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Recent questions tagged tifr2010

1 votes
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1
TIFR ECE 2010 | Question: 1
A linear system could be a composition of Two non-linear systems a non-causal non-linear system and a linear system a time varying non-linear system and a time varying linear system All of the above None of the above
A linear system could be a composition ofTwo non-linear systemsa non-causal non-linear system and a linear systema time varying non-linear system and a time varying linea...
1 votes
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2
TIFR ECE 2010 | Question: 2
For $x \in[0, \pi / 2], \alpha$ for which $\sin (x) \geq x-\alpha x^{3}$ is $\alpha>1 /(2 \pi)$ $\alpha \geq 1 / 6$ $\alpha \leq 1 /(2 \pi)$ $\alpha=1 / 4$ None of the above
For $x \in[0, \pi / 2], \alpha$ for which $\sin (x) \geq x-\alpha x^{3}$ is$\alpha>1 /(2 \pi)$$\alpha \geq 1 / 6$$\alpha \leq 1 /(2 \pi)$$\alpha=1 / 4$None of the above
1 votes
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6
TIFR ECE 2010 | Question: 6
If we convolve $\sin (t) / t$ with itself, then we get $C \sin (t) / t$ for some constant $C$ $C \cos (t) / t$ for some constant $C$ $C \cos (t) / t^{2}$ for some constant $C$ $C_{1} \sin (t) / t^{2}+C_{2} \cos (t) / t^{2}$ for some constants $C_{1}, C_{2}$ None of the above
If we convolve $\sin (t) / t$ with itself, then we get$C \sin (t) / t$ for some constant $C$$C \cos (t) / t$ for some constant $C$$C \cos (t) / t^{2}$ for some constant $...
1 votes
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9
TIFR ECE 2010 | Question: 9
The $z$-transform of a sequence $\left\{x_{n}\right\}_{n=-\infty}^{\infty}$ is defined to be $X(z)=\sum_{n=-\infty}^{\infty} x_{n} z^{-n}$. The $z$-transform of the sequence $y_{n}=x_{2 n+1}$ is $Y(z)=z(X(z)-X(-z)) / 2$ ... $Y(z)=z(X(\sqrt{z})-X(-\sqrt{z})) / 2$ $Y(z)=(X(\sqrt{z})-X(-\sqrt{z})) / 2$
The $z$-transform of a sequence $\left\{x_{n}\right\}_{n=-\infty}^{\infty}$ is defined to be $X(z)=\sum_{n=-\infty}^{\infty} x_{n} z^{-n}$. The $z$-transform of the seque...
1 votes
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12
TIFR ECE 2010 | Question: 12
Consider a system with input $x(t)$ and the output $y(t)$ is given by \[ y(t)=x(t)-\sin (t) x(t-1)-0.5 x(t+2)+1 . \] The system is Non-linear Non-causal Time varying All of the above None of the above
Consider a system with input $x(t)$ and the output $y(t)$ is given by\[y(t)=x(t)-\sin (t) x(t-1)-0.5 x(t+2)+1 .\]The system isNon-linearNon-causalTime varyingAll of the a...
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13
TIFR ECE 2010 | Question: 13
Output of a linear system with input $x(t)$ is given by \[ y(t)=\int_{-\infty}^{\infty} h(t, \tau) x(\tau) . \] The system is time invariant if $h(t, \tau)=h(t-\tau)$ $h(t, \tau)=h(\tau)$ $h(t, \tau)=h(t)$ $h(t, \tau)=$ constant $h(t, \tau)$ is a continuous function of $t$
Output of a linear system with input $x(t)$ is given by\[y(t)=\int_{-\infty}^{\infty} h(t, \tau) x(\tau) .\]The system is time invariant if$h(t, \tau)=h(t-\tau)$$h(t, \ta...
1 votes
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14
TIFR ECE 2010 | Question: 14
Define $\operatorname{sign}(x)=0$ for $x=0, \operatorname{sign}(x)=1$ for $x>0$ and $\operatorname{sign}(x)=-1$ for $x<0$. For $n \geq 0$ ... $-1,1,-1,1, \ldots$. $0,1,-1,1,-1, \ldots$ $0,1,1,1,-1,1,-1,1, \ldots$ None of the above
Define $\operatorname{sign}(x)=0$ for $x=0, \operatorname{sign}(x)=1$ for $x>0$ and $\operatorname{sign}(x)=-1$ for $x<0$. For $n \geq 0$, let\[Y_{n}=\operatorname{sign}\...
1 votes
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15
TIFR ECE 2010 | Question: 15
Let $\imath=\sqrt{-1}$. Then $\imath^{\imath}$ could be $\exp (\pi / 2)$ $\exp (\pi / 4)$ Can't determine Takes infinite values Is a complex number
Let $\imath=\sqrt{-1}$. Then $\imath^{\imath}$ could be$\exp (\pi / 2)$$\exp (\pi / 4)$Can't determineTakes infinite valuesIs a complex number
1 votes
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17
TIFR ECE 2010 | Question: 17
Let $a_{1} \geq a_{2} \geq \cdots \geq a_{k} \geq 0$. Then the limit \[ \lim _{n \rightarrow \infty}\left(\sum_{i=1}^{k} a_{i}^{n}\right)^{1 / n} \] is $0$ $\infty$ $a_{k}$ $a_{1}$ $\left(\sum_{i=1}^{k} a_{k}\right) / k$
Let $a_{1} \geq a_{2} \geq \cdots \geq a_{k} \geq 0$. Then the limit\[\lim _{n \rightarrow \infty}\left(\sum_{i=1}^{k} a_{i}^{n}\right)^{1 / n}\]is$0$$\infty$$a_{k}$$a_{1...
1 votes
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19
TIFR ECE 2010 | Question: 19
Let us define an interval $A(n)$ as a function of $n$ as $A(n)=(-1 / n, 1 / n)$. Then the set of points that lie in the intersection of $A_{n}{ }^{\prime} s, n=1, \ldots, \infty$ is an interval is a single point is an empty set cannot be determined has two disjoint intervals
Let us define an interval $A(n)$ as a function of $n$ as $A(n)=(-1 / n, 1 / n)$. Then the set of points that lie in the intersection of $A_{n}{ }^{\prime} s, n=1, \ldots,...
1 votes
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20
TIFR ECE 2010 | Question: 20
The function $f(t)$ is a convolution of $t^{2}$ with $\exp \left(-t^{2} / 2\right) / \sqrt{2 \pi}$. Its derivative is $2 t$ $t^{2}$ $2 t+t e^{-t^{2} / 2}$ Does not have a simple closed form expression None of the above
The function $f(t)$ is a convolution of $t^{2}$ with $\exp \left(-t^{2} / 2\right) / \sqrt{2 \pi}$. Its derivative is$2 t$$t^{2}$$2 t+t e^{-t^{2} / 2}$Does not have a sim...
0 votes
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21
TIFR ECE 2010: 1
41. A linear system could be a composition of Two non-linear systems a non-causal non-linear system and a linear system a time varying non-linear system and a time varying linear system All of the above None of the above
41. A linear system could be a composition ofTwo non-linear systemsa non-causal non-linear system and a linear systema time varying non-linear system and a time varying l...
0 votes
0 answers
22
TIFR ECE 2010: 2
42. For $x \in[0, \pi / 2], \alpha$ for which $\sin (x) \geq x-\alpha x^{3}$ is $\alpha>1 /(2 \pi)$ $\alpha \geq 1 / 6$ $\alpha \leq 1 /(2 \pi)$ $\alpha=1 / 4$ None of the above
42. For $x \in[0, \pi / 2], \alpha$ for which $\sin (x) \geq x-\alpha x^{3}$ is$\alpha>1 /(2 \pi)$$\alpha \geq 1 / 6$$\alpha \leq 1 /(2 \pi)$$\alpha=1 / 4$None of the abo...
0 votes
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23
TIFR ECE 2010: 3
43. Consider two independent random variables $X$ and $Y$ having probability density functions uniform in the interval $[0,1]$. When $\alpha \geq 1$, the probability that $\max (X, Y)>\alpha \min (X, Y)$ is $1 /(2 \alpha)$ $\exp (1-\alpha)$ $1 / \alpha$ $1 / \alpha^{2}$ $1 / \alpha^{3}$
43. Consider two independent random variables $X$ and $Y$ having probability density functions uniform in the interval $[0,1]$. When $\alpha \geq 1$, the probability that...
0 votes
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26
TIFR ECE 2010: 6
46. If we convolve $\sin (t) / t$ with itself, then we get $C \sin (t) / t$ for some constant $C$ $C \cos (t) / t$ for some constant $C$ $C \cos (t) / t^{2}$ for some constant $C$ $C_{1} \sin (t) / t^{2}+C_{2} \cos (t) / t^{2}$ for some constants $C_{1}, C_{2}$ None of the above
46. If we convolve $\sin (t) / t$ with itself, then we get$C \sin (t) / t$ for some constant $C$$C \cos (t) / t$ for some constant $C$$C \cos (t) / t^{2}$ for some consta...
0 votes
0 answers
29
TIFR ECE 2010: 9
49. The $z$-transform of a sequence $\left\{x_{n}\right\}_{n=-\infty}^{\infty}$ is defined to be $X(z)=\sum_{n=-\infty}^{\infty} x_{n} z^{-n}$. The $z$-transform of the sequence $y_{n}=x_{2 n+1}$ is $Y(z)=z(X(z)-X(-z)) / 2$ ... $Y(z)=z(X(\sqrt{z})-X(-\sqrt{z})) / 2$ $Y(z)=(X(\sqrt{z})-X(-\sqrt{z})) / 2$
49. The $z$-transform of a sequence $\left\{x_{n}\right\}_{n=-\infty}^{\infty}$ is defined to be $X(z)=\sum_{n=-\infty}^{\infty} x_{n} z^{-n}$. The $z$-transform of the s...
0 votes
0 answers
32
TIFR ECE 2010: 12
52. Consider a system with input $x(t)$ and the output $y(t)$ is given by \[ y(t)=x(t)-\sin (t) x(t-1)-0.5 x(t+2)+1 . \] The system is Non-linear Non-causal Time varying All of the above None of the above
52. Consider a system with input $x(t)$ and the output $y(t)$ is given by\[y(t)=x(t)-\sin (t) x(t-1)-0.5 x(t+2)+1 .\]The system isNon-linearNon-causalTime varyingAll of t...
0 votes
0 answers
33
TIFR ECE 2010: 13
53. Output of a linear system with input $x(t)$ is given by \[ y(t)=\int_{-\infty}^{\infty} h(t, \tau) x(\tau) . \] The system is time invariant if $h(t, \tau)=h(t-\tau)$ $h(t, \tau)=h(\tau)$ $h(t, \tau)=h(t)$ $h(t, \tau)=$ constant $h(t, \tau)$ is a continuous function of $t$
53. Output of a linear system with input $x(t)$ is given by\[y(t)=\int_{-\infty}^{\infty} h(t, \tau) x(\tau) .\]The system is time invariant if$h(t, \tau)=h(t-\tau)$$h(t,...
0 votes
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34
TIFR ECE 2010: 14
54. Define $\operatorname{sign}(x)=0$ for $x=0, \operatorname{sign}(x)=1$ for $x>0$ and $\operatorname{sign}(x)=-1$ for $x
54. Define $\operatorname{sign}(x)=0$ for $x=0, \operatorname{sign}(x)=1$ for $x>0$ and $\operatorname{sign}(x)=-1$ for $x
0 votes
0 answers
35
TIFR ECE 2010: 15
55. Let $\imath=\sqrt{-1}$. Then $t^{2}$ could be $\exp (\pi / 2)$ $\exp (\pi / 4)$ Can't determine Takes infinite values Is a complex number
55. Let $\imath=\sqrt{-1}$. Then $t^{2}$ could be$\exp (\pi / 2)$$\exp (\pi / 4)$Can't determineTakes infinite valuesIs a complex number
0 votes
0 answers
36
TIFR ECE 2010: 16
56. Consider two independent random variables $X$ and $Y$ having probability density functions uniform in the interval $[0,1]$. The probability that $X+Y>1.5$ is $1 / 4$ $1 / 8$ $1 / 3$ $\operatorname{Pr}\{X+Y
56. Consider two independent random variables $X$ and $Y$ having probability density functions uniform in the interval $[0,1]$. The probability that $X+Y>1.5$ is$1 / 4$$1...
0 votes
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37
TIFR ECE 2010: 17
57. Let $a_{1} \geq a_{2} \geq \cdots \geq a_{k} \geq 0$. Then the limit \[ \lim _{n \rightarrow \infty}\left(\sum_{i=1}^{k} a_{i}^{n}\right)^{1 / n} \] is 0 $\infty$ $a_{k}$ $a_{1}$ $\left(\sum_{i=1}^{k} a_{k}\right) / k$
57. Let $a_{1} \geq a_{2} \geq \cdots \geq a_{k} \geq 0$. Then the limit\[\lim _{n \rightarrow \infty}\left(\sum_{i=1}^{k} a_{i}^{n}\right)^{1 / n}\]is0$\infty$$a_{k}$$a_...
0 votes
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38
TIFR ECE 2010: 18
58. Under what conditions is the following inequality true for $a, b>0$ \[ \log _{e}(a+b) \geq \lambda \log _{e}(a / \lambda)+(1-\lambda) \log _{e}(b /(1-\lambda)) \] $\lambda=0.5$ $00$ $a / \lambda>0,0
58. Under what conditions is the following inequality true for $a, b>0$\[\log _{e}(a+b) \geq \lambda \log _{e}(a / \lambda)+(1-\lambda) \log _{e}(b /(1-\lambda))\]$\lambd...
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39
TIFR ECE 2010: 19
59. Let us define an interval $A(n)$ as a function of $n$ as $A(n)=(-1 / n, 1 / n)$. Then the set of points that lie in the intersection of $A_{n}{ }^{\prime} s, n=1, \ldots, \infty$ is an interval is a single point is an empty set cannot be determined has two disjoint intervals
59. Let us define an interval $A(n)$ as a function of $n$ as $A(n)=(-1 / n, 1 / n)$. Then the set of points that lie in the intersection of $A_{n}{ }^{\prime} s, n=1, \ld...
0 votes
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40
TIFR ECE 2010: 20
60. The function $f(t)$ is a convolution of $t^{2}$ with $\exp \left(-t^{2} / 2\right) / \sqrt{2 \pi}$. Its derivative is $2 t$ $t^{2}$ $2 t+t e^{-t^{2} / 2}$ Does not have a simple closed form expression None of the above
60. The function $f(t)$ is a convolution of $t^{2}$ with $\exp \left(-t^{2} / 2\right) / \sqrt{2 \pi}$. Its derivative is$2 t$$t^{2}$$2 t+t e^{-t^{2} / 2}$Does not have a...
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