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161
TIFR ECE 2014 | Question: 11
It is known that the signal $x(t)$, where $t$ denotes time, belongs to the following class: \[ \left\{A \sin \left(2 \pi f_{0} t+\theta\right): f_{0}=1 \mathrm{~Hz}, 0 \leq A \leq 1,0<\theta \leq \pi\right\} \] If you ... how many samples are required to determine the signal? $1$ sample. $2$ samples. $1$ sample per second. $2$ samples per second. None of the above.
It is known that the signal $x(t)$, where $t$ denotes time, belongs to the following class:\[\left\{A \sin \left(2 \pi f_{0} t+\theta\right): f_{0}=1 \mathrm{~Hz}, 0 \leq...
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Dec 14, 2022
Others
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162
TIFR ECE 2014 | Question: 12
Assume that $Y, Z$ are independent, zero-mean, continuous random variables with variances $\sigma_{Y}^{2}$ and $\sigma_{Z}^{2},$ respectively. Let $X=Y+Z$. The optimal value of $\alpha$ which minimizes $\mathbb{E}\left[(X-\alpha Y)^{2}\right]$ ... $1$ $\frac{\sigma_{Y}^{2}}{\sigma_{Z}^{2}}$ None of the above.
Assume that $Y, Z$ are independent, zero-mean, continuous random variables with variances $\sigma_{Y}^{2}$ and $\sigma_{Z}^{2},$ respectively. Let $X=Y+Z$. The optimal va...
126
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asked
Dec 14, 2022
Probability and Statistics
tifr2014
probability-and-statistics
probability
random-variable
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163
TIFR ECE 2014 | Question: 13
Let function $f: \mathbf{R} \rightarrow \mathbf{R}$ be convex, i.e., for $x, y \in \mathbf{R}, \alpha \in[0,1], f(\alpha x+(1-\alpha) y) \leq$ $\alpha f(x)+(1-\alpha) f(y)$. Then which of the following is $\text{TRUE?}$ $f(x) \leq f(y)$ whenever ... $f$ and $g$ are both convex, then $\min \{f, g\}$ is also convex. For a random variable $X, E(f(X)) \geq f(E(X))$.
Let function $f: \mathbf{R} \rightarrow \mathbf{R}$ be convex, i.e., for $x, y \in \mathbf{R}, \alpha \in[0,1], f(\alpha x+(1-\alpha) y) \leq$ $\alpha f(x)+(1-\alpha) f(y...
97
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asked
Dec 14, 2022
Calculus
tifr2014
calculus
functions
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1
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164
TIFR ECE 2014 | Question: 14
Suppose that a random variable $X$ has a probability density function \[ \begin{aligned} f(x) & =c(x-4) \quad \text { for } 4 \leq x \leq 6 \\ & =0 \quad \text { for all other } x \end{aligned} \] for some constant $c$. What is the expected value of $X$ given that $X \geq 5?$ $5 \frac{5}{9}$ $5 \frac{1}{2}$ $5 \frac{3}{4}$ $5 \frac{1}{4}$ $5 \frac{5}{8}$
Suppose that a random variable $X$ has a probability density function\[\begin{aligned}f(x) & =c(x-4) \quad \text { for } 4 \leq x \leq 6 \\& =0 \quad \text { for all othe...
102
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asked
Dec 14, 2022
Probability and Statistics
tifr2014
probability-and-statistics
probability
probability-density-function
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0
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165
TIFR ECE 2014 | Question: 15
You are allotted a rectangular room of a fixed height. You have decided to paint the three walls and put wallpaper on the fourth one. Walls can be painted at a cost of Rs. $10$ per meter and the wall paper can be put at the rate of Rs $20$ per meter for that ... $200$ square meter room? $400 \times \sqrt{3} $ $400$ $400 \times \sqrt{2}$ $200 \times \sqrt{3}$ $500$
You are allotted a rectangular room of a fixed height. You have decided to paint the three walls and put wallpaper on the fourth one. Walls can be painted at a cost of Rs...
87
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asked
Dec 14, 2022
Others
tifr2014
quantitative-aptitude
geometry
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166
TIFR ECE 2014 | Question: 16
A fair dice (with faces numbered $1, \ldots, 6$ ) is independently rolled twice. Let $X$ denote the maximum of the two outcomes. The expected value of $X$ is $4 \frac{1}{2}$ $3 \frac{1}{2}$ $5$ $4 \frac{17}{36} $ $4 \frac{3}{4}$
A fair dice (with faces numbered $1, \ldots, 6$ ) is independently rolled twice. Let $X$ denote the maximum of the two outcomes. The expected value of $X$ is$4 \frac{1}{2...
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Dec 14, 2022
Probability and Statistics
tifr2014
probability-and-statistics
probability
expectation
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167
TIFR ECE 2014 | Question: 17
Let $X$ be a Gaussian random variable with mean $\mu_{1}$ and variance $\sigma_{1}^{2}$. Now, suppose that $\mu_{1}$ itself is a random variable, which is also Gaussian distributed with mean $\mu_{2}$ and variance $\sigma_{2}^{2}$. Then the distribution ... variable with mean $\mu_{2}$ and variance $\sigma_{1}^{2}+\sigma_{2}^{2}$. Has no known form. None of the above.
Let $X$ be a Gaussian random variable with mean $\mu_{1}$ and variance $\sigma_{1}^{2}$. Now, suppose that $\mu_{1}$ itself is a random variable, which is also Gaussian d...
92
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Dec 14, 2022
Probability and Statistics
tifr2014
probability-and-statistics
probability
normal-distribution
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168
TIFR ECE 2014 | Question: 18
A non-negative loss in a car accident is distributed with the following probability density function \[ f(x)=\frac{1}{10} \exp (-x / 10) \] for $x \geq 0$. Suppose that first $5$ units of loss is incurred by the insured and the remaining loss if any is covered by the ... $5+10 \exp \left(-\frac{1}{2}\right)$ $15 \exp \left(-\frac{1}{2}\right)$
A non-negative loss in a car accident is distributed with the following probability density function\[f(x)=\frac{1}{10} \exp (-x / 10)\]for $x \geq 0$. Suppose that first...
114
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asked
Dec 14, 2022
Probability and Statistics
tifr2014
probability-and-statistics
probability
probability-density-function
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169
TIFR ECE 2014 | Question: 19
Consider a $2^{k} \times N$ binary matrix $A=\left\{a_{\ell, k}\right\}, a_{\ell, k} \in\{0,1\}$. For rows $i$ and $j$, let the Hamming distance be $d_{i, j}=\sum_{\ell=1}^{N}\left|a_{i, \ell}-a_{j, \ell}\right|$. Let $D_{\min }=\min _{i, j} d_{i, j}$. ... $D_{\min } \leq N-k+1$. $D_{\min } \leq N-k$. $D_{\min } \leq N-k-1$. $D_{\min } \leq N-k-2$. None of the above.
Consider a $2^{k} \times N$ binary matrix $A=\left\{a_{\ell, k}\right\}, a_{\ell, k} \in\{0,1\}$. For rows $i$ and $j$, let the Hamming distance be $d_{i, j}=\sum_{\ell=1...
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Dec 14, 2022
Others
tifr2014
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170
TIFR ECE 2014 | Question: 20
What is \[ \lim _{n \rightarrow \infty} \cos \frac{\pi}{2^{2}} \cos \frac{\pi}{2^{3}} \cdots \cos \frac{\pi}{2^{n}} ? \] $0$ $\pi / 2$ $1 / \sqrt{2}$ $2 / \pi$ None of the above.
What is\[\lim _{n \rightarrow \infty} \cos \frac{\pi}{2^{2}} \cos \frac{\pi}{2^{3}} \cdots \cos \frac{\pi}{2^{n}} ?\]$0$$\pi / 2$$1 / \sqrt{2}$$2 / \pi$None of the above....
84
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Dec 14, 2022
Calculus
tifr2014
calculus
limits
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171
TIFR ECE 2013 | Question: 1
The unit step response of a discrete-time, linear, time-invariant system is \[ y[n]=\left\{\begin{array}{rl} 0, & n<0 \\ 1, & n \geq 0 \text { and } n \text { even } \\ -1, & n \geq 0 \text { and } ... the system is bounded-input, bounded-output $\text{(BIBO)}$ stable there is not enough information to determine $\text{(BIBO)}$ stability none of the above
The unit step response of a discrete-time, linear, time-invariant system is\[y[n]=\left\{\begin{array}{rl}0, & n<0 \\1, & n \geq 0 \text { and } n \text { even } \\-1, & ...
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Dec 12, 2022
Others
tifr2013
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172
TIFR ECE 2013 | Question: 2
The output $\{y(n)\}$ of a discrete time system with input $\{x(n)\}$ is given by \[ y(n)=\sum_{k=0}^{N-1} a^{k} x(n-k) . \] The difference equation for the inverse system is given by $y(n)=x(n)-a x(n-1)$ ... $(a)$ above, otherwise the inverse does not exist If $|a|<1$, then the answer is $(b)$ above, otherwise the inverse does not exist None of the above
The output $\{y(n)\}$ of a discrete time system with input $\{x(n)\}$ is given by\[y(n)=\sum_{k=0}^{N-1} a^{k} x(n-k) .\]The difference equation for the inverse system is...
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Dec 12, 2022
Others
tifr2013
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173
TIFR ECE 2013 | Question: 3
$X$ and $Y$ are jointly Gaussian random variables with zero mean. A constant-pdf contour is where the joint density function takes on the same value. If the constant-pdf contours of $X, Y$ are as shown above, which of the following could their covariance matrix $\mathbf{K}$ ... $\mathbf{K}=\left[\begin{array}{cc}1 & -0.5 \\ -0.5 & 2\end{array}\right]$
$X$ and $Y$ are jointly Gaussian random variables with zero mean.A constant-pdf contour is where the joint density function takes on the same value. If the constant-pdf c...
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Others
tifr2013
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174
TIFR ECE 2013 | Question: 4
Consider a fair coin that has probability $1 / 2$ of showing heads $(\text{H})$ in a toss and $1 / 2$ of showing tails $(\text{T})$. Suppose we independently flip a fair coin over and over again. What is the probability that $\text{HT}$ sequence occurs before $\text{TT}?$ $3 / 4$ $1 / 2$ $2 / 3$ $1 / 3$ $1 / 4$
Consider a fair coin that has probability $1 / 2$ of showing heads $(\text{H})$ in a toss and $1 / 2$ of showing tails $(\text{T})$. Suppose we independently flip a fair ...
85
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Dec 12, 2022
Probability and Statistics
tifr2013
probability-and-statistics
probability
conditional-probability
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175
TIFR ECE 2013 | Question: 5
Let $x(n)=\sin (2 \pi k n / N), n=0,1, \ldots, N-1$, where $2 k \neq N$ and $0<k \leq N-1$. Then the circular convolution of $\{x(n)\}$ with itself is $N \cos (4 \pi k n / N)$ $N \sin (4 \pi k n / N)$ $-N \cos (2 \pi k n / N) / 2$ $-N \sin (2 \pi k n / N) / 2$ None of the above
Let $x(n)=\sin (2 \pi k n / N), n=0,1, \ldots, N-1$, where $2 k \neq N$ and $0<k \leq N-1$. Then the circular convolution of $\{x(n)\}$ with itself is$N \cos (4 \pi k n /...
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Others
tifr2013
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176
TIFR ECE 2013 | Question: 6
The two-dimensional Fourier transform of a function $f(t, s)$ is given by \[ F(\omega, \theta)=\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(t, s) \exp (-j \omega t) \exp (-j \theta s) d t d s . \] Let $\delta(t)$ be the delta function and let $u(t)=0$ ... $\exp (-(t+s)) u(t+s)$ $\exp (-t) u(t) \delta(s)$ $\exp (-t) \delta(t+s)$ None of the above
The two-dimensional Fourier transform of a function $f(t, s)$ is given by\[F(\omega, \theta)=\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(t, s) \exp (-j \omega t) \e...
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Others
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177
TIFR ECE 2013 | Question: 7
The $Z$-transform of $\{x(n)\}$ is defined as $X(z)=\sum_{n} x(n) z^{-n}$ (for those $z$ for which the series converges). Let $u(n)=1$ for $n \geq 0$ and $u(n)=0$ for $n<0$. The inverse $Z$-transform of $X(z)=$ ... is (a), otherwise the inverse is not well-defined If $|a|<1$, then the answer is (b), otherwise the inverse is not well-defined None of the above
The $Z$-transform of $\{x(n)\}$ is defined as $X(z)=\sum_{n} x(n) z^{-n}$ (for those $z$ for which the series converges). Let $u(n)=1$ for $n \geq 0$ and $u(n)=0$ for $n<...
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Others
tifr2013
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178
TIFR ECE 2013 | Question: 8
The following circuit with an ideal operational amplifier is A low pass filter A high pass filter A bandpass filter A bandstop filter An all pass amplifier
The following circuit with an ideal operational amplifier isA low pass filterA high pass filterA bandpass filterA bandstop filterAn all pass amplifier
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Dec 12, 2022
Others
tifr2013
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179
TIFR ECE 2013 | Question: 9
Let $X$ and $Y$ be two zero mean independent continuous random variables. Let $Z_{1}=\max (X, Y)$, and $Z_{2}=\min (X, Y)$. Then which of the following is TRUE. $Z_{1}$ and $Z_{2}$ are uncorrelated. $Z_{1}$ and $Z_{2}$ are independent. $P\left(Z_{1}=Z_{2}\right)=\frac{1}{2}$. Both $(a)$ and $(c)$ Both $(a)$ and $(b)$
Let $X$ and $Y$ be two zero mean independent continuous random variables. Let $Z_{1}=\max (X, Y)$, and $Z_{2}=\min (X, Y)$. Then which of the following is TRUE.$Z_{1}$ an...
83
views
asked
Dec 12, 2022
Probability and Statistics
tifr2013
probability-and-statistics
probability
random-variable
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180
TIFR ECE 2013 | Question: 10
Consider the following series of square matrices: \[ \begin{array}{l} H_{1}=[1], \\ H_{2}=\left[\begin{array}{cc} 1 & 1 \\ 1 & -1 \end{array}\right], \end{array} \] and for $k=2,3, \ldots$, the $2^{k} \times 2^{k}$ matrix $H_{2^{k}}$ is recursively defined as \[ H_{2^{k}}=\ ... is $H_{2^{k}} H_{2^{k}}^{T}?)$ $0$ $2^{k}$ $2^{k / 2}$ $2^{k 2^{k-1}}$ $2^{k 2^{k}}$
Consider the following series of square matrices:\[\begin{array}{l}H_{1}= , \\H_{2}=\left[\begin{array}{cc}1 & 1 \\1 & -1\end{array}\right],\end{array}\]and for $k=2,3, \...
42
views
asked
Dec 12, 2022
Linear Algebra
tifr2013
linear-algebra
determinant
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1
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0
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181
TIFR ECE 2013 | Question: 11
Two matrices $A$ and $B$ are called similar if there exists another matrix $S$ such that $S^{-1} A S=B$. Consider the statements: If $A$ and $B$ are similar then they have identical rank. If $A$ and $B$ ... Both $\text{I}$ and $\text{II}$ but not $\text{III}$. All of $\text{I}, \text{II}$ and $\text{III}$.
Two matrices $A$ and $B$ are called similar if there exists another matrix $S$ such that $S^{-1} A S=B$. Consider the statements:If $A$ and $B$ are similar then they have...
91
views
asked
Dec 12, 2022
Linear Algebra
tifr2013
linear-algebra
rank-of-matrix
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0
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182
TIFR ECE 2013 | Question: 12
Let $A$ be a Hermitian matrix and let $I$ be the Identity matrix with same dimensions as $A$. Then for a scalar $\alpha>0, A+\alpha I$ has the same eigenvalues as of $A$ but different eigenvectors the same eigenvalues and eigenvectors as of ... those of $A$ and same eigenvectors as of $A$ eigenvalues and eigenvectors with no relation to those of $A$ None of the above
Let $A$ be a Hermitian matrix and let $I$ be the Identity matrix with same dimensions as $A$. Then for a scalar $\alpha>0, A+\alpha I$ hasthe same eigenvalues as of $A$ b...
42
views
asked
Dec 12, 2022
Linear Algebra
tifr2013
linear-algebra
eigen-values
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1
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0
answers
183
TIFR ECE 2013 | Question: 13
Let $A$ be a square matrix and $x$ be a vector whose dimensions match $A$. Let $B^{\dagger}$ be the conjugate transpose of $B$. Then which of the following is not true: $x^{\dagger} A^{2} x$ is always non-negative $x^{\dagger} A x$ ... $A=A^{\dagger}$ then $x^{\dagger} A y$ is complex for some vector $y$ with same dimensions as $x$
Let $A$ be a square matrix and $x$ be a vector whose dimensions match $A$. Let $B^{\dagger}$ be the conjugate transpose of $B$. Then which of the following is not true:$x...
42
views
asked
Dec 12, 2022
Linear Algebra
tifr2013
linear-algebra
matrices
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1
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0
answers
184
TIFR ECE 2013 | Question: 14
$X, Y, Z$ are integer valued random variables with the following two properties: $X$ and $Y$ are independent. For all integer $x$, conditioned on the event $\{X=x\}$, we have that $Y$ and $Z$ are independent (in other words, conditioned on ... and $Z$ are independent Conditioned on $Z$, the random variables $X$ and $Y$ are independent All of the above None of the above
$X, Y, Z$ are integer valued random variables with the following two properties:$X$ and $Y$ are independent.For all integer $x$, conditioned on the event $\{X=x\}$, we ha...
43
views
asked
Dec 12, 2022
Probability and Statistics
tifr2013
probability-and-statistics
probability
random-variable
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0
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185
TIFR ECE 2013 | Question: 15
Consider a sequence of non-negative numbers $\left\{x_{n}: n=1,2, \ldots\right\}$. Which of the following statements cannot be true? $\sum_{n=1}^{\infty} x_{n}=\infty$ but $x_{n}$ decreases to zero as $n$ increases. $\sum_{n=1}^{\infty} x_{n}<\infty$ ... and each $x_{n} \leq 1 / n^{2}$. $\sum_{n=1}^{\infty} x_{n}<\infty$ and each $x_{n}>x_{n+1}$.
Consider a sequence of non-negative numbers $\left\{x_{n}: n=1,2, \ldots\right\}$. Which of the following statements cannot be true?$\sum_{n=1}^{\infty} x_{n}=\infty$ but...
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Dec 12, 2022
Others
tifr2013
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186
TIFR ECE 2013 | Question: 16
A surprise quiz contains three multiple choice questions; question $1$ has $3$ suggested answers, question $2$ has four, and question $3$ has two. A completely unprepared student decides to choose the answers at random. If $X$ is the number of questions the student answers ... expected number of correct answers is $15 / 12$ $7 / 12$ $13 / 12$ $18 / 12$ None of the above
A surprise quiz contains three multiple choice questions; question $1$ has $3$ suggested answers, question $2$ has four, and question $3$ has two. A completely unprepared...
73
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asked
Dec 12, 2022
Probability and Statistics
tifr2013
probability-and-statistics
probability
random-variable
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0
answers
187
TIFR ECE 2013 | Question: 17
Consider four coins, three of which are fair, that is they have heads on one side and tails on the other and both are equally likely to occur in a toss. The fourth coin has heads on both sides. Given that one coin amongst the four is picked at random and is tossed, and the ... is the probability that its other side is tails? $1 / 2$ $3 / 8$ $3 / 5$ $3 / 4$ $5 / 7$
Consider four coins, three of which are fair, that is they have heads on one side and tails on the other and both are equally likely to occur in a toss. The fourth coin h...
41
views
asked
Dec 12, 2022
Probability and Statistics
tifr2013
probability-and-statistics
probability
random-variable
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0
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188
TIFR ECE 2013 | Question: 18
Consider a coin tossing game between Santa and Banta. Both of them toss two coins sequentially, first Santa tosses a coin then Banta and so on. Santa tosses a fair coin: Probability of heads is $1 / 2$ and probability of tails is $1 / 2$. Banta's coin probabilities depend on ... the two trials conducted by each of them? $1 / 2$ $5 / 16$ $3 / 16$ $1 / 4$ $1 / 3$
Consider a coin tossing game between Santa and Banta. Both of them toss two coins sequentially, first Santa tosses a coin then Banta and so on. Santa tosses a fair coin: ...
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views
asked
Dec 12, 2022
Probability and Statistics
tifr2013
probability-and-statistics
probability
conditional-probability
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0
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189
TIFR ECE 2013 | Question: 19
Which of the following is true for polynomials defined over real numbers $\mathbb{R}$. Every odd degree polynomial has a real root. Every odd degree polynomial has at least one complex root. Every even degree polynomial has at least one complex root. Every even degree polynomial has a real root. None of the above
Which of the following is true for polynomials defined over real numbers $\mathbb{R}$.Every odd degree polynomial has a real root.Every odd degree polynomial has at least...
41
views
asked
Dec 12, 2022
Calculus
tifr2013
calculus
polynomials
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0
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190
TIFR ECE 2013 | Question: 20
A function $f: \mathbb{R} \rightarrow \mathbb{R}$ is convex if for $x, y \in \mathbb{R}, \alpha \in[0,1], f(\alpha x+(1-\alpha) y) \leq \alpha f(x)+(1-\alpha) f(y)$. Which of the following is not convex: $x^{2}$ $x^{3}$ $x$ $x^{4}$ $\mathrm{e}^{x}$
A function $f: \mathbb{R} \rightarrow \mathbb{R}$ is convex if for $x, y \in \mathbb{R}, \alpha \in[0,1], f(\alpha x+(1-\alpha) y) \leq \alpha f(x)+(1-\alpha) f(y)$.Which...
92
views
asked
Dec 12, 2022
Calculus
tifr2013
calculus
functions
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0
answers
191
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}\...
117
views
asked
Dec 8, 2022
Calculus
tifr2012
calculus
maxima-minima
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0
answers
192
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$$\infty$$\...
90
views
asked
Dec 8, 2022
Calculus
tifr2012
calculus
limits
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0
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193
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...
149
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Dec 8, 2022
Others
tifr2012
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194
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...
72
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Dec 8, 2022
Others
tifr2012
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195
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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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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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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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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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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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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