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162
TIFR ECE 2014 | Question: 2
Evaluate the limit \[ \lim _{n \rightarrow \infty}\left(2 n^{4}\right)^{\frac{1}{3 n}} . \] $e$ $1$ $2^{\frac{1}{3}}$ $0$ None of the above
Evaluate the limit\[\lim _{n \rightarrow \infty}\left(2 n^{4}\right)^{\frac{1}{3 n}} .\]$e$$1$$2^{\frac{1}{3}}$$0$None of the above
1 votes
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163
TIFR ECE 2014 | Question: 3
For a non-negative continuous random variable $X$, which of the following is TRUE? $E\{X\}=\int_{0}^{\infty} P(X>x) d x$. $E\{X\}=\int_{0}^{\infty} P(X \leq x) d x$. $P(X<x) \leq \frac{E\{X\}}{x}$. $(a)$ and $(c)$. None of the above.
For a non-negative continuous random variable $X$, which of the following is TRUE?$E\{X\}=\int_{0}^{\infty} P(X>x) d x$.$E\{X\}=\int_{0}^{\infty} P(X \leq x) d x$.$P(X<x)...
1 votes
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164
TIFR ECE 2014 | Question: 4
A system accepts a sequence of real numbers $x[n]$ as input and outputs \[ y[n]=\left\{\begin{array}{ll} 0.5 x[n]-0.25 x[n-1], & n \text { even } \\ 0.75 x[n], & n \text { odd } \end{array}\right. \] The system is non-linear. non-causal. time-invariant. All of the above. None of the above.
A system accepts a sequence of real numbers $x[n]$ as input and outputs\[y[n]=\left\{\begin{array}{ll}0.5 x[n]-0.25 x[n-1], & n \text { even } \\0.75 x[n], & n \text { od...
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165
TIFR ECE 2014 | Question: 5
The matrix \[ A=\left(\begin{array}{ccc} 1 & a_{1} & a_{1}^{2} \\ 1 & a_{2} & a_{2}^{2} \\ 1 & a_{3} & a_{3}^{2} \end{array}\right) \] is invertible when $a_{1}>a_{2}>a_{3}$ $a_{1}<a_{2}<a_{3}$ $a_{1}=3, a_{2}=2, a_{3}=4$ All of the above None of the above
The matrix\[A=\left(\begin{array}{ccc}1 & a_{1} & a_{1}^{2} \\1 & a_{2} & a_{2}^{2} \\1 & a_{3} & a_{3}^{2}\end{array}\right)\]is invertible when$a_{1}>a_{2}>a_{3}$$a_{1}...
1 votes
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166
TIFR ECE 2014 | Question: 6
Let $g:[0, \pi] \rightarrow \mathbb{R}$ be continuous and satisfy \[ \int_{0}^{\pi} g(x) \sin (n x) d x=0 \] for all integers $n \geq 2$. Then which of the following can you say about $g?$ $g$ must be identically zero. $g(\pi / 2)=1$. $g$ need not be identically zero. $g(\pi)=0$. None of the above.
Let $g:[0, \pi] \rightarrow \mathbb{R}$ be continuous and satisfy\[\int_{0}^{\pi} g(x) \sin (n x) d x=0\]for all integers $n \geq 2$. Then which of the following can you ...
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170
TIFR ECE 2014 | Question: 10
Consider the two quadrature amplitude modulation $\text{(QAM)}$ constellations below. Suppose that the channel has additive white Gaussian noise channel and no intersymbol interference. The constellation points are picked equally likely. Let $P\text{(QAM)}$ denote the ... .
Consider the two quadrature amplitude modulation $\text{(QAM)}$ constellations below. Suppose that the channel has additive white Gaussian noise channel and no intersymbo...
1 votes
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176
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...
1 votes
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180
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....
1 votes
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185
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 /...
1 votes
0 answers
188
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
1 votes
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200
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...
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