# 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:

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

$y=\theta^{x}$

$y=x^{2}$

$y=e^{-x^{2}}$

There are 5 questions to complete.