# 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.