GATE EC 2007

Question 1
If E denotes expectation, the variance of a random variable X is given by
A
E[X^{2}]-E^{2}[X]
B
E[X^{2}]+E^{2}[X]
C
E[X^{2}]
D
E^{2}[X]
Communication Systems   Random Processes
Question 1 Explanation: 
\begin{array}{l} \sigma_{X}^{2}=E\left[X^{2}\right]-E^{2}[X] \\ \text { AC. Power }=\text { Total power - DC power } \end{array}
Question 2
The following plot shows a function which varies linearly with x. The value of the integral I=\int_{1}^{2} ydx is
A
1
B
2.5
C
4
D
5
Engineering Mathematics   Calculus
Question 2 Explanation: 
\begin{aligned} y &=x+1 \\ I &=\int_{1}^{2} y d x \\ &=\int_{1}^{2}(x+1) d x=\left.\frac{(x+1)^{2}}{2}\right|_{1} \\ &=\frac{1}{2}(9-4)=2.5 \end{aligned}
Question 3
For |x| \lt \lt 1, \; coth (x) can be approximated as
A
x
B
x^{2}
C
\frac{1}{x}
D
\frac{1}{x^{2}}
Engineering Mathematics   Calculus
Question 3 Explanation: 
\cot h x=\frac{\cos h x}{\sin h x}=\frac{1}{x}
Question 4
\lim_{\theta \rightarrow 0}\frac{sin(\frac{\theta }{2})}{\theta } is
A
0.5
B
1
C
2
D
not defined
Engineering Mathematics   Calculus
Question 4 Explanation: 
\lim _{\theta \rightarrow 0} \frac{\frac{1}{2} \times \sin \left(\frac{\theta}{2}\right)}{\theta \times \frac{1}{2}}=\frac{1}{2} \lim _{\theta \rightarrow 0} \frac{\sin \frac{\theta}{2}}{\frac{\theta}{2}}=\frac{1}{2}=0.5
Question 5
Which one of following functions is strictly bounded?
A
1/x^{2}
B
e^{x}
C
x^{2}
D
e^{-x^{2}}
Engineering Mathematics   Calculus
Question 5 Explanation: 


y=\frac{1}{x^{2}}


y=\theta^{x}


y=x^{2}


y=e^{-x^{2}}
Question 6
For the function e^{-x}, the linear approximation around x=2 is
A
(3-x)e^{-2}
B
1-x
C
[3+2\sqrt{2}-(1+\sqrt{2})x]e^{2}
D
e^{-2}
Engineering Mathematics   Numerical Methods
Question 7
An independent voltage source in series with an impedance Z_{s}=R_{s}+jX_{s}, delivers a maximum average power to a load impedance Z_{L} when
A
Z_{L}=R_{s}+jX_{s}
B
Z_{L}=R_{s}
C
Z_{L}=jX_{s}
D
Z_{L}=R_{s}-jX_{s}
Network Theory   Network Theorems
Question 7 Explanation: 
Z_{L}=R_{s}-j X_{s}
For maximum power transfer
\begin{array}{l} Z_{L}=Z_{S}^{*} \\ Z_{L}=R_{s}-j X_{s} \end{array}
Question 8
The RC circuit shown in the figure is
A
a low-pass filter
B
a high-pass filter
C
a band-pass filter
D
a band-reject filter
Network Theory   Network Functions
Question 8 Explanation: 
At \omega \rightarrow \infty , Capacitor \rightarrow short circuited
Circuit looks like,


at\omega \rightarrow 0, Capacitor \rightarrow open circuited
Circuit looks like


So frequency response of the circuit will be,

So the circuit is Band pass filter
Question 9
The electron and hole concentrations in an intrinsic semiconductor are n_{i} per cm^{3} at 300 K. Now, if acceptor impurities are introduced with a concentration of N_{A} per cm^{3} (where N_{A}\gt \gt n_{i} ), the electron concentration per cm^{3} at 300 K will be
A
n_{i}
B
n_{i}+N_{A}
C
N_{A}-n_{i}
D
\frac{{n_{i}}^2}{N_{A}}
Electronic Devices   Basic Semiconductor Physics
Question 9 Explanation: 
By the law of electrical neutrality
\begin{aligned} p+N_{0} &=n+N_{A} \\ \text{as}\quad\quad N_{0} &=0\\ N_{A}&>>n_{i} \cong 0 \quad p=N_{A}\\ \end{aligned}
using mass action law \mathrm{np}=n_{i}^{2}
So. \quad n=\frac{n_{i}^{2}}{p}=\frac{n_i^{2}}{N_{A}}
Question 10
In a p^{+}n junction diode under reverse biased the magnitude of electric field is maximum at
A
the edge of the depletion region on the p-side
B
the edge of the depletion region on the n-side
C
the p^{+}n junction
D
the centre of the depletion region on the n-side
Electronic Devices   PN-Junction Diodes and Special Diodes
Question 10 Explanation: 
Electrical field is always maximum at the junction.
There are 10 questions to complete.
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