# GATE EE 2016 SET 2

 Question 1
The output expression for the Karnaugh map shown below is
 A $A+\bar{B}$ B $A+\bar{C}$ C $\bar{A}+\bar{C}$ D $\bar{A}+C$
Digital Electronics   Boolean Algebra and Minimization
Question 1 Explanation:

$F=A+\bar{C}$
 Question 2
The circuit shown below is an example of a
 A low pass filter. B band pass filter C high pass filter. D notch filter.
Analog Electronics   Operational Amplifiers
Question 2 Explanation:

$\frac{V_{out}}{V_{in}}=-\left [\frac{\frac{ R_2\cdot \frac{1}{j\omega C}}{R_2+\frac{1}{j\omega C}}}{R_1} \right ]$
$\frac{V_{out}}{V_{in}}=-\left [ \frac{R_2}{R_1(R_2j\omega C+1)} \right ]$
So the system is a low pass filter.
 Question 3
The following figure shows the connection of an ideal transformer with primary to secondary turns ratio of 1:100. The applied primary voltage is 100 V (rms), 50 Hz, AC. The rms value of the current I, in ampere, is __________.
 A 100 B 10 C 20 D 50
Electric Circuits   Magnetically Coupled Circuits, Network Topology and Filters
Question 3 Explanation:
The above circuit can be drown by transferring secondary circuit to primary side.

$I=\frac{100V}{(8+10j-4j)\Omega }$
$\;\;=\frac{100V}{(8+6j)\Omega }$
So the rms value of I will be 10 A.
 Question 4
Consider a causal LTI system characterized by differential equation $\frac{dy(t)}{dt}+\frac{1}{6}y(t)=3x(t)$. The response of the system to the input $x(t)=3e^{-\frac{t}{3}}u(t)$. where u(t) denotes the unit step function, is
 A $9e^{-\frac{t}{3}}u(t)$ B $9e^{-\frac{t}{6}}u(t)$ C $9e^{-\frac{t}{3}}u(t)-6e^{-\frac{t}{6}}u(t)$ D $54e^{-\frac{t}{6}}u(t)-54e^{-\frac{t}{3}}u(t)$
Signals and Systems   Linear Time Invariant Systems
Question 4 Explanation:
The differential equation
\begin{aligned} \frac{dy(t)}{dt} &+\frac{1}{6}y(t)=3x(t) \\ \text{So, }sY(s)&+\frac{1}{6}Y(s) =3X(s) \\ Y(s) &=\frac{3X(s)}{\left ( s+\frac{1}{6} \right )} \\ X(s) &=\frac{9}{\left ( s+\frac{1}{3} \right )} \\ \text{So, } Y(s)&=\frac{9}{\left ( s+\frac{1}{3} \right )\left ( s+\frac{1}{6} \right )} \\ &=\frac{54}{\left ( s+\frac{1}{6} \right )} -\frac{54}{\left ( s+\frac{1}{3} \right )}\\ \text{So, }y(t) &= (54e^{-1/6t}-54e^{-1/3t})u(t) \end{aligned}
 Question 5
Suppose the maximum frequency in a band-limited signal $x(t)$ is 5 kHz. Then, the maximum frequency in $x(t)\cos (2000\pi t)$, in kHz, is ________.
 A 5 B 6 C 7 D 8
Signals and Systems   Fourier Transform
Question 5 Explanation:
Maximum possible frequency of $x(t)(2000 \pi t)=f_1+f_2=5+1=6kHz$
 Question 6
Consider the function $f(z)=z+z^*$ where z is a complex variable and $z^*$ denotes its complex conjugate. Which one of the following is TRUE?
 A f(z) is both continuous and analytic B f(z) is continuous but not analytic C f(z) is not continuous but is analytic D f(z) is neither continuous nor analytic
Engineering Mathematics   Complex Variables
Question 6 Explanation:
\begin{aligned} f(z) &=z+z^* \\ f(z)&=2x \text{ is continuous (polynomial)} \\ u &=2x, v=0 \\ u_x &=2, u_y=0 \\ v_x&=0, v_y=0 \end{aligned}
C.R. equation not satisfied.
Therefore, no where analytic.
 Question 7
A 3 x 3 matrix P is such that, $P^{3} = P$. Then the eigenvalues of P are
 A 1, 1, -1 B 1, 0.5 + j0.866, 0.5 - j0.866 C 1, -0.5 + j0.866, -0.5 - j0.866 D 0, 1, -1
Engineering Mathematics   Linear Algebra
Question 7 Explanation:
By Calyey Hamilton theorem,
$\lambda ^3=\lambda$
$\lambda =0, 1, -1$
 Question 8
The solution of the differential equation, for $t \gt 0, y''(t)+2y'(t)+y(t)=0$ with initial conditions y(0) = 0 and y'(0) = 1, is (u(t) denotes the unit step function),
 A $te^{-t}u(t)$ B $(e^{-t}-te^{-t})u(t)$ C $(-e^{-t}+te^{-t})u(t)$ D $e^{-t}u(t)$
Engineering Mathematics   Differential Equations
Question 8 Explanation:
The differentail equation is
$y'(t)+2y'(t)+y(t)=0$
So, $(s^2y(s)-sy(0)-y'(0))+2[sy(s)-y(0)]+y(s) =0$
So, $y(s)=\frac{sy(0)+y'(0)+2y(0)}{s^2+2s+1}$
Given that $y'(0)=1, y(0)=0$
So, $y(s)=\frac{1}{(s+1)^2}$
So, $y(t)=te^{-t}u(t)$
 Question 9
The value of the line integral
$\int_{c}(2xy^{2}dx+2x^{2}ydy+dz)$
along a path joining the origin (0,0,0) and the point (1,1,1) is
 A 0 B 2 C 4 D 6
Engineering Mathematics   Calculus
Question 9 Explanation:
$\int _C\bar{F}\cdot \bar{dr}$
where,
$\bar{F}=xy^2\bar{i}+2x^2y\bar{j}+\bar{k}$
$\bigtriangledown \times \vec{F}=\vec{O}$
$(\vec{F}$ is irrotational $\Rightarrow \vec{F}$ is conservative)
\begin{aligned} \vec{F}&=\bigtriangledown \phi \\ \phi _x&=2xy^2 \\ \phi _y&=2x^2y \\ \phi _z&= 1\\ \Rightarrow \; \phi &=x^2y^2+z+C \end{aligned}
where, $\vec{F}$ is conservative
$\int _C \bar{F}\bar{dr}=\int_{(0,0,0)}^{(1,1,1)}d\phi =[x^2y^2+z]_{(0,0,0)}^{(1,1,1,)}=2$
 Question 10
Let f(x) be a real, periodic function satisfying f(-x)=-f(x). The general form of its Fourier series representation would be
 A $f(x)=a_{0}+\sum_{k=1}^{\infty }a_{k}cos(kx)$ B $f(x)=\sum_{k=1}^{\infty }b_{k}sin(kx)$ C $f(x)=a_{0}+\sum_{k=1}^{\infty }a_{2k}cos(kx)$ D $f(x)=\sum_{k=0}^{\infty }a_{2k+1}sin(2k+1)x$
Signals and Systems   Fourier Series
Question 10 Explanation:
Given that,
$f(-x)=-f(x)$
So, function is an odd function.
So, the fourier series will have sine term only. So,
$f(x)=\sum_{k=1}^{\infty }b_x \sin (kx)$
There are 10 questions to complete.