# GATE 2016 EC – SET 3 (Session 4) – Complete Solutions

Q1. Consider a 2 × 2 square matrix

where x is unknown. If the eigenvalues of the matrix A are (σ + jω ) and (σ − jω ) , then x is equal to

(A) + jω                          (B) − jω                       (C) +ω                       (D) −ω

Solution: (D)

Q2. For $$f(z)=\frac{\sin (z)}{z^{2}}$$ the residue of the pole at z = 0 is __________

Solution: Key = 1

Q3. The probability of getting a “head” in a single toss of a biased coin is 0.3. The coin is tossed repeatedly till a “head” is obtained. If the tosses are independent, then the probability of getting “head” for the first time in the fifth toss is __________

Solution: Key = 0.07 to 0.08

Q4. The integral  is equal to __________

Solution: Key = 2

Q5. Consider the first order initial value problem

y’ = y + 2x − x2, y(0) = 1, (0 ≤ x < ∞)

with exact solution y(x) = x2 + ex. For x = 0.1, the percentage difference between the exact solution and the solution obtained using a single iteration of the second-order Runge-Kutta method with step-size h = 0.1 is __________

Solution: Key = 0.06 to 0.063

Q6. Consider the signal x(t) = cos(6πt) + sin(8πt), where t is in seconds. The Nyquist sampling rate (in samples/second) for the signal y(t) = x(2t + 5) is

(A) 8                    (B) 12                       (C) 16                      (D) 32

Solution: (C)

Q7. If the signal  with ∗ denoting the convolution operation, then x(t) is equal to

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

Solution: (A)

Q8. A discrete-time signal x[n] = δ[n − 3] + 2δ[n − 5] has z-transform X(z). If Y(z) = X(−z) = X(−z) is the z-transform of another signal y[n], then

(A) y[n] = x[n]                (B) y[n] = x[−n]                (C) y[n] = −x[n]                  (D)  y[n] = −x[−n]

Solution: (C)

Q9. In the RLC circuit shown in the figure, the input voltage is given by

vi(t) = 2 cos(200t) + 4sin(500t).

The output voltage v0(t) is

(A) cos(200t) + 2sin(500t)                  (B) 2cos(200t) + 4sin(500t)

(C) sin(200t) + 2 cos(500t)                 (D) 2 sin(200t) + 4 cos(500t)

Solution: (B)

Q10. The I-V characteristics of three types of diodes at the room temperature, made of semiconductors X, Y and Z, are shown in the figure. Assume that the diodes are uniformly doped and identical in all respects except their materials. If EgX, EgY and EgZ are the band gaps of X, Y and Z, respectively, then

(A)               (B)                (C)

(D)  no relationship among these band gaps exists.

Solution: (C)

Q11. The figure shows the band diagram of a Metal Oxide Semiconductor (MOS). The surface region of this MOS is in

(A) inversion                     (B) accumulation                        (C) depletion                      (D) flat band

Solution: (A)

Q12. The figure shows the I-V characteristics of a solar cell illuminated uniformly with solar light of power 100 mW/cm2. The solar cell has an area of 3 cm2 and a fill factor of 0.7. The maximum efficiency (in %) of the device is __________

Solution: Key = 20.5 to 21.5

Q13. The diodes D1 and D2 in the figure are ideal and the capacitors are identical. The product RC is very large compared to the time period of the ac voltage. Assuming that the diodes do not breakdown in the reverse bias, the output voltage VO (in volt) at the steady state is __________

Solution: Key = 0

Q14. Consider the circuit shown in the figure. Assuming VBE1 = VEB2 = 0.7 volt, the value of the dc voltage VC2 (in volt) is __________

Solution: Key = 0.45 to 0055

Q15. In the astable multivibrator circuit shown in the figure, the frequency of oscillation (in kHz) at the output pin 3 is __________

Solution: Key = 5.55 to 5.75

Q16. In an 8085 microprocessor, the contents of the accumulator and the carry flag are A7 (in hex) and 0, respectively. If the instruction RLC is executed, then the contents of the accumulator (in hex) and the carry flag, respectively, will be

(A) 4E and 0                (B) 4E and 1                (C) 4F and 0                 (D) 4F and 1

Solution: (D)

Q17. The logic functionality realized by the circuit shown below is

(A)

Solution: (D)

Q18. The minimum number of 2-input NAND gates required to implement a 2-input XOR gate is

(A) 4                        (B) 5                     (C) 6                   (D) 7

Solution: (A)

Q19. The block diagram of a feedback control system is shown in the figure. The overall closed-loop gain G of the system is

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

Solution: (B)

Q20. For the unity feedback control system shown in the figure, the open-loop transfer function G(s) is given as

The steady state error ess due to a unit step input is

(A) 0                 (B) 0.5                 (C) 1.0                (D)  ∞

Solution: (A)

Q21. For a superheterodyne receiver, the intermediate frequency is 15 MHz and the local oscillator frequency is 3.5 GHz. If the frequency of the received signal is greater than the local oscillator frequency, then the image frequency (in MHz) is __________

Solution: Key = 3485

Q22. An analog baseband signal, bandlimited to 100 Hz, is sampled at the Nyquist rate. The samples are quantized into four message symbols that occur independently with probabilities p1 = p4 = 0.125 and p2 = p3. The information rate (bits/sec) of the message source is __________

Solution: Key = 360 to 363

Q23. A binary baseband digital communication system employs the signal

for transmission of bits. The graphical representation of the matched filter output y(t) for this signal will be

(A)                (B)

(C)                    (D)

Solution: (C)

Q24. If a right-handed circularly polarized wave is incident normally on a plane perfect conductor, then the reflected wave will be

(A) right-handed circularly polarized                               (B) right-handed circularly polarized

(C) elliptically polarized with a tilt angle of 45°             (D) horizontally polarized

Solution: (B)

Q25. Faraday’s law of electromagnetic induction is mathematically described by which one of the following equations?

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

Solution: (C)

Q26. The particular solution of the initial value problem given below is

with y (0) = 3 and

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

Solution: (A)

Q27. If the vectors e1 = (1, 0, 2), e2 = (0, 1, 0) and e3 = (−2, 0, 1) form an orthogonal basis of the threedimensional real space R3, then vetor u = (4, 3, −3)  ∈  R2 can be expressed as

(A)               (B)

(C)               (D)

Solution: (D)

Q28. A triangle in the xy-plane is bounded by the straight lines 2x = 3y, y = 0 and x = 3. The volume above the triangle and under the plane x + y + z = 6 is __________

Solution: Key = 10

Q29. The values of the integral  along a closed contour c in anti-clockwise direction for

(i) the point z0 = 2 inside the contour c, and
(ii) the point z0 = 2 outside the contour c, respectively, are

(A) (i) 2.72, (ii) 0             (B) (i) 7.39, (ii) 0               (C) (i) 0, (ii) 2.72               (D) (i) 0, (ii) 7.39

Solution: (B)

Q30. A signal  is the input to an LTI system with the transfer function

H(s) = es + e−s.

If Ck denotes the kth coefficient in the exponential Fourier series of the output signal, then C3 is equal to

(A) 0                 (B) 1                     (C) 2                      (D) 3

Solution: (B)

Q31. The ROC (region of convergence) of the z-transform of a discrete-time signal is represented by the shaded region in the z-plane. If the signal x[n] (2.0)|n|, −∞ < n < +∞, then the ROC of its z-transform is represented by

(A)

(B)

(C)

(D)

Solution: (D)

Q32. Assume that the circuit in the figure has reached the steady state before time t = 0 when the 3Ω resistor suddenly burns out, resulting in an open circuit. The current i(t) (in ampere) at t = 0+ is _______

Solution: Key = 0.98 to 1.02 ; -1.02 to -0.98 ; 1.96 to 2.04 ; -2.04 to -1.96

Q33. In the figure shown, the current i (in ampere) is __________

Solution: Key = -1.05 to -0.95

Q34. The z-parameter matrix  for the two-port network shown is

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

Solution: (A)

Q35. A continuous-time speech signal xa(t) is sampled at a rate of 8 kHz and the samples are subsequently grouped in blocks, each of size N. The DFT of each block is to be computed in real time using the radix-2 decimation-in-frequency FFT algorithm. If the processor performs all operations sequentially, and takes 20 μs for computing each complex multiplication (including multiplications by 1 and −1) and the time required for addition/subtraction is negligible, then the maximum value of N is __________

Solution: Key = 4096

Q36. The direct form structure of an FIR (finite impulse response) filter is shown in the figure.

The filter can be used to approximate a

(A) low-pass filter           (B) high-pass filter               (C) band-pass filter          (D) band-stop filter

Solution: (C)

Q37. The injected excess electron concentration profile in the base region of an npn BJT, biased in the active region, is linear, as shown in the figure. If the area of the emitter-base junction is 0.001 cm2, μn = 800 cm2 (V-s) in the base region and depletion layer widths are negligible, then the collector current IC (in mA) at room temperature is __________

(Given: thermal voltage VT = 26 mV at room temperature, electronic charge q = 1.6 × 10−19 C)

Solution: Key = 6.55 to 6.75

Q38. Figures I and II show two MOS capacitors of unit area. The capacitor in Figure I has insulator materials X (of thickness t1 = 1 nm and dielectric constant ε1 = 4) and Y (of thickness t2 = 3 nm and dielectric constant ε2 = 20). The capacitor in Figure II has only insulator material X of thickness tEq. If the capacitors are of equal capacitance, then the value of tEq (in nm) is __________

Solution: Key = 1.55 to 1.65

Q39. The I-V characteristics of the zener diodes D1 and D2 are shown in Figure I. These diodes are used in the circuit given in Figure II. If the supply voltage is varied from 0 to 100 V, then breakdown occurs in

(A) D1 only              (B) D2 only              (C) both D1and D2               (D) none of D1 and D2

Solution: (A)

Q40. For the circuit shown in the figure, R1 = R2 = R3 = 1Ω L= 1 μH and C = 1 μF. If the input Vin = cos(106t), then the overall voltage gain (Vout /Vin) of the circuit is __________

Solution: Key = -1.0

Q41. In the circuit shown in the figure, the channel length modulation of all transistors is non-zero (λ ≠ 0). Also, all transistors operate in saturation and have negligible body effect. The ac small signal voltage gain (Vo/Vin) of the circuit is

(A)                              (B)

(C)                (D)

Solution: (C)

Q42. In the circuit shown in the figure, transistor M1 is in saturation and has transconductance gm = 0.01 siemens. Ignoring internal parasitic capacitances and assuming the channel length modulation λ to be zero, the small signal input pole frequency (in kHz) is __________

Solution: Key = 56 to 63

Q43. Following is the K-map of a Boolean function of five variables P, Q, R, S and X. The minimum sum-of-product (SOP) expression for the function is

(A)                       (B)

(C)                                                              (D)

Solution: (B)

Q44. For the circuit shown in the figure, the delays of NOR gates, multiplexers and inverters are 2 ns, 1.5 ns and 1 ns, respectively. If all the inputs P, Q, R, S and T are applied at the same time instant, the maximum propagation delay (in ns) of the circuit is __________

Solution: Key = 6

Q45. For the circuit shown in the figure, the delay of the bubbled NAND gate is 2 ns and that of the counter is assumed to be zero.

If the clock (Clk) frequency is 1 GHz, then the counter behaves as a

(A) mod-5 counter              (B) mod-6 counter              (C) mod-7 counter               (D) mod-8 counter

Solution: (D)

Q46. The first two rows in the Routh table for the characteristic equation of a certain closed-loop control system are given as

The range of K for which the system is stable is

(A) −2.0 < K < 0.5           (B) 0 < K < 0.5                 (C) 0 < K < ∞              (D) 0.5 < K < ∞

Solution: (D)

Q47. A second-order linear time-invariant system is described by the following state equations

where x1(t) and x2(t) are the two state variables and u(t) denotes the input. If the output c(t) = x1(t), then the system is

(A) controllable but not observable                      (B) observable but not controllable

(C) both controllable and observable                   (D) neither controllable nor observable

Solution: (A)

Q48. The forward-path transfer function and the feedback-path transfer function of a single loop negative feedback control system are given as

respectively. If the variable parameter K is real positive, then the location of the breakaway point on the root locus diagram of the system is __________

Solution: Key = -3.45 to -3.35

Q49. A wide sense stationary random process X(t) passes through the LTI system shown in the figure. If the autocorrelation function of X(t) is RX(τ), then the autocorrelation function RY (τ) of the output Y(t) is equal to

(A)                            (B)

(C)                                                  (D)

Solution: (B)

Q50. A voice-grade AWGN (additive white Gaussian noise) telephone channel has a bandwidth of 4.0 kHz and two-sided noise power spectral density  If information at the rate of 52 kbps is to be transmitted over this channel with arbitrarily small bit error rate, then the
minimum bit-energy Eb (in mJ/bit) necessary is __________

Solution: Key = 30 to 33

Q51. The bit error probability of a memoryless binary symmetric channel is 10−5. If 105bits are sent over this channel, then the probability that not more than one bit will be in error is __________

Solution: Key = 0.70 : 0.75

Q52. Consider an air-filled rectangular waveguide with dimensions a = 2.286 cm and b = 1.016 cm. At 10 GHz operating frequency, the value of the propagation constant (per meter) of the corresponding propagating mode is __________

Solution: Key = 155.0 : 162.0

Q53. Consider an air-filled rectangular waveguide with dimensions a = 2.286 cm and b = 1.016 cm. The increasing order of the cut-off frequencies for different modes is

(A)                       (B)

(C)                       (D)

Solution: (C)

Q54. A radar operating at 5 GHz uses a common antenna for transmission and reception. The antenna has a gain of 150 and is aligned for maximum directional radiation and reception to a target 1 km away having radar cross-section of 3 m2. If it transmits 100 kW, then the received power (in μW) is _________

Solution: Key = 0.01 : 0.02

Q55. Consider the charge profile shown in the figure. The resultant potential distribution is best described by

(A)              (B)

(C)            (D)

Solution: (D)