# GATE EC 2017 SET-1

 Question 1
The clock frequency of an 8085 microprocessor is 5 MHz. If the time required to execute an instruction is 1.4 $\mu s$, then the number of T-states needed for executing the instruction is
 A 1 B 6 C 7 D 8
Microprocessors   Instructions of 8085 Microprocessor
Question 1 Explanation:
Given than,
$f_{\mathrm{CLK}}=5 \mathrm{MHz}$
Execution time $=1.4 \mu \mathrm{s}$
Execution time $=n(T-\text { state })$
$n=$ number of T-states required to execute the instruction
T- state (or) $T_{\mathrm{CLK}}=\frac{1}{f_{\mathrm{CLK}}}=0.2 \mu \mathrm{s}$
So, $\quad n=\frac{1.4 \mu \mathrm{s}}{T_{\mathrm{CLK}}}=\frac{1.4}{0.2}=7$
 Question 2
Consider a single input single output discrete-time system with x[n] as input and y[n] as output, where the two are related as
$y[n]=\left\{\begin{matrix} n|x[n]| & for 0\leq n\leq 10\\ x[n]-x[n-1] & othrwise \end{matrix}\right.$
Which one of the following statements is true about the system?
 A It is causal and stable B It is causal but not stable C It is not causal but stable D It is neither causal nor stable
Signals and Systems   Basics of Signals and Systems
Question 2 Explanation:
Since present output does not depend upon future values of input, the system is causal and also every bounded input produces bounded output, so it is stable.

 Question 3
Consider the following statement about the linear dependence of the real valued functions $y_{1}=1, \; y_{2}=x \; and \; y_{3}=x^{2}$ , over the field of real numbers.
I. $y_{1}, \; y_{2} \; and \; y_{3}$ are linearly independent on $-1 \leq x \leq 0$
II. $y_{1}, \; y_{2} \; and \; y_{3}$ are linearly dependent on $0 \leq x \leq 1$
III. $y_{1}, \; y_{2} \; and \; y_{3}$ are linearly independent on $0 \leq x \leq 1$
IV. $y_{1}, \; y_{2} \; and \; y_{3}$ are linearly dependent on $-1 \leq x \leq 0$
Which one among the following is correct ?
 A Both I and II are true B Both I and III are true C Both II and IV are true D Both III and IV are true
Engineering Mathematics   Calculus
Question 3 Explanation:
Any of the given three functions cannot be written as the linear combination of other two functions. Hence, the statements I and III are correct
 Question 4
Consider the 5 x 5 matrix
A=$\begin{bmatrix} 1 &2 &3 & 4 & 5\\ 5 & 1& 2& 3& 4\\ 4& 5&1 & 2&3 \\ 3 & 4 & 5 & 1 & 2\\ 2&3 & 4&5 & 1 \end{bmatrix}$
It is given that A has only one real eigen value. Then the real eigen value of A is
 A -2.5 B 0 C 15 D 25
Engineering Mathematics   Linear Algebra
Question 4 Explanation:
\begin{aligned} |A-\lambda I|&=0 \\ \begin{array}{|ccccc|} 1-\lambda & 2 & 3 & 4 & 5 \\ 5 & 1-\lambda & 2 & 3 & 4 \\ 4 & 5 & 1-\lambda & 2 & 3 \\ 3 & 4 & 5 & 1-\lambda & 2 \\ 2 & 3 & 4 & 5 & 1-\lambda \end{array}&=0\\ \end{aligned}
sum of all elements in every one row must be zero.
\begin{aligned} \text{i.e. }\quad 15-\lambda&=0\\ \lambda&=15 \end{aligned}
 Question 5
The voltage of an electromagnetic wave propagating in a coaxial cable with uniform characteristic impedance is $V(\iota )=e^{-y \iota +j\omega t}$ volts, Where $\iota$ is the distance along the length of the cable in meters. $\gamma =(0.1+j40)m^{-1}$ is the complex propagation constant, and $\omega = 2\pi \times 10^{9} rad/ s$ is the angular frequency. The absolute value of the attenuation in the cable in dB/meter is __________.
 A 0.5 B 0.86 C 0.34 D 0.94
Electromagnetics   Transmission Lines
Question 5 Explanation:
\begin{aligned} V(l) &=V_{O} e^{-\alpha l} e^{-j\beta l} e^{j\omega t} \\ \text { Attenuation } &=\frac{\mid \text { Input } \mid}{\mid \text { Output } \mid}=\frac{\left|V_{O}(0)\right|}{\left|V_{o}(l)\right|} \end{aligned}
Attenuation per meter $=\frac{\left|V_{0}\right|}{\left|V_{0}(1 \mathrm{m})\right|}=e^{\alpha}$
Attenuation in $\mathrm{dB} / \mathrm{m}=\left(20 \log e^{\alpha}\right) \mathrm{dB} / \mathrm{m}$
$=20(0.1) \log e=0.868 \mathrm{dB} / \mathrm{m}$

There are 5 questions to complete.