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Recent questions tagged tifrece2021
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TIFR ECE 2021 | Question: 1
Consider a system with input $x(t)$ and output $y(t)$ such that \[y(t)=t \;x(t) .\] Consider the following statements: The system is linear. The system is time-invariant. The system is causal. Then which of the following is $\text{TRUE?}$ Only ... Only statement $3$ is correct. Only statements $1$ and $3$ are correct. All three statements $1, 2,$ and $3$ are correct.
Consider a system with input $x(t)$ and output $y(t)$ such that\[y(t)=t \;x(t) .\]Consider the following statements:The system is linear.The system is time-invariant.The ...
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46.4k
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Nov 30, 2022
Others
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2
TIFR ECE 2021 | Question: 2
Given a fixed perimeter of $1,$ among the following shapes, which one has the largest area? Square A regular pentagon A regular hexagon A regular septagon A regular octagon
Given a fixed perimeter of $1,$ among the following shapes, which one has the largest area?SquareA regular pentagonA regular hexagonA regular septagonA regular octagon
admin
46.4k
points
75
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admin
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Nov 30, 2022
Quantitative Aptitude
tifrece2021
quantitative-aptitude
geometry
area
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–
1
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0
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3
TIFR ECE 2021 | Question: 3
Consider the following statements: $\lim _{x \rightarrow 0} \frac{\sin x}{x}=1$. $\lim _{x \rightarrow 0} \frac{1-\cos x}{x^{2}}=1$. $\lim _{x \rightarrow 0} \frac{1-\cos x}{x}=1$. Which of the following is $\text{TRUE?}$ Only Statement $1$ ... $1$ and $3$ are correct. All of Statements $1, 2,$ and $3$ are correct. None of the three Statements $1,2,$ and $3$ are correct.
Consider the following statements:$\lim _{x \rightarrow 0} \frac{\sin x}{x}=1$.$\lim _{x \rightarrow 0} \frac{1-\cos x}{x^{2}}=1$.$\lim _{x \rightarrow 0} \frac{1-\cos x}...
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46.4k
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94
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Calculus
tifrece2021
calculus
limits
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1
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TIFR ECE 2021 | Question: 4
The first-order differential equation $\frac{d y(t)}{d t}+2 y(t)=x(t)$ describes a particular continuous-time system initially at rest at origin i.e., $x(0)=0$. Consider the following statements? System is memoryless. System is causal. System is stable. Which of the ... correct. All $(1), (2)$ and $(3)$ are correct. Only $(2)$ and $(3)$ are correct. None of the above
The first-order differential equation $\frac{d y(t)}{d t}+2 y(t)=x(t)$ describes a particular continuous-time system initially at rest at origin i.e., $x(0)=0$. Consider ...
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46.4k
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108
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Nov 30, 2022
Differential Equations
tifrece2021
differential-equations
first-order-differential-equation
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1
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TIFR ECE 2021 | Question: 5
Recall that \[\operatorname{sinc}(t)=\frac{\sin (\pi t)}{\pi t}\] and convolution of functions $x(t)$ and $y(t)$ is defined as \[x(t) \star y(t)=\int_{-\infty}^{\infty} x(t-\tau) y(\tau) d \tau .\] What is the necessary and sufficient condition on positive real ... \quad \text { for all real } t \text {. }\] $f<a$ $f>a$ $f<a^{-1}$ $f>a^{-1}$ None of the above
Recall that\[\operatorname{sinc}(t)=\frac{\sin (\pi t)}{\pi t}\]and convolution of functions $x(t)$ and $y(t)$ is defined as\[x(t) \star y(t)=\int_{-\infty}^{\infty} x(t-...
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TIFR ECE 2021 | Question: 6
Consider a fair coin (i.e., both heads and tails have equal probability of appearing). Suppose we toss the coin repeatedly until both sides have been seen. What is the expected number of times we would have seen heads? $1$ $5 / 4$ $3 / 2$ $2$ None of the above
Consider a fair coin (i.e., both heads and tails have equal probability of appearing). Suppose we toss the coin repeatedly until both sides have been seen. What is the ex...
admin
46.4k
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70
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admin
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Nov 30, 2022
Others
tifrece2021
probability-and-statistics
probability
conditional-probability
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1
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TIFR ECE 2021 | Question: 7
Consider the function \[f(y)=\int_{1}^{y} \frac{1}{1+x^{2}} d x-\log _{e}(1+y)\] where $\log _{e}(x)$ denotes the natural logarithm of $x$. Which of the following is true: The function $f(y)$ ... $y \geq 1$. The derivative of function $f(y)$ does not exist at $y=1$.
Consider the function\[f(y)=\int_{1}^{y} \frac{1}{1+x^{2}} d x-\log _{e}(1+y)\]where $\log _{e}(x)$ denotes the natural logarithm of $x$.Which of the following is true:Th...
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46.4k
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93
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admin
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Nov 30, 2022
Calculus
tifrece2021
calculus
definite-integrals
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1
votes
0
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8
TIFR ECE 2021 | Question: 8
The maximum area of a parallelogram inscribed in the ellipse (i.e. all the vertices of the parallelogram are on the ellipse) $x^{2}+4 y^{2}=1$ is: $2$ $4$ $1$ $5$ $3$
The maximum area of a parallelogram inscribed in the ellipse (i.e. all the vertices of the parallelogram are on the ellipse) $x^{2}+4 y^{2}=1$ is:$2$$4$$1$$5$$3$
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46.4k
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98
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admin
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Nov 30, 2022
Quantitative Aptitude
tifrece2021
quantitative-aptitude
geometry
area
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1
votes
0
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9
TIFR ECE 2021 | Question: 9
A stick of length $1$ is broken at a point chosen uniformly at random. Which of the following is false? Twice the length of the smaller piece is greater than the length of the larger piece with positive probability. One half of the length of the ... . The product of the length of the smaller piece and the larger piece is greater than $1 / 4$ with positive probability.
A stick of length $1$ is broken at a point chosen uniformly at random. Which of the following is false?Twice the length of the smaller piece is greater than the length of...
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46.4k
points
41
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Nov 30, 2022
Probability and Statistics
tifrece2021
probability-and-statistics
probability
uniform-distribution
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1
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0
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TIFR ECE 2021 | Question: 10
Suppose $\vec{u}, \vec{v}_{1}, \vec{v}_{2} \in \mathbb{R}^{n}$. Let the real number $a_{1}^{*}$ be such that it solves the following optimization problem \[d_{1}=\min _{a_{1} \in \mathbb{R}}\left\|\vec{u}-a_{1} \vec{v}_{1}\right\|,\] where we denote the length ... $\left\|\vec{u}-\left(\vec{p}_{2}-\vec{p}_{1}\right)\right\|$ $0$
Suppose $\vec{u}, \vec{v}_{1}, \vec{v}_{2} \in \mathbb{R}^{n}$. Let the real number $a_{1}^{*}$ be such that it solves the following optimization problem\[d_{1}=\min _{a_...
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46.4k
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79
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admin
asked
Nov 30, 2022
Calculus
tifrece2021
vector-analysis
vector-in-planes
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1
votes
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answers
11
TIFR ECE 2021 | Question: 11
Suppose that $X_{1}$ and $X_{2}$ denote the output of rolls of two independent dices that can each take integer values $\{1,2,3,4,5,6\}$ with probability $1 / 6$ for each outcome. Further, $U$ denotes a continuous random variable that is independent of $X_{1}$ and $X_{2}$ ... on this sum what is the probability that $X_{1}$ equals $2?$ $2.21$ $3$ $1 / 6$ $1 / 5$ $1 / 3$
Suppose that $X_{1}$ and $X_{2}$ denote the output of rolls of two independent dices that can each take integer values $\{1,2,3,4,5,6\}$ with probability $1 / 6$ for each...
admin
46.4k
points
87
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admin
asked
Nov 30, 2022
Probability and Statistics
tifrece2021
probability-and-statistics
probability
random-variable
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1
votes
0
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12
TIFR ECE 2021 | Question: 12
An ant does a random walk in a two dimensional plane starting at the origin at time $0.$ At every integer time greater than $0,$ it moves one centimeter away from its earlier position in a random direction independent of its past. After $4$ steps, what is the expected square of the distance (measured in centimeters) from its starting point? $4$ $1$ $2$ $\pi$ $0$
An ant does a random walk in a two dimensional plane starting at the origin at time $0.$ At every integer time greater than $0,$ it moves one centimeter away from its ear...
admin
46.4k
points
103
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admin
asked
Nov 30, 2022
Quantitative Aptitude
tifrece2021
quantitative-aptitude
geometry
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1
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0
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13
TIFR ECE 2021 | Question: 13
Consider a unit Euclidean ball in $4$ dimensions, and let $V_{n}$ be its volume and $S_{n}$ its surface area. Then $S_{n} / V_{n}$ is equal to: $1$ $4$ $5$ $2$ $3$
Consider a unit Euclidean ball in $4$ dimensions, and let $V_{n}$ be its volume and $S_{n}$ its surface area. Then $S_{n} / V_{n}$ is equal to:$1$$4$$5$$2$$3$
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14
TIFR ECE 2021 | Question: 14
A tourist starts by taking one of the $n$ available paths, denoted by $1,2, \cdots, n$. An hour into the journey, the path $i$ subdivides into further $1+i$ subpaths, only one of which leads to the destination. The tourist has no map and makes random choices of the path and the ... $\frac{10}{36}$ $\frac{11}{36}$ $\frac{12}{36}$ $\frac{13}{36}$ $\frac{14}{36}$
A tourist starts by taking one of the $n$ available paths, denoted by $1,2, \cdots, n$. An hour into the journey, the path $i$ subdivides into further $1+i$ subpaths, onl...
admin
46.4k
points
70
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admin
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Nov 30, 2022
Probability and Statistics
tifrece2021
probability-and-statistics
probability
random-variable
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1
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0
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15
TIFR ECE 2021 | Question: 15
We have the sequence, $a_{n}=\frac{1}{n \log ^{2} n}, n \geq 2$, where log is the logarithm to the base $2$ and let $A=\sum_{n=2}^{\infty} a_{n}$ ... $H(X)?$ $H(X) \leq 3$ $H(X) \in(3,5]$ $H(X) \in(5,10]$ $H(X)>10$ but finite $H(X)$ is unbounded
We have the sequence, $a_{n}=\frac{1}{n \log ^{2} n}, n \geq 2$, where log is the logarithm to the base $2$ and let $A=\sum_{n=2}^{\infty} a_{n}$ be the sum of the sequen...
admin
46.4k
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73
views
admin
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Nov 30, 2022
Probability and Statistics
tifrece2021
probability-and-statistics
probability
random-variable
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