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201

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

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202

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 $\text{(iii)}$

admin asked in Probability and Statistics Nov 29, 2022

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203

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

admin asked in Linear Algebra Nov 29, 2022

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204

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

admin asked in Calculus Nov 29, 2022

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205

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

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206

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

admin asked in Calculus Nov 29, 2022

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207

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 $Y=\max \left\{X_{1}, X_{2}\right\}$ ... $\operatorname{Pr}[Z=1]>\operatorname{Pr}[Z=2]=\frac{1}{2}$ $\operatorname{Pr}[Z=1]<\operatorname{Pr}[Z=2]$

admin asked in Probability and Statistics Nov 29, 2022

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208

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

admin asked in Probability and Statistics Nov 29, 2022

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209

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

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210

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

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211

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

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212

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 $X+Y?$

admin asked in Probability and Statistics Nov 29, 2022

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213

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

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214

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 ... $1$ $\frac{2}{3}$ $\frac{1}{2}$ $\frac{1}{3}$ $\frac{1}{4}$

admin asked in Probability and Statistics Nov 29, 2022

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215

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 $2 / 3?$ $\frac{2}{9}$ $\frac{1}{27}$ $\frac{13}{27}$ $\frac{1}{3}$ $\frac{7}{18}$

admin asked in Probability and Statistics Nov 29, 2022

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216

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 $X \geq 5?$ $5 \frac{1}{2}$ $7$ $10$ $8 \frac{1}{2}$ $6$

admin asked in Probability and Statistics Nov 29, 2022

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217

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 ... $H\left[R_{513}, C_{513} \mid R_{1}, R_{2}, \ldots, R_{512}\right]?$ $\log _{2} 513$ $9$ $10$ $19$ $81$

admin asked in Probability and Statistics Nov 29, 2022

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218

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

admin asked in Linear Algebra Nov 29, 2022

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219

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

admin asked in Linear Algebra Nov 29, 2022

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220

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 \[ x^{2}+y^{2}=1 ? \] $1$ $\sqrt{2}$ $2$ $3$ $4$

admin asked in Linear Algebra Nov 29, 2022

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221

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? $40 \; \mathrm{H}$ $20 \; \mathrm{H}$ $60 \; \mathrm{H}$ $42 \; \mathrm{H}$

admin asked in Others Oct 17, 2022

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222

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 $-87.5^{\circ}$

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223

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 $\frac{4}{81}$ $\frac{1}{9}$ $\frac{5}{81}$ $\frac{2}{81}$

admin asked in Others Oct 17, 2022

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224

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 $-0.6$ $-1$ $0.6$ $0$

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225

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 $e^{-j2w} [\cos 4 w+2 \cos 2 w+2]$ $[\cos 2 w+2 \cos w+2]$ $e^{-jw} [\cos 2 w+2 \cos w+2]$ $e^{-j2w} [\cos 2 w+2 \cos w+2]$

admin asked in Others Oct 17, 2022

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226

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

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227

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

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228

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

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229

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

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230

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

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231

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 $0 \; \Omega$ $25 \; \Omega$ $50 \; \Omega$ $100 \; \Omega$

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232

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 $1.5\; \mathrm{GHz}$ $2.0\; \mathrm{GHz}$ $2.5\; \mathrm{GHz}$ $3.0\; \mathrm{GHz}$

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233

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

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234

GATE ECE 1997 | Question 1.9
A probability density fucntion is given by $p(x)=\mathrm{K} e^{-x^2 / 2}-\infty<x<\infty$. The value of $K$ should be $\frac{1}{\sqrt{2 \pi}}$ $\sqrt{\frac{2}{\pi}}$ $\frac{1}{2 \sqrt{\pi}}$ $\frac{1}{\pi \sqrt{2}}$

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235

GATE ECE 1997 | Question 1.10
A deterministic signal has the power spectrum given in the figure is, The minimum sampling rate needed to completely represent this signal is $1 \; \mathrm{kHz}$ $2 \; \mathrm{kHz}$ $3 \; \mathrm{kHz}$ None of the above

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236

GATE ECE 1997 | Question 1.11
The voltage $V$ in the figure is $10 \mathrm{~V}$ $15 \mathrm{~V}$ $5 \mathrm{~V}$ None of the above

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237

GATE ECE 1997 | Question 2.1
In the $\text{BJT}$ amplifier shown in the figure is the transistor is based in the forward active region. Putting a capacitor across $R_E$ will decrease the voltage gain and decrease the input impedance increase the voltage gain ... the input impedance decrease the voltage gain and increase the input impedance increase the voltage gain and increase the input impedance

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238

GATE ECE 1997 | Question 2.2
A cascade amplifier stage is equivalent to a common emitter stage followed by a common base stage a common base stage followed by an emitter follower an emitter follower stage followed by a common base stage a common base stage followed by a common emitter stage

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239

GATE ECE 1997 | Question 2.3
For a $\text{MOS}$ capacitor fabricated on a $p$-type semiconductor, strong inversion occurs when surface potential is equal to Fermi potential surface potential is zero surface potential is negative and equal to Fermi potential in magnitude surface potential is positive and equal to twice the Fermi potential

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240

GATE ECE 1997 | Question 2.4
In a common emitter $\text{BJT}$ amplifier, the maximum usable supply voltage is limited by Avalanche breakdown of Base-Emitter junction Collector-Base breakdown voltage with emitter open $\left(\mathrm{BV}_{\mathrm{CBO}}\right)$ Collector-Emitter breakdown ... with base open $\left(\mathrm{BV}_{\mathrm{CBO}}\right)$ Zener breakdown voltage of the Emitter-Base junction

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