GATE 2018
General Aptitude (GA) Set5
Q. 1 – Q. 5 carry one mark each. Q.1
“By giving him the last ______ of the cake, you will ensure lasting _____ in our house today.” The words that best fill the blanks in the above sentence are (A) peas, piece (C) peace, piece
Q.2
(B) piece, peace (D) peace, peas
“Even though there is a vast scope for its ________, tourism has remained a/an _________ area.” The words that best fill the blanks in the above sentence are (A) improvement, neglected (B) rejection, approved (C) fame, glum (D) interest, disinterested
Q.3
If the number 715∎423 is divisible by 3 (∎ denotes the missing digit in the thousandths place), then the smallest whole number in the place of ∎ is _________. (A) 0
Q.4
Q.5
1
1
4
16
What is the value of 1 + + (A) 2
(C) 5
(B) 2
(B)
7 4
+
1 64
+
1 256
(D) 6
+ ⋯?
(C)
3 2
(D)
4 3
A 1.5 m tall person is standing at a distance of 3 m from a lamp post. The light from the lamp at the top of the post casts her shadow. The length of the shadow is twice her height. What is the height of the lamp post in meters? (A) 1.5
(B) 3
(C) 4.5
(D) 6
Q. 6 – Q. 10 carry two marks each. Q.6
Leila aspires to buy a car worth Rs. 10,00,000 after 5 years. What is the minimum amount in Rupees that she should deposit now in a bank which offers 10% annual rate of interest, if the interest was compounded annually? (A) 5,00,000 (C) 6,66,667
GA
(B) 6,21,000 (D) 7,50,000 1/3
GATE 2018 Q.7
Two alloys A and B contain gold and copper in the ratios of 2:3 and 3:7 by mass, respectively. Equal masses of alloys A and B are melted to make an alloy C. The ratio of gold to copper in alloy C is ______. (A) 5:10
Q.8
General Aptitude (GA) Set5
(B) 7:13
(C) 6:11
(D) 9:13
The Cricket Board has long recognized John’s potential as a leader of the team. However, his onfield temper has always been a matter of concern for them since his junior days. While this aggression has filled stadia with diehard fans, it has taken a toll on his own batting. Until recently, it appeared that he found it difficult to convert his aggression into big scores. Over the past three seasons though, that picture of John has been replaced by a cerebral, calculative and successful batsmancaptain. After many years, it appears that the team has finally found a complete captain. Which of the following statements can be logically inferred from the above paragraph? (i) Even as a junior cricketer, John was considered a good captain. (ii) Finding a complete captain is a challenge. (iii) Fans and the Cricket Board have differing views on what they want in a captain. (iv) Over the past three seasons John has accumulated big scores. (A) (i), (ii) and (iii) only (B) (iii) and (iv) only (C) (ii) and (iv) only (D) (i), (ii), (iii) and (iv)
Q.9
A cab was involved in a hit and run accident at night. You are given the following data about the cabs in the city and the accident. (i) 85% of cabs in the city are green and the remaining cabs are blue. (ii) A witness identified the cab involved in the accident as blue. (iii) It is known that a witness can correctly identify the cab colour only 80% of the time. Which of the following options is closest to the probability that the accident was caused by a blue cab? (A) 12%
GA
(B) 15%
(C) 41%
(D) 80%
2/3
GATE 2018 General Aptitude (GA) Set5 Q.10 A coastal region with unparalleled beauty is home to many species of animals. It is dotted with coral reefs and unspoilt white sandy beaches. It has remained inaccessible to tourists due to poor connectivity and lack of accommodation. A company has spotted the opportunity and is planning to develop a luxury resort with helicopter service to the nearest major city airport. Environmentalists are upset that this would lead to the region becoming crowded and polluted like any other major beach resorts. Which one of the following statements can be logically inferred from the information given in the above paragraph? (A) The culture and tradition of the local people will be influenced by the tourists. (B) The region will become crowded and polluted due to tourism. (C) The coral reefs are on the decline and could soon vanish. (D) Helicopter connectivity would lead to an increase in tourists coming to the region.
END OF THE QUESTION PAPER
GA
3/3
GATE 2018
Electronics and Communication Engineering
Q. 1 – Q. 25 carry one mark each. Q.1
Two identical nMOS transistors M1 and M2 are connected as shown below. The circuit is used as an amplifier with the input connected between G and S terminals and the output taken between D and S terminals. Vbias and VD are so adjusted that both transistors are in saturation. The transconductance of this combination is defined as 𝑔𝑚 resistance is 𝑟𝑜
=
𝜕𝑣𝐷𝑆 𝜕𝑖𝐷
=
𝜕𝑖𝐷
𝜕𝑣𝐺𝑆
while the output
, where 𝑖𝐷 is the current flowing into the drain of M2 . Let 𝑔𝑚1 ,
𝑔𝑚2 be the transconductances and 𝑟𝑜1 , 𝑟𝑜2 be the output resistances of transistors M1 and M2 , respectively.
Which of the following statements about estimates for 𝑔𝑚 and 𝑟𝑜 is correct? (A) (B) (C) (D) Q.2
𝑔𝑚 ≈ 𝑔𝑚1 . 𝑔𝑚2 . 𝑟𝑜2 and 𝑟𝑜 ≈ 𝑟𝑜1 + 𝑟𝑜2 . 𝑔𝑚 ≈ 𝑔𝑚1 + 𝑔𝑚2 and 𝑟𝑜 ≈ 𝑟𝑜1 + 𝑟𝑜2 . 𝑔𝑚 ≈ 𝑔𝑚1 and 𝑟𝑜 ≈ 𝑟𝑜1 . 𝑔𝑚2 . 𝑟𝑜2. 𝑔𝑚 ≈ 𝑔𝑚1 and 𝑟𝑜 ≈ 𝑟𝑜2 .
In the circuit shown below, the opamp is ideal and Zener voltage of the diode is 2.5 volts. At the input, unit step voltage is applied, i.e. 𝑣𝐼𝑁 (𝑡) = 𝑢(𝑡) volts. Also, at 𝑡 = 0, the voltage across each of the capacitors is zero.
The time 𝑡, in milliseconds, at which the output voltage 𝑣𝑂𝑈𝑇 crosses −10 V is (A) 2.5 Q.3
(C) 7.5
(D) 10
A good transimpedance amplifier has (A) (B) (C) (D)
EC
(B) 5
low input impedance and high output impedance. high input impedance and high output impedance. high input impedance and low output impedance. low input impedance and low output impedance.
1/17
GATE 2018
Q.4
Electronics and Communication Engineering
Let the input be u and the output be y of a system, and the other parameters are real constants. Identify which among the following systems is not a linear system: (A)
d3y d2y dy du d 2u a1 2 a2 a3 y b3 u b2 b1 2 (with initial rest conditions) dt 3 dt dt dt dt t
(B)
y t e t u d 0
(C) (D) Q.5
y au b, b 0 y au
The Nyquist stability criterion and the Routh criterion both are powerful analysis tools for determining the stability of feedback controllers. Identify which of the following statements is FALSE: (A) Both the criteria provide information relative to the stable gain range of the system. (B) The general shape of the Nyquist plot is readily obtained from the Bode magnitude plot for all minimumphase systems. (C) The Routh criterion is not applicable in the condition of transport lag, which can be readily handled by the Nyquist criterion. (D) The closedloop frequency response for a unity feedback system cannot be obtained from the Nyquist plot.
Q.6
Consider 𝑝(𝑠) = 𝑠 3 + 𝑎2 𝑠 2 + 𝑎1 𝑠 + 𝑎0 with all real coefficients. It is known that its derivative 𝑝′(𝑠) has no real roots. The number of real roots of 𝑝(𝑠) is (A) 0
Q.7
(B) 1
(C) 2
(D) 3
In a pn junction diode at equilibrium, which one of the following statements is NOT TRUE? (A) The hole and electron diffusion current components are in the same direction. (B) The hole and electron drift current components are in the same direction. (C) On an average, holes and electrons drift in opposite direction. (D) On an average, electrons drift and diffuse in the same direction.
EC
2/17
GATE 2018
Q.8
Electronics and Communication Engineering
The logic function f ( X , Y ) realized by the given circuit is VDD
X
X
Y
Y f (X , Y )
(A) NOR
Q.9
(B) AND
(C) NAND
(D) XOR
A function 𝐹(𝐴, 𝐵, 𝐶) defined by three Boolean variables A, B and C when expressed as sum of products is given by 𝐹 = 𝐴̅ ⋅ 𝐵̅ ⋅ 𝐶̅ + 𝐴̅ ⋅ 𝐵 ⋅ 𝐶̅ + 𝐴 ⋅ 𝐵̅ ⋅ 𝐶̅ ̅ ̅ ̅ where, 𝐴, 𝐵 , and 𝐶 are the complements of the respective variables. The product of sums (POS) form of the function F is (A) 𝐹 = (𝐴 + 𝐵 + 𝐶) ⋅ (𝐴 + 𝐵̅ + 𝐶) ⋅ (𝐴̅ + 𝐵 + 𝐶) (B) 𝐹 = (𝐴̅ + 𝐵̅ + 𝐶̅ ) ⋅ (𝐴̅ + 𝐵 + 𝐶̅ ) ⋅ (𝐴 + 𝐵̅ + 𝐶̅ ) (C) 𝐹 = (𝐴 + 𝐵 + 𝐶̅ ) ⋅ (𝐴 + 𝐵̅ + 𝐶̅ ) ⋅ (𝐴̅ + 𝐵 + 𝐶̅ ) ⋅ (𝐴̅ + 𝐵̅ + 𝐶) ⋅ (𝐴̅ + 𝐵̅ + 𝐶̅ ) (D) 𝐹 = (𝐴̅ + 𝐵̅ + 𝐶) ⋅ (𝐴̅ + 𝐵 + 𝐶) ⋅ (𝐴 + 𝐵̅ + 𝐶) ⋅ (𝐴 + 𝐵 + 𝐶̅ ) ⋅ (𝐴 + 𝐵 + 𝐶)
EC
3/17
GATE 2018
Electronics and Communication Engineering
Q.10 The points P, Q, and R shown on the Smith chart (normalized impedance chart) in the following figure represent:
P
Q
R
(A) P: Open Circuit, Q: Short Circuit, R: Matched Load (B) P: Open Circuit, Q: Matched Load, R: Short Circuit (C) P: Short Circuit, Q: Matched Load, R: Open Circuit (D) P: Short Circuit, Q: Open Circuit, R: Matched Load
Q.11 Let M be a real 4 4 matrix. Consider the following statements: S1: M has 4 linearly independent eigenvectors. S2: M has 4 distinct eigenvalues. S3: M is nonsingular (invertible). Which one among the following is TRUE? (A) S1 implies S2 (C) S2 implies S1
Q.12
(B) S1 implies S3 (D) S3 implies S2
ax 2 by 2 f f , where a and b are constants. If Let f ( x, y ) at x 1 and y 2 , then xy x y the relation between a and b is (A) a
b 4
(B) a
b 2
(C) a 2b
(D) a 4b
Q.13 A discretetime allpass system has two of its poles at 0.250 and 230 . Which one of the following statements about the system is TRUE? (A) It has two more poles at 0.530 and 40 . (B) It is stable only when the impulse response is twosided. (C) It has constant phase response over all frequencies. (D) It has constant phase response over the entire zplane.
EC
4/17
GATE 2018
Electronics and Communication Engineering
Q.14 Let 𝑥(𝑡) be a periodic function with period 𝑇 = 10. The Fourier series coefficients for this series are denoted by 𝑎𝑘 , that is ∞
2𝜋
𝑥(𝑡) = ∑ 𝑎𝑘 𝑒 𝑗𝑘 𝑇 𝑡 𝑘=−∞
The same function 𝑥(𝑡) can also be considered as a periodic function with period 𝑇 ′ = 40. Let 𝑏𝑘 be the Fourier series coefficients when period is taken as 𝑇′. If ∑∞ 𝑘=−∞𝑎𝑘  = 16, then ∞ ∑𝑘=−∞𝑏𝑘  is equal to (A) 256
(B) 64
(C) 16
(D) 4
Q.15 Consider the following amplitude modulated signal: 𝑠(𝑡) = cos(2000 𝜋𝑡) + 4 cos(2400 𝜋𝑡) + cos(2800 𝜋𝑡). The ratio (accurate to three decimal places) of the power of the message signal to the power of the carrier signal is __________.
Q.16 Consider a binary channel code in which each codeword has a fixed length of 5 bits. The Hamming distance between any pair of distinct codewords in this code is at least 2. The maximum number of codewords such a code can contain is _________.
Q.17 A binary source generates symbols 𝑋 ∈ {−1,1} which are transmitted over a noisy channel. The probability of transmitting 𝑋 = 1 is 0.5. Input to the threshold detector is 𝑅 = 𝑋 + 𝑁. The probability density function 𝑓𝑁 (𝑛) of the noise 𝑁 is shown below.
𝑓𝑁 (𝑛) 0.5
−2
2
𝑛
If the detection threshold is zero, then the probability of error (correct to two decimal places) is __________.
Q.18 A pn step junction diode with a contact potential of 0.65 V has a depletion width of 1μm at equilibrium. The forward voltage (in volts, correct to two decimal places) at which this width reduces to 0.6 μm is _______.
EC
5/17
GATE 2018
Electronics and Communication Engineering
Q.19 A traffic signal cycles from GREEN to YELLOW, YELLOW to RED and RED to GREEN. In each cycle, GREEN is turned on for 70 seconds, YELLOW is turned on for 5 seconds and the RED is turned on for 75 seconds. This traffic light has to be implemented using a finite state machine (FSM). The only input to this FSM is a clock of 5 second period. The minimum number of flipflops required to implement this FSM is _______.
Q.20 There are two photolithography systems: one with light source of wavelength λ1 = 156 nm (System 1) and another with light source of wavelength λ2 = 325 nm (System 2). Both photolithography systems are otherwise identical. If the minimum feature sizes that can be realized using System1 and System2 are Lmin1 and Lmin2 respectively, the ratio Lmin1/Lmin2 (correct to two decimal places) is__________. Q.21 A lossy transmission line has resistance per unit length R 0.05 / m . The line is distortionless and has characteristic impedance of 50 . The attenuation constant (in Np/m, correct to three decimal places) of the line is _______. Q.22
2k k x Consider matrix A 2 and vector x 1 . The number of distinct real values 2 k k k x2 of k for which the equation Ax 0 has infinitely many solutions is _______.
Q.23 Let X 1 , X 2 , X 3 and X 4 be independent normal random variables with zero mean and unit variance. The probability that X 4 is the smallest among the four is _______.
Q.24
𝑡2
Taylor series expansion of 𝑓(𝑥) =
𝑥 −( ) ∫0 𝑒 2 𝑑𝑡
around 𝑥 = 0 has the form
𝑓(𝑥) = 𝑎0 + 𝑎1 𝑥 + 𝑎2 𝑥 2 + ⋯ The coefficient 𝑎2 (correct to two decimal places) is equal to _______.
EC
6/17
GATE 2018
Electronics and Communication Engineering
Q.25 The ABCD matrix for a twoport network is defined by:
V1 A B V2 I C D I 2 1
I1
2
2
V1
I2
5
V2
The parameter B for the given twoport network (in ohms, correct to two decimal places) is _______.
EC
7/17
GATE 2018
Electronics and Communication Engineering
Q. 26 – Q. 55 carry two marks each. Q.26 The circuit shown in the figure is used to provide regulated voltage (5 V) across the 1k resistor. Assume that the Zener diode has a constant reverse breakdown voltage for a current range, starting from a minimum required Zener current, I Z min 2 mA to its maximum allowable current. The input voltage VI may vary by 5% from its nominal value of 6 V. The resistance of the diode in the breakdown region is negligible.
R VI
5V
1kΩ
The value of R and the minimum required power dissipation rating of the diode, respectively, are (A) 186 and 10 mW
(B) 100 and 40 mW
(C) 100 and 10 mW
(D) 186 and 40 mW
Q.27 Let c(t ) Ac cos(2 f c t ) and m(t ) cos(2 f mt ). It is given that f c 5 f m . The signal c(t ) m(t ) is applied to the input of a nonlinear device, whose output vo (t ) is related to 2 the input vi (t ) as vo (t ) avi (t ) bvi (t ) , where a and b are positive constants. The output of the nonlinear device is passed through an ideal bandpass filter with center
frequency f c and bandwidth 3 f m , to produce an amplitude modulated (AM) wave. If it is desired to have the sideband power of the AM wave to be half of the carrier power, then a b is (A) 0.25
(B) 0.5
(C) 1
(D) 2
Q.28 Consider a white Gaussian noise process 𝑁(𝑡) with twosided power spectral density 2 𝑆𝑁 (𝑓) = 0.5 W/Hz as input to a filter with impulse response 0.5𝑒 −𝑡 /2 (where 𝑡 is in seconds) resulting in output 𝑌(𝑡). The power in 𝑌(𝑡) in watts is (A) 0.11
EC
(B) 0.22
(C) 0.33
(D) 0.44
8/17
GATE 2018
Electronics and Communication Engineering
Q.29 The state equation and the output equation of a control system are given below: 4 1.5 2 x x u , 0 4 0 y 1.5 0.625 x .
The transfer function representation of the system is (A) (C)
3s 5 s 4s 6 4s 1.5 2 s 4s 6
(B)
2
(D)
3s 1.875 s 2 4s 6 6s 5 2 s 4s 6
Q.30 Red (R), Green (G) and Blue (B) Light Emitting Diodes (LEDs) were fabricated using pn junctions of three different inorganic semiconductors having different bandgaps. The builtin voltages of red, green and blue diodes are VR, VG and VB, respectively. Assume donor and acceptor doping to be the same (NA and ND, respectively) in the p and n sides of all the three diodes. Which one of the following relationships about the builtin voltages is TRUE? (A) (B) (C) (D)
VR > VG > VB VR < VG < VB VR = VG = VB VR > VG < VB
Q.31 A fourvariable Boolean function is realized using 4 1 multiplexers as shown in the figure.
I0
I1
VCC I2 I3
I0
4 1
I1
MUX
I2
S1
S0
U
V
I3
F (U , V ,W , X ) 4 1 MUX S1
S0
W X
The minimized expression for F (U , V ,W , X ) is
EC
(A) (U V U V ) W
(B) (U V U V )(W X W X )
(C) (U V U V ) W
(D) (U V U V )(W X W X )
9/17
GATE 2018
Electronics and Communication Engineering
Q.32 A 2 2 ROM array is built with the help of diodes as shown in the circuit below. Here W0 and W1 are signals that select the word lines and B0 and B1 are signals that are output of the sense amps based on the stored data corresponding to the bit lines during the read operation. B0
B1 Sense amps
B0 B1
W0
W0
D00 D01
W1
D10 D11
W1
Bits stored in the ROM Array
VDD
During the read operation, the selected word line goes high and the other word line is in a high impedance state. As per the implementation shown in the circuit diagram above, what are the bits corresponding to Dij (where i = 0 or 1 and j = 0 or 1) stored in the ROM? (A)
1
0
0
1
(B)
0
1
1
0
(C)
1
0
1
0
(D)
1
1
0
0
Q.33 The distance (in meters) a wave has to propagate in a medium having a skin depth of 0.1 m so that the amplitude of the wave attenuates by 20 dB, is (A) 0.12
(B) 0.23
(C) 0.46
(D) 2.3
Q.34 A curve passes through the point x 1, y 0 and satisfies the differential equation
dy x 2 y 2 y . The equation that describes the curve is dx 2y x 2 (A) ln 1 y2 x 1
x
y (C) ln 1 x 1 x
EC
(B)
1 y2 ln 1 x 1 2 x2
(D)
1 y ln 1 x 1 2 x
10/17
GATE 2018
Electronics and Communication Engineering
Q.35 For the circuit given in the figure, the voltage VC (in volts) across the capacitor is
(A) 1.25 2 sin(5t 0.25 )
(B) 1.25 2 sin(5t 0.125 )
(C) 2.5 2 sin(5t 0.25 )
(D) 2.5 2 sin(5t 0.125 )
Q.36 For the circuit given in the figure, the magnitude of the loop current (in amperes, correct to three decimal places) 0.5 second after closing the switch is _______.
1V
1
1
1H
Q.37 A dc current of 26 μA flows through the circuit shown. The diode in the circuit is forward biased and it has an ideality factor of one. At the quiescent point, the diode has a junction capacitance of 0.5 nF . Its neutral region resistances can be neglected. Assume that the room temperature thermal equivalent voltage is 26 mV.
5 sin( t ) mV
100
V
For 2 106 rad/s , the amplitude of the smallsignal component of diode current (in μA, correct to one decimal place) is _______.
EC
11/17
GATE 2018
Electronics and Communication Engineering
Q.38 An opamp based circuit is implemented as shown below.
31 k
+15 V 1 k
–
+
A
+
1V +
vo
–
–
–15 V
In the above circuit, assume the opamp to be ideal. The voltage (in volts, correct to one decimal place) at node A, connected to the negative input of the opamp as indicated in the figure is _________. Q.39 The input 4sinc(2𝑡) is fed to a Hilbert transformer to obtain 𝑦(𝑡), as shown in the figure below:
Here sinc(𝑥) =
sin(𝜋𝑥) 𝜋𝑥
∞ 𝑦(𝑡)2 𝑑𝑡 . The value (accurate to two decimal places) of ∫−∞
is _______.
Q.40
A random variable 𝑋 takes values −0.5 and 0.5 with probabilities
1 4
and
3 4
, respectively.
The noisy observation of 𝑋 is 𝑌 = 𝑋 + 𝑍, where 𝑍 has uniform probability density over the interval (−1, 1). 𝑋 and 𝑍 are independent. If the MAP rule based detector outputs 𝑋̂ as −0.5, 𝑋̂ = { 0.5,
𝑌<𝛼 𝑌 ≥ 𝛼,
then the value of 𝛼 (accurate to two decimal places) is _______.
EC
12/17
GATE 2018
Electronics and Communication Engineering
Q.41 For a unity feedback control system with the forward path transfer function
G s
K s s 2
The peak resonant magnitude M r of the closedloop frequency response is 2. The corresponding value of the gain K (correct to two decimal places) is _________.
Q.42 The figure below shows the Bode magnitude and phase plots of a stable transfer function
𝐺 (𝑠 ) =
𝑛0 3 2 𝑠 +𝑑2 𝑠 +𝑑1 𝑠+𝑑0
.
Consider the negative unity feedback configuration with gain 𝑘 in the feedforward path. The closed loop is stable for 𝑘 < 𝑘0 . The maximum value of 𝑘0 is ______.
Q.43 In the circuit shown below, the (𝑊/𝐿) value for M2 is twice that for M1. The two nMOS transistors are otherwise identical. The threshold voltage 𝑉𝑇 for both transistors is 1.0V. Note that 𝑉𝐺𝑆 for M2 must be > 1.0 V. Current through the nMOS transistors can be modeled as 𝑊 1 2 𝐼𝐷𝑆 = 𝜇𝐶𝑜𝑥 ( ) ((𝑉𝐺𝑆 − 𝑉𝑇 )𝑉𝐷𝑆 − 𝑉𝐷𝑆 ) for 𝑉𝐷𝑆 ≤ 𝑉𝐺𝑆 − 𝑉𝑇 𝐿 2 𝑊 𝐼𝐷𝑆 = 𝜇𝐶𝑜𝑥 ( ) (𝑉𝐺𝑆 − 𝑉𝑇 )2 /2 𝐿
for 𝑉𝐷𝑆 ≥ 𝑉𝐺𝑆 − 𝑉𝑇
The voltage (in volts, accurate to two decimal places) at 𝑉𝑥 is _______. EC
13/17
GATE 2018
Electronics and Communication Engineering
Q.44 A solar cell of area 1.0 cm 2 , operating at 1.0 sun intensity, has a short circuit current of 20 mA, and an open circuit voltage of 0.65 V. Assuming room temperature operation and thermal equivalent voltage of 26 mV, the open circuit voltage (in volts, correct to two decimal places) at 0.2 sun intensity is _______. Q.45 A junction is made between p Si with doping density NA1 = 1015 cm3 and p Si with doping density NA2 = 1017 cm3. Given: Boltzmann constant k = 1.38 x 1023 JK1, electronic charge q = 1.6 x 1019 C. Assume 100% acceptor ionization. At room temperature (T = 300K), the magnitude of the builtin potential (in volts, correct to two decimal places) across this junction will be _________________.
Q.46 In the circuit shown below, a positive edgetriggered D FlipFlop is used for sampling input data 𝐷𝑖𝑛 using clock 𝐶𝐾. The XOR gate outputs 3.3 volts for logic HIGH and 0 volts for logic LOW levels. The data bit and clock periods are equal and the value of Δ𝑇/𝑇𝐶𝐾 = 0.15, where the parameters Δ𝑇 and 𝑇𝐶𝐾 are shown in the figure. Assume that the FlipFlop and the XOR gate are ideal.
If the probability of input data bit (𝐷𝑖𝑛 ) transition in each clock period is 0.3, the average value (in volts, accurate to two decimal places) of the voltage at node 𝑋, is _______.
EC
14/17
GATE 2018
Electronics and Communication Engineering
Q.47 The logic gates shown in the digital circuit below use strong pulldown nMOS transistors for LOW logic level at the outputs. When the pulldowns are off, highvalue resistors set the output logic levels to HIGH (i.e. the pullups are weak). Note that some nodes are intentionally shorted to implement “wired logic”. Such shorted nodes will be HIGH only if the outputs of all the gates whose outputs are shorted are HIGH.
The number of distinct values of 𝑋3 𝑋2 𝑋1 𝑋0 (out of the 16 possible values) that give 𝑌 = 1 is _______.
Q.48 The cutoff frequency of TE01 mode of an air filled rectangular waveguide having inner dimensions a cm b cm ( a b ) is twice that of the dominant TE10 mode. When the waveguide is operated at a frequency which is 25% higher than the cutoff frequency of the dominant mode, the guide wavelength is found to be 4 cm. The value of b (in cm, correct to two decimal places) is _______.
Q.49 A uniform plane wave traveling in free space and having the electric field 𝐸⃗ = (√2𝑎̂𝑥 − 𝑎̂𝑧 ) cos[6√3𝜋 × 108 𝑡 − 2𝜋(𝑥 + √2𝑧)] 𝑉/𝑚 is incident on a dielectric medium (relative permittivity > 1, relative permeability = 1) as shown in the figure and there is no reflected wave. z Free space
Dielectric Medium (r >1) x
The relative permittivity (correct to two decimal places) of the dielectric medium is ___________.
EC
15/17
GATE 2018
Electronics and Communication Engineering
Q.50 The position of a particle 𝑦(𝑡) is described by the differential equation: 𝑑2𝑦 𝑑𝑦 5𝑦 = − − . 𝑑𝑡 2 𝑑𝑡 4 The initial conditions are 𝑦(0) = 1 and
𝑑𝑦

𝑑𝑡 𝑡=0
= 0. The position (accurate to two
decimal places) of the particle at 𝑡 = 𝜋 is _______.
Q.51 The contour C given below is on the complex plane z x jy , where j 1 . y C
1
The value of the integral
1
j C
1
x
dz is _______. z 1 2
Q.52 Let 𝑟 = 𝑥 2 + 𝑦 − 𝑧 and 𝑧 3 − 𝑥𝑦 + 𝑦𝑧 + 𝑦 3 = 1. Assume that 𝑥 and 𝑦 are independent variables. At (𝑥, 𝑦, 𝑧) = (2, −1,1), the value (correct to two decimal places) of
𝜕𝑟 𝜕𝑥
is
_________ .
Q.53 Consider the network shown below with 𝑅1 = 1Ω, 𝑅2 = 2Ω and 𝑅3 = 3Ω. The network is connected to a constant voltage source of 11V.
The magnitude of the current (in amperes, accurate to two decimal places) through the source is _______.
EC
16/17
GATE 2018
Electronics and Communication Engineering
Q.54 A band limited lowpass signal 𝑥(𝑡) of bandwidth 5 kHz is sampled at a sampling rate 𝑓𝑠 . The signal 𝑥(𝑡) is reconstructed using the reconstruction filter 𝐻(𝑓) whose magnitude response is shown below:
The minimum sampling rate 𝑓𝑠 (in kHz) for perfect reconstruction of 𝑥(𝑡) is _______.
Q.55 Let 𝑋[𝑘] = 𝑘 + 1, 0 ≤ 𝑘 ≤ 7 be 8point DFT of a sequence 𝑥[𝑛], where 𝑋[𝑘] = ∑𝑁−1 𝑛=0 𝑥[𝑛]𝑒
−𝑗2𝜋𝑛𝑘/𝑁
.
The value (correct to two decimal places) of ∑3𝑛=0 𝑥[2𝑛] is ______.
END OF THE QUESTION PAPER
EC
17/17