Recent questions tagged tifrece2019 / GO Electronics

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TIFR ECE 2019 | 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, \\ \alpha y[2 n-1]+\beta ... -linear, but time-invariant only if $\alpha=0$ (parameters $\beta$ and $\gamma$ can take arbitrary values) 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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TIFR ECE 2019 | Question: 2
Let $A$ and $B$ be two square matrices that have full rank. Let $\lambda_{A}$ be an eignevalue of $A$ and $\lambda_{B}$ an eigenvalue of $B$. Which of the following is always $\text{TRUE}?$ $A B$ has full rank $A-B$ ... an eigenvalue of $A B$ $A+B$ has full rank At least one of $\lambda_{A}$ or $\lambda_{B}$ is an eigenvalue of $A B$

Let $A$ and $B$ be two square matrices that have full rank. Let $\lambda_{A}$ be an eignevalue of $A$ and $\lambda_{B}$ an eigenvalue of $B$. Which of the following is al...

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TIFR ECE 2019 | Question: 3
Consider a function $f: \mathbf{R} \rightarrow \mathbf{R}$ such that $f(x)=1$ if $x$ is rational, and $f(x)=1-\epsilon,$ where $0<\epsilon<1$, if $x$ is irrational. Which of the following is $\text{TRUE}?$ $\lim _{x \rightarrow \infty} f(x)=1$ ... $1-\epsilon$ $\max _{x \geq 1} f(x)=1$ None of the above

Consider a function $f: \mathbf{R} \rightarrow \mathbf{R}$ such that $f(x)=1$ if $x$ is rational, and $f(x)=1-\epsilon,$ where $0<\epsilon<1$, if $x$ is irrational. Which...

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4

TIFR ECE 2019 | Question: 4
Let $f(x)=\sqrt{x^{2}-4 x+4},$ for $x \in(-\infty, \infty)$. Here, $\sqrt{y}$ denotes the non-negative square root of $y$ when $y$ is non-negative. Then, which of the following is $\text{TRUE}?$ $f(x)$ is ... differentiable $f(x)$ is continuous and differentiable $f(x)$ is continuous but not differentiable $f(x)$ is neither continuous nor differentiable None of the above

Let $f(x)=\sqrt{x^{2}-4 x+4},$ for $x \in(-\infty, \infty)$. Here, $\sqrt{y}$ denotes the non-negative square root of $y$ when $y$ is non-negative. Then, which of the fol...

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TIFR ECE 2019 | Question: 5
Consider the function $f(x)=e^{x^{2}}-8 x^{2}$ for all $x$ on the real line. For how many distinct values of $x$ do we have $f(x)=0?$ $1$ $4$ $2$ $3$ $5$

Consider the function $f(x)=e^{x^{2}}-8 x^{2}$ for all $x$ on the real line. For how many distinct values of $x$ do we have $f(x)=0?$ $1$$4$$2$$3$$5$

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6

TIFR ECE 2019 | Question: 6
Suppose that $X_{1}$ and $X_{2}$ denote the random outcomes of independent rolls of two dice. Each of the dice takes each of the six values $1,2,3,4,5$, and $6$ with equal probability. What is the value of the conditional expectation \[\mathbf{E}\left[\max \left(X_{1}, X_{2}\right) \mid \min \left(X_{1}, X_{2}\right)=3\right] ?\] $33 / 7$ $4$ $5$ $9 / 2$ $19 / 4$

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

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7

TIFR ECE 2019 | Question: 7
Consider two random variables $X$ and $Y$ which take values in a finite set $S$. Let $p_{X, Y}$ represent their joint probability mass function (p.m.f.) and let $p_{X}$ and $p_{Y}$, respectively, be the marginal p.m.f.'s of $X$ and $Y$, respectively. Which of ... None of the above

Consider two random variables $X$ and $Y$ which take values in a finite set $S$. Let $p_{X, Y}$ represent their joint probability mass function (p.m.f.) and let $p_{X}$ a...

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8

TIFR ECE 2019 | Question: 8
Let $K$ be a cube of side $1$ in $\mathbb{R}^{3}$, with its centre at the origin, and its sides parallel to the co-ordinate axes. For $t \geq 0$, let $K_{t}$ be the set of all points in $\mathbb{R}^{3}$ whose Euclidean distance to $K$ is less than or equal to $t$ ... $V \leq \frac{4}{3} \pi\left(\frac{\sqrt{3}}{2}+t\right)^{3}$ $V \geq(1+2 t)^{3}$

Let $K$ be a cube of side $1$ in $\mathbb{R}^{3}$, with its centre at the origin, and its sides parallel to the co-ordinate axes. For $t \geq 0$, let $K_{t}$ be the set o...

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9

TIFR ECE 2019 | Question: 9
Consider a coin which comes up heads with probability $p$ and tails with probability $1-p$, where $0 < p < 1.$ Suppose we keep tossing the coin until we have seen both sides of the coin. What is the expected number of times we would have seen tails? (Hint: the expected number of ... $(1/p.)$ $\frac{1}{p}$ $1+\frac{1}{1-p}$ $p+\frac{1}{p}-1$ $2$ None of the above

Consider a coin which comes up heads with probability $p$ and tails with probability $1-p$, where $0 < p < 1.$ Suppose we keep tossing the coin until we have seen both si...

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10

TIFR ECE 2019 | Question: 10
Let $X, Z_{1}$, and $Z_{2}$ be independent random variables taking values in the set $\{0,1\}$. $X$ is uniformly distributed in $\{0,1\}$, while the distributions of $Z_{1}$ and $Z_{2}$ are such that if we define $Y_{1}=X+Z_{1}$ and $Y_{2}=X+Z_{2}$, where addition ... $\left(1 / p_{1}+1 / p_{2}\right)^{-1}$ $\left(1+1 / p_{1}+1 / p_{2}\right)^{-1}$ None of the above

Let $X, Z_{1}$, and $Z_{2}$ be independent random variables taking values in the set $\{0,1\}$. $X$ is uniformly distributed in $\{0,1\}$, while the distributions of $Z_{...

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11

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}\...

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12

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...

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13

TIFR ECE 2019 | Question: 13
For $t>0$, let $S_{t}$ denote the ball of radius $t$ centered at the origin in $\mathbb{R}^{n}$. That is, \[S_{t}=\left\{\mathbf{x} \in \mathbb{R}^{n} \mid \sum_{i=1}^{n} x_{i}^{2} \leq t^{2}\right\} .\] Let $N_{t}$ be the number of ... $\lim _{t \rightarrow \infty} \frac{N_{t}}{V_{t}}=1$ $N_{t}$ is a monotonically decreasing function of $t$

For $t>0$, let $S_{t}$ denote the ball of radius $t$ centered at the origin in $\mathbb{R}^{n}$. That is,\[S_{t}=\left\{\mathbf{x} \in \mathbb{R}^{n} \mid \sum_{i=1}^{n} ...

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14

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...

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15

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...

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