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Recent questions tagged tifr2012
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TIFR ECE 2012 | Question: 1
The minimum value of $f(x)=\ln \left(1+\exp \left(x^{2}-3 x+2\right)\right)$ for $x \geq 0$, where $\ln (\cdot)$ denotes the natural logarithm, is $\ln \left(1+e^{-1 / 4}\right)$ $\ln (5 / 3)$ $0$ $\ln \left(1+e^{2}\right)$ None of the above
The minimum value of $f(x)=\ln \left(1+\exp \left(x^{2}-3 x+2\right)\right)$ for $x \geq 0$, where $\ln (\cdot)$ denotes the natural logarithm, is$\ln \left(1+e^{-1 / 4}\...
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Dec 8, 2022
Calculus
tifr2012
calculus
maxima-minima
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1
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TIFR ECE 2012 | Question: 2
Let $\alpha_{1}, \alpha_{2}, \cdots, \alpha_{k}$ be complex numbers. Then \[ \lim _{n \rightarrow \infty}\left|\sum_{i=1}^{k} \alpha_{i}^{n}\right|^{1 / n} \] is $0$ $\infty$ $\alpha_{k}$ $\alpha_{1}$ $\max _{j}|\alpha_{j}|$
Let $\alpha_{1}, \alpha_{2}, \cdots, \alpha_{k}$ be complex numbers. Then\[\lim _{n \rightarrow \infty}\left|\sum_{i=1}^{k} \alpha_{i}^{n}\right|^{1 / n}\]is$0$$\infty$$\...
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46.4k
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90
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Dec 8, 2022
Calculus
tifr2012
calculus
limits
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1
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3
TIFR ECE 2012 | Question: 3
A sequence of numbers $\left(x_{n}: n=1,2,3, \ldots\right)$ is said to have a limit $x$, if given any number $\epsilon>0$, there exists an integer $n_{\epsilon}$ ... $6$ and has a limit that equals $6$ . None of the above statements are true.
A sequence of numbers $\left(x_{n}: n=1,2,3, \ldots\right)$ is said to have a limit $x$, if given any number $\epsilon>0$, there exists an integer $n_{\epsilon}$ such tha...
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46.4k
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149
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Dec 8, 2022
Others
tifr2012
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1
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4
TIFR ECE 2012 | Question: 4
The signal $x_{n}=0$ for $n<0$ and $x_{n}=a^{n} / n$ ! for $n \geq 0$. Its $z$-transform $X(z)=\sum_{n=-\infty}^{\infty} x_{n} z^{-n}$ is $1 /\left(z^{-1}-a\right)$, region of convergence $\text{(ROC)}$: $|z| \leq 1 / a$ ... $|z|>a$ Item $(a)$ if $a>1$, Item $(b)$ if $a<1$ $\exp \left(a z^{-1}\right)$, $\text{ROC}$: entire complex plane.
The signal $x_{n}=0$ for $n<0$ and $x_{n}=a^{n} / n$ ! for $n \geq 0$. Its $z$-transform $X(z)=\sum_{n=-\infty}^{\infty} x_{n} z^{-n}$ is$1 /\left(z^{-1}-a\right)$, regio...
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Dec 8, 2022
Others
tifr2012
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1
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5
TIFR ECE 2012 | Question: 5
Consider a periodic square wave $f(t)$ with a period of $1$ second such that $f(t)=1$ for $t \in[0,1 / 2)$ and $f(t)=-1$ for $t \in[1 / 2,1)$. It is passed through an ideal low-pass filter with cutoff at $2 \mathrm{~Hz}$. Then the output is $\sin (2 \pi t)$ ... $\sin (2 \pi t)-\cos (2 \pi t)$ None of the above
Consider a periodic square wave $f(t)$ with a period of $1$ second such that $f(t)=1$ for $t \in[0,1 / 2)$ and $f(t)=-1$ for $t \in[1 / 2,1)$. It is passed through an ide...
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Dec 8, 2022
Others
tifr2012
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1
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TIFR ECE 2012 | Question: 6
Let $u(t)$ be the unit step function that takes value $1$ for $t \geq 0$ and is zero otherwise. Let $f(t)=e^{-t} u(t)$ and $g(t)=u(t) u(1-t)$. Then the convolution of $f(t)$ and $g(t)$ is $(e-1) e^{-t} u(t)$ $1-e^{-t}$ for $0 \leq t \leq 1,(e-1) e^{-t}$ for $t \geq 1$ and zero otherwise $t e^{-t} u(t)$ The convolution integral is not well defined None of the above
Let $u(t)$ be the unit step function that takes value $1$ for $t \geq 0$ and is zero otherwise. Let $f(t)=e^{-t} u(t)$ and $g(t)=u(t) u(1-t)$. Then the convolution of $f(...
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Dec 8, 2022
Others
tifr2012
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TIFR ECE 2012 | Question: 7
A linear time-invariant system has a transfer function $H(s)=1 /(1+s)$. If the input to the system is $\cos (t)$, the output is $\left(e^{j t}+e^{-j t}\right) / 2$ where $j=\sqrt{-1}$ $\cos (t) / 2$ $(\cos (t)+\sin (t)) / 2 \sqrt{ }$ $\sin (t) / 2$. The system is unstable and the output is not well-defined.
A linear time-invariant system has a transfer function $H(s)=1 /(1+s)$. If the input to the system is $\cos (t)$, the output is$\left(e^{j t}+e^{-j t}\right) / 2$ where $...
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46.4k
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admin
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Dec 8, 2022
Others
tifr2012
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TIFR ECE 2012 | Question: 8
The input to a series $\text{RLC}$ circuit is a sinusoidal voltage source and the output is the current in the circuit. Which of the following is true about the magnitude frequency response of this system? Dependending on the values of $\text{R, L}$ ... $1 /(2 \pi \sqrt{\text{LC}})$.
The input to a series $\text{RLC}$ circuit is a sinusoidal voltage source and the output is the current in the circuit. Which of the following is true about the magnitude...
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46.4k
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96
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Dec 8, 2022
Others
tifr2012
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TIFR ECE 2012 | Question: 9
$x(t)$ is a signal of bandwidth $4 \mathrm{~kHz}$. It was sampled at a rate of $16 \mathrm{~kHz}$. \[ x_{n}=x(n T), \quad n \text { integer, } \quad T=\frac{1}{16} \mathrm{~ms} . \] Due to a data handling error alternate samples were erased ... $y(t)$ over a low pass filter of bandwidth $4\text{ KHz}$ any of the above none of the above
$x(t)$ is a signal of bandwidth $4 \mathrm{~kHz}$. It was sampled at a rate of $16 \mathrm{~kHz}$.\[x_{n}=x(n T), \quad n \text { integer, } \quad T=\frac{1}{16} \mathrm{...
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46.4k
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90
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Dec 8, 2022
Others
tifr2012
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10
TIFR ECE 2012 | Question: 10
Suppose three dice are rolled independently. Each dice can take values $1$ to $6$ with equal probability. Find the probability that the second highest outcome equals the average of the other two outcomes. Here, the ties may be resolved arbitrarily. $1 / 6$ $1 / 9$ $39 / 216$ $7 / 36$ $43 / 216$
Suppose three dice are rolled independently. Each dice can take values $1$ to $6$ with equal probability. Find the probability that the second highest outcome equals the ...
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46.4k
points
87
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Dec 8, 2022
Probability and Statistics
tifr2012
probability-and-statistics
probability
independent-events
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1
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0
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TIFR ECE 2012 | Question: 11
A Poisson random variable $X$ is given by $\operatorname{Pr}\{X=k\}=\mathrm{e}^{-\lambda} \lambda^{k} / k !, k=0,1,2, \ldots$ for $\lambda>0$. The variance of $X$ scales as $\lambda$ $\lambda^{2}$ $\lambda^{3}$ $\sqrt{\lambda}$ None of the above
A Poisson random variable $X$ is given by $\operatorname{Pr}\{X=k\}=\mathrm{e}^{-\lambda} \lambda^{k} / k !, k=0,1,2, \ldots$ for $\lambda>0$. The variance of $X$ scales ...
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46.4k
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76
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Dec 8, 2022
Probability and Statistics
tifr2012
probability-and-statistics
probability
poisson-distribution
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1
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TIFR ECE 2012 | Question: 12
In modeling the number of health insurance claims filed by an individual during a three year period, an analyst makes a simplifying assumption that for all non-negative integer up to $5$. \[ p_{n+1}=\frac{1}{2} p_{n} \] where $p_{n}$ denotes the probability that a ... files more than two claims in this period? $7 / 31$ $29 / 125$ $1 / 3$ $13 / 125$ None of the above
In modeling the number of health insurance claims filed by an individual during a three year period, an analyst makes a simplifying assumption that for all non-negative i...
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46.4k
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96
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admin
asked
Dec 8, 2022
Probability and Statistics
tifr2012
probability-and-statistics
probability
conditional-probability
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1
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TIFR ECE 2012 | Question: 13
Consider a single amoeba that at each time slot splits into two with probability $p$ or dies otherwise with probability $1-p$. This process is repeated independently infinitely at each time slot, i.e. if there are any amoebas left at time slot $t$, then they all split independently into ... $\min \left\{\frac{1 \pm \sqrt{1-4 p(1-p)}}{2(1-p)}\right\}$ None of the above
Consider a single amoeba that at each time slot splits into two with probability $p$ or dies otherwise with probability $1-p$. This process is repeated independently infi...
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46.4k
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94
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Dec 8, 2022
Probability and Statistics
tifr2012
probability-and-statistics
probability
independent-events
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1
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0
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14
TIFR ECE 2012 | Question: 14
Let $X$ and $Y$ be indepedent, identically distributed standard normal random variables, i.e., the probability density function of $X$ is \[f_{X}(x)=\frac{1}{\sqrt{2 \pi}} \exp \left(-\frac{x^{2}}{2}\right),-\infty<x<\infty. \] The random variable $Z$ is defined ... none of the above
Let $X$ and $Y$ be indepedent, identically distributed standard normal random variables, i.e., the probability density function of $X$ is\[f_{X}(x)=\frac{1}{\sqrt{2 \pi}}...
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46.4k
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88
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admin
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Dec 8, 2022
Probability and Statistics
tifr2012
probability-and-statistics
probability
normal-distribution
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1
votes
0
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15
TIFR ECE 2012 | Question: 15
Consider a string of length $1 \mathrm{~m}$. Two points are chosen independently and uniformly random on it thereby dividing the string into three parts. What is the probability that the three parts can form the sides of a triangle? $1 / 4$ $1 / 3$ $1 / 2$ $2 / 3$ $3 / 4$
Consider a string of length $1 \mathrm{~m}$. Two points are chosen independently and uniformly random on it thereby dividing the string into three parts. What is the prob...
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46.4k
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167
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admin
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Dec 8, 2022
Probability and Statistics
tifr2012
probability-and-statistics
probability
uniform-distribution
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1
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16
TIFR ECE 2012 | Question: 16
Let $P$ be a $n \times n$ matrix such that $P^{k}=\mathbf{0}$, for some $k \in \mathbb{N}$ and where $\mathbf{0}$ is an all zeros matrix. Then at least how many eigenvalues of $P$ are zero $1$ $n-1$ $n$ $0$ None of the above
Let $P$ be a $n \times n$ matrix such that $P^{k}=\mathbf{0}$, for some $k \in \mathbb{N}$ and where $\mathbf{0}$ is an all zeros matrix. Then at least how many eigenvalu...
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46.4k
points
88
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admin
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Dec 8, 2022
Linear Algebra
tifr2012
linear-algebra
eigen-values
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1
votes
0
answers
17
TIFR ECE 2012 | Question: 17
Let $A=U \Lambda U^{\dagger}$ be a $n \times n$ matrix, where $U U^{\dagger}=I$. Which of the following statements is TRUE. The matrix $I+A$ has non-negative eigen values The matrix $I+A$ is symmetic $\operatorname{det}(I+A)=\operatorname{det}(I+\Lambda)$ $(a)$ and $(c)$ $(b)$ and $(c)$ $(a), (b)$ and $(c)$
Let $A=U \Lambda U^{\dagger}$ be a $n \times n$ matrix, where $U U^{\dagger}=I$. Which of the following statements is TRUE.The matrix $I+A$ has non-negative eigen valuesT...
admin
46.4k
points
94
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admin
asked
Dec 8, 2022
Linear Algebra
tifr2012
linear-algebra
eigen-values
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1
votes
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answers
18
TIFR ECE 2012 | Question: 18
Under a certain coordinate transformation from $(x, y)$ to $(u, v)$ the circle $x^{2}+y^{2}=1$ shown below on the left side was transformed into the ellipse shown on the right side. If the transformation is of the form \[ \left[\begin{array}{l} u \\ v \end{array}\right]=\mathbf{A}\ ... \] $A_{1}$ only $A_{2}$ only $A_{1}$ or $A_{2}$ $A_{1}$ or $A_{3}$ $A_{2}$ or $A_{3}$
Under a certain coordinate transformation from $(x, y)$ to $(u, v)$ the circle $x^{2}+y^{2}=1$ shown below on the left side was transformed into the ellipse shown on the ...
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46.4k
points
83
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admin
asked
Dec 8, 2022
Linear Algebra
tifr2012
linear-algebra
matrices
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1
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0
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19
TIFR ECE 2012 | Question: 19
$X$ and $Y$ are two $3$ by $3$ matrices. If \[ X Y=\left(\begin{array}{rrr} 1 & 3 & -2 \\ -4 & 2 & 5 \\ 2 & -8 & -1 \end{array}\right) \] then $X$ has rank $2$ at least one of $X, Y$ is not invertible $X$ can't be an invertible matrix $X$ and $Y$ could both be invertible. None of the above
$X$ and $Y$ are two $3$ by $3$ matrices. If\[X Y=\left(\begin{array}{rrr}1 & 3 & -2 \\-4 & 2 & 5 \\2 & -8 & -1\end{array}\right)\]then$X$ has rank $2$at least one of $X, ...
admin
46.4k
points
116
views
admin
asked
Dec 8, 2022
Linear Algebra
tifr2012
linear-algebra
determinant
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–
1
votes
0
answers
20
TIFR ECE 2012 | Question: 20
Let $A$ be a $2 \times 2$ matrix with all entries equal to $1.$ Define $B=\sum_{n=0}^{\infty} A^{n} / n !$. Then $B=e^{2} A / 2$ $B=\left(\begin{array}{cc}1+e & e \\e & 1+e\end{array}\right)$ ... $B=\left(\begin{array}{cc}1+e^{2} & e^{2} \\e^{2} & 1+e^{2}\end{array}\right)$ None of the above
Let $A$ be a $2 \times 2$ matrix with all entries equal to $1.$ Define $B=\sum_{n=0}^{\infty} A^{n} / n !$. Then$B=e^{2} A / 2$$B=\left(\begin{array}{cc}1+e & e \\e & 1+e...
admin
46.4k
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89
views
admin
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Dec 8, 2022
Linear Algebra
tifr2012
linear-algebra
matrices
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