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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

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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

admin asked in Calculus Dec 15, 2022

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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

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4

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 $C$

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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

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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

admin asked in Linear Algebra Dec 15, 2022

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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

admin asked in Linear Algebra Dec 15, 2022

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8

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

admin asked in Probability and Statistics Dec 15, 2022

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9

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

admin asked in Probability and Statistics Dec 15, 2022

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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

admin asked in Probability and Statistics Dec 15, 2022

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11

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

admin asked in Quantitative Aptitude Dec 15, 2022

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12

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 $z^{*}$.

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13

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

admin asked in Linear Algebra Dec 15, 2022

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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

admin asked in Probability and Statistics Dec 15, 2022

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15

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.

admin asked in Probability and Statistics Dec 15, 2022

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16

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 $1 /(2 \alpha)$. $\exp (1-\alpha)$ $1-\alpha$ $(1-\alpha)^{2}$ $1-\alpha^{2}$

admin asked in Probability and Statistics Dec 14, 2022

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17

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

admin asked in Calculus Dec 14, 2022

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18

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.

admin asked in Probability and Statistics Dec 14, 2022

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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.

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20

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

admin asked in Linear Algebra Dec 14, 2022

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21

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.

admin asked in Calculus Dec 14, 2022

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22

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.

admin asked in Linear Algebra Dec 14, 2022

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23

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.

admin asked in Calculus Dec 14, 2022

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24

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 $x(t)=\sin (2 t), y(t)=\sin (t) \cos (t) \operatorname{could~be.~}$

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25

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 ... .

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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.

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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.

admin asked in Probability and Statistics Dec 14, 2022

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28

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 $X, E(f(X)) \geq f(E(X))$.

admin asked in Calculus Dec 14, 2022

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29

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 $X \geq 5?$ $5 \frac{5}{9}$ $5 \frac{1}{2}$ $5 \frac{3}{4}$ $5 \frac{1}{4}$ $5 \frac{5}{8}$

admin asked in Probability and Statistics Dec 14, 2022

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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} $ $400$ $400 \times \sqrt{2}$ $200 \times \sqrt{3}$ $500$

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31

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} $ $4 \frac{3}{4}$

admin asked in Probability and Statistics Dec 14, 2022

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32

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.

admin asked in Probability and Statistics Dec 14, 2022

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33

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 ... $5+10 \exp \left(-\frac{1}{2}\right)$ $15 \exp \left(-\frac{1}{2}\right)$

admin asked in Probability and Statistics Dec 14, 2022

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34

TIFR ECE 2014 | Question: 19
Consider a $2^{k} \times N$ binary matrix $A=\left\{a_{\ell, k}\right\}, a_{\ell, k} \in\{0,1\}$. For rows $i$ and $j$, let the Hamming distance be $d_{i, j}=\sum_{\ell=1}^{N}\left|a_{i, \ell}-a_{j, \ell}\right|$. Let $D_{\min }=\min _{i, j} d_{i, j}$. ... $D_{\min } \leq N-k+1$. $D_{\min } \leq N-k$. $D_{\min } \leq N-k-1$. $D_{\min } \leq N-k-2$. None of the above.

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35

TIFR ECE 2014 | Question: 20
What is \[ \lim _{n \rightarrow \infty} \cos \frac{\pi}{2^{2}} \cos \frac{\pi}{2^{3}} \cdots \cos \frac{\pi}{2^{n}} ? \] $0$ $\pi / 2$ $1 / \sqrt{2}$ $2 / \pi$ None of the above.

admin asked in Calculus Dec 14, 2022

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36

TIFR ECE 2013 | Question: 1
The unit step response of a discrete-time, linear, time-invariant system is \[ y[n]=\left\{\begin{array}{rl} 0, & n<0 \\ 1, & n \geq 0 \text { and } n \text { even } \\ -1, & n \geq 0 \text { and } ... the system is bounded-input, bounded-output $\text{(BIBO)}$ stable there is not enough information to determine $\text{(BIBO)}$ stability none of the above

admin asked in Others Dec 12, 2022

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37

TIFR ECE 2013 | Question: 2
The output $\{y(n)\}$ of a discrete time system with input $\{x(n)\}$ is given by \[ y(n)=\sum_{k=0}^{N-1} a^{k} x(n-k) . \] The difference equation for the inverse system is given by $y(n)=x(n)-a x(n-1)$ ... $(a)$ above, otherwise the inverse does not exist If $|a|<1$, then the answer is $(b)$ above, otherwise the inverse does not exist None of the above

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38

TIFR ECE 2013 | Question: 3
$X$ and $Y$ are jointly Gaussian random variables with zero mean. A constant-pdf contour is where the joint density function takes on the same value. If the constant-pdf contours of $X, Y$ are as shown above, which of the following could their covariance matrix $\mathbf{K}$ ... $\mathbf{K}=\left[\begin{array}{cc}1 & -0.5 \\ -0.5 & 2\end{array}\right]$

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39

TIFR ECE 2013 | Question: 4
Consider a fair coin that has probability $1 / 2$ of showing heads $(\text{H})$ in a toss and $1 / 2$ of showing tails $(\text{T})$. Suppose we independently flip a fair coin over and over again. What is the probability that $\text{HT}$ sequence occurs before $\text{TT}?$ $3 / 4$ $1 / 2$ $2 / 3$ $1 / 3$ $1 / 4$

admin asked in Probability and Statistics Dec 12, 2022

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40

TIFR ECE 2013 | Question: 5
Let $x(n)=\sin (2 \pi k n / N), n=0,1, \ldots, N-1$, where $2 k \neq N$ and $0<k \leq N-1$. Then the circular convolution of $\{x(n)\}$ with itself is $N \cos (4 \pi k n / N)$ $N \sin (4 \pi k n / N)$ $-N \cos (2 \pi k n / N) / 2$ $-N \sin (2 \pi k n / N) / 2$ None of the above

admin asked in Others Dec 12, 2022

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