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81
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.
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82
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.
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83
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.
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84
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.
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85
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
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86
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.
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87
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.
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88
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.
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89
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.
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90
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.
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91
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
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92
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 $(d)$.
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93
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.
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94
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.
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95
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
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96
TIFR ECE 2010 | Question: 8
Consider a discrete time channel with binary inputs and binary outputs. Let $x_{n}$ denote the input bit at time $n$ and $y_{k}$ denote the output bit at time $\text{k}$. The channel operation is such that to produce the output $y_{n}$ it drops one ... we do not make any error If $R<1 / 2$, then there exists a scheme with zero error All of the above None of the above
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97
TIFR ECE 2010 | Question: 1
A linear system could be a composition of Two non-linear systems a non-causal non-linear system and a linear system a time varying non-linear system and a time varying linear system All of the above None of the above
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98
TIFR ECE 2010 | Question: 17
Let $a_{1} \geq a_{2} \geq \cdots \geq a_{k} \geq 0$. Then the limit \[ \lim _{n \rightarrow \infty}\left(\sum_{i=1}^{k} a_{i}^{n}\right)^{1 / n} \] is $0$ $\infty$ $a_{k}$ $a_{1}$ $\left(\sum_{i=1}^{k} a_{k}\right) / k$
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99
TIFR ECE 2010 | Question: 20
The function $f(t)$ is a convolution of $t^{2}$ with $\exp \left(-t^{2} / 2\right) / \sqrt{2 \pi}$. Its derivative is $2 t$ $t^{2}$ $2 t+t e^{-t^{2} / 2}$ Does not have a simple closed form expression None of the above
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100
TIFR ECE 2010 | Question: 7
A voltage source with internal resistance $\text{R}$ is connected to an inductor $\text{L}$ and a capacitor $\text{C}$ connected in parallel. The output is the common voltage across the inductor and the capacitor. What is the nature of the transfer ... depending upon the values of $\text{L}$ and $\text{C}$. The circuit is not stable and no transfer function exists.
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101
TIFR ECE 2010 | Question: 13
Output of a linear system with input $x(t)$ is given by \[ y(t)=\int_{-\infty}^{\infty} h(t, \tau) x(\tau) . \] The system is time invariant if $h(t, \tau)=h(t-\tau)$ $h(t, \tau)=h(\tau)$ $h(t, \tau)=h(t)$ $h(t, \tau)=$ constant $h(t, \tau)$ is a continuous function of $t$
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102
TIFR ECE 2010 | Question: 18
Under what conditions is the following inequality true for $a, b>0$ $ \log _e(a+b) \geq \lambda \log _e(a / \lambda)+(1-\lambda) \log _e(b /(1-\lambda)) $ $\lambda=0.5$ $0<a / \lambda \leq 1, b /(1-\lambda)>0$ $a / \lambda>0,0<b /(1-\lambda) \leq 1$ All of the above None of the above
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quantitative-aptitude
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103
TIFR ECE 2010 | Question: 5
Let $Y(t)=\sum_{n=-\infty}^{\infty} x_{n} h(t-n T)$. We sample $Y(t)$ at time instants $n T / 2$ and let $Y_{n}=Y(n T / 2)$. Which of the following is true? $\left\{Y_{n}\right\}$ can be interpreted as the output of a discrete time, ... of a discrete time, linear, time-invariant system with input $\left\{X_{n}\right\}$. Both $a)$ and $b)$ above Both $b)$ and $c)$ above
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104
TIFR ECE 2010 | Question: 16
Consider two independent random variables $\text{X}$ and $\text{Y}$ having probability density functions uniform in the interval $[0,1]$. The probability that $\text{X + Y}>1.5$ is $1 / 4$ $1 / 8$ $1 / 3$ $\operatorname{Pr}\{\text{X + Y} <0.25\}$ None of the above
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105
TIFR ECE 2010 | Question: 19
Let us define an interval $A(n)$ as a function of $n$ as $A(n)=(-1 / n, 1 / n)$. Then the set of points that lie in the intersection of $A_{n}{ }^{\prime} s, n=1, \ldots, \infty$ is an interval is a single point is an empty set cannot be determined has two disjoint intervals
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106
TIFR ECE 2010 | Question: 2
For $x \in[0, \pi / 2], \alpha$ for which $\sin (x) \geq x-\alpha x^{3}$ is $\alpha>1 /(2 \pi)$ $\alpha \geq 1 / 6$ $\alpha \leq 1 /(2 \pi)$ $\alpha=1 / 4$ None of the above
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107
TIFR ECE 2010 | Question: 12
Consider a system with input $x(t)$ and the output $y(t)$ is given by \[ y(t)=x(t)-\sin (t) x(t-1)-0.5 x(t+2)+1 . \] The system is Non-linear Non-causal Time varying All of the above None of the above
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108
TIFR ECE 2010 | Question: 3
Consider two independent random variables $\text{X}$ and $\text{Y}$ having probability density functions uniform in the interval $[0,1]$. When $\alpha \geq 1$, the probability that $\max (\text{X, Y})>\alpha \min (\text{X, Y})$ is $1 /(2 \alpha)$ $\exp (1-\alpha)$ $1 / \alpha$ $1 / \alpha^{2}$ $1 / \alpha^{3}$
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109
TIFR ECE 2010 | Question: 4
Let $Y_{n}=s_{n}+W_{n}$ where $\left\{s_{n}\right\}$ is the desired signal bandlimited to $[-W, W]$ and $\left\{W_{n}\right\}$ is a noise component, which is sparse (that is, only few samples are non-zero), bursty (that is, runs of non-zero samples are ... of $\left\{Y_{n+k}\right\}_{k=-K}^{K}$ for suitably chosen $K$ Both $a)$ and $b)$ are better than the other options
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110
TIFR ECE 2010 | Question: 10
$\text{H}$ is a circulant matrix (row $n$ is obtained by circularly shifting row $1$ to the right by $n$ positions) and $\text{F}$ is the $\text{DFT}$ matrix. Which of the following is true? $F H F^{H}$ is circulant, where $F^{H}$ is the inverse $\text{DFT}$ matrix. $F H F^{H}$ is tridiagonal $F H F^{H}$ is diagonal $F H F^{H}$ has real entries None of the above
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111
TIFR ECE 2010 | Question: 11
Consider \[ \text{F}=\frac{1}{2}\left[\begin{array}{cccc} 1 & 1 & 1 & 1 \\ 1 & 1 & -1 & -1 \\ 1 & -1 & -1 & 1 \\ 1 & -1 & 1 & -1 \end{array}\right], \quad x=\left[\begin{array}{l} 2.1 \\ 1.2 \\ ... 2 \\ -1 \end{array}\right] \] The inner product between $\text{F}x$ and $\text{F}y$ is $0$ $1$ $-1$ $-1.2$ None of the above
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112
TIFR ECE 2010 | Question: 6
If we convolve $\sin (t) / t$ with itself, then we get $C \sin (t) / t$ for some constant $C$ $C \cos (t) / t$ for some constant $C$ $C \cos (t) / t^{2}$ for some constant $C$ $C_{1} \sin (t) / t^{2}+C_{2} \cos (t) / t^{2}$ for some constants $C_{1}, C_{2}$ None of the above
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113
TIFR ECE 2010 | Question: 9
The $z$-transform of a sequence $\left\{x_{n}\right\}_{n=-\infty}^{\infty}$ is defined to be $X(z)=\sum_{n=-\infty}^{\infty} x_{n} z^{-n}$. The $z$-transform of the sequence $y_{n}=x_{2 n+1}$ is $Y(z)=z(X(z)-X(-z)) / 2$ ... $Y(z)=z(X(\sqrt{z})-X(-\sqrt{z})) / 2$ $Y(z)=(X(\sqrt{z})-X(-\sqrt{z})) / 2$
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114
TIFR ECE 2010 | Question: 14
Define $\operatorname{sign}(x)=0$ for $x=0, \operatorname{sign}(x)=1$ for $x>0$ and $\operatorname{sign}(x)=-1$ for $x<0$. For $n \geq 0$, let \[ Y_{n}=\operatorname{sign}\left(X(n)-Z_{n}\right), \] where $Z_{n}=\sum_{k \leq n} Y_{k}, Z_{0}=0, X(t)=t$ ... $1$'s, followed by $-1,1,-1,1, \ldots$. $0,1,-1,1,-1, \ldots$ $0,1,1,1,-1,1,-1,1, \ldots$ None of the above
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115
TIFR ECE 2010 | Question: 15
Let $\imath=\sqrt{-1}$. Then $\imath^{\imath}$ could be $\exp (\pi / 2)$ $\exp (\pi / 4)$ Can't determine Takes infinite values Is a complex number
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116
TIFR ECE 2022 | Question: 8
Let $a, b, c$ be real numbers such that the following system of equations has a solution \[\begin{aligned} x+2 y+3 z &=a & & (1)\\ 8 x+10 y+12 z &=b & & (2)\\ 7 x+8 y+9 z &=c-1 & & (3) \end{aligned}\] Let $A$ be a ... 1 & 0 \\ -1 & 0 & 1 \end{array}\right]\] What is the value of $\operatorname{det}(A)$? $1$ $2$ $3$ $4$ $5$
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117
TIFR ECE 2022 | Question: 1
Suppose that a random variable $X$ can take $5$ values $\{1,2,3,4,5\}$ with probabilities that depend upon $n \geq 0$ and are given by \[P(X=k)=\frac{e^{k n}}{e^{n}+e^{2 n}+e^{3 n}+e^{4 n}+e^{5 n}}\] for $k=1,2,3,4,5$. ... $1$ as $n \rightarrow \infty$ It converges to $5$ as $n \rightarrow \infty$ It converges to $0$ as $n \rightarrow \infty$
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118
TIFR ECE 2021 | Question: 2
Given a fixed perimeter of $1,$ among the following shapes, which one has the largest area? Square A regular pentagon A regular hexagon A regular septagon A regular octagon
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119
TIFR ECE 2021 | Question: 3
Consider the following statements: $\lim _{x \rightarrow 0} \frac{\sin x}{x}=1$. $\lim _{x \rightarrow 0} \frac{1-\cos x}{x^{2}}=1$. $\lim _{x \rightarrow 0} \frac{1-\cos x}{x}=1$. Which of the following is $\text{TRUE?}$ Only Statement $1$ ... $1$ and $3$ are correct. All of Statements $1, 2,$ and $3$ are correct. None of the three Statements $1,2,$ and $3$ are correct.
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120
TIFR ECE 2022 | Question: 4
Evaluate the value of \[\max \left(x^{2}+(1-y)^{2}\right),\] where the maximisation above is over $x$ and $y$ such that $0 \leq x \leq y \leq 1$. $0$ $2$ $1 / 2$ $1 / 4$ $1$
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