Recent questions tagged tifrece2018 / GO Electronics

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TIFR ECE 2018 | Question: 1
Consider a discrete-time system which in response to input sequence $x[n] \;( n$ integer) outputs the sequence $y[n]$ such that \[y[n]=\left\{\begin{array}{ll} 0, & n=-1,-2,-3, \ldots, \\ \frac{1}{2 ... describes the system? Linear, time-invariant Linear, time-variant Non-linear, time-invariant Non-linear, time-variant Cannot be determined from the information given

Consider a discrete-time system which in response to input sequence $x[n] \;( n$ integer) outputs the sequence $y[n]$ such that\[y[n]=\left\{\begin{array}{ll}0, & n=-1,-2...

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2

TIFR ECE 2018 | Question: 2
A hotel has $n$ rooms numbered $1,2, \ldots, n$. For each room there is one spare key labeled with the room number. The hotel manager keeps all the spare keys in a box. Her mischievous son got hold of the box and permuted the labels uniformly at random. What is the ... Use linearity of expectation] $1$ $\frac{n-1}{n}$ $\frac{n}{n-1}$ $\frac{n}{2}$ None of the above

A hotel has $n$ rooms numbered $1,2, \ldots, n$. For each room there is one spare key labeled with the room number. The hotel manager keeps all the spare keys in a box. H...

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TIFR ECE 2018 | Question: 3
Let $\lim _{n \rightarrow \infty} f(n)=\infty$ and $\lim _{n \rightarrow \infty} g(n)=\infty$. Then which of the following is necessarily $\text{TRUE.}$ $\lim _{n \rightarrow \infty}|f(n)-g(n)|=\infty$ $\lim _{n \rightarrow \infty}|f(n)-g(n)|=0$ $\lim _{n \rightarrow \infty}|f(n) / g(n)|=\infty$ $\lim _{n \rightarrow \infty}|f(n) / g(n)|=1$ None of the above

Let $\lim _{n \rightarrow \infty} f(n)=\infty$ and $\lim _{n \rightarrow \infty} g(n)=\infty$. Then which of the following is necessarily $\text{TRUE.}$$\lim _{n \rightar...

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4

TIFR ECE 2018 | Question: 4
Consider \[f(x)=\frac{(x \log x+x)^{5}(1+2 / x)^{x}}{(x+1 / x)^{5}(\log x+1 / \log x)^{6}}\] What can we say about $\lim _{x \rightarrow \infty} f(x)$ ? The function $f(x)$ does not have a limit as $x \rightarrow \infty$ ... $\lim _{x \rightarrow \infty} f(x)=e^{1 / 2}$ $\lim _{x \rightarrow \infty} f(x)=0$ $\lim _{x \rightarrow \infty} f(x)=\infty$

Consider\[f(x)=\frac{(x \log x+x)^{5}(1+2 / x)^{x}}{(x+1 / x)^{5}(\log x+1 / \log x)^{6}}\]What can we say about $\lim _{x \rightarrow \infty} f(x)$ ?The function $f(x)$ ...

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TIFR ECE 2018 | Question: 5
Suppose $\vec{u}, \vec{v}_{1}, \vec{v}_{2} \in \mathbb{R}^{n}$ are linearly independent vectors. Let the pair of real numbers $\left(a_{1}^{*}, a_{2}^{*}\right)$ be such that they solve the following optimization problem \[d=\min _{a_{1}, a_{2} \in \mathbb{R}}\left\ ... $\left\|\vec{v}_{*}\right\|^{2}-\|\vec{u}\|^{2}$ None of the above

Suppose $\vec{u}, \vec{v}_{1}, \vec{v}_{2} \in \mathbb{R}^{n}$ are linearly independent vectors. Let the pair of real numbers $\left(a_{1}^{*}, a_{2}^{*}\right)$ be such ...

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6

TIFR ECE 2018 | Question: 6
Consider the system shown below. If $K>0$, which of the following describes the system? Stable, causal Stable, non-causal Unstable, non-causal Unstable, causal Cannot be determined from the information given

Consider the system shown below.If $K>0$, which of the following describes the system?Stable, causalStable, non-causalUnstable, non-causalUnstable, causalCannot be determ...

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7

TIFR ECE 2018 | Question: 7
Let $X_{1}, X_{2}$ and $X_{3}$ be independent random variables with uniform distribution over $[0, \theta]$. Consider the following statements. $E\left[\max \left\{X_{1}, X_{2}, X_{3}\right\}\right]=\frac{3}{4} \theta$ ... $\text{(i)}$ Only $\text{(ii)}$ Only $\text{(iii)}$ Only $\text{(iv)}$ All of $\text{(i) - (iv)}$

Let $X_{1}, X_{2}$ and $X_{3}$ be independent random variables with uniform distribution over $[0, \theta]$. Consider the following statements.$E\left[\max \left\{X_{1}, ...

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8

TIFR ECE 2018 | Question: 8
Let $A$ be an $n \times n$ real matrix for which two distinct non-zero $n$-dimensional real column vectors $v_{1}, v_{2}$ satisfy the relation $A v_{1}=A v_{2}$. Consider the following statements. At least one eigenvalue of $A$ is zero. $A$ ... $\text{(i)}$ Only $\text{(ii)}$ Only $\text{(iii)}$ Only $\text{(iv)}$ All of $\text{(i) - (iv)}$

Let $A$ be an $n \times n$ real matrix for which two distinct non-zero $n$-dimensional real column vectors $v_{1}, v_{2}$ satisfy the relation $A v_{1}=A v_{2}$. Consider...

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9

TIFR ECE 2018 | Question: 9
Let $X$ and $Y$ be two independent and identically distributed binary random variables that take values $\{-1,+1\}$ each with probability $1 / 2$. Let $Z_{1}=\max (X, Y)$, and $Z_{2}=\min (X, Y)$. Consider the following statements. $Z_{1}$ and $Z_{2}$ are uncorrelated ... $\text{(iii)}$ Both $\text{(i) and (ii), but not (iii)}$ All of $\text{(i), (ii) and (iii)}$

Let $X$ and $Y$ be two independent and identically distributed binary random variables that take values $\{-1,+1\}$ each with probability $1 / 2$. Let $Z_{1}=\max (X, Y)$...

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10

TIFR ECE 2018 | Question: 10
Suppose that $X_{1}$ and $X_{2}$ denote the random outcomes of independent rolls of two dice each of which takes six values $1,2,3,4,5,6$ with equal probability. What is the conditional expectation \[E\left[X_{1} \mid \max \left(X_{1}, X_{2}\right)=5\right]\] $3$ $4$ $35 / 9$ $5 / 2$ $15 / 4$

Suppose that $X_{1}$ and $X_{2}$ denote the random outcomes of independent rolls of two dice each of which takes six values $1,2,3,4,5,6$ with equal probability. What is ...

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11

TIFR ECE 2018 | Question: 11
Assume the following well known result: If a coin is flipped independently many times and its probability of heads $(H)$ is $p \in(0,1)$ and probability of tails $(T)$ is $(1-p)$, then the expected number of coin flips till the first time a heads is observed is $1 / p$. What is the ... $\frac{1}{1-(1-p)^{2}}(4+1 / p)$ $\frac{1}{p}+\frac{1}{1-p}$

Assume the following well known result: If a coin is flipped independently many times and its probability of heads $(H)$ is $p \in(0,1)$ and probability of tails $(T)$ is...

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12

TIFR ECE 2018 | Question: 12
Suppose that Amitabh Bachchan has ten coins in his pocket. $3$ coins have tails on both sides. $4$ coins have heads on both sides. $3$ coins have heads on one side and tails on the other and both the outcomes are equally likely when that coin is flipped. In a bet with Dharmendra ... that the other side of this coin is heads? $1 / 2$ $3 / 10$ $1 / 4$ $0.3$ $1 / 3$

Suppose that Amitabh Bachchan has ten coins in his pocket. $3$ coins have tails on both sides. $4$ coins have heads on both sides. $3$ coins have heads on one side and ta...

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13

TIFR ECE 2018 | Question: 13
Consider five distinct binary vectors $X_{1}, \ldots, X_{5}$ each of length $10$. Let \[d_{i j}=\sum_{k=1}^{10}\left(X_{i k} \text { XOR } X_{j k}\right),\] (i.e., $d_{i j}$ is the number of coordinates where $X_{i}$ ... to $X_{5}$, and argue about the result noting that there are five binary vectors.] $d=10$ $d=9$ $d=8$ $d<8$ Information is not sufficient

Consider five distinct binary vectors $X_{1}, \ldots, X_{5}$ each of length $10$. Let\[d_{i j}=\sum_{k=1}^{10}\left(X_{i k} \text { XOR } X_{j k}\right),\](i.e., $d_{i j}...

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14

TIFR ECE 2018 | Question: 14
Define the $\ell_{p}$ ball in two dimensions as the set of points $(x, y)$ such that $|x|^{p}+|y|^{p} \leq 1$. Which of the following is $\text{FALSE:}$ The $\ell_{2}$ ball is contained in the $\ell_{3}$ ball The $\ell_{2}$ ball is contained in ... $\ell_{2}$ ball is contained in the $\ell_{5}$ ball The $\ell_{1}$ ball is contained in the $\ell_{3}$ ball

Define the $\ell_{p}$ ball in two dimensions as the set of points $(x, y)$ such that $|x|^{p}+|y|^{p} \leq 1$. Which of the following is $\text{FALSE:}$The $\ell_{2}$ bal...

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15

TIFR ECE 2018 | Question: 15
Consider real-valued continuous functions $f:[0,2] \rightarrow(-\infty, \infty)$ and let \[A=\int_{0}^{1}|f(x)| d x \quad \text { and } B=\int_{1}^{2}|f(x)| d x .\] Which of the following is $\text{TRUE}?$ There exists an $f$ so that \[A+B<\int_{0}^{2} f(x) ... such that $\int_{0}^{1} f(x) d x=3$ There does not exist an $f$ so that \[A+B \leq-\int_{0}^{2} f(x) d x\]

Consider real-valued continuous functions $f:[0,2] \rightarrow(-\infty, \infty)$ and let\[A=\int_{0}^{1}|f(x)| d x \quad \text { and } B=\int_{1}^{2}|f(x)| d x .\]Which o...

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