Recent questions without answers / GO Electronics

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TIFR ECE 2017 | Question: 4
A Schmitt trigger circuit is a comparator circuit with a hysteresis. Consider the Schmitt trigger circuit in the figure implemented using an opamp. What are the trigger levels for this circuit? $\pm \frac{R_{1}}{R_{2}} V_{s}$ $\pm \frac{R_{2}}{R_{1}} V_{s}$ $\pm \frac{R_{1}}{R_{1}+R_{2}} V_{s}$ $\pm \frac{R_{2}}{R_{1}+R_{2}} V_{s}$ None of the above

A Schmitt trigger circuit is a comparator circuit with a hysteresis. Consider the Schmitt trigger circuit in the figure implemented using an opamp. What are the trigger l...

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TIFR ECE 2017 | Question: 5
Consider the inequality \[ n-\frac{1}{n} \geq \sqrt{n^{2}-1}, \] where $n$ is an integer $\geq 1$. Which of the following statements is $\text{TRUE?}$ This inequality holds for all integers $n \geq 1$ ... not hold for any integer $n \geq 1$ $n-\frac{1}{n}=\sqrt{n^{2}-1}$ for infinitely many integers $n \geq 1$

Consider the inequality\[n-\frac{1}{n} \geq \sqrt{n^{2}-1},\]where $n$ is an integer $\geq 1$. Which of the following statements is $\text{TRUE?}$This inequality holds fo...

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323

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...

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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...

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TIFR ECE 2017 | Question: 8
Consider the two positive integer sequences, defined for a fixed positive integer $c \geq 2$ \[f(n)=\frac{1}{n}\left\lfloor\frac{n}{c}\right\rfloor, \quad g(n)=n\left\lfloor\frac{c}{n}\right\rfloor\] where $\lfloor t\rfloor$ denotes the ... $0$ The first sequence converges to $1 / c$, while the second sequence converges to $c$

Consider the two positive integer sequences, defined for a fixed positive integer $c \geq 2$\[f(n)=\frac{1}{n}\left\lfloor\frac{n}{c}\right\rfloor, \quad g(n)=n\left\lflo...

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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...

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327

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...

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328

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...

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329

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...

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330

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}...

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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...

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332

TIFR ECE 2017 | Question: 15
Suppose that $f(x)$ is a real valued continuous function such that $f(x) \rightarrow \infty$ as $x \rightarrow \infty$. Further, let \[a_{n}=\sum_{j=1}^{n} 1 / j\] and \[b_{n}=\sum_{j=1}^{n} 1 / j^{2} .\] Which of the following statements is true ... any number $M$ so that $f\left(b_{n}\right)$ and $f\left(a_{n}\right)$ are greater than $M$ for all $n$ None of the above

Suppose that $f(x)$ is a real valued continuous function such that $f(x) \rightarrow \infty$ as $x \rightarrow \infty$. Further, let\[a_{n}=\sum_{j=1}^{n} 1 / j\]and\[b_{...

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333

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} ...

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334

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...

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335

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...

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336

TIFR ECE 2016 | Question: 4
Consider a system which in response to input $x(t)$ outputs \[y(t)=x\left(t^{2}\right) .\] Which of the following describes the system? linear, time-invariant, causal linear, time-invariant, non-causal linear, time-variant non-linear, time-invariant non-linear, time-variant

Consider a system which in response to input $x(t)$ outputs\[y(t)=x\left(t^{2}\right) .\]Which of the following describes the system?linear, time-invariant, causallinear,...

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337

TIFR ECE 2016 | Question: 5
Consider the opamp circuit in the figure. Approximately what is $V_{0}$? $-\left(\frac{V_{1}}{2}+V_{2}\right)$ $-\left(\frac{V_{1}}{4}+\frac{V_{2}}{2}\right)$ $-\left(V_{1}+2 V_{2}\right)$ $-\left(4 V_{1}+2 V_{2}\right)$ None of the above

Consider the opamp circuit in the figure. Approximately what is $V_{0}$?$-\left(\frac{V_{1}}{2}+V_{2}\right)$$-\left(\frac{V_{1}}{4}+\frac{V_{2}}{2}\right)$$-\left(V_{1}+...

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338

TIFR ECE 2016 | Question: 6
What is the Laplace transform $F(s)$ of the signal $f(t), t \geq 0$ defined below? In $t \in[0,1),$ ... $\frac{1}{s\left(1-e^{-s / 2}\right)}$ $\frac{1}{s\left(1+e^{-s / 2}\right)}$

What is the Laplace transform $F(s)$ of the signal $f(t), t \geq 0$ defined below? In $t \in[0,1),$\[f(t)=\left\{\begin{array}{ll}1, & t \in\left[0, \frac{1}{2}\right) \\...

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339

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 $X+Y?$

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TIFR ECE 2016 | Question: 8
In terms of their frequency responses, which of the following is the odd one out? All four circuits are equivalent

In terms of their frequency responses, which of the following is the odd one out? All four circuits are equivalent

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341

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...

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342

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 ...

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343

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...

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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...

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345

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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346

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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347

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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348

GATE ECE 2005 | Question: 81b
Statement for Linked Answer Questions $81 a$ and $81 b$: Consider an $8085$ microprocessor system If in addition following code exists from $019 \mathrm{H}$ onwards, $\text{ORI 40 H}$ $\text{ADD M}$ What will be the result in the accumulator after the last instruction is executed? $40 \; \mathrm{H}$ $20 \; \mathrm{H}$ $60 \; \mathrm{H}$ $42 \; \mathrm{H}$

Statement for Linked Answer Questions $81 a$ and $81 b$:Consider an $8085$ microprocessor systemIf in addition following code exists from $019 \mathrm{H}$ onwards,$\text{...

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349

GATE ECE 2005 | Question: 82b
Statement for Linked Answer Questions $82a$ and $82b:$ The dopen loop transfer function of a unity feedback system is given by \[\mathrm{G}(s)=\frac{3 e^{-2}}{s(s+2)}\] Based on the above results, the gain and phase margins of the system will be $-7.09$ ... $87.5^{\circ}$ $7.09 \mathrm{~dB}$ and $-87.5^{\circ}$ $-7.09 \mathrm{~dB}$ and $-87.5^{\circ}$

Statement for Linked Answer Questions $82a$ and $82b:$The dopen loop transfer function of a unity feedback system is given by\[\mathrm{G}(s)=\frac{3 e^{-2}}{s(s+2)}\]Base...

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350

GATE ECE 2005 | Question: 83b
Statement for Linked Answer Questions 83a and 83b Asymmetric three-level midtread quantizer is to be designed assuming equiprobable occurence of all quantization levels. The quantization noise power for the quantization region between $-a$ and $+a$ in the figure is $\frac{4}{81}$ $\frac{1}{9}$ $\frac{5}{81}$ $\frac{2}{81}$

Statement for Linked Answer Questions 83a and 83bAsymmetric three-level midtread quantizer is to be designed assuming equiprobable occurence of all quantization levels.Th...

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351

GATE ECE 2005 | Question: 84b
Statement of Linked Answer Questions $84a$ and $84b$ Voltage standing wave pattern in a lossless transmission line with characteristic impedance $50 \mathrm{W}$ and a resistive load is shown in the figure. The reflection coefficient is given by $-0.6$ $-1$ $0.6$ $0$

Statement of Linked Answer Questions $84a$ and $84b$Voltage standing wave pattern in a lossless transmission line with characteristic impedance $50 \mathrm{W}$ and a resi...

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352

GATE ECE 2005 | Question: 85b
Statement of Linked Answer Questions $85 a$ and $85b$ A sequence $x(n)$ has non-zero values as shown in the figure The Fourier transform of $y(2 n)$ will be $e^{-j2w} [\cos 4 w+2 \cos 2 w+2]$ $[\cos 2 w+2 \cos w+2]$ $e^{-jw} [\cos 2 w+2 \cos w+2]$ $e^{-j2w} [\cos 2 w+2 \cos w+2]$

Statement of Linked Answer Questions $85 a$ and $85b$A sequence $x(n)$ has non-zero values as shown in the figureThe Fourier transform of $y(2 n)$ will be$e^{-j2w} [\cos ...

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353

GATE ECE 1997 | Question 1.1
The current $i_4$ in the circuit of the figure is equal to $12 \mathrm{~A}$ $-12 \mathrm{~A}$ $4 \mathrm{~A}$ None of these

The current $i_4$ in the circuit of the figure is equal to$12 \mathrm{~A}$$-12 \mathrm{~A}$$4 \mathrm{~A}$None of these

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GATE ECE 1997 | Question: 1.2
The voltage $\mathrm{V}$ in the figure is equal to $3 \mathrm{~V}$ $-3 \mathrm{~V}$ $5 \mathrm{~V}$ None of these

The voltage $\mathrm{V}$ in the figure is equal to$3 \mathrm{~V}$$-3 \mathrm{~V}$$5 \mathrm{~V}$None of these

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355

GATE ECE 1997 | Question 1.3
The voltage $\mathrm{V}$ in the figure is always equal to $9 \mathrm{~V}$ $5 \mathrm{~V}$ $1 \mathrm{~V}$ None of the above

The voltage $\mathrm{V}$ in the figure is always equal to$9 \mathrm{~V}$$5 \mathrm{~V}$$1 \mathrm{~V}$None of the above

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GATE ECE 1997 | Question 1.4
The function $f(t)$ has the Fourier Transform $g(\omega)$. The Fourier Transform $f f(t) g(t)\left(=\int_{-\infty}^{\infty} g(t) e^{-j \omega t} d t\right)$ is $\frac{1}{2 \pi} f(\omega)$ $\frac{1}{2 \pi} f(-\omega)$ $2 \pi f(-\omega)$ None of the above

The function $f(t)$ has the Fourier Transform $g(\omega)$. The Fourier Transform$f f(t) g(t)\left(=\int_{-\infty}^{\infty} g(t) e^{-j \omega t} d t\right)$ is$\frac{1}{2 ...

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GATE ECE 1997 | Question 1.5
The Laplace Transform of $e \alpha t \cos (\alpha \; t)$ is equal to $\frac{(s-\alpha)}{(s-\alpha)^2+\alpha^2}$ $\frac{(s+\alpha)}{(s-\alpha)^2+\alpha^2}$ $\frac{1}{(s-\alpha)^2}$ None of the above

The Laplace Transform of $e \alpha t \cos (\alpha \; t)$ is equal to$\frac{(s-\alpha)}{(s-\alpha)^2+\alpha^2}$$\frac{(s+\alpha)}{(s-\alpha)^2+\alpha^2}$$\frac{1}{(s-\alph...

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GATE ECE 1997 | Question 1.6
A transmission line of $50 \; \Omega$ characteristic impedance is terminated with a $100 \; \Omega$ resistance. The minimum impedance measured on the line is equal to $0 \; \Omega$ $25 \; \Omega$ $50 \; \Omega$ $100 \; \Omega$

A transmission line of $50 \; \Omega$ characteristic impedance is terminated with a $100 \; \Omega$ resistance. The minimum impedance measured on the line is equal to$0 ...

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359

GATE ECE 1997 | Question 1.7
A rectangular air-filled waveguide has cross section of $4 \mathrm{~cm} \times 10 \mathrm{~cm}$. The minimum frequency which can propagate in the waveguide is $1.5\; \mathrm{GHz}$ $2.0\; \mathrm{GHz}$ $2.5\; \mathrm{GHz}$ $3.0\; \mathrm{GHz}$

A rectangular air-filled waveguide has cross section of $4 \mathrm{~cm} \times 10 \mathrm{~cm}$. The minimum frequency which can propagate in the waveguide is$1.5\; \math...

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GATE ECE 1997 | Question 1.8
The line code that has zero $\text{dc}$ component for pulse transmission of random binary data is Non-return to zero $\text{(NRZ)}$ Retrun to zero $\text{(RZ)}$ Alternate Mark Inversion $\text{(AM)}$ None of the above

The line code that has zero $\text{dc}$ component for pulse transmission of random binary data isNon-return to zero $\text{(NRZ)}$Retrun to zero $\text{(RZ)}$Alternate Ma...

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