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# GATE 2016 EC - SET 2 (Session 3) - Complete Solutions

Q1. The value of x for which the matrix

has zero as an eigenvalue is ________

Solution: Key = 0.95 to 1.05

Q2. Consider the complex valued function f(z) = 2z3 + b|z|3where z is a complex variable. The value of b for which the function f(z) is analytic is ______

Solution: Key = 0

Q3. As x varies from −1 to +3, which one of the following describes the behaviour of the function f(x) = x3 − 3x2 + 1?

(A) f(x) increases monotonically.

(B) f(x) increases, then decreases and increases again.

(C) f(x) decreases, then increases and decreases again.

(D) f(x) increases and then decreases.

Solution: (B)

Q4. How many distinct values of x satisfy the equation sin(x) = x/2, where x is in radians?

(A) 1               (B) 2                (C) 3               (D) 4 or more

Solution: (C)

Q5. Consider the time-varying vector $I=\hat{x}15\cos (\omega t)+\hat{y}\sin (\omega t)$ in Cartesian coordinates, where ω > 0 is a constant. When the magnitude |I| is at its minimum value, the angle θ that I makes with the x axis (in degrees, such that 0 ≤ θ ≤ 180) is _____

Solution: Key = 90

Q6. In the circuit shown below, Vs is a constant voltage source and IL is a constant current load.

The value of IL that maximizes the power absorbed by the constant current load is

(A) $\frac{V_{s}}{4R}$        (B) $\frac{V_{s}}{2R}$     (C) $\frac{V_{s}}{R}$       (D) $\infty$

Solution: (B)

Q7. The switch has been in position 1 for a long time and abruptly changes to position 2 at t = 0.

If time t is in seconds, the capacitor voltage VC (in volts) for t > 0 is given by

(A) 4(1 − exp(−t/0.5))        (B) 10 − 6 exp(−t/0.5)        (C) 4(1 − exp(-t/0.6))       (D) 10 − 6 exp(−t/0.6)

Solution: (D)

Q8. The figure shows an RLC circuit with a sinusoidal current source.

At resonance, the ratio |IL|/|IR|, i.e., the ratio of the magnitudes of the inductor current phasor and the resistor current phasor, is _______

Solution: Key = 0.30 to 0.34

Q9. The z-parameter matrix for the two-port network shown is

where the entries are in Ω. Suppose Zb(jω) = Rb + jω.

Then the value of Rb (in Ω) equals ______

Solution: Key = 2.98 to 3.02

Q10. The energy of the signal $x(t)=\frac{\sin (4\pi t)}{4\pi t}$ is ________

Solution: Key = 0.24 to 0.26

Q11. The Ebers-Moll model of a BJT is valid

(A) only in active mode                                     (B) only in active and saturation modes

(C) only in active and cut-off modes              (D) in active, saturation and cut-off modes

Solution: (D)

Q12. A long-channel NMOS transistor is biased in the linear region with VDS = 50 mV and is used as a resistance. Which one of the following statements is NOT correct?

(A) If the device width W is increased, the resistance decreases.

(B) If the threshold voltage is reduced, the resistance decreases.

(C) If the device length L is increased, the resistance increases.

(D) If VGS is increased, the resistance increases.

Solution: (D)

Q13. Assume that the diode in the figure has Von = 0.7 V, but is otherwise ideal.

The magnitude of the current i2 (in mA) is equal to _______

Solution: Key = 0.25

Q14. Resistor R1 in the circuit below has been adjusted so that I1= 1 mA. The bipolar transistors Q1 and Q2 are perfectly matched and have very high current gain, so their base currents are negligible. The supply voltage Vcc is 6 V. The thermal voltage kT/q is 26 mV.

The value of R2 (in Ω) for which I2 = 100 μA is ______

Solution: Key = 570 to 610

Q15. Which one of the following statements is correct about an ac-coupled common-emitter amplifier operating in the mid-band region?

(A) The device parasitic capacitances behave like open circuits, whereas coupling and bypass capacitances behave like short circuits.

(B) The device parasitic capacitances, coupling capacitances and bypass capacitances behave like open circuits.

(C) The device parasitic capacitances, coupling capacitances and bypass capacitances behave like short circuits.

(D) The device parasitic capacitances behave like short circuits, whereas coupling and bypass capacitances behave like open circuits.

Solution: (A)

Q16. Transistor geometries in a CMOS inverter have been adjusted to meet the requirement for worst case charge and discharge times for driving a load capacitor C. This design is to be converted to that of a NOR circuit in the same technology, so that its worst case charge and discharge times while driving the same capacitor are similar. The channel lengths of all transistors are to be kept unchanged. Which one of the following statements is correct?

(A) Widths of PMOS transistors should be doubled, while widths of NMOS transistors should be halved.

(B) Widths of PMOS transistors should be doubled, while widths of NMOS transistors should not be changed.

(C) Widths of PMOS transistors should be halved, while widths of NMOS transistors should not be changed.

(D) Widths of PMOS transistors should be unchanged, while widths of NMOS transistors should be halved.

Solution: (B)

Q17. Assume that all the digital gates in the circuit shown in the figure are ideal, the resistor R = 10 kΩ and the supply voltage is 5 V. The D flip-flops D1, D2, D3, D4 and D5 are initialized with logic values 0,1,0,1 and 0, respectively. The clock has a 30% duty cycle.

The average power dissipated (in mW) in the resistor R is ______

Solution: Key = 1.45 to 1.55

Q18. A 4:1 multiplexer is to be used for generating the output carry of a full adder. A and B are the bits to be added while Cin is the input carry and Cout is the output carry. A and B are to be used as the select bits with A being the more significant select bit.

Which one of the following statements correctly describes the choice of signals to be connected to the inputs I0, I1, I2and I3 so that the output is Cout?

(A) $I_{0}=0$, $I_{1}=C_{in}$, $I_{2}=C_{in}$ and $I_{3}=1$

(B) $I_{0}=1$, $I_{1}=C_{in}$, $I_{2}=C_{in}$ and $I_{3}=1$

(C) $I_{0}=C_{in}$, $I_{1}=0$, $I_{2}=1$ and $I_{3}=C_{in}$

(D) $I_{0}=0$, $I_{1}=C_{in}$, $I_{2}=1$ and $I_{3}=C_{in}$

Solution: (A)

Q19. The response of the system $G(s)=\frac{s-2}{(s+1)(s+3)}$ to the unit step input u(t) is y(t).

The value of $\frac{\mathrm{d} y}{\mathrm{d} t}$ at $t=0^{+}$ is ________

Solution: Key = 0.96 to 1.04

Q20. The number and direction of encirclements around the point −1 + j0 in the complex plane by the Nyquist plot of $G(s)=\frac{1-s}{4+2s}$ is

(A) zero.                                  (B) one, anti-clockwise.

(C) one, clockwise.                (D) two, clockwise.

Solution: (A)

Q21. A discrete memoryless source has an alphabet {a1, a2, a3, a4} with corresponding probabilities  The minimum required average codeword length in bits to represent this source for error-free reconstruction is ________

Solution: Key = 1.74 to 1.76

Q22. A speech signal is sampled at 8 kHz and encoded into PCM format using 8 bits/sample. The PCM data is transmitted through a baseband channel via 4-level PAM. The minimum bandwidth (in kHz) required for transmission is ________

Solution: Key = 15.9 to 16.1

Q23. A uniform and constant magnetic field  exists in the  direction in vacuum. A particle of mass m with a small charge q is introduced into this region with an initial velocity  Given that B, m, q, vx and vz are all non-zero, which one of the following describes the eventual trajectory of the particle?

(A) Helical motion in the direction.              (B) Circular motion in the xy plane.

(C) Linear motion in the direction.               (C) Linear motion in the direction.

Solution: (A)

Q24. Let the electric field vector of a plane electromagnetic wave propagating in a homogenous medium be expressed as  , where the propagation constant β is a function of the angular frequency ω. Assume that β(ω) and Ex, are known and are real. From the information available, which one of the following CANNOT be determined?

(A) The type of polarization of the wave.           (B) The group velocity of the wave.

(C) The phase velocity of the wave.                     (D) The power flux through the z = 0 plane.

Solution: (D)

Q25. Light from free space is incident at an angle θi to the normal of the facet of a step-index large core optical fibre. The core and cladding refractive indices are n1 = 1.5 and n2 = 1.4, respectively.

The maximum value of θi (in degrees) for which the incident light will be guided in the core of the fibre is ________

Solution: Key = 32 to 33

Q26. The ordinary differential equation

with x (0) = 1

is to be solved using the forward Euler method. The largest time step that can be used to solve the equation without making the numerical solution unstable is ________

Solution: Key = 0.6 to 0.7

Q27. Suppose C is the closed curve defined as the circle x2 + y2= 1 with C oriented anti-clockwise. The value of ∮(xy2dx + x2y dy) over the curve C equals________

Solution: Key = -0.03 to 0.03

Q28. Two random variables X and Y are distributed according to

The probability P(X + Y ≤ 1) is ______

Solution: Key = 0.32 to 0.34

Q29. The matrix

has det(A) = 100 and trace (A) = 14.

The value of |a − b| is _______

Solution: Key = 2.9 to 3.1

Q30. In the given circuit, each resistor has a value equal to 1 Ω.

What is the equivalent resistance across the terminals a and b?

(A) 1/6 Ω            (B) 1/3 Ω             (C) 9/20 Ω               (D) 8/15 Ω

Solution: (D)

Q31. In the circuit shown in the figure, the magnitude of the current (in amperes) through R2 is _____

Solution: Key = 4.9 to 5.1

Q32. A continuous-time filter with transfer function $H(s)=\frac{2s+6}{s^{2}+6s+8}$ is converted to a discrete-time filter with transfer function $G(s)=\frac{2z^{2}-0.5032z}{z^{2}-0.5032z+k}$ so that the impulse response of the continuous-time filter, sampled at 2 Hz, is identical at the sampling instants to the impulse response of the discrete time filter. The value of k is _______

Solution: Key = 0.04 to 0.06

Q33. The Discrete Fourier Transform (DFT) of the 4-point sequence

x[n] = {x[0], x[1], x[2], x[3]} = {3, 2, 3, 4} is

X[k] = {X[0], X[1], X[2], X[3]} = {12, 2j, 0, −2j}.

If X1[k] is the DFT of the 12-point sequence x1[n] = {3, 0, 0, 2, 0, 0, 3, 0, 0, 4, 0, 0}, the value of  is ______

Solution: Key = 5.9 to 6.1

Q34. The switch S in the circuit shown has been closed for a long time. It is opened at time t = 0 and remains open after that. Assume that the diode has zero reverse current and zero forward voltage drop.

The steady state magnitude of the capacitor voltage VC (in volts) is _____

Solution: Key = 99 to 101

Q35. A voltage VG is applied across a MOS capacitor with metal gate and p-type silicon substrate at T = 300 K. The inversion carrier density (in number of carriers per unit area) for VG = 0.8 V is 2 × 1011 cm−2. For VG = 1.3 V, the inversion carrier is 4 × 1011 cm−2. What is the value of the inversion carrier density for VG = 1.8 V?

(A) $4.5\times 10^{11}cm^{-2}$                  (B) $6.0\times 10^{11}cm^{-2}$

(C) $7.2\times 10^{11}cm^{-2}$                  (D) $8.4\times 10^{11}cm^{-2}$

Solution: (B)

Q36. Consider avalanche breakdown in a silicon p+ n junction. The n-region is uniformly doped with a donor density ND. Assume that breakdown occurs when the magnitude of the electric field at any point in the device becomes equal to the critical field Ecrit. Assume Ecrit to be independent of ND. If the built-in voltage of the p+ n junction is much smaller than the breakdown voltage, VBR, the relationship between VBR and ND is given by

(A) $V_{BR}\times \sqrt{N_{D}}$ = constant

(B) $N_{D}\times \sqrt{V_{BR}}$ = constant

(C) $N_{D}\times V_{BR}$ = constant

(D) $N_{D}\V_{BR}$ = constant

Solution: (C)

Q37. Consider a region of silicon devoid of electrons and holes, with an ionized donor density of Nd+= 1017 cm−3. The electric field at x = 0 is V/cm and the electric field at x = L is 50 kV/cm in the positive x direction. Assume that the electric field is zero in the y and z directions at all points.

Given q = 1.6 × 10−19 coulomb, ∈0 = 8.85 × 10−14 F/cm, ∈r = 11.7 for silicon, the value of L in nm is ______

Solution: Key = 30 to 34

Q38. Consider a long-channel NMOS transistor with source and body connected together. Assume that the electron mobility is independent of VGS and VDS. Given,

gm = 0.5 μA/V for VDS = 50 mV and VGS = 2 V,

gd = 8 μA/V for VGS = 2 V and VDS = 0 V,

The threshold voltage (in volts) of the transistor is ________

Solution: Key = 1.18 to 1.22

Q39. The figure shows a half-wave rectifier with a 475 μF filter capacitor. The load draws a constant current I0 = 1 A from the rectifier. The figure also shows the input voltage Vi, the output voltage Vc and the peak-to-peak voltage ripple u on Vc. The input voltage Vi is a triangle-wave with an amplitude of 10 V and a period of 1 ms.

The value of the ripple u (in volts) is _______

Solution: Key = 1.9 to 2.2

Q40. In the opamp circuit shown, the Zener diodes Z1 and Z2 clamp the output voltage V0 to +5V or −5 V. The switch S is initially closed and is opened at time t = 0.

The time t = t1 (in seconds) at which V0 changes state is _______

Solution: Key = 0.7 to 0.9

Q41. An opamp has a finite open loop voltage gain of 100. Its input offset voltage Vios (= +5mV) is modeled as shown in the circuit below. The amplifier is ideal in all other respects. Vinput is 25 mV.

The output voltage (in millivolts) is ________

Solution: Key = 400 to 425

Q42. An 8 Kbyte ROM with an active low Chip Select input  is to be used in an 8085 microprocessor based system. The ROM should occupy the address range 1000H to 2FFFH. The address lines are designated as A15 to A0. where A15 is the most significant address bit. Which one of the following logic expressions will generate the correct  signal for this ROM?

(A)

(B)

(C)

(D)

Solution: (A)

Q43. In an N bit flash ADC, the analog voltage is fed simultaneously to 2N − 1 comparators. The output of the comparators is then encoded to a binary format using digital circuits. Assume that the analog voltage source Vin (whose output is being converted to digital format) has a source resistance of 75 Ω as shown in the circuit diagram below and the input capacitance of each comparator is 8 pF. The input must settle to an accuracy of 1/2 LSB even for a full scale input change for proper conversion. Assume that the time taken by the thermometer to binary encoder is negligible.

If the flash ADC has 8 bit resolution, which one of the following alternatives is closest to the maximum sampling rate ?

(A) 1 megasamples per second                       (B) 6 megasamples per second

(C) 64 megasamples per second                    (D) 256 megasamples per second

Solution: (A)

Q44. The state transition diagram for a finite state machine with states A, B and C, and binary inputs X, Y and Z, is shown in the figure.

Which one of the following statements is correct?

(A) Transitions from State A are ambiguously defined.

(B) Transitions from State B are ambiguously defined.

(C) Transitions from State C are ambiguously defined.

(D) All of the state transitions are defined unambiguously.

Solution: (C)

Q45. In the feedback system shown below $G(s)=\frac{1}{s^{2}+2s}$ .

The step response of the closed-loop system should have minimum settling time and have no overshoot.

The required value of gain k to achieve this is _______

Solution: Key = 0.95 to 1.05

Q46. In the feedback system shown below $G(s)=\frac{1}{(s+1)(s+2)(s+3)}$

The positive value of k for which the gain margin of the loop is exactly 0 dB and the phase margin of the loop is exactly zero degree is ________

Solution: Key = 59.5 to 60.5

Q47. The asymptotic Bode phase plot of $G(s)=\frac{k}{(s+0.1)(s+10)(s+p_{1})}$ with k and p1 both positive, is shown below.

The value of p1 is ______

Solution: Key = 0.95 to 1.05

Q48. An information source generates a binary sequence {αn}. αncan take one of the two possible values −1 and +1 with equal probability and are statistically independent and identically distributed. This sequence is precoded to obtain another sequence {βn}, as βn = α+ kαn-3 where

If there is a null at  in the power spectral density of X(t), then k is ______

Solution: Key = -1.01 to -0.99

Q49. An ideal band-pass channel 500 Hz – 2000 Hz is deployed for communication. A modem is designed to transmit bits at the rate of 4800 bits/s using 16-QAM. The roll-off factor of a pulse with a raised cosine spectrum that utilizes the entire frequency band is ________

Solution: Key =0.24 to 0.26

Q50. Consider a random process X(t) = 3V(t) − 8, where V(t) is a zero mean stationary random process with autocorrelationRv(τ) = 4e−5|τ|. The power in X(t) is _____

Solution: Key = 99 to 101

Q51. A binary communication system makes use of the symbols “zero” and “one”. There are channel errors. Consider the following events:

x0 : a “zero” is transmitted

x1 : a “one” is transmitted

y0 : a “zero” is received

y1 : a “one” is received

The following probabilities are given:  and  The information in bits that you obtain when you learn which symbol has been received (while you know that a “zero” has been transmitted) is ________

Solution: Key = 0.80 to 0.82

Q52. The parallel-plate capacitor shown in the figure has movable plates. The capacitor is charged so that the energy stored in it is E when the plate separation is d. The capacitor is then isolated electrically and the plates are moved such that the plate separation becomes 2d.

At this new plate separation, what is the energy stored in the capacitor, neglecting fringing effects?

(A) 2E             (B) √ 2E            (C) E           (D) E/2

Solution: (A)

Q53. A lossless microstrip transmission line consists of a trace of width w. It is drawn over a practically infinite ground plane and is separated by a dielectric slab of thickness t and relative permittivity εr >  1. The inductance per unit length and the characteristic impedance of this line are L and Z0, respectively.

Which one of the following inequalities is always satisfied?

(A)              (B)              (C)                (D)

Solution: (B)

Q54. A microwave circuit consisting of lossless transmission lines T1 and T2 is shown in the figure. The plot shows the magnitude of the input reflection coefficient Γ as a function of frequency f. The phase velocity of the signal in the transmission lines is 2 × 108 m/s.

The length L (in meters) of T2 is ______

Solution: Key = 0.09 to 0.11

Q55. A positive charge q is placed at x = 0 between two infinite metal plates placed at x = −d and at x = +d respectively. The metal plates lie in the yz plane.

The charge is at rest at t = 0 when a voltage +V is applied to the plate at −d and voltage −V is applied to the plate at x = +d. Assume that the quantity of the charge q is small enough that it does not perturb the field set up by the metal plates. The time that the charge q takes to reach the right plate is proportional to

(A)           (B)              (C)           (D)

Solution: (C)

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