# GATE EE 2017 SET 1

 Question 1
Consider $g(t)=\left\{\begin{matrix} t-\left \lceil t \right \rceil, & t\geq 0 \\ t-\left \lceil t \right \rceil , & otherwise \end{matrix}\right. , \; \; where\; t \in \mathbb{R}$
Here, $\left \lfloor t \right \rfloor$ represents the largest integer less than or equal to t and $\left \lceil t \right \rceil$ denotes the smallest integer greater than or equal to t. The coefficient of the second harmonic component of the Fourier series representing g(t) is _________.
 A 0 B 1 C 2 D 3
Signals and Systems   Fourier Series
Question 1 Explanation:
Given that, $g(t)=\left\{\begin{matrix} t-\left \lfloor t \right \rfloor, & t \geq 0\\ t-\left \lceil t \right \rceil & \text{otherwise} \end{matrix}\right.$
where,
$\left \lfloor t \right \rfloor=$ greatest integer less than or equal to 't'.
$\left \lceil t \right \rceil=$ smallest integer greater than or equal to 't'.
Now,

Since, g(t) is nonperiodic. So, there is no Fourier series expansion of this signal and hence no need to calculate harmonic here.
 Question 2
A source is supplying a load through a 2-phase, 3-wire transmission system as shown in figure below. The instantaneous voltage and current in phase-a are $V_{an}=220sin(100\pi t)$V and $i_{a}=10sin (100\pi t)$A, respectively. Similarly for phase-b the instantaneous voltage and current are $V_{bn}=220cos (100\pi t)$V and $i_{b}=10cos( 100\pi t$A, respectively

The total instantaneous power flowing form the source to the load is
 A 2200W B $2200 sin^{2}(100\pi t)W$ C 440W D $2200 sin(100\pi t) cos(100 \pi t) W$
Power Systems   Performance of Transmission Lines, Line Parameters and Corona
Question 2 Explanation:
\begin{aligned} V_{an}&=220 \sin (100 \pi t)V\\ i_a&=10 \sin (100 \pi t)A\\ V_{bn}&=220 \cos (100 \pi t)V\\ i_b&=10 \cos (100 \pi t)A\\ p&=V_{an}i_a+V_{bn}i_b\\ &=2200 W \end{aligned}
 Question 3
A three-phase, 50Hz, star-connected cylindrical-rotor synchronous machine is running as a motor. The machine is operated from a 6.6 kV grid and draws current at unity power factor (UPF). The synchronous reactance of the motor is 30 $\Omega$ per phase. The load angle is $30^{\circ}$. The power delivered to the motor in kW is _______.
 A 2520.5 B 1640.8 C 838.3 D 400.8
Electrical Machines   Synchronous Machines
Question 3 Explanation:
For synchronous motor,
$\bar{V_t}=\bar{E_f}+j\bar{I_a}X_s$
As currrent is drawn at unity power factors.
Therefore,
\begin{aligned} E_f \cos \delta &=V_t \\ E_f&= \frac{V_t}{\cos \delta }=\frac{6.6kV}{\sqrt{3}} \times \frac{1}{\cos 30^{\circ}}\\ E_f&= \frac{6.6 \times 10^3}{\sqrt{3} \times \sqrt{3}} \times 2V\\ V_t &=\frac{6.6 \times 10^3}{\sqrt{3}} \\ \therefore \;\;P_e &=\frac{3V_{ph}E_{ph}}{X_s} \sin \delta \\ &= \frac{3 \times \frac{6.6 \times 10^3}{\sqrt{3}} \times \frac{6.6 \times 10^3}{\sqrt{3} \times \sqrt{3}} \times 2 }{30} \times \sin 30^{\circ}\\ &=838.3kW \end{aligned}
 Question 4
For a complex number z, $\lim_{z\rightarrow i}\frac{z^{2}+1}{z^{3}+2z-i(z^{2}+2)}$ is
 A -2i B -i C i D 2i
Engineering Mathematics   Complex Variables
Question 4 Explanation:
\begin{aligned} \lim_{z \to i}&=\frac{z^2+1}{z^3+2z-i(z^2+2)}\\ \lim_{z \to i}&=\frac{2z}{3z^2+2-i(2z)}\\ &=\frac{2i}{3i^2+2-i(2i)}\\ &=\frac{2i}{-3+2+2}\\ &=\frac{2i}{-3+4}=2i \end{aligned}
 Question 5
Consider an electron, a neutron and a proton initially at rest and placed along a straight line such that the neutron is exactly at the center of the line joining the electron and proton. At t=0, the particles are released but are constrained to move along the same straight line. Which of these will collide first?
 A The particles will never collide B All will collide together C Proton and Neutron D Electron and Neutron
Electromagnetic Fields   Electrostatic Fields
Question 5 Explanation:
Given that electron, neutron and proton are in straight line.

The electron will move towards proton and proton will move towards electron and force will be same $F=\frac{q_1q_2}{4 \pi \in _0 R^2}$. But acceleration of electron will be more than proton as mass of electron $\lt$ mass of proton. Since neutron are neutral they will not move. Thus electron will hit neutron first.
 Question 6
Let z(t)=x(t) * y(t) , where "*" denotes convolution. Let c be a positive real-valued constant. Choose the correct expression for z(ct).
 A c x(ct)*y(ct) B x(ct)*y(ct) C c x(t)*y(ct) D c x(ct)*y(t)
Signals and Systems   Linear Time Invariant Systems
Question 6 Explanation:
Time scaling property of convolution.
If, $x(t)*y(t)=z(t)$
Then, $x(ct)*y(ct)=\frac{1}{c} z(ct)$
$z(ct)=c \times x(ct) * y(ct)$
 Question 7
A 3-bus power system is shown in the figure below, where the diagonal elements of Y-bus matrix are $Y_{11}=-j12pu$, $\; Y_{22}=-j15pu$ and $Y_{33}=-j7pu$

The per unit values of the line reactances p, q and r shown in the figure are
 A p=-0.2, q=-0.1, r=-0.5 B p=0.2, q=0.1, r=0.5 C p=-5, q=-10, r=-2 D p=5, q=10, r=2
Power Systems   Load Flow Studies
Question 7 Explanation:
Given,
\begin{aligned} Y_{11} &=-j12 \; p.u. \\ Y_{22} &=-j15 \; p.u.\\ Y_{33} &=-j7 \; p.u. \\ &\text{We know that,} \\ Y_{11} &=y_{12}+y_{13}=-j12 \; p.u. \;\;...(i) \\ Y_{22} &=y_{12}+y_{23}=-j15 \; p.u. \;\;...(ii) \\ Y_{33} &=y_{13}+y_{23}=-j7 \; p.u. \;\;...(iii) \\ &\text{From eq. (i) and (ii)} \\ y_{13}-y_{23} &=j3\; p.u. \\ y_{13}+y_{23} &=-j7 \; p.u. \\ y_{13}&= -j2\; p.u.\\ y_{23}&= -j5\; p.u.\\ y_{12}&= -j10\; p.u. \end{aligned}
The p.u. values of line reactances p, q and r are
\begin{aligned} jr&=\frac{1}{-j2}=j0.5\; p.u.\\ jp&=\frac{1}{-j5}=j0.2\; p.u.\\ jq&=\frac{1}{-j10}=j0.1\; p.u.\\ \therefore \; p&=0.2, q=0.1, r=0.5 \end{aligned}
 Question 8
The equivalent resistance between the terminals A and B is ______ $\Omega$.
 A 2.2 B 1.2 C 1 D 3
Electric Circuits   Basics
Question 8 Explanation:
Consider the following circuit diagram,

After rearrangement we get

Now, $R_{AB}=1+\frac{6}{5}+0.8=3\Omega$
 Question 9
The Boolean expression $AB + A\bar{C} + BC$ simplifies to
 A $BC+A\bar{C}$ B $AB+A\bar{C}+B$ C $AB+A\bar{C}$ D $AB+BC$
Digital Electronics   Boolean Algebra and Minimization
Question 9 Explanation:

$BC+A\bar{C}$
 Question 10
The following measurements are obtained on a single phase load:
V = 220V$\pm$1%, I = 5.0A$\pm$1% and W=555W$\pm$ 2%.
If the power factor is calculated using these measurements, the worst case error in the calculated power factor in percent is ________.
 A 20% B 40% C 4% D 0.40%
Electrical and Electronic Measurements   Characteristics of Instruments and Measurement Systems
Question 10 Explanation:
\begin{aligned} V &= 220 \pm 1\% \\ I&= 5 \pm 1 \%\\ W &=555 \pm 2 \% \\ W&= VI \cos (\phi )\\ p.f. &=\cos (\phi )=\frac{W}{VI} \\ &= \frac{555 \pm 2 \%}{(220 \pm 1\%)(5 \pm 1 \%)}\\ &= \frac{555}{220 \times 5} \pm 4\%\\ p.f.&= 0.5 \pm 4\% \end{aligned}
There are 10 questions to complete.