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TIFR ECE 2023 | Question: 1
Consider a fair coin with probability of heads and tails equal to $1 / 2$. Moreover consider two dice, first $\mathrm{D}_{1}$ that has three faces numbered $1,3,5$ and second $\mathrm{D}_{2}$ that has three faces numbered $2,4,6$ ... rolled dice in the experiment. What is $\mathbb{E}[X]$ ? $\frac{7}{2}$ 4 3 $\frac{9}{2}$ None of the above

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TIFR ECE 2023 | Question: 2
$\begin{array}{rlr}a^*=\max _{x, y} & x^2+y^2-8 x+7 \\ \text { s.t. } & x^2+y^2 \leq 1 \\ & y \geq 0\end{array}$ Then $a^{\star}$ is $16$ $14$ $12$ $10$ None of the above

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TIFR ECE 2023 | Question: 3
Let \[ \mathcal{P}=\left\{(x, y): x+y \geq 1,2 x+y \geq 2, x+2 y \geq 2,(x-1)^{2}+(y-1)^{2} \leq 1\right\} . \] Compute \[ \min _{(x, y) \in \mathcal{P}} 2 x+3 y \] $2$ $3$ $4$ $6$ None of the above

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TIFR ECE 2023 | Question: 4
Recall that the entropy (in bits) of a random variable $\mathrm{X}$ which takes values in $\mathbb{N}$ ... random variable which denotes the number of tosses made. What is the entropy of $\mathrm{X}$ in bits? $1$ $2$ $4$ Infinity None of the above

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5

TIFR ECE 2023 | Question: 5
Let $\mathrm{B}$ denote the unit ball in $\mathbb{R}^{2}$, and $\mathrm{Q}$ a square of side length $2$. Let $\mathrm{K}$ be the set of all vectors $z$ such that for some $x \in \mathrm{B}$ and some $y \in \mathrm{Q}, z=x+y$. The area of $\mathrm{K}$ is $4+\pi$ $6+\pi$ $8+\pi$ $10+\pi$ $12+\pi$

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6

TIFR ECE 2023 | Question: 6
An ant in the plane travels in a spiral such that its position $(x(t), y(t))$ at time $t \geq 0$ is $\left(e^{t} \cos t, e^{t} \sin t\right)$. At time $t=1$, find the real part of $\ln (x(t)+i y(t))$. $-2$ $1$ $0$ $-1$ $2$

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TIFR ECE 2023 | Question: 7
Let $f(x)$ be a positive continuous function on the real line that is the density of a random variable $X$. The differential entropy of $X$ is defined to be $-\int_{-\infty}^{\infty} f(x) \ln f(x) d x$. In which case does $X$ have the least differential entropy? You may use these facts: The ... $f(x):=(1 / 4) e^{-|x| / 2}$. $f(x):=e^{-2|x|}$.

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TIFR ECE 2023 | Question: 8
Suppose a bag contains 5 red balls, 3 blue balls, and 2 black balls. Balls are drawn without replacement until the bag is empty. Let $X_{i}$ be a random variable which takes value 1 if the $i$-th ball drawn is red, value 2 if that ball is blue, and 3 if it is black. Let the ... Only (i) and (ii) Only (i) and (iii) All of (i), (ii), and (iii) None of (i), (ii), or (iii)

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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. $1$ is an eigenvalue of $A$ 2. The magnitude of any eigenvalue of $A$ ... statements $1$ and $3$ are correct Only statements $2$ and $3$ are correct All statements $1,2$ , and $3$ are correct

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10

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 $f(t) * g(t)$ ... $\frac{1}{2}(\exp (-t)+\sin (t)-2 \cos (t)) u(t)$ $\frac{1}{2}(\exp (-t)-\sin (t)+2 \cos (t)) u(t)$

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11

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? $-1 \leq x^{*}<-0.5$ $-0.5 \leq x^{*}<0$ $x^{*}=0$ $0<x^* \leq 0.5$ $0.5<x^* \leq 1$

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12

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

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13

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 $X$ with the following probability density function (p.d.f.): \[ f_{N}(n)=\left\{\begin{array} ... $0$ $1 / 8$ $1 / 4$ $1 / 2$ None of the above

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14

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: $F(1)$ $1$ $\frac{1}{2}$ $\frac{1}{\sqrt{2 \pi}}$ $\frac{\pi}{4}$

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15

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 ... is 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.

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16

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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17

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

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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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19

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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21

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

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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

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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

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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

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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

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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

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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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28

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

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29

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

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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.

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31

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

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32

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

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33

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.

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34

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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35

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

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36

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.

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37

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.

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38

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.

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39

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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40

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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