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# GATE 2016 EC - SET 1 - Complete Solutions

Q1. Let M4 = I, (where I denotes the identity matrix) and M ≠ I, M2 ≠ I and M3 ≠ I. Then, for any natural number k, M−1equals:

(A) $M^{4k+1}$           (B) $M^{4k+2}$               (C) $M^{4k+3}$             (D)$M^{4k}$

Solution: (C)

Q2. The second moment of a Poisson-distributed random variable is 2. The mean of the random variable is _______

Solution: Key = 0.9 to 1.1

Q3. Given the following statements about a function f : R → R, select the right option:

(P) If f(x) is not continuous at x = x0, then it is also differentiable at x = x0.

(Q) If f(x) is continuous at x = x0, then it may not be differentiable at x = x0.

(R) If f(x) is differentiable at x = x0, then it is also continuous at x = x0.

(A) P is true, Q is false, R is false                  (B) P is false, Q is true, R is true

(C) P is false, Q is true, R is false                  (D) P is true, Q is false, R is true

Solution: (B)

Q4. Which one of the following is a property of the solutions to the Laplace equation: ∇2f = 0?

(A) The solutions have neither maxima nor minima anywhere except at the boundaries.

(B) The solutions are not separable in the coordinates.

(C) The solutions are not continuous.

(D) The solutions are not dependent on the boundary conditions.

Solution: (A)

Q5. Consider the plot of f(x) versus x as shown below.

Suppose  Which one of the following is a graph of F(x)?

(A)

(C)              (D)

Solution: (C)

Q6. Which one of the following is an eigen function of the class of all continuous-time, linear, timeinvariant systems (u(t) denotes the unit-step function)?

(A) $e^{j\omega _{0}t}u(t)$           (B) $\cos(\omega _{0}t)$

(C) $e^{j\omega _{0}t}$                  (D) $\sin(\omega _{0}t)$

Solution: (C)

Q7. A continuous-time function x(t) is periodic with period T. The function is sampled uniformly with a sampling period Ts. In which one of the following cases is the sampled signal periodic?

(A) $T=\sqrt{2}T_{s}$          (B) $T=1.2T_{s}$             (C) Always             (D) Never

Solution: (B)

Q8. Consider the sequence x[n] = anu[n] + bnu[n], where u[n] denotes the unit-step sequence and 0 < |a| < |b| < 1. The region of convergence (ROC) of the z-transform of x[n] is

(A) |z| > |a|               (B) |z| > |b|              (C) |z| < |a|               (D) |a| < |z| < |b|

Solution: (B)

Q9. Consider a two-port network with the transmission matrix:  If the network is reciproal, then

(A) $T^{-1}=1$           (B) $T^{2}=1$           (C) Determinant (T) = 0          (D) Determinant (T) = 1

Solution: (D)

Q10. A continuous-time sinusoid of frequency 33 Hz is multiplied with a periodic Dirac impulse train of frequency 46 Hz. The resulting signal is passed through an ideal analog low-pass filter with a cutoff frequency of 23 Hz. The fundamental frequency (in Hz) of the output is _________

Solution: Key = 12 to 14

Q11. A small percentage of impurity is added to an intrinsic semiconductor at 300 K. Which one of the following statements is true for the energy band diagram shown in the following figure?

(A) Intrinsic semiconductor doped with pentavalent atoms to form n-type semiconductor

(B) Intrinsic semiconductor doped with trivalent atoms to form n-type semiconductor

(C) Intrinsic semiconductor doped with pentavalent atoms to form p-type semiconductor

(D) Intrinsic semiconductor doped with trivalent atoms to form p-type semiconductor

Solution: (A)

Q12. Consider the following statements for a metal oxide semiconductor field effect transistor (MOSFET):

P: As channel length reduces, OFF-state current increases.

Q: As channel length reduces, output resistance increases.

R: As channel length reduces, threshold voltage remains constant.

S: As channel length reduces, ON current increases.

Which of the above statements are INCORRECT?

(A) P and Q           (B) P and S            (C) Q and R             (D) R and S

Solution: (C)

Q13. Consider the constant current source shown in the figure below. Let β represent the current gain of the transistor.

The load current I0 through RL is

(A)

(B)

(C)

(D)

Solution: (B)

Q14. The following signal Vi of peak voltage 8 V is applied to the non-inverting terminal of an ideal opamp. The transistor has VBE = 0.7 V, β = 100; VLED = 1.5 V, VCC  = 10 V and −VCC= −10 V.

The number of times the LED glows is ________

Solution: Key = 2.9 to 3.1

Q15. Consider the oscillator circuit shown in the figure. The function of the network (shown in dotted lines) consisting of the 100 kΩ resistor in series with the two diodes connected back-to-back is to:

(A) introduce amplitude stabilization by preventing the op amp from saturating and thus producing sinusoidal oscillations of fixed amplitude

(B) introduce amplitude stabilization by forcing the opamp to swing between positive and negative saturation and thus producing square wave oscillations of fixed amplitude

(C) introduce frequency stabilization by forcing the circuit to oscillate at a single frequency

(D) enable the loop gain to take on a value that produces square wave oscillations

Solution: (A)

Q16. The block diagram of a frequency synthesizer consisting of a Phase Locked Loop (PLL) and a divide-by-N counter (comprising ÷ 2, ÷ 4, ÷8, ÷16 outputs) is sketched below. The synthesizer is excited with a 5 kHz signal (Input 1). The free-running frequency of the PLL is set to 20 kHz. Assume that the commutator switch makes contacts repeatedly in the order 1-2-3-4.

The corresponding frequencies synthesized are:

(A) 10 kHz, 20 kHz, 40 kHz, 80 kHz                    (B) 20 kHz, 40 kHz, 80 kHz, 160 kHz

(C) 80 kHz, 40 kHz, 20 kHz, 10 kHz                    (D) 160 kHz, 80 kHz, 40 kHz, 20 kHz

Solution: (A)

Q17. The output of the combinational circuit given below is

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

Solution: (C)

Q18. What is the voltage Vout in the following circuit?

(A) 0V

(B)

(C) Switching threshold of inverter

(D) $V_{DD}$

Solution: (C)

Q19. Match the inferences X, Y, and Z, about a system, to the corresponding properties of the elements of first column in Routh’s Table of the system characteristic equation.

X: The system is stable …
Y: The system is unstable …
Z: The test breaks down …
P: … when all elements are positive
Q: … when any one element is zero
R: … when there is a change in sign of coefficients

(A) X→P, Y→Q, Z→R                 (B) X→Q, Y→P, Z→R

(C) X→R, Y→Q, Z→P                 (D) X→P, Y→R, Z→Q

Solution: (D)

Q20. A closed-loop control system is stable if the Nyquist plot of the corresponding open-loop transfer function

(A) encircles the s-plane point (−1 + j0) in the counterclockwise direction as many times as the number of right-half s-plane poles.

(B) encircles the s-plane point (0 − j1) in the clockwise direction as many times as the number of right-half s-plane poles.

(C) encircles the s-plane point (−1 + j0) in the counterclockwise direction as many times as the number of left-half s-plane poles.

(D) encircles the s-plane point (−1 + j0) in the counterclockwise direction as many times as the number of right-half s-plane zeros.

Solution: (A)

Q21. Consider binary data transmission at a rate of 56 kbps using baseband binary pulse amplitude modulation (PAM) that is designed to have a raised-cosine spectrum. The transmission bandwidth (in kHz) required for a roll-off factor of 0.25 is ________

Solution: Key = 34.5 to 35.5

Q22. A superheterodyne receiver operates in the frequency range of 58 MHz − 68 MHz. The intermediate frequency fIF and local oscillator frequency fLO are chosen such that fIF ≤ fLO . It is required that the image frequencies fall outside t he 5 8 M Hz − 68 MHz band. The minimum required fIF (in MHZ) is _________

Solution: Key = 4.9 to 5.1

Q23. The amplitude of a sinusoidal carrier is modulated by a single sinusoid to obtain the amplitude modulated signal s(t) = 5 cos 1600πt + 20 cos 1800πt + 5 cos 2000πt. The value of the modulation index is __________

Solution: Key = 0.49 to 0.51

Q24. Concentric spherical shells of radii 2 m, 4 m, and 8 m carry uniform surface charge densities of 20 nC/m2, −4 nC/m2and ρs, respectively. The value of ρs (nC/m2) required to ensure that the electric flux density  at radius 10 m is _______

Solution: Key = -0.28 to -0.22

Q25. The propagation constant of a lossy transmission line is (2 + j5) m−1 and its characteristic impedance is (5 + j0) Ω at ω = 106 rad s−1. The value of the line constants L, C, R, G are, respectively,

(A) L = 200 μH/m, C = 0.1 μF/m, R = 50 Ω/m, G= 0.02 S/m

(B) L = 250 μH/m, C = 0.1 μF/m, R = 100 Ω/m, G= 0.04 S/m

(C) L = 200 μH/m, C = 0.2 μF/m, R = 100 Ω/m, G= 0.02 S/m

(D) L = 250 μH/m, C = 0.2 μF/m, R = 50 Ω/m, G= 0.04 S/m

Solution: (B)

Q26. The integral $\frac{1}{2\pi }\iint_{D}(x+y+10)dx dy$ where D denotes the disc, x2 + y2 ≤ 4, evaluates to _______

Solution: Key = 18 to 22

Q27. A sequence x[n] is specified as

The initial conditions are x[0] = 1, x[1] = 1, and x[n] = 0 for n < 0. The value of x[12] is _________

Solution: Key = 230 to 240

Q28. In the following integral, the contour C encloses the points 2πj and − 2πj

The value of the integral is ________

Solution: Key = -136 to -132

Q29. The region specified by

in cylindrical coordinates has volume of _______

Solution: Key = 4.66 to 4.76

Q29. The Laplace transform of the causal periodic square wave of period T shown in the figure below is

(A)               (B)          (C)        (D

Solution: MTA

Q31. A network consisting of a finite number of linear resistor (R), inductor (L), and capacitor (C) elements, connected all in series or all in parallel, is excited with a source of the form

The source has nonzero impedance. Which one of the following is a possible form of the output measured across a resistor in the network?

(A)

(B)

(C)               (D)

Solution: (A)

Q32. A first-order low-pass filter of time constant T is excited with different input signals (with zero initial conditions up to t = 0). Match the excitation signals X, Y, Z with the corresponding time responses for t ≥ 0:

X: Impulse                 P:1 − e−t/T

Y: Unit step               Q: t − T(1 − e−t/T)

Z: Ramp                    R: e−t/T

(A) X→R, Y→Q, Z→P               (B) X→Q, Y→P, Z→R

(C) X→R, Y→P, Z→Q               (D) X→P, Y→R, Z→Q

Solution: (C)

Q33. An AC voltage source V = 10 sin(t) volts is applied to the following network. Assume that R1 = 3 kΩ, R2 = 6 kΩ and R3= 9 kΩ, and that the diode is ideal.

RMS current Irms (in mA) through the diode is ________

Solution: Key = 0.9 to 1.1

Q34. In the circuit shown in the figure, the maximum power (in watt) delivered to the resistor R is __________

Solution: Key = 0.78 to 0.82

Q35. Consider the signal

x[n] = 6 δ[n + 2] + 3δ[n + 1] + 8δ[n] + 7δ[n − 1] + 4δ[n − 2]

If X(e) is the discrete-time Fourier transform of x[n]

then  is equal to __________

Solution: Key = 7.9 to 8.1

Q36. Consider a silicon p-n junction with a uniform acceptor doping concentration of 1017 cm−3 on the p-side and a uniform donor doping concentration of 1016  cm−3 on the n-side. No external voltage is applied to the diode. Given: kT/q = 26 mV, ni = 1.5 × 1010 cm−3 , εSi = 12 ε= 8.85 × 10−14 F/m, and q = 1.6 × 10−19 C.

The charge per unit junction area (nC cm−2) in the depletion region on the p-side is ___________

Solution: Key = -5.0 to -4.6

Q37. Consider an n-channel metal oxide semiconductor field effect transistor (MOSFET) with a gate-tosource voltage of 1.8 V. Assume that W/L= 4 μNCox = 70 × 10−6 AV−2, the threshold voltage is 0.3V, and the channel length modulation parameter is 0.09 V−1. In the saturation region, the drain conductance (in micro seimens) is ________

Solution: Key = 28 to 29

Q38. The figure below shows the doping distribution in a p-type semiconductor in log scale.

The magnitude of the electric field (in kV/cm) in the semiconductor due to non uniform doping is _________

Solution: Key = 1.10 to 1.25

Q39. Consider a silicon sample at T = 300 K, with a uniform donor density N= 5 × 1016 cm−2 cm−3 s−1 throughout the sample. The incident radiation is turned off at t = 0. Assume low-level injection to be valid and ignore surface effects. The carrier lifetimes are τpo = 0.1 μs and τno = 0.5 μs.

The hole concentration at t = 0 and the hole concentration at t = 0.3 μs, respectively, are

(A) $1.5\times 10^{13}cm^{-3}$ and $7.47\times 10^{11}cm^{-3}$

(B) $1.5\times 10^{13}cm^{-3}$ and $8.23\times 10^{11}cm^{-3}$

(C) $7.5\times 10^{13}cm^{-3}$ and $3.73\times 10^{11}cm^{-3}$

(D) $7.5\times 10^{13}cm^{-3}$ and $4.12\times 10^{11}cm^{-3}$

Solution: (A)

Q40. An ideal opamp has voltage sources V1, V3, V5, …, VN-1connected to the non-inverting input and V2, V4, V6, …, VN connected to the inverting input as shown in the figure below (+VCC = 15 volt, −VCC = −15 volt). The voltages V1, V2, V3, V4, V5, V6,… are 1, − 1/2, 1/3, −1/4, 1/5, −1/6, … volt, respectively. As N approaches infinity, the output voltage (in volt) is ___________

Solution: Key = 14.9 to 15.5

Q41. A p-i-n photodiode of responsivity 0.8A/W is connected to the inverting input of an ideal opamp as shown in the figure, +Vcc = 15 V, −Vcc = −15V, Load resistor R= 10 kΩ. If 10 μW of power is incident on the photodiode, then the value of the photocurrent (in μA) through the load is ________

Solution: Key = 790 to 810; -810 to -790

Q42.  Identify the circuit below.

(A) Binary to Gray code converter                 (B) Binary to XS3 converter

(C) Gray to Binary converter                           (D) XS3 to Binary converter

Solution: MTA

Q43. The functionality implemented by the circuit below is

(A) 2-to-1 multiplexer

(B) 4-to-1 multiplexer

(C)  7-to-1 multiplexer

(D) 6-to-1 multiplexer

Solution: (B)

Q44. In an 8085 system, a PUSH operation requires more clock cycles than a POP operation. Which one of the following options is the correct reason for this?

(A) For POP, the data transceivers remain in the same direction as for instruction fetch (memory to processor), whereas for PUSH their direction has to be reversed.

(B) Memory write operations are slower than memory read operations in an 8085 based system.

(C) The stack pointer needs to be pre-decremented before writing registers in a PUSH, whereas a POP operation uses the address already in the stack pointer.

(D) Order of registers has to be interchanged for a PUSH operation, whereas POP uses their natural order.

Solution: (C)

Q45. The open-loop transfer function of a unity-feedback control system is

The value of K at the breakaway point of the feedback control system’s root-locus plot is ________

Solution: Key = 1.2 to 1.3; -1.3 to -1.2

Q46. The open-loop transfer function of a unity-feedback control system is given by

For the peak overshoot of the closed-loop system to a unit step input to be 10%, the value of K is ____________

Solution: Key = 2.7 to 3.0

Q47. The transfer function of a linear time invariant system is given by
H(s) = 2s4 − 5s2+ 5s − 2
The number of zeros in the right half of the s-plane is ________

Solution: Key =3

Q48. Consider a discrete memoryless source with alphabet S = {s0, s1, s2, s3, s4, ….} and respective probabilities of occurrence  The entropy of the source (in bits) is _______

Solution: Key = 1.8 to 2.2

Q49. A digital communication system uses a repetition code for channel encoding/decoding. During transmission, each bit is repeated three times (0 is transmitted as 000, and 1 is transmitted as 111). It is assumed that the source puts out symbols independently and with equal probability. The decoder operates as follows: In a block of three received bits, if the number of zeros exceeds the number of ones, the decoder decides in favor of a 0, and if the number of ones exceeds the number of zeros, the decoder decides in favor of a 1. Assuming a binary symmetric channel with crossover probability p = 0.1, the average probability of error is ________

Solution: Key = 0.025 to 0.030

Q50. An analog pulse s(t) is transmitted over an additive white Gaussian noise (AWGN) channel. The received signal is r(t) = s(t) + n(t), where n(t) is additive white Gaussian noise with power spectral density  The received signal is passed through a filter with impulse response h(t). Let Es and Eh denote the energies of the pulse s(t) and the filter h(t), respectively. When the signal to-noise ratio (SNR) is maximized at the output of the filter (SNRmax), which of the following holds?

(A)                 (B)

(C)                 (D)

Solution: (A)

Q51. The current density in a medium is given by

The total current and the average current density flowing through the portion of a spherical surface r = 0.8 m,  are given, respectively, by

(A)           (B)

(C)              (D)

Solution: MTA

Q52. An antenna pointing in a certain direction has a noise temperature of 50 K. The ambient temperature is 290 K. The antenna is connected to a pre-amplifier that has a noise figure of 2 dB and an available gain of 40 dB over an effective bandwidth of 12 MHz. The effective input noise temperature Te for the amplifier and the noise power Pao at the output of the preamplifier, respectively, are

(A)                  (B)

(C)                    (D)

Solution: (A)

Q53. Two lossless X-band horn antennas are separated by a distance of 200λ.  The amplitude reflection coefficients at the terminals of the transmitting and receiving antennas are 0.15 and 0.18, respectively. The maximum directivities of the transmitting and receiving antennas (over the isotropic antenna) are 18 dB and 22 dB, respectively. Assuming that the input power in the lossless transmission line connected to the antenna is 2 W, and that the antennas are perfectly aligned and polarization matched, the power ( in mW) delivered to the load at the receiver is ________

Solution: Key = 2.7 to 3.3

Q54. The electric field of a uniform plane wave travelling along the negative z direction is given by the following equation:

This wave is incident upon a receiving antenna placed at the origin and whose radiated electric field towards the incident wave is given by the following equation:

The polarization of the incident wave, the polarization of the antenna and losses due to the polarization mismatch are, respectively,

(A) Linear, Circular (clockwise), −5dB               (B) Circular (clockwise), Linear, −5dB

(C) Circular (clockwise), Linear, −3dB               (D) Circular (anti clockwise), Linear, −3dB

Solution: (C) and (D)

Q55. The far-zone power density radiated by a helical antenna is approximated as:

The radiated power density is symmetrical with respect to ϕ and exists only in the upper hemisphere:  Cis a constant. The power radiated by the antenna (in watts) and the maximum directivity of the antenna, respectively, are

(A) $1.5C_{0}, 10dB$            (B) $1.256C_{0}, 10dB$

(C) $1.256C_{0}, 12dB$        (D) $1.56C_{0}, 12dB$

Solution: (B)

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