# GATE Electronics and Communication 2020

 Question 1
If $v_1, v_2,..., v_6$ are six vectors in $\mathbb{R}^4$, which one of the following statements is False?
 A It is not necessary that these vectors span $\mathbb{R}^4$. B These vectors are not linearly independent. C Any four of these vectors form a basis for $\mathbb{R}^4$. D If {$v_1, v_3,v_5, v_6$} spans $\mathbb{R}^4$, then it forms a basis for $\mathbb{R}^4$.
Engineering Mathematics   Calculus
Question 1 Explanation:
$v_1, v_2,..., v_6$ are six vectors in $\mathbb{R}^4$.
For a 4-dimensional vector space,
(i) any four linearly independent vectors form a basis (or)
(ii) Any set of four vectors in $\mathbb{R}^4$ spans $\mathbb{R}^4$, then it forms a basis.
Therefore, clearly options (A), (B), (D) are true.
Option (C) is FALSE
 Question 2
For a vector field $\vec{A}$, which one of the following is False?
 A $\vec{A}$ is solenoidal if $\bigtriangledown \cdot \vec{A}=0$ B $\bigtriangledown \times \vec{A}$ is another vector field. C $\vec{A}$ is irrotational if $\bigtriangledown ^2 \vec{A}=0$. D $\bigtriangledown \times(\bigtriangledown \times \vec{A})=\bigtriangledown (\bigtriangledown \cdot \vec{A})-\bigtriangledown ^2 \vec{A}$
Engineering Mathematics   Calculus
Question 2 Explanation:
Divergence and curl operator is performed on a vector field $\vec{A}$
Curl operation provides a vector orthogonal to the given vector field $\vec{A}$
$\bigtriangledown \times(\bigtriangledown \times \vec{A})=\bigtriangledown (\bigtriangledown \cdot \vec{A})-\bigtriangledown ^2 \vec{A}$
If a vector field is irrortational then $\bigtriangledown \times \vec{A}=0$
If a vector field is solenoidal then $\bigtriangledown \cdot \vec{A}=0$
If a field is scalar A, then $\bigtriangledown ^2 \vec{A}=0$, is a laplacian equation.
Hence option (C) is incorrect
 Question 3
The partial derivative of the function

$f(x,y,z)=e^{1-x \cos y}+xze^{-1/(1+y^2)}$

with respect to x at the point (1,0,e) is
 A -1 B 0 C 1 D $\frac{1}{e}$
Engineering Mathematics   Calculus
Question 3 Explanation:
\begin{aligned} \text{Given, } f(x,y,z)&=e^{1-x\cos y}+xze^{-1/(1+y^{2})} \\ \frac{\partial f}{\partial x}&=e^{1-x\cos y}(0-\cos y)+ze^{-1/1+y^{2}} \\ \left ( \frac{\partial f }{\partial x} \right )_{(1,0,e)}&=e^{0}(0-1)+e\cdot e^{-1/(1+0)} \\ &=-1+1=0 \end{aligned}
 Question 4
The general solution of $\frac{d^2y}{dx^2}-6\frac{dy}{dx}+9y=0$ is
 A $y=C_1e^{3x}+C_2e^{-3x}$ B $y=(C_1+C_2x)e^{-3x}$ C $y=(C_1+C_2x)e^{3x}$ D $y=C_1e^{3x}$
Engineering Mathematics   Differential Equations
Question 4 Explanation:
Taking $\frac{\mathrm{d} }{\mathrm{d} x}=D$
Given, $D^{2}-6D+9=0$
$(D-3)^2=0$
$D=3,3$
So, Solution of the given Differential equation
$y=(C_{1}+C_{2}x)e^{3x}$
 Question 5
The output y[n] of a discrete-time system for an input x[n] is

$y[n]=\begin{matrix} max\\ -\infty \leq k\leq n \end{matrix}\; |x[k]|$.

The unit impulse response of the system is
 A 0 for all n B 1 for all n C unit step signal u[n]. D unit impulse signal $\delta$[n].
Signals and Systems   LTI Systems Continuous and Discrete
 Question 6
A single crystal intrinsic semiconductor is at a temperature of 300 K with effective density of states for holes twice that of electrons. The thermal voltage is 26 mV. The intrinsic Fermi level is shifted from mid-bandgap energy level by
 A 18.02 meV B 9.01 meV C 13.45 meV D 26.90 meV
Electronic Devices   Basic Semiconductor Physics
Question 6 Explanation:
$\frac{E_{c}+E_{v}}{2}-E_{F_{i}}=\frac{KT}{2}\ln \left ( \frac{N_{C}}{N_{V}}\right )\, \, \, \, \, \, \left ( \because N_{C}=\frac{N_{V}}{2} \right )$
$=\frac{0.026}{2}\ln 0.5=-9.01\, meV$
 Question 7
Consider the recombination process via bulk traps in a forward biased $pn$ homojunction diode. The maximum recombination rate is $U_{max}$. If the electron and the hole capture cross-section are equal, which one of the following is False?
 A With all other parameters unchanged, $U_{max}$ decreases if the intrinsic carrier density is reduced. B $U_{max}$ occurs at the edges of the depletion region in the device. C $U_{max}$ depends exponentially on the applied bias. D With all other parameters unchanged,$U_{max}$ increases if the thermal velocity of the carriers increases.
Electronic Devices   PN-Junction Diodes and Special Diodes
 Question 8
The components in the circuit shown below are ideal. If the op-amp is in positive feedback and the input voltage $V_i$ is a sine wave of amplitude 1 V, the output voltage $V_o$ is A a non-inverted sine wave of 2 V amplitude B an inverted sine wave of 1 V amplitude C a square wave of 5 V amplitude D a constant of either +5 or -5V
Analog Circuits   Operational Amplifiers
Question 8 Explanation: Given circuit is a Schmitt trigger of non-inverting type.
$V_{o}=\pm 5\, V$
$V^{+}=\frac{V_{o}\times 1+V_{i}\times 1}{1+1}=\frac{V_{o}+V}{2}$
let,$V_{o}=-5\, V,\, \, \, \,V^{+}=\frac{-5+V_{i}}{2}$
$V_{o}$ can change from -5 V to +5 V if $V^{+} \gt 0$
i.e. $\frac{-5+V_{i}}{2} \gt 0\Rightarrow V_{i} \gt 5\, V$
similarly, $V_{o}$ can change from -5 V to +5 V if $V_{i} \lt -5\, V$
But given input has peak value 1 V. Hence output cannot change from +5 V to -5 V or -5 V to +5 V.
Output remain constant at +5 V or -5 V.
 Question 9
In the circuit shown below, the Thevenin voltage $V_{TH}$ is A 2.4 V B 2.8 V C 3.6 V D 4.5 V
Network Theory   Network Theorems
Question 9 Explanation:
By applying the Source Transformation Question 10
The figure below shows a multiplexer where $S_1 \; and \; S_0$ are the select lines, $I_0 \; to \; I_3$ are the input data lines, EN is the enable line, and $F(P, Q, R)$ is the output, F is A $PQ+\bar{Q}R$ B $P+Q\bar{R}$ C $P\bar{Q}R+\bar{P}Q$ D $\bar{Q}+PR$
Digital Circuits   Combinational Circuits
Question 10 Explanation:
Output,$F=\bar{P}\bar{Q}R+P\bar{Q}R+PQ\, \, \, \,$
$F=\bar{Q}R+PQ$ There are 10 questions to complete.