GATE ECE 2014 Set 1
GATE ECE
1
For the amplifier shown in the figure, the BJT parameters are VBE = 0.7 V, $$\beta $$ = 200, and thermal voltage VT = 25 mV. The voltage gain (Vo/Vi of the amplifier is ______.
2
A BJT is baised in forward active mode Assume VBE = 0.7 V, kT/q = 25 mV and reverse saturation current Is = 10-13A. The transconductance of the BJT (in mA/V) is
3
In the low - pass filter shown in the figure, for a cut - off frequency of 5 KHz, the value of R2 (in k$$\Omega $$ ) is _________
4
A good current buffer has
5
In the voltage regulator circuit shown in the figure, the op - amp is ideal. The BJT has VBE = 0.7 V and $$\beta $$ = 100, and the Zener voltage is 4.7 V. For a regulated output of 9 V, the value of R (in $$\Omega $$) is ______
6
In the circuit shown, the op-amp has finite input impedance, infinite voltage gain and zero input offset voltage. The output voltage Vout is
7
In the ac equivalent circuit shown in the figure, if i$${_F}$$ is very large, the type of feedback is
8
In a code - division multiple access (CDMA) system with N = 8 chips, the maximum number of users who can be assigned mutually orthogonal signature sequences is _______.
9
Consider a random process $$X\left( t \right) = \sqrt 2 \sin \left( {2\pi t + \varphi } \right),$$ where the random phase $$\varphi $$ is uniformly distributed in the interval $$\left[ {0,\,\,2\pi } \right].$$ The auto - correlation $$E\left[ {X\left( {{t_1}} \right)X\left( {{t_2}} \right)} \right]$$ is
10
The capacity of a Binary Symmetric Channel (BSC) with cross - over probability 0.5 is ______ .
11
A fair coin is tossed repeatedly until a 'Head' appears for the first time. Let L be the number of tosses to get this first 'Head'. The entropy H (L) in bits is _______________.
12
Let $$Q\left( {\sqrt y } \right)$$ be the BER of a BPSK system over an AWGN channel with two - sided noise power spectral density N0/2. The parameter 𝛾 is a function of bit energy and noise power spectral density. A system with two independent and identical AWGN channels with noise power spectral density N0/2 is shown in the figure. The BPSK demodulator receives the sum of outputs of both the channels
If the BER of this system is $$Q\left( {b\sqrt y } \right),$$ then the value of b is -----------.
13
The forward path transfer function of a unity negative feedback system is given by
$$$G\left(s\right)\;=\;\frac k{\left(s\;+\;2\right)\left(s\;-\;1\right)}$$$
The value of K which will place both the poles of the closed-loop system at the same
location, is ______.
14
For the following feedback system $$G\left(s\right)=\frac1{\left(s+1\right)\left(s+2\right)}$$.The 2% settling time of the step
response is required to be less than 2 seconds.
Which one of the following compensators C(s) achieves this?
Which one of the following compensators C(s) achieves this?15
Consider the feedback system shown in the figure. The Nyquist plot of G(s) is also shown. Which
one of the following conclusions is correct?
16
The phase margin in degrees of G(s)=$${{10} \over {\left( {s + 0.1} \right)\left( {s + 1} \right)\left( {s + 10} \right)}},$$
using the asymptotic Bode plot is ______
17
Consider the state space model of a system, as given below
The system is
18
Consider the Boolean function,
F(w,z,y,z)=wy+ xy +$$\overline w \,xyz + \overline w \,\overline x y\, + xz + \,\overline {x\,} \,\overline y \,$$ $$\overline z $$
Which one of the following is the complete set of essential prime implicants?
19
The Boolean expression
(X+Y)(X+$$\overline Y $$)+($$\overline {(X\overline Y ) + \overline X } $$ simplifies to
20
Five JK flip - flops are cascaded to form circuit shown in figure. Clock pulses at a frequency of 1 MHz are applied as shown. The frequency (in kHz) of the waveform at Q3 is ______.
21
The digital logic shown in the figure satisfies the given state diagram when Q1 is connected to
input A of the XOR gate.
Suppose the XOR gate is replaced by an XNOR gate. Which one of the following options preserves the state diagram?
22
The output F in the digital logic circuit shown in the figure is
23
For a parallel plate transmission line, let v be the speed of propagation and Z be the characteristic impedance. Neglecting fringe effects, a redution of the spacing between the plates by a factor of two results in
24
A two - port network has scattering parameters given by $$[S]$$ $$ = \left[ {\matrix{
{{s_{11}}} & {{s_{12}}} \cr
{{s_{21}}} & {{s_{22}}} \cr
} } \right].$$
If the port - 2 of the two - port is short circuited, the $${{s_{11}}}$$ parameter for the resultant one - port network is
If the port - 2 of the two - port is short circuited, the $${{s_{11}}}$$ parameter for the resultant one - port network is
25
In spherical coordinates, let $${{{\widehat a}_{_0}},\,{{\widehat a}_{_\phi }}}$$ denote unit vectors along the $$\theta ,\,\,\phi $$directions.
$$$E = {{100} \over r}\sin \,\theta \cos \left( {\omega t - \beta r} \right){\widehat a_{_\theta }}\,\,V/m$$$
and $$$H = {{0.265} \over r}\sin \,\theta \cos \left( {\omega t - \beta r} \right){\widehat a_{_\phi }}\,\,A/m$$$
and $$$H = {{0.265} \over r}\sin \,\theta \cos \left( {\omega t - \beta r} \right){\widehat a_{_\phi }}\,\,A/m$$$
Represent the electric and magnetic field components of the EM wave at large distances $$r$$ from a dipole antenna, in free space. The average power $$(W)$$ crossing the hemispherical shell located at $$r = 1\,\,km$$, $$0 \le \theta \le \pi /2$$ _______.
26
A depletion type N -channel MOSFET is biased in its linear region for use as a voltage controlled resistor. Assume threshold voltage VTH = -0.5 V, VGS = 2.0 V, VDS = 5 V, W/L=100, COX=10-8 F/cm2 and $${\mu _n}$$ = 800 cm2/V-s. The value of the resistance of the voltage controlled resistor (in $$\Omega $$ ) is _____.
27
When the optical power incident on a photodiode is 10μW and the responsivity is 0.8 A/W,
the photocurrent generated (in μA) is ________.
28
In the ac equivalent circuit shown in the figure, if in is the input current and RF is very large, the type of feedback
29
In the following circuit employing pass transistor logic, all NMOS transistors are identical
with a threshold voltage of 1 V. Ignoring the body-effect, the output voltages at P, Q and R
are,
30
If fixed positive charges are present in the gate oxide of an n-channel enhancement type
MOSFET, it will lead to
31
Consider the matrix $${J_6} = \left[ {\matrix{
0 & 0 & 0 & 0 & 0 & 1 \cr
0 & 0 & 0 & 0 & 1 & 0 \cr
0 & 0 & 0 & 1 & 0 & 0 \cr
0 & 0 & 1 & 0 & 0 & 0 \cr
0 & 1 & 0 & 0 & 0 & 0 \cr
1 & 0 & 0 & 0 & 0 & 0 \cr
} } \right]$$
Which is obtained by reversing the order of the columns of the identity matrix $${{\rm I}_6}$$. Let $$P = {{\rm I}_6} + \alpha \,\,{J_6},$$ where $$\alpha $$ is a non $$-$$ negative real number. The value of $$\alpha $$ for which det $$(P)=0$$ is _______.
32
For matrices of same dimension $$M,N$$ and scalar $$c,$$ which one of these properties DOES NOT ALWAYS hold ?
33
$$A$$ real $$\left( {4\,\, \times \,\,4} \right)$$ matrix $$A$$ satisfies the equation $${A^2} = {\rm I},$$ where $${\rm I}$$ is the $$\left( {4\,\, \times \,\,4} \right)$$ identity matrix. The positive eigen value of $$A$$ is _______.
34
The Taylor series expansion of $$3$$ $$sin$$ $$x$$ $$+2cos$$ $$x$$ is
35
The volume under the surface $$z\left( {x,y} \right) = x + y$$ and above the triangle in the $$xy$$ plane defined by $$\left\{ {0 \le y \le x} \right.$$ and $$\,\left. {0 \le x \le 12} \right\}$$ is _________.
36
Let $$\,{X_1},\,\,{X_2}\,\,$$ and $$\,{X_3}\,$$ be independent and identically distributed random variables with the uniform distribution on $$\left[ {0,1} \right].$$ The probability $$p$$ {$${X_1}$$ is the largest} is __________.
37
Let $$X$$ be a real - valued random variable with $$E\left[ X \right]$$ and $$E\left[ {{X^2}} \right]$$ denoting the mean values of $$X$$ and $${{X^2}}$$, respectively. The relation which always holds true is
38
In a housing society, half of the families have a single child per family, while the remaining half have two children per family. The probability that a child picked at random, has a sibling is _______.
39
$$C$$ is a closed path in the $$z-$$plane given by
$$\left| z \right| = 3.$$ The value of the integral
$$\oint\limits_c {{{{z^2} - z + 4j} \over {z + 2j}}dz} $$ is
$$\left| z \right| = 3.$$ The value of the integral
$$\oint\limits_c {{{{z^2} - z + 4j} \over {z + 2j}}dz} $$ is
40
A system is described by the following differential equation, where $$u(t)$$ is the input to the system and $$y(t)$$ is the output of the system.
$$$\mathop y\limits^ \bullet \left( t \right) + 5y\left( t \right) = u\left( t \right)$$$
When $$y(0)=1$$ and $$u(t)$$ is a unit step function, $$y(t)$$ is
41
In the circuit shown in the figure, the value of capacitor C (in mF) needed to have critically damped response i(t) is ______.
42
A periodic variable x is shown in the figure as a function of time. The root - mean - square (rms) value of x is _______ .
43
For maximum power transfer between two cascaded sections of an electrical network, the relationship between the output impedance Z1 of the first section to the input impedance Z2 of the second section is
44
A Y-network has resistances of 10Ω each in two of its arms, while the third arm has a resistance of 11Ω in the equivalent ∆ − network, the lowest value (in Ω) among the three resistances is ______________.
45
A 230 V rms source supplies power to two loads connected in parallel. The first load draws
10 kW at 0.8 leading power factor and the second one draws 10 kVA at 0.8 lagging power
factor. The complex power delivered by the source is
46
Consider the configuration shown in the figure which is a portion of a larger electrical network
For R = 1 Ω and currents i1 = 2A, i4 = -1A, i5 = -4A, which one of the following is TRUE?
47
For a function g(t), it is given that $$\int_{ - \infty }^\infty {g(t){e^{ - j\omega t}}dt = \omega {e^{ - 2{\omega ^2}}}} $$ for any real value $$\omega $$. If y(t)=$$\int_{ - \infty }^t {g(\tau )d\tau ,\,then\,\int_{ - \infty }^\infty {y(t)\,dt} \,} $$ is
48
A discrete - time signal x[n] = $${\rm{sin(}}\,{\pi ^2}n)$$, n being an integer is
49
Consider two real valued signals, $$x\left( t \right)$$ band - limited to $$\,\left[ { - 500Hz,\,\,500Hz} \right]$$ and $$y\left( t \right)$$ band - limited to $$\,\left[ { - 1\,\,kHz,\,\,1kHz} \right].$$ For $$z\left( t \right)\,\, = \,x\left( t \right).y\left( t \right),$$ the Nyquist sampling frequency $$\left( {in\,\,kHz} \right)$$ is________.
50
A system is described by the following differential equation, where u(t) is the input to the system and y(t) is output of the system $$\mathop y\limits^ \bullet \left( t \right) + 5y\left( t \right) = u\left( t \right)$$
When y(0) = 1 and u(t) is a unit step function, y(t) is
51
Consider a discrete time periodic signal x$$\left[ n \right]$$= $$\sin \left( {{{\pi n} \over 5}} \right)$$. Let ak be the complex Fourier serier coefficients of x$$\left[ n \right]$$. The coefficients $$\left\{ {{a_k}} \right\}$$ are non- zero when k = Bm $$ \pm $$ 1, where m is any integer. The value of B is _________________.
52
A continuous, linear time - invariant fiilter has an impulse response h(t) described by $$h\left( t \right) = \left\{ {\matrix{
{3\,for\,0 \le t \le 3} \cr
{0\,otherwise} \cr
} } \right.$$
Whjen a constant input of value 5 is applied to this filter, the steady state output is ____.
53
Let x $$\left[ n\right]$$= $${\left( { - {1 \over 9}} \right)^n}\,u(n) - {\left( { - {1 \over 3}} \right)^n}u( - n - 1).$$ The region of Convergence (ROC) of the z-tansform of x$$\left[ n \right]$$
General Aptitude
1
You are given three coins: one has heads on both faces, the second has tails on both faces,
and the third has a head on one face and a tail on the other. You choose a coin at random and
toss it, and it comes up heads. The probability that the other face is tails is
2
Find the odd one from the following group:
W,E,K,O
I,Q,W,A
F,N,T,X
N,V,B,D
W,E,K,O
I,Q,W,A
F,N,T,X
N,V,B,D
3
For submitting tax returns, all resident males with annual income below Rs 10 lakh should fill
up Form P and all resident females with income below Rs 8 lakh should fill up Form All
people with incomes above Rs 10 lakh should fill up Form R, except non residents with
income above Rs 15 lakhs, who should fill up Form S. All others should fill Form T. An
example of a person who should fill Form T is
4
A train that is 280 meters long, traveling at a uniform speed, crosses a platform in 60 seconds
and passes a man standing on the platform in 20 seconds. What is the length of the platform
in meters?
5
The exports and imports (in crores of Rs.) of a country from 2000 to 2007 are given in the
following bar chart. If the trade deficit is defined as excess of imports over exports, in which
year is the trade deficit 1/5th of the exports?
