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Most answered questions in Engineering Mathematics
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121
TIFR ECE 2019 | Question: 11
Let $X$ and $Y$ be independent Gaussian random variables with means $1$ and $2$ and variances $3$ and $4$ respectively. What is the minimum possible value of $\mathbf{E}\left[(X+Y-t)^{2}\right]$, when $t$ varies over all real numbers? $7$ $5$ $1.5$ $3.5$ $2.5$
Let $X$ and $Y$ be independent Gaussian random variables with means $1$ and $2$ and variances $3$ and $4$ respectively. What is the minimum possible value of $\mathbf{E}\...
admin
46.4k
points
36
views
admin
asked
Nov 30, 2022
Probability and Statistics
tifrece2019
probability-and-statistics
probability
expectation
+
–
1
votes
0
answers
122
TIFR ECE 2019 | Question: 12
Consider an urn with $a$ red and $b$ blue balls. Balls are drawn out one-by-one, without replacement and uniformly at random, until the first red ball is drawn. What is the expected total number of balls drawn by this process? (Hint: Consider deriving an appropriate recurrence.) $\frac{a+b}{a+1}$ $\frac{a+b+1}{a}$ $\frac{a+b}{a}$ $\frac{a+b+1}{a+1}$ $a$
Consider an urn with $a$ red and $b$ blue balls. Balls are drawn out one-by-one, without replacement and uniformly at random, until the first red ball is drawn. What is t...
admin
46.4k
points
32
views
admin
asked
Nov 30, 2022
Probability and Statistics
tifrece2019
probability-and-statistics
probability
expectation
+
–
1
votes
0
answers
123
TIFR ECE 2019 | Question: 14
Consider the circle of radius $1$ centred at the origin in two dimensions. Choose two points $x$ and $y$ independently at random so that both are uniformly distributed on the circle. Let the vectors joining the origin to $x$ and $y$ be $X$ and $Y$, respectively. Let $\theta$ be ... $\mathbf{E}\left[|x-y|^{2}\right]=\sqrt{3}$ $\mathbf{E}\left[|x-y|^{2}\right]=1$
Consider the circle of radius $1$ centred at the origin in two dimensions. Choose two points $x$ and $y$ independently at random so that both are uniformly distributed on...
admin
46.4k
points
27
views
admin
asked
Nov 30, 2022
Probability and Statistics
tifrece2019
probability-and-statistics
probability
uniform-distribution
+
–
1
votes
0
answers
124
TIFR ECE 2019 | Question: 15
Anu reached a bus stop at $\text{9:00 AM.}$ She knows that the number of minutes after $\text{9:00 AM}$ when the bus will arrive is distributed with probability density function (p.d.f.) $f$ where \[f(x)=\frac{1}{10} \exp (-x / 10)\] for $x \geq 0$, ... time, measured in minutes after $\text{9:00 AM,}$ would Anu expect the bus to arrive? $12.5$ $15$ $7.5$ $10$ $12.5$
Anu reached a bus stop at $\text{9:00 AM.}$ She knows that the number of minutes after $\text{9:00 AM}$ when the bus will arrive is distributed with probability density f...
admin
46.4k
points
34
views
admin
asked
Nov 30, 2022
Probability and Statistics
tifrece2019
probability-and-statistics
probability
probability-density-function
+
–
1
votes
0
answers
125
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...
admin
46.4k
points
90
views
admin
asked
Nov 29, 2022
Probability and Statistics
tifrece2018
probability-and-statistics
probability
expectation
+
–
1
votes
0
answers
126
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...
admin
46.4k
points
124
views
admin
asked
Nov 29, 2022
Calculus
tifrece2018
calculus
limits
+
–
1
votes
0
answers
127
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 $f(x)$ ...
admin
46.4k
points
82
views
admin
asked
Nov 29, 2022
Calculus
tifrece2018
calculus
limits
+
–
1
votes
0
answers
128
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 ...
admin
46.4k
points
94
views
admin
asked
Nov 29, 2022
Vector Analysis
tifrece2018
vector-analysis
vector-in-planes
+
–
1
votes
0
answers
129
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}, ...
admin
46.4k
points
94
views
admin
asked
Nov 29, 2022
Probability and Statistics
tifrece2018
probability-and-statistics
probability
uniform-distribution
+
–
1
votes
0
answers
130
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...
admin
46.4k
points
92
views
admin
asked
Nov 29, 2022
Linear Algebra
tifrece2018
linear-algebra
matrices
+
–
1
votes
0
answers
131
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 $Z_{1}=\max (X, Y)$...
admin
46.4k
points
117
views
admin
asked
Nov 29, 2022
Probability and Statistics
tifrece2018
probability-and-statistics
probability
random-variable
+
–
1
votes
0
answers
132
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 ...
admin
46.4k
points
100
views
admin
asked
Nov 29, 2022
Probability and Statistics
tifrece2018
probability-and-statistics
probability
expectation
+
–
1
votes
0
answers
133
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...
admin
46.4k
points
95
views
admin
asked
Nov 29, 2022
Probability and Statistics
tifrece2018
probability-and-statistics
probability
expectation
+
–
1
votes
0
answers
134
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...
admin
46.4k
points
113
views
admin
asked
Nov 29, 2022
Probability and Statistics
tifrece2018
probability-and-statistics
probability
conditional-probability
+
–
1
votes
0
answers
135
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...
admin
46.4k
points
101
views
admin
asked
Nov 29, 2022
Calculus
tifrece2018
calculus
definite-integrals
+
–
1
votes
0
answers
136
TIFR ECE 2017 | Question: 6
Let $a, b \in\{0,1\}$. Consider the following statements where $*$ is the $\text{AND}$ operator, $\oplus$ is $\text{EXCLUSIVE-OR,}$ and ${ }^{c}$ denotes the complement function. $\max \left\{a * b, b \oplus a^{\mathrm{c}}\right\}=1$ ... $\text{(iii)}$ only $\text{(iii)}$ and $\text{(iv)}$ only $\text{(iv)}$ and $\text{(i)}$ only None of the above
Let $a, b \in\{0,1\}$. Consider the following statements where $*$ is the $\text{AND}$ operator, $\oplus$ is $\text{EXCLUSIVE-OR,}$ and ${ }^{c}$ denotes the complement f...
admin
46.4k
points
134
views
admin
asked
Nov 29, 2022
Calculus
tifrece2017
calculus
functions
+
–
1
votes
0
answers
137
TIFR ECE 2017 | Question: 7
A circulant matrix is a square matrix whose each row is the preceding row rotated to the right by one element, e.g., the following is a $3 \times 3$ circulant matrix. \[\left(\begin{array}{lll} 1 & 2 & 3 \\ 3 & 1 & 2 \\ 2 & 3 & 1 \ ... $j=\sqrt{-1}$ A vector whose $k$-th element is $\sinh \left(\frac{2 \pi k}{n}\right)$ None of the above
A circulant matrix is a square matrix whose each row is the preceding row rotated to the right by one element, e.g., the following is a $3 \times 3$ circulant matrix.\[\l...
admin
46.4k
points
88
views
admin
asked
Nov 29, 2022
Linear Algebra
tifrece2017
linear-algebra
eigen-values
+
–
1
votes
0
answers
138
TIFR ECE 2017 | Question: 9
Recall that for a random variable $X$ which takes values in $\mathbb{N}$, the set of natural numbers, its entropy in bits is defined as \[H(X)=\sum_{n=1}^{\infty} p_{n} \log _{2} \frac{1}{p_{n}},\] where, for $n \in \mathbb{N}, p_{n}$ denotes the ... entropy of $X$ in bits? $1$ $1.5$ $\frac{1+\sqrt{5}}{2} \approx 1.618$ (the golden ratio) $2$ None of the above
Recall that for a random variable $X$ which takes values in $\mathbb{N}$, the set of natural numbers, its entropy in bits is defined as\[H(X)=\sum_{n=1}^{\infty} p_{n} \l...
admin
46.4k
points
74
views
admin
asked
Nov 29, 2022
Probability and Statistics
tifrece2017
probability-and-statistics
probability
random-variable
+
–
1
votes
0
answers
139
TIFR ECE 2017 | Question: 10
Consider a single coin where the probability of heads is $p \in(0,1)$ and probability of tails is $1-p$. Suppose that this coin is flipped an infinite number of times. Let $N_{1}$ denote the number of flips till we see heads for the first time. Let $N_{2}$ denote the number of flips after ... $\frac{2}{p}$ $\frac{1}{p^{2}+(1-p)^{2}}$ $\frac{2}{p(1-p)}$
Consider a single coin where the probability of heads is $p \in(0,1)$ and probability of tails is $1-p$. Suppose that this coin is flipped an infinite number of times. Le...
admin
46.4k
points
92
views
admin
asked
Nov 29, 2022
Probability and Statistics
tifrece2017
probability-and-statistics
probability
expectation
+
–
1
votes
0
answers
140
TIFR ECE 2017 | Question: 11
Consider a unit length interval $[0,1]$ and choose a point $X$ on it with uniform distribution. Let $L_{1}=X$ and $L_{2}=1-X$ be the length of the two sub-intervals created by this point on the unit interval. Let $L=\max \left\{L_{1}, L_{2}\right\}$. Consider ... $\text{(ii)}$ Only $\text{(i)}$ and $\text{(iii)}$ Only $\text{(ii)}$ and $\text{(iv)}$ None of the above
Consider a unit length interval $[0,1]$ and choose a point $X$ on it with uniform distribution. Let $L_{1}=X$ and $L_{2}=1-X$ be the length of the two sub-intervals creat...
admin
46.4k
points
79
views
admin
asked
Nov 29, 2022
Probability and Statistics
tifrece2017
probability-and-statistics
probability
uniform-distribution
+
–
1
votes
0
answers
141
TIFR ECE 2017 | Question: 12
Consider a signal $X$ that can take two values, $-1$ with probability $p$ and $+1$ with probability $1-p$. Let $Y=X+N$, where $N$ is mean zero random noise that has probability density function symmetric about $0.$ Given $p$ and on observing $Y$, the detection problem is ... $\text{(iii)}$ Only $\text{(i)}$ and $\text{(ii)}$ Only $\text{(i)}$ and $\text{(iii)}$
Consider a signal $X$ that can take two values, $-1$ with probability $p$ and $+1$ with probability $1-p$. Let $Y=X+N$, where $N$ is mean zero random noise that has proba...
admin
46.4k
points
83
views
admin
asked
Nov 29, 2022
Probability and Statistics
tifrece2017
probability-and-statistics
probability
probability-density-function
+
–
1
votes
0
answers
142
TIFR ECE 2017 | Question: 13
Let $A$ be an $n \times n$ matrix. Consider the following statements. $A$ can have full-rank even if there exists two vectors $v_{1} \neq v_{2}$ such that $A v_{1}=A v_{2}$. $A$ can be similar to the identity matrix, when $A$ is not the identity matrix. Recall that ... $\text{(ii)}$ Only $\text{(iii)}$ $\text{(i), (ii),}$ and $\text{(iii)}$ None of the above
Let $A$ be an $n \times n$ matrix. Consider the following statements.$A$ can have full-rank even if there exists two vectors $v_{1} \neq v_{2}$ such that $A v_{1}=A v_{2}...
admin
46.4k
points
86
views
admin
asked
Nov 29, 2022
Linear Algebra
tifrece2017
linear-algebra
matrices
+
–
1
votes
0
answers
143
TIFR ECE 2017 | Question: 14
Consider the positive integer sequence \[x_{n}=n^{50} e^{-(\log (n))^{3 / 2}}, \quad n=1,2,3, \ldots\] Which of the following statements is $\text{TRUE?}$ For every $M>0$, there exists an $n$ such that $x_{n}>M$ ... and then increases with $n \geq 1$ Sequence $\left\{x_{n}\right\}$ eventually converges to zero as $n \rightarrow \infty$ None of the above
Consider the positive integer sequence\[x_{n}=n^{50} e^{-(\log (n))^{3 / 2}}, \quad n=1,2,3, \ldots\]Which of the following statements is $\text{TRUE?}$For every $M>0$, t...
admin
46.4k
points
88
views
admin
asked
Nov 29, 2022
Calculus
tifrece2017
calculus
maxima-minima
+
–
1
votes
0
answers
144
TIFR ECE 2016 | Question: 1
Suppose $f(x)=c x^{-\alpha}$ for some $c>0$ and $\alpha>0$ such that $\int_{1}^{\infty} f(x) \mathrm{d} x=1$. Then, which of the following is possible? $\int_{1}^{\infty} x f(x) \mathrm{d} x=\infty$ ... $\int_{1}^{\infty} \frac{f(x)}{1+\ln x} \mathrm{~d} x=\infty$ None of the above
Suppose $f(x)=c x^{-\alpha}$ for some $c>0$ and $\alpha>0$ such that $\int_{1}^{\infty} f(x) \mathrm{d} x=1$. Then, which of the following is possible?$\int_{1}^{\infty} ...
admin
46.4k
points
87
views
admin
asked
Nov 29, 2022
Calculus
tifrece2016
calculus
definite-integrals
+
–
1
votes
0
answers
145
TIFR ECE 2016 | Question: 2
Let $X_{1}$ and $X_{2}$ be two independent continuous real-valued random variables taking values in the unit interval $[0,1]$. Let $Y=\max \left\{X_{1}, X_{2}\right\}$ ... $\operatorname{Pr}[Z=1]>\operatorname{Pr}[Z=2]=\frac{1}{2}$ $\operatorname{Pr}[Z=1]<\operatorname{Pr}[Z=2]$
Let $X_{1}$ and $X_{2}$ be two independent continuous real-valued random variables taking values in the unit interval $[0,1]$. Let $Y=\max \left\{X_{1}, X_{2}\right\}$ an...
admin
46.4k
points
83
views
admin
asked
Nov 29, 2022
Probability and Statistics
tifrece2016
probability-and-statistics
probability
random-variable
+
–
1
votes
0
answers
146
TIFR ECE 2016 | Question: 3
Let $(X, Y)$ be a pair of independent random variables. Suppose $X$ takes values in $\{1, \ldots, 6\}$ with equal probability, and $Y$ takes values in $\{2,3\}$ with $\operatorname{Pr}[Y=2]=p$. Let $Z=(X \bmod Y)+1$ ... $\operatorname{Pr}[Z=1]=\frac{1}{2}$ for $p=\frac{1}{2}$ $\operatorname{Pr}[Z=1]=p(1-p)$ None of the above
Let $(X, Y)$ be a pair of independent random variables. Suppose $X$ takes values in $\{1, \ldots, 6\}$ with equal probability, and $Y$ takes values in $\{2,3\}$ with $\op...
admin
46.4k
points
100
views
admin
asked
Nov 29, 2022
Probability and Statistics
tifrece2016
probability-and-statistics
probability
random-variable
+
–
1
votes
0
answers
147
TIFR ECE 2016 | Question: 7
Suppose $X$ and $Y$ are independent Gaussian random variables, whose pdfs are represented below. Which of the following describes the pdf of the $X+Y?$
Suppose $X$ and $Y$ are independent Gaussian random variables, whose pdfs are represented below. Which of the following describes the pdf of the $X+Y?$
admin
46.4k
points
100
views
admin
asked
Nov 29, 2022
Probability and Statistics
tifrece2016
+
–
1
votes
0
answers
148
TIFR ECE 2016 | Question: 9
Suppose $Y=X+Z$, where $X$ and $Z$ are independent zero-mean random variables each with variance $1.$ Let $\hat{X}(Y)=a Y$ be the optimal linear least-squares estimate of $X$ from $Y$, i.e., $a$ is chosen such that $E\left[(X-a Y)^{2}\right]$ is minimized. What is the resulting ... $1$ $\frac{2}{3}$ $\frac{1}{2}$ $\frac{1}{3}$ $\frac{1}{4}$
Suppose $Y=X+Z$, where $X$ and $Z$ are independent zero-mean random variables each with variance $1.$ Let $\hat{X}(Y)=a Y$ be the optimal linear least-squares estimate of...
admin
46.4k
points
78
views
admin
asked
Nov 29, 2022
Probability and Statistics
tifrece2016
probability-and-statistics
probability
random-variable
+
–
1
votes
0
answers
149
TIFR ECE 2016 | Question: 10
Let $U_{1}, U_{2}, U_{3}$ be independent random variables that are each uniformly distributed between zero and one. What is the probability that the second highest value amongst the three lies between $1 / 3$ and $2 / 3?$ $\frac{2}{9}$ $\frac{1}{27}$ $\frac{13}{27}$ $\frac{1}{3}$ $\frac{7}{18}$
Let $U_{1}, U_{2}, U_{3}$ be independent random variables that are each uniformly distributed between zero and one. What is the probability that the second highest value ...
admin
46.4k
points
82
views
admin
asked
Nov 29, 2022
Probability and Statistics
tifrece2016
probability-and-statistics
probability
uniform-distribution
+
–
1
votes
0
answers
150
TIFR ECE 2016 | Question: 11
Suppose that a random variable $X$ has a probability density function (pdf) given by \[f(x)=c \exp (-2 x)\] for $x \geq 1$, and $f(x)=0$, for $x<1$, where $c$ is an appropriate constant so that $f(x)$ is a valid pdf. What is the expected value of $X$ given that $X \geq 5?$ $5 \frac{1}{2}$ $7$ $10$ $8 \frac{1}{2}$ $6$
Suppose that a random variable $X$ has a probability density function (pdf) given by\[f(x)=c \exp (-2 x)\]for $x \geq 1$, and $f(x)=0$, for $x<1$, where $c$ is an appropr...
admin
46.4k
points
31
views
admin
asked
Nov 29, 2022
Probability and Statistics
tifrece2016
probability-and-statistics
probability
expectation
+
–
1
votes
0
answers
151
TIFR ECE 2016 | Question: 12
Recall that the Shannon entropy of a random variables $X$ taking values in a finite set $S$ is given by \[H[X]=-\sum_{x \in S} \operatorname{Pr}[X=x] \log _{2} \operatorname{Pr}[X=x] .\] (We set $0 \log _{2} 0=0$.) For a pair of random variables $(X, Y)$ taking ... $H\left[R_{513}, C_{513} \mid R_{1}, R_{2}, \ldots, R_{512}\right]?$ $\log _{2} 513$ $9$ $10$ $19$ $81$
Recall that the Shannon entropy of a random variables $X$ taking values in a finite set $S$ is given by\[H[X]=-\sum_{x \in S} \operatorname{Pr}[X=x] \log _{2} \operatorna...
admin
46.4k
points
28
views
admin
asked
Nov 29, 2022
Probability and Statistics
tifrece2016
probability-and-statistics
probability
random-variable
+
–
1
votes
0
answers
152
TIFR ECE 2016 | Question: 13
Suppose $m$ and $n$ are positive integers, $m \neq n$, and $A$ is an $m \times n$ matrix with real entires. Consider the following statements. $\operatorname{rank}\left(A A^{T}\right)=\operatorname{rank}\left(A^{T} A\right)$ ... Which of the above statements is true for all such $A?$ Only (i) Only (ii) Only (iii) (i) and (iii) None of them
Suppose $m$ and $n$ are positive integers, $m \neq n$, and $A$ is an $m \times n$ matrix with real entires. Consider the following statements.$\operatorname{rank}\left(A ...
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Linear Algebra
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matrices
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TIFR ECE 2016 | Question: 14
Consider matrices $A \in \mathbb{R}^{n \times m}, B \in \mathbb{R}^{m \times m}$, and $C \in \mathbb{R}^{m \times n}$. Let $r=\operatorname{rank}(A B C)$. Which of the following must be true? $\min (m, n) \leq r \leq \max (m, n)$ ... $\min (m, n) \leq r \leq \max (\operatorname{rank}(A), \operatorname{rank}(B), \operatorname{rank}(C))$ None of the above
Consider matrices $A \in \mathbb{R}^{n \times m}, B \in \mathbb{R}^{m \times m}$, and $C \in \mathbb{R}^{m \times n}$. Let $r=\operatorname{rank}(A B C)$. Which of the fo...
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Linear Algebra
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linear-algebra
rank-of-matrix
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154
TIFR ECE 2016 | Question: 15
What is \[ \max _{x, y}\left[\begin{array}{ll} x & y \end{array}\right]\left[\begin{array}{cc} 3 & \sqrt{2} \\ \sqrt{2} & 2 \end{array}\right]\left[\begin{array}{l} x \\ y \end{array}\right] \] subject to \[ x^{2}+y^{2}=1 ? \] $1$ $\sqrt{2}$ $2$ $3$ $4$
What is\[\max _{x, y}\left[\begin{array}{ll}x & y\end{array}\right]\left[\begin{array}{cc}3 & \sqrt{2} \\\sqrt{2} & 2\end{array}\right]\left[\begin{array}{l}x \\y\end{arr...
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Linear Algebra
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linear-algebra
system-of-equations
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155
GATE ECE 2009 | Question: 1
The order of the differential equation $\dfrac{d^{2} y}{d t^{2}}+\left(\dfrac{d y}{d t}\right)^{3}+y^{4}=e^{-t} \quad$ is $1$ $2$ $3$ $4$
The order of the differential equation $\dfrac{d^{2} y}{d t^{2}}+\left(\dfrac{d y}{d t}\right)^{3}+y^{4}=e^{-t} \quad$ is$1$$2$$3$$4$
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Differential Equations
gate2009-ec
differential-equations
second-order-differential-equation
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156
GATE ECE 2010 | Question: 1
The eigenvalues of a skew-symmetric matrix are always zero always pure imaginary either zero or pure imaginary always real
The eigenvalues of a skew-symmetric matrix arealways zeroalways pure imaginaryeither zero or pure imaginaryalways real
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Linear Algebra
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linear-algebra
eigen-values
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157
GATE ECE 2010 | Question: 3
A function $n(x)$ satisfies the differential equation $\frac{d^{2} n(x)}{d x^{2}}-\frac{n(x)}{L^{2}}=0$ where $L$ is a constant. The boundary conditions are: $n(0)=K$ and $n(\infty)=0$. The solution to this equation is $n(x)=K \exp (x / L)$ $n(x)=K \exp (-x / \sqrt{L})$ $n(x)=K^{2} \exp (-x / L)$ $n(x)=K \exp (-x / L)$
A function $n(x)$ satisfies the differential equation $\frac{d^{2} n(x)}{d x^{2}}-\frac{n(x)}{L^{2}}=0$ where $L$ is a constant. The boundary conditions are: $n(0)=K$ and...
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46.4k
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Differential Equations
gate2010-ec
differential-equations
second-order-differential-equation
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1
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0
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158
GATE ECE 2010 | Question: 26
If $e^{y}=x^{\frac{1}{x}}$, then $y$ has a maximum at $x=e$ minimum at $x=e$ maximum at $x=e^{-1}$ minimum at $x=e^{-1}$
If $e^{y}=x^{\frac{1}{x}}$, then $y$ has amaximum at $x=e$minimum at $x=e$maximum at $x=e^{-1}$minimum at $x=e^{-1}$
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Calculus
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calculus
maxima-minima
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159
GATE ECE 2010 | Question: 27
A fair coin is tossed independently four times. The probability of the event "the number of times heads show up is more than the number of times tails show up" is $\frac{1}{16}$ $\frac{1}{8}$ $\frac{1}{4}$ $\frac{5}{16}$
A fair coin is tossed independently four times. The probability of the event "the number of times heads show up is more than the number of times tails show up" is$\frac{1...
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Probability and Statistics
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probability-and-statistics
probability
independent-events
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1
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0
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160
GATE ECE 2010 | Question: 30
Consider a differential equation $\dfrac{d y(x)}{d x}-y(x)=x$ with the initial condition $y(0)=0$. Using Euler's first order method with a step size of $0.1$, the value of $y(0.3)$ is $0.01$ $0.031$ $0.0631$ $0.1$
Consider a differential equation $\dfrac{d y(x)}{d x}-y(x)=x$ with the initial condition $y(0)=0$. Using Euler's first order method with a step size of $0.1$, the value o...
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Differential Equations
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differential-equations
first-order-differential-equation
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