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121
TIFR ECE 2023 | Question: 9
Consider an $n \times n$ matrix $A$ with the property that each element of $A$ is non-negative and the sum of elements of each row is $1$. Consider the following statements. $1$ is an eigenvalue of $A$ The magnitude of any eigenvalue of $A$ is at ... statements $1$ and $3$ are correct Only statements $2$ and $3$ are correct All statements $1,2$ , and $3$ are correct
Consider an $n \times n$ matrix $A$ with the property that each element of $A$ is non-negative and the sum of elements of each row is $1$.Consider the following statement...
admin
46.4k
points
113
views
admin
asked
Mar 14, 2023
Linear Algebra
tifrece2023
engineering-mathematics
linear-algebra
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0
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0
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122
TIFR ECE 2023 | Question: 10
Convolution between two functions $f(t)$ and $g(t)$ is defined as follows: $f(t) * g(t)=\int_{-\infty}^{\infty} f(\tau) g(t-\tau) d \tau$ Let $u(t)$ be the unit-step function, i.e., $u(t)=1$ for $t \geq 0$ and $u(t)=0$ for $t<0$. What is $f(t) * g(t)$ ... $\frac{1}{2}(\exp (-t)+\sin (t)-2 \cos (t)) u(t)$ $\frac{1}{2}(\exp (-t)-\sin (t)+2 \cos (t)) u(t)$
Convolution between two functions $f(t)$ and $g(t)$ is defined as follows:$$f(t) * g(t)=\int_{-\infty}^{\infty} f(\tau) g(t-\tau) d \tau$$Let $u(t)$ be the unit-step func...
admin
46.4k
points
131
views
admin
asked
Mar 14, 2023
Calculus
tifrece2023
engineering-mathematics
calculus
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0
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0
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123
TIFR ECE 2023 | Question: 11
Consider the function $f(x)=x e^{|x|}+4 x^{2}$ for values of $x$ which lie in the interval $[-1,1]$. In this domain, suppose the function attains the minimum value at $x^{*}$. Which of the following is true? $-1 \leq x^{*}<-0.5$ $-0.5 \leq x^{*}<0$ $x^{*}=0$ $0<x^* \leq 0.5$ $0.5<x^* \leq 1$
Consider the function$$f(x)=x e^{|x|}+4 x^{2}$$for values of $x$ which lie in the interval $[-1,1]$. In this domain, suppose the function attains the minimum value at $x^...
admin
46.4k
points
125
views
admin
asked
Mar 14, 2023
Linear Algebra
tifrece2023
engineering-mathematics
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0
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0
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124
TIFR ECE 2023 | Question: 12
Consider a disk $D$ of radius $1$ centered at the origin. Let $X$ be a point uniformly distributed on $D$ and let the distance of $X$ from the origin be $R$. Let $A$ be the (random) area of the disk with radius $R$ centered at the origin. Then $\mathbb{E}[A]$ is $\frac{\pi}{3}$ $\frac{\pi}{6}$ $\frac{\pi}{4}$ $\frac{\pi}{2}$ None of the above
Consider a disk $D$ of radius $1$ centered at the origin. Let $X$ be a point uniformly distributed on $D$ and let the distance of $X$ from the origin be $R$. Let $A$ be t...
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46.4k
points
117
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admin
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Mar 14, 2023
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125
TIFR ECE 2023 | Question: 13
Let $X$ be a random variable which takes values $1$ and $-1$ with probability $1 / 2$ each. Suppose $Y=X+N$, where $N$ is a random variable independent of $X$ ... $0$ $1 / 8$ $1 / 4$ $1 / 2$ None of the above
Let $X$ be a random variable which takes values $1$ and $-1$ with probability $1 / 2$ each. Suppose $Y=X+N$, where $N$ is a random variable independent of $X$ with the fo...
admin
46.4k
points
127
views
admin
asked
Mar 14, 2023
Probability and Statistics
tifrece2023
engineering-mathematics
probability
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0
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0
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126
TIFR ECE 2023 | Question: 14
Suppose that $Z \sim \mathcal{N}(0,1)$ is a Gaussian random variable with mean zero and variance $1$. Let $F(z) \equiv \mathbb{P}(Z \leq z)$ be the cumulative distribution function $\operatorname{(CDF)}$ of $Z$. Define a new random variable $Y$ as $Y=F(Z)$. This means that the ... of $\mathbb{E}[Y]$ is: $F(1)$ $1$ $\frac{1}{2}$ $\frac{1}{\sqrt{2 \pi}}$ $\frac{\pi}{4}$
Suppose that $Z \sim \mathcal{N}(0,1)$ is a Gaussian random variable with mean zero and variance $1$. Let $F(z) \equiv \mathbb{P}(Z \leq z)$ be the cumulative distributio...
admin
46.4k
points
132
views
admin
asked
Mar 14, 2023
Vector Analysis
tifrece2023
engineering-mathematics
gausss-theorem
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0
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0
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127
TIFR ECE 2023 | Question: 15
Let $\left\{x_{n}\right\}_{n \geq 0}$ be a sequence of real numbers which satisfy $x_{n+1}\left(1+x_{n+1}\right) \leq x_{n}\left(1+x_{n}\right), \quad n \geq 0 .$ Choose the correct option from the following. ... always bounded but does not necessarily converge. The sequence always converges to a non-zero limit. The sequence always converges to zero. None of the above.
Let $\left\{x_{n}\right\}_{n \geq 0}$ be a sequence of real numbers which satisfy$$x_{n+1}\left(1+x_{n+1}\right) \leq x_{n}\left(1+x_{n}\right), \quad n \geq 0 .$$Choose ...
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46.4k
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106
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Mar 14, 2023
Others
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128
TIFR ECE 2015 | Question: 1
For a time-invariant system, the impulse response completely describes the system if the system is causal and non-linear non-causal and non-linear causal and linear All of the above None of the above
For a time-invariant system, the impulse response completely describes the system if the system iscausal and non-linearnon-causal and non-linearcausal and linearAll of th...
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46.4k
points
105
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Dec 15, 2022
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129
TIFR ECE 2015 | Question: 2
Let $x[n]=a^{\lfloor n \mid}$, ( $a$ is real, $0<a<1$ ) and the discrete time Fourier transform $\text{(DTFT)}$ of $x[n]$ is given by $X(\omega)=\sum_{n=-\infty}^{\infty} x[n] e^{-j \omega n}$. Then the $\text{DTFT}$ ... zero only at one value of $\omega \in[-\pi, \pi]$ Its maximum value is larger than $1$ Its minimum value is less than $-1$ None of the above
Let $x[n]=a^{\lfloor n \mid}$, ( $a$ is real, $0<a<1$ ) and the discrete time Fourier transform $\text{(DTFT)}$ of $x[n]$ is given by $X(\omega)=\sum_{n=-\infty}^{\infty}...
admin
46.4k
points
99
views
admin
asked
Dec 15, 2022
Calculus
tifr2015
calculus
discrete-fourier-transform
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1
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130
TIFR ECE 2015 | Question: 3
Let $h(t)$ be the impulse response of an ideal low-pass filter with cut-off frequency $5 \mathrm{kHz} .\; \mathrm{Let}\; g[n]= h(n T)$, for integer $n$, be a sampled version of $h(t)$ ... -time filter with $g[n]$ as its unit impulse response is a low-pass filter high-pass filter band-pass filter band-stop filter all-pass filter
Let $h(t)$ be the impulse response of an ideal low-pass filter with cut-off frequency $5 \mathrm{kHz} .\; \mathrm{Let}\; g[n]= h(n T)$, for integer $n$, be a sampled vers...
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46.4k
points
73
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asked
Dec 15, 2022
Others
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131
TIFR ECE 2015 | Question: 4
The capacity of a certain additive white Gaussian noise channel of bandwidth $1 \mathrm{~MHz}$ is $\mathrm{known}$ to be $8 \text{ Mbps}$ when the average transmit power constraint is $50 \mathrm{~mW}$. Which of the following statements can we make about the capacity $C$ ... $C=8$ $8 < C < 16$ $C=16$ $C>16$ There is not enough information to determine $C$
The capacity of a certain additive white Gaussian noise channel of bandwidth $1 \mathrm{~MHz}$ is $\mathrm{known}$ to be $8 \text{ Mbps}$ when the average transmit power ...
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46.4k
points
43
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Dec 15, 2022
Others
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132
TIFR ECE 2015 | Question: 5
What is the following passive circuit? Low-pass filter High-pass filter Band-pass filter Band-stop filter All-pass filter
What is the following passive circuit?Low-pass filterHigh-pass filterBand-pass filterBand-stop filterAll-pass filter
admin
46.4k
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122
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admin
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Dec 15, 2022
Others
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133
TIFR ECE 2015 | Question: 6
$\textbf{A}$ is an $n \times n$ square matrix of reals such that $\mathbf{A y}=\mathbf{A}^{T} \mathbf{y}$, for all real vectors $\mathbf{y}$. Which of the following can we conclude? $\mathbf{A}$ is invertible $\mathbf{A}^{T}=\mathbf{A}$ $\mathbf{A}^{2}=\mathbf{A}$ Only (i) Only (ii) Only (iii) Only (i) and (ii) None of the above
$\textbf{A}$ is an $n \times n$ square matrix of reals such that $\mathbf{A y}=\mathbf{A}^{T} \mathbf{y}$, for all real vectors $\mathbf{y}$. Which of the following can w...
admin
46.4k
points
85
views
admin
asked
Dec 15, 2022
Linear Algebra
tifr2015
linear-algebra
matrices
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1
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0
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134
TIFR ECE 2015 | Question: 7
Let $A$ be an $8 \times 8$ matrix of the form \[ \left[\begin{array}{cccc} 2 & 1 & \ldots & 1 \\ 1 & 2 & \ldots & 1 \\ \vdots & \vdots & \ddots & \vdots \\ 1 & 1 & \ldots & 2 \end{array}\ ... $\operatorname{det}(A)=9$ $\operatorname{det}(A)=18$ $\operatorname{det}(A)=14$ $\operatorname{det}(A)=27$ None of the above
Let $A$ be an $8 \times 8$ matrix of the form\[\left[\begin{array}{cccc}2 & 1 & \ldots & 1 \\1 & 2 & \ldots & 1 \\\vdots & \vdots & \ddots & \vdots \\1 & 1 & \ldots & 2\e...
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46.4k
points
111
views
admin
asked
Dec 15, 2022
Linear Algebra
tifr2015
linear-algebra
determinant
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1
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0
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135
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...
admin
46.4k
points
96
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admin
asked
Dec 15, 2022
Probability and Statistics
tifr2015
probability-and-statistics
probability
random-variable
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136
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...
admin
46.4k
points
94
views
admin
asked
Dec 15, 2022
Probability and Statistics
tifr2015
probability-and-statistics
probability
random-variable
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137
TIFR ECE 2015 | Question: 10
Let $X$ be a uniform random variable between $[0,1]$. And let \[ M=\min _{m X \geq 1, m \in \mathbb{N}} m . \] Then which of the following is true? $E(M)=\infty$ $E(M) \in[5,10]$ $E(M)=\exp (1)$ $E(M)=\pi$ None of the above
Let $X$ be a uniform random variable between $[0,1]$. And let\[M=\min _{m X \geq 1, m \in \mathbb{N}} m .\]Then which of the following is true?$E(M)=\infty$$E(M) \in[5,10...
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46.4k
points
84
views
admin
asked
Dec 15, 2022
Probability and Statistics
tifr2015
probability-and-statistics
probability
random-variable
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138
TIFR ECE 2015 | Question: 11
For $x>0$, for which range of values of $\alpha$ is the following inequality true? \[ x \log _{e}(x) \geq x-\alpha \] $\alpha \geq 1 / 2$ $\alpha \geq 0$ $\alpha \leq 2$ $\alpha \geq 1$ None of the above
For $x>0$, for which range of values of $\alpha$ is the following inequality true?\[x \log _{e}(x) \geq x-\alpha\]$\alpha \geq 1 / 2$$\alpha \geq 0$$\alpha \leq 2$$\alpha...
admin
46.4k
points
97
views
admin
asked
Dec 15, 2022
Quantitative Aptitude
tifr2015
quantitative-aptitude
logarithms
inequality
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1
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0
answers
139
TIFR ECE 2015 | Question: 12
Consider the following optimization problem \[ \max (2 x+3 y) \] subject to the following three constraints \[ \begin{aligned} x+y & \leq 5, \\ x+2 y & \leq 10, \text { and } \\ x & <3 . \end{aligned} \] Let $z^{*}$ be the ... $(x, y)$ that satisfy the above three constraints such that $2 x+3 y$ equals $z^{*}$.
Consider the following optimization problem\[\max (2 x+3 y)\]subject to the following three constraints\[\begin{aligned}x+y & \leq 5, \\x+2 y & \leq 10, \text { and } \\x...
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46.4k
points
83
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Dec 15, 2022
Others
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140
TIFR ECE 2015 | Question: 13
Let \[ A=\left(\begin{array}{ccc} 1 & 1+\varepsilon & 1 \\ 1+\varepsilon & 1 & 1+\varepsilon \\ 1 & 1+\varepsilon & 1 \end{array}\right) \] Then for $\varepsilon=10^{-6}, A$ has only negative eigenvalues only non-zero eigenvalues only positive eigenvalues one negative and one positive eigenvalue None of the above
Let\[A=\left(\begin{array}{ccc}1 & 1+\varepsilon & 1 \\1+\varepsilon & 1 & 1+\varepsilon \\1 & 1+\varepsilon & 1\end{array}\right)\]Then for $\varepsilon=10^{-6}, A$ haso...
admin
46.4k
points
92
views
admin
asked
Dec 15, 2022
Linear Algebra
tifr2015
linear-algebra
eigen-values
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141
TIFR ECE 2015 | Question: 14
Consider a frog that lives on two rocks $A$ and $B$ and moves from one rock to the other randomly. If it is at Rock $A$ at any time, irrespective of which rocks it occupied in the past, it jumps back to Rock $A$ with probability $2 / 3$ and instead jumps to Rock ... of $n$ jumps as $n \rightarrow \infty?$ $1 / 2 $ $2 / 3$ $1$ The limit does not exist None of the above
Consider a frog that lives on two rocks $A$ and $B$ and moves from one rock to the other randomly. If it is at Rock $A$ at any time, irrespective of which rocks it occupi...
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46.4k
points
89
views
admin
asked
Dec 15, 2022
Probability and Statistics
tifr2015
probability-and-statistics
probability
conditional-probability
limits
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1
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0
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142
TIFR ECE 2015 | Question: 15
Let $x_{1}=-1$ and $x_{2}=1$ be two signals that are transmitted with equal probability. If signal $x_{i}, i \in$ $\{1,2\}$ is transmitted, the received signal is $y=x_{i}+n_{i}$, where $n_{i}$ ... $\theta^{\star}$ to minimize the probability of error is $\leq 0$ None of the above.
Let $x_{1}=-1$ and $x_{2}=1$ be two signals that are transmitted with equal probability. If signal $x_{i}, i \in$ $\{1,2\}$ is transmitted, the received signal is $y=x_{i...
admin
46.4k
points
78
views
admin
asked
Dec 15, 2022
Probability and Statistics
tifr2015
probability-and-statistics
probability
normal-distribution
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0
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143
TIFR ECE 2014 | Question: 1
Consider two independent and identically distributed random variables $X$ and $Y$ uniformly distributed in $[0,1]$. For $\alpha \in[0,1]$, the probability that $\alpha \max (X, Y)<\min (X, Y)$ is $1 /(2 \alpha)$. $\exp (1-\alpha)$ $1-\alpha$ $(1-\alpha)^{2}$ $1-\alpha^{2}$
Consider two independent and identically distributed random variables $X$ and $Y$ uniformly distributed in $[0,1]$. For $\alpha \in[0,1]$, the probability that $\alpha \m...
admin
46.4k
points
112
views
admin
asked
Dec 14, 2022
Probability and Statistics
tifr2014
probability-and-statistics
probability
uniform-distribution
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144
TIFR ECE 2014 | Question: 2
Evaluate the limit \[ \lim _{n \rightarrow \infty}\left(2 n^{4}\right)^{\frac{1}{3 n}} . \] $e$ $1$ $2^{\frac{1}{3}}$ $0$ None of the above
Evaluate the limit\[\lim _{n \rightarrow \infty}\left(2 n^{4}\right)^{\frac{1}{3 n}} .\]$e$$1$$2^{\frac{1}{3}}$$0$None of the above
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46.4k
points
84
views
admin
asked
Dec 14, 2022
Calculus
tifr2014
calculus
limits
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145
TIFR ECE 2014 | Question: 3
For a non-negative continuous random variable $X$, which of the following is TRUE? $E\{X\}=\int_{0}^{\infty} P(X>x) d x$. $E\{X\}=\int_{0}^{\infty} P(X \leq x) d x$. $P(X<x) \leq \frac{E\{X\}}{x}$. $(a)$ and $(c)$. None of the above.
For a non-negative continuous random variable $X$, which of the following is TRUE?$E\{X\}=\int_{0}^{\infty} P(X>x) d x$.$E\{X\}=\int_{0}^{\infty} P(X \leq x) d x$.$P(X<x)...
admin
46.4k
points
96
views
admin
asked
Dec 14, 2022
Probability and Statistics
tifr2014
probability-and-statistics
probability
random-variable
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146
TIFR ECE 2014 | Question: 4
A system accepts a sequence of real numbers $x[n]$ as input and outputs \[ y[n]=\left\{\begin{array}{ll} 0.5 x[n]-0.25 x[n-1], & n \text { even } \\ 0.75 x[n], & n \text { odd } \end{array}\right. \] The system is non-linear. non-causal. time-invariant. All of the above. None of the above.
A system accepts a sequence of real numbers $x[n]$ as input and outputs\[y[n]=\left\{\begin{array}{ll}0.5 x[n]-0.25 x[n-1], & n \text { even } \\0.75 x[n], & n \text { od...
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46.4k
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93
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Others
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147
TIFR ECE 2014 | Question: 5
The matrix \[ A=\left(\begin{array}{ccc} 1 & a_{1} & a_{1}^{2} \\ 1 & a_{2} & a_{2}^{2} \\ 1 & a_{3} & a_{3}^{2} \end{array}\right) \] is invertible when $a_{1}>a_{2}>a_{3}$ $a_{1}<a_{2}<a_{3}$ $a_{1}=3, a_{2}=2, a_{3}=4$ All of the above None of the above
The matrix\[A=\left(\begin{array}{ccc}1 & a_{1} & a_{1}^{2} \\1 & a_{2} & a_{2}^{2} \\1 & a_{3} & a_{3}^{2}\end{array}\right)\]is invertible when$a_{1}>a_{2}>a_{3}$$a_{1}...
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46.4k
points
83
views
admin
asked
Dec 14, 2022
Linear Algebra
tifr2014
linear-algebra
matrices
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0
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148
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 ...
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46.4k
points
95
views
admin
asked
Dec 14, 2022
Calculus
tifr2014
calculus
definite-integrals
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0
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149
TIFR ECE 2014 | Question: 7
Let $A$ be an $n \times n$ real matrix. It is known that there are two distinct $n$-dimensional real column vectors $v_{1}, v_{2}$ such that $A v_{1}=A v_{2}$. Which of the following can we conclude about $A?$ All eigenvalues of $A$ are non-negative. $A$ is not full rank. $A$ is not the zero matrix. $\operatorname{det}(A) \neq 0$. None of the above.
Let $A$ be an $n \times n$ real matrix. It is known that there are two distinct $n$-dimensional real column vectors $v_{1}, v_{2}$ such that $A v_{1}=A v_{2}$. Which of t...
admin
46.4k
points
113
views
admin
asked
Dec 14, 2022
Linear Algebra
tifr2014
linear-algebra
eigen-values
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1
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0
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150
TIFR ECE 2014 | Question: 8
Consider a square pulse $g(t)$ of height $1$ and width $1$ centred at $1 / 2$. Define $f_{n}(t)=\frac{1}{n}\left(g(t) *^{n} g(t)\right),$ where $*^{n}$ stands for $n$-fold convolution. Let $f(t)=\lim _{n \rightarrow \infty} f_{n}(t)$. Then, which ... $\infty$. $f(t)$ has width $\infty$ and height $1$ . $f(t)$ has width $0$ and height $\infty$. None of the above.
Consider a square pulse $g(t)$ of height $1$ and width $1$ centred at $1 / 2$. Define $f_{n}(t)=\frac{1}{n}\left(g(t) *^{n} g(t)\right),$ where $*^{n}$ stands for $n$-fol...
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46.4k
points
93
views
admin
asked
Dec 14, 2022
Calculus
tifr2014
calculus
limits
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0
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151
TIFR ECE 2014 | Question: 9
Consider the following input $x(t)$ and output $y(t)$ pairs for two different systems. $x(t)=\sin (t), y(t)=\cos (t),$ $x(t)=t+\sin (t), y(t)=2 t+\sin (t-1).$ Which of these systems could possibly be linear and time invariant? Choose the most appropriate answer ... i) nor (ii). neither, but a system with $x(t)=\sin (2 t), y(t)=\sin (t) \cos (t) \operatorname{could~be.~}$
Consider the following input $x(t)$ and output $y(t)$ pairs for two different systems.$x(t)=\sin (t), y(t)=\cos (t),$$x(t)=t+\sin (t), y(t)=2 t+\sin (t-1).$Which of these...
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46.4k
points
101
views
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Others
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152
TIFR ECE 2014 | Question: 10
Consider the two quadrature amplitude modulation $\text{(QAM)}$ constellations below. Suppose that the channel has additive white Gaussian noise channel and no intersymbol interference. The constellation points are picked equally likely. Let $P\text{(QAM)}$ denote the ... .
Consider the two quadrature amplitude modulation $\text{(QAM)}$ constellations below. Suppose that the channel has additive white Gaussian noise channel and no intersymbo...
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46.4k
points
77
views
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Dec 14, 2022
Others
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153
TIFR ECE 2014 | Question: 11
It is known that the signal $x(t)$, where $t$ denotes time, belongs to the following class: \[ \left\{A \sin \left(2 \pi f_{0} t+\theta\right): f_{0}=1 \mathrm{~Hz}, 0 \leq A \leq 1,0<\theta \leq \pi\right\} \] If you ... how many samples are required to determine the signal? $1$ sample. $2$ samples. $1$ sample per second. $2$ samples per second. None of the above.
It is known that the signal $x(t)$, where $t$ denotes time, belongs to the following class:\[\left\{A \sin \left(2 \pi f_{0} t+\theta\right): f_{0}=1 \mathrm{~Hz}, 0 \leq...
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46.4k
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73
views
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154
TIFR ECE 2014 | Question: 12
Assume that $Y, Z$ are independent, zero-mean, continuous random variables with variances $\sigma_{Y}^{2}$ and $\sigma_{Z}^{2},$ respectively. Let $X=Y+Z$. The optimal value of $\alpha$ which minimizes $\mathbb{E}\left[(X-\alpha Y)^{2}\right]$ ... $1$ $\frac{\sigma_{Y}^{2}}{\sigma_{Z}^{2}}$ None of the above.
Assume that $Y, Z$ are independent, zero-mean, continuous random variables with variances $\sigma_{Y}^{2}$ and $\sigma_{Z}^{2},$ respectively. Let $X=Y+Z$. The optimal va...
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155
TIFR ECE 2014 | Question: 13
Let function $f: \mathbf{R} \rightarrow \mathbf{R}$ be convex, i.e., for $x, y \in \mathbf{R}, \alpha \in[0,1], f(\alpha x+(1-\alpha) y) \leq$ $\alpha f(x)+(1-\alpha) f(y)$. Then which of the following is $\text{TRUE?}$ $f(x) \leq f(y)$ whenever ... $f$ and $g$ are both convex, then $\min \{f, g\}$ is also convex. For a random variable $X, E(f(X)) \geq f(E(X))$.
Let function $f: \mathbf{R} \rightarrow \mathbf{R}$ be convex, i.e., for $x, y \in \mathbf{R}, \alpha \in[0,1], f(\alpha x+(1-\alpha) y) \leq$ $\alpha f(x)+(1-\alpha) f(y...
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Calculus
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156
TIFR ECE 2014 | Question: 14
Suppose that a random variable $X$ has a probability density function \[ \begin{aligned} f(x) & =c(x-4) \quad \text { for } 4 \leq x \leq 6 \\ & =0 \quad \text { for all other } x \end{aligned} \] for some constant $c$. What is the expected value of $X$ given that $X \geq 5?$ $5 \frac{5}{9}$ $5 \frac{1}{2}$ $5 \frac{3}{4}$ $5 \frac{1}{4}$ $5 \frac{5}{8}$
Suppose that a random variable $X$ has a probability density function\[\begin{aligned}f(x) & =c(x-4) \quad \text { for } 4 \leq x \leq 6 \\& =0 \quad \text { for all othe...
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Probability and Statistics
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157
TIFR ECE 2014 | Question: 15
You are allotted a rectangular room of a fixed height. You have decided to paint the three walls and put wallpaper on the fourth one. Walls can be painted at a cost of Rs. $10$ per meter and the wall paper can be put at the rate of Rs $20$ per meter for that ... $200$ square meter room? $400 \times \sqrt{3} $ $400$ $400 \times \sqrt{2}$ $200 \times \sqrt{3}$ $500$
You are allotted a rectangular room of a fixed height. You have decided to paint the three walls and put wallpaper on the fourth one. Walls can be painted at a cost of Rs...
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Others
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158
TIFR ECE 2014 | Question: 16
A fair dice (with faces numbered $1, \ldots, 6$ ) is independently rolled twice. Let $X$ denote the maximum of the two outcomes. The expected value of $X$ is $4 \frac{1}{2}$ $3 \frac{1}{2}$ $5$ $4 \frac{17}{36} $ $4 \frac{3}{4}$
A fair dice (with faces numbered $1, \ldots, 6$ ) is independently rolled twice. Let $X$ denote the maximum of the two outcomes. The expected value of $X$ is$4 \frac{1}{2...
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Probability and Statistics
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159
TIFR ECE 2014 | Question: 17
Let $X$ be a Gaussian random variable with mean $\mu_{1}$ and variance $\sigma_{1}^{2}$. Now, suppose that $\mu_{1}$ itself is a random variable, which is also Gaussian distributed with mean $\mu_{2}$ and variance $\sigma_{2}^{2}$. Then the distribution ... variable with mean $\mu_{2}$ and variance $\sigma_{1}^{2}+\sigma_{2}^{2}$. Has no known form. None of the above.
Let $X$ be a Gaussian random variable with mean $\mu_{1}$ and variance $\sigma_{1}^{2}$. Now, suppose that $\mu_{1}$ itself is a random variable, which is also Gaussian d...
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Probability and Statistics
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160
TIFR ECE 2014 | Question: 18
A non-negative loss in a car accident is distributed with the following probability density function \[ f(x)=\frac{1}{10} \exp (-x / 10) \] for $x \geq 0$. Suppose that first $5$ units of loss is incurred by the insured and the remaining loss if any is covered by the ... $5+10 \exp \left(-\frac{1}{2}\right)$ $15 \exp \left(-\frac{1}{2}\right)$
A non-negative loss in a car accident is distributed with the following probability density function\[f(x)=\frac{1}{10} \exp (-x / 10)\]for $x \geq 0$. Suppose that first...
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