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.