be indepedent, identically distributed standard normal random variables, i.e., the probability density function of $X$ is \[f_{X}(x)=\frac{1}{\sqrt{2 \pi}} \exp \left(-\frac{x^{2}}{2}\right),-\infty<x<\infty. \] The random variable $Z$ is defined ... none of the above
Let $X$ and $Y$ be indepedent, identically distributed standard normal random variables, i.e., the probability density function of $X$ is\[f_{X}(x)=\frac{1}{\sqrt{2 \pi}}...
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46.4k
points
80
views
admin
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Dec 8, 2022
Probability and Statistics
tifr2012
probability-and-statistics
probability
normal-distribution
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215
TIFR ECE 2012 | Question: 15
Consider a string of length $1 \mathrm{~m}$. Two points are chosen independently and uniformly random on it thereby dividing the string into three parts. What is the probability that the three parts can form the sides of a triangle? $1 / 4$ $1 / 3$ $1 / 2$ $2 / 3$ $3 / 4$
Consider a string of length $1 \mathrm{~m}$. Two points are chosen independently and uniformly random on it thereby dividing the string into three parts. What is the prob...
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46.4k
points
156
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admin
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Dec 8, 2022
Probability and Statistics
tifr2012
probability-and-statistics
probability
uniform-distribution
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216
TIFR ECE 2012 | Question: 16
Let $P$ be a $n \times n$ matrix such that $P^{k}=\mathbf{0}$, for some $k \in \mathbb{N}$ and where $\mathbf{0}$ is an all zeros matrix. Then at least how many eigenvalues of $P$ are zero $1$ $n-1$ $n$ $0$ None of the above
Let $P$ be a $n \times n$ matrix such that $P^{k}=\mathbf{0}$, for some $k \in \mathbb{N}$ and where $\mathbf{0}$ is an all zeros matrix. Then at least how many eigenvalu...
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46.4k
points
77
views
admin
asked
Dec 8, 2022
Linear Algebra
tifr2012
linear-algebra
eigen-values
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217
TIFR ECE 2012 | Question: 17
Let $A=U \Lambda U^{\dagger}$ be a $n \times n$ matrix, where $U U^{\dagger}=I$. Which of the following statements is TRUE. The matrix $I+A$ has non-negative eigen values The matrix $I+A$ is symmetic $\operatorname{det}(I+A)=\operatorname{det}(I+\Lambda)$ $(a)$ and $(c)$ $(b)$ and $(c)$ $(a), (b)$ and $(c)$
Let $A=U \Lambda U^{\dagger}$ be a $n \times n$ matrix, where $U U^{\dagger}=I$. Which of the following statements is TRUE.The matrix $I+A$ has non-negative eigen valuesT...
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46.4k
points
88
views
admin
asked
Dec 8, 2022
Linear Algebra
tifr2012
linear-algebra
eigen-values
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218
TIFR ECE 2012 | Question: 18
Under a certain coordinate transformation from $(x, y)$ to $(u, v)$ the circle $x^{2}+y^{2}=1$ shown below on the left side was transformed into the ellipse shown on the right side. If the transformation is of the form \[ \left[\begin{array}{l} u \\ v \end{array}\right]=\mathbf{A}\ ... \] $A_{1}$ only $A_{2}$ only $A_{1}$ or $A_{2}$ $A_{1}$ or $A_{3}$ $A_{2}$ or $A_{3}$
Under a certain coordinate transformation from $(x, y)$ to $(u, v)$ the circle $x^{2}+y^{2}=1$ shown below on the left side was transformed into the ellipse shown on the ...
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46.4k
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76
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admin
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Dec 8, 2022
Linear Algebra
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linear-algebra
matrices
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219
TIFR ECE 2012 | Question: 19
$X$ and $Y$ are two $3$ by $3$ matrices. If \[ X Y=\left(\begin{array}{rrr} 1 & 3 & -2 \\ -4 & 2 & 5 \\ 2 & -8 & -1 \end{array}\right) \] then $X$ has rank $2$ at least one of $X, Y$ is not invertible $X$ can't be an invertible matrix $X$ and $Y$ could both be invertible. None of the above
$X$ and $Y$ are two $3$ by $3$ matrices. If\[X Y=\left(\begin{array}{rrr}1 & 3 & -2 \\-4 & 2 & 5 \\2 & -8 & -1\end{array}\right)\]then$X$ has rank $2$at least one of $X, ...
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46.4k
points
107
views
admin
asked
Dec 8, 2022
Linear Algebra
tifr2012
linear-algebra
determinant
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0
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220
TIFR ECE 2012 | Question: 20
Let $A$ be a $2 \times 2$ matrix with all entries equal to $1.$ Define $B=\sum_{n=0}^{\infty} A^{n} / n !$. Then $B=e^{2} A / 2$ $B=\left(\begin{array}{cc}1+e & e \\e & 1+e\end{array}\right)$ ... $B=\left(\begin{array}{cc}1+e^{2} & e^{2} \\e^{2} & 1+e^{2}\end{array}\right)$ None of the above
Let $A$ be a $2 \times 2$ matrix with all entries equal to $1.$ Define $B=\sum_{n=0}^{\infty} A^{n} / n !$. Then$B=e^{2} A / 2$$B=\left(\begin{array}{cc}1+e & e \\e & 1+e...
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46.4k
points
81
views
admin
asked
Dec 8, 2022
Linear Algebra
tifr2012
linear-algebra
matrices
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1
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0
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221
TIFR ECE 2011 | Question: 1
Output of a linear system with input $x(t)$ is given by \[y(t)=\int_{-\infty}^{\infty} h(t, \tau) x(\tau)+1.\] The system is linear if $h(t, \tau)=h(t-\tau)$ $h(t, \tau)=h(\tau)$ $h(t, \tau)=h(t)$ $h(t, \tau)=$ constant None of the above.
Output of a linear system with input $x(t)$ is given by\[y(t)=\int_{-\infty}^{\infty} h(t, \tau) x(\tau)+1.\]The system is linear if$h(t, \tau)=h(t-\tau)$$h(t, \tau)=h(\t...
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46.4k
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Others
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222
TIFR ECE 2011 | Question: 2
The minimum number of unit delay elements required for realizing an infinite impulse response $\text{(IIR)}$ filter is/are $0$ $1$ $\infty$. $>1$. None of the above.
The minimum number of unit delay elements required for realizing an infinite impulse response $\text{(IIR)}$ filter is/are$0$$1$$\infty$.$>1$.None of the above.
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46.4k
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Others
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TIFR ECE 2011 | Question: 3
The Fourier transform of \[x(t)=\frac{t^{n-1}}{(n-1) !} \mathrm{e}^{-a t} u(t), \quad a>0\] $(\jmath=\sqrt{-1}, u(t)=1$ for $t \geq 0, u(t)=0, t<0)$ is $(a+\jmath \omega)^{n}$ $\sum_{k=1}^{n} \frac{(a+\jmath \omega)^{k}}{k !}$ $na\jmath \omega$ $\frac{1}{(a+\jmath \omega)^{n}}$ None of the above.
The Fourier transform of\[x(t)=\frac{t^{n-1}}{(n-1) !} \mathrm{e}^{-a t} u(t), \quad a>0\]$(\jmath=\sqrt{-1}, u(t)=1$ for $t \geq 0, u(t)=0, t<0)$ is$(a+\jmath \omega)^{n...
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46.4k
points
95
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Dec 5, 2022
Others
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TIFR ECE 2011 | Question: 4
Let $\lim _{n \rightarrow \infty} x_{n}=x$. Then which of the following is $\text{TRUE.}$ There exists an $n_{0}$, such that for all $n>n_{0},\left|x_{n}-x\right|=0$. There exists an $n_{0}$ ... $n>n_{0},\left|\frac{x_{n}}{x}\right| \leq \epsilon$ for any $\epsilon>0$. None of the above.
Let $\lim _{n \rightarrow \infty} x_{n}=x$. Then which of the following is $\text{TRUE.}$There exists an $n_{0}$, such that for all $n>n_{0},\left|x_{n}-x\right|=0$.There...
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46.4k
points
70
views
admin
asked
Dec 5, 2022
Calculus
tifr2011
calculus
limits
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225
TIFR ECE 2011 | Question: 5
Consider a system with input $x(t)$ and the output $y(t)$ is given by \[y(t)=x(t)-0.5 x(t-1)-0.5 x(t-2)+1 .\] The system is Linear Non-causal Time varying All of the above None of the above
Consider a system with input $x(t)$ and the output $y(t)$ is given by\[y(t)=x(t)-0.5 x(t-1)-0.5 x(t-2)+1 .\]The system isLinearNon-causalTime varyingAll of the aboveNone ...
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46.4k
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83
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Others
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226
TIFR ECE 2011 | Question: 6
Let $\mathrm{H}(\mathrm{z})$ be the $z$-transform of the transfer function corresponding to an input output relation $y(n)-\frac{1}{2} y(n-1)=x(n)+\frac{1}{3} x(n-1)$. Then which of the following is TRUE The $\operatorname{ROC}$ ... $|z|<\frac{1}{2}$. $\operatorname{Both}$ (a) and (b). System is necessarily causal. None of the above.
Let $\mathrm{H}(\mathrm{z})$ be the $z$-transform of the transfer function corresponding to an input output relation $y(n)-\frac{1}{2} y(n-1)=x(n)+\frac{1}{3} x(n-1)$. Th...
admin
46.4k
points
70
views
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asked
Dec 5, 2022
Others
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227
TIFR ECE 2011 | Question: 7
Assume you are using a binary code error correcting code $C$. If the minimum Hamming distance between any two codewords of $C$ is $3$. Then We can correct and detect $2$ bit errors. We can correct $1$ bit errors and detect $2$ bit errors. We can correct $2$ bit errors and detect $1$ bit errors. We can correct $1$ bit errors and detect $1$ bit errors. None of the above.
Assume you are using a binary code error correcting code $C$. If the minimum Hamming distance between any two codewords of $C$ is $3$. ThenWe can correct and detect $2$ b...
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46.4k
points
75
views
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asked
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Others
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228
TIFR ECE 2011 | Question: 8
Let $f(x, y)$ be a function in two variables $x, y$. Then which of the following is true $\max _{x} \min _{y} f(x, y) \leq \min _{y} \max _{x} f(x, y)$. $\max _{x} \min _{y} f(x, y) \geq \min _{y} \max _{x} f(x, y)$ ... $\max _{x} \min _{y} f(x, y)=\min _{y} \max _{x} f(x, y)+\min _{y} \min _{x} f(x, y)$. None of the above.
Let $f(x, y)$ be a function in two variables $x, y$. Then which of the following is true$\max _{x} \min _{y} f(x, y) \leq \min _{y} \max _{x} f(x, y)$.$\max _{x} \min _{y...
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46.4k
points
88
views
admin
asked
Dec 5, 2022
Calculus
tifr2011
calculus
maxima-minima
+
–
1
votes
0
answers
229
TIFR ECE 2011 | Question: 9
Consider two independent random variables $X$ and $Y$ having probability density functions uniform in the interval $[-1,1]$. The probability that $X^{2}+Y^{2}>1$ is $\pi / 4$ $1-\pi / 4$ $\pi / 2-1$ Probability that $X^{2}+Y^{2}<0.5$ None of the above
Consider two independent random variables $X$ and $Y$ having probability density functions uniform in the interval $[-1,1]$. The probability that $X^{2}+Y^{2}>1$ is$\pi /...
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46.4k
points
90
views
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asked
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Probability and Statistics
tifr2011
probability-and-statistics
probability
probability-density-function
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0
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230
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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46.4k
points
98
views
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asked
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Calculus
tifr2011
calculus
continuity-and-differentiability
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1
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0
answers
231
TIFR ECE 2011 | Question: 11
What is the value of $\lambda$ such that $\operatorname{Prob}\{X>\operatorname{mean}\{X\}\}=1 / e$, where $\text{PDF}$ of $X$ is $p_{X}(x)=\lambda e^{-\lambda x}, x \geq 0, \lambda>0?$ $1$ $1 / e$ $1 / \sqrt{e}$ $1 / e^{2}$ All of the above
What is the value of $\lambda$ such that $\operatorname{Prob}\{X>\operatorname{mean}\{X\}\}=1 / e$, where $\text{PDF}$ of $X$ is $p_{X}(x)=\lambda e^{-\lambda x}, x \geq ...
admin
46.4k
points
82
views
admin
asked
Dec 5, 2022
Probability and Statistics
tifr2011
probability-and-statistics
probability
poisson-distribution
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0
answers
232
TIFR ECE 2011 | Question: 12
Consider two communication systems $C_{1}$ and $C_{2}$ that use pulse amplitude modulation $\text{(PAM)}$, $P A M_{1}$ and $P A M_{2}$. Let the distance between any two points of $P A M_{1}$ be $d$, and $P A M_{2}$ be $2 d$, respectively. Assume that $C_{1}$ ... $P_{1}=P_{2}$. $P_{1} < P_{2}$ $P_{1}>P_{2}$. $P_{1}=P_{2}+\frac{1}{2}$ None of the above.
Consider two communication systems $C_{1}$ and $C_{2}$ that use pulse amplitude modulation $\text{(PAM)}$, $P A M_{1}$ and $P A M_{2}$. Let the distance between any two p...
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46.4k
points
82
views
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Others
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233
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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46.4k
points
105
views
admin
asked
Dec 5, 2022
Calculus
tifr2011
calculus
limits
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234
TIFR ECE 2011 | Question: 14
In household electrical wiring which configuration is used to connect different electrical equipments. Series. Parallel Combination of series and parallel. Any of the above. None of the above.
In household electrical wiring which configuration is used to connect different electrical equipments.Series.ParallelCombination of series and parallel.Any of the above.N...
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TIFR ECE 2011 | Question: 15
Consider a channel where $x_{n} \in\{0,1\}$ is the input and $y_{n}=x_{n} * z_{n}$ is the output, where $*$ is $\text{EX-OR}$ operation, and $P\left(z_{n}=x_{n-1}\right)=P\left(z_{n}=y_{n-1}\right)=\frac{1}{2}$ ... $\frac{1}{2}.$ $1$. $<1.$ $\geq 0.$ Both $(c)$ and $(d)$.
Consider a channel where $x_{n} \in\{0,1\}$ is the input and $y_{n}=x_{n} * z_{n}$ is the output, where $*$ is $\text{EX-OR}$ operation, and $P\left(z_{n}=x_{n-1}\right)=...
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TIFR ECE 2011 | Question: 16
Consider a triangular shaped pulse $x$ of base $2 T$ and unit height centered at $0$ , i.e. $x(t)=0$ for $|t|>T, x(t)=1-|t|$ for $t \in[-T, T]$. Then if $x$ is convolved with itself, the output is Square shape. Triangular shape. Bell shape. Inverted $\text{U}$ shape. None of the above.
Consider a triangular shaped pulse $x$ of base $2 T$ and unit height centered at $0$ , i.e. $x(t)=0$ for $|t|>T, x(t)=1-|t|$ for $t \in[-T, T]$. Then if $x$ is convolved ...
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TIFR ECE 2011 | Question: 17
Let $x[n]$ and $y[n]$ be the input and output of a linear time invariant $\text{(LTI)}$ system. Then which of following system is $\text{LTI}$. $z[n]=y[n]+c$ for a constant $c$. $z[n]=x[n] y[n]$. $z[n]=y[n]+x[n]+c$ for a constant $c$. $z[n]=y[n]+x[n]$. None of the above.
Let $x[n]$ and $y[n]$ be the input and output of a linear time invariant $\text{(LTI)}$ system. Then which of following system is $\text{LTI}$.$z[n]=y[n]+c$ for a constan...
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TIFR ECE 2011 | Question: 18
Which of the following statements is TRUE. The cascade of a non-causal linear time invariant $\text{(LTI)}$ system with a causal $\text{LTI}$ system can be causal. If $h[n] \leq 2$ for all $n$, then the $\text{LTI}$ system with $h[n]$ as its impulse response is stable and ... $u[n]=1, n \geq 0, u[n]=0, n<0$, the $\text{LTI}$ system is stable. Both $(c)$ and $(d)$.
Which of the following statements is TRUE.The cascade of a non-causal linear time invariant $\text{(LTI)}$ system with a causal $\text{LTI}$ system can be causal.If $h[n]...
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TIFR ECE 2011 | Question: 19
Let $R_{X}(\tau)$ be the autocorrelation function of a zero mean stationary random process $X(t)$. Which of following statements is FALSE. If $R_{X}(\tau)=0, \forall \tau, X(n)$ and $X(m), n \neq m$ are independent. $R_{X}(\tau)=R_{X}(-\tau)$. $R_{X}(0)=E\left[X^{2}\right]$, where $E$ denotes the expectation. $R_{X}(0) \geq R_{X}(\tau), \forall \tau.$ None of the above.
Let $R_{X}(\tau)$ be the autocorrelation function of a zero mean stationary random process $X(t)$. Which of following statements is FALSE.If $R_{X}(\tau)=0, \forall \tau,...
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TIFR ECE 2011 | Question: 20
Let $x(t)$ be a signal whose Fourier transform $X(f)$ is zero for $|f|>W$. Using a sampler with sampling frequency $4 W$, which of the following filters can be used to exactly reconstruct $x(t)$ ... $\text{[-3W} \;5 \mathrm{W}]$. All the above. None of the above.
Let $x(t)$ be a signal whose Fourier transform $X(f)$ is zero for $|f|>W$. Using a sampler with sampling frequency $4 W$, which of the following filters can be used to ex...
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