# GATE 2015 EC – SET 1 – Complete Solutions

Q1. Consider a system of linear equations :

x – 2y + 3z = –1,

x – 3y + 4z = 1, and

–2x + 4y – 6z = k.

The value of k for which the system has infinitely many solutions is _______.

Solution: Key = 2

Q2. A function f(x) = 1 – x2 + x3= is defined in the closed interval [–1, 1]. The value of x, in the open interval (–1, 1) for which the mean value theorem is satisfied, is

(A) –1/2              (B) –1/3               (C) 1/3                (D) 1/2

Solution: (B)

Q3. Suppose A and B are two independent events with probabilities P(A) ≠ 0 and P(B) ≠ 0. Let  be their complements. Which one of the following statements is FALSE?

(A) P(A ∩ B) = P(A) P(B)                 (B) P(A|B) = P(A)

(C) P(A ∪ B) = P(A) + P(B)              (D)

Solution: (C)

Q4. Let z = x + iy be a complex variable. Consider that contour integration is performed along the unit circle in anticlockwise direction. Which one of the following is NOT TRUE?

(A) The residue of at z = 1 is 1/2               (B)

(C)                                                     (D)

Solution: (D)

Q5. The value of p such that the vector  is an eigenvector of the matrix  is

Solution: Key = 16.5 to 17.5

Q6. In the circuit shown, at resonance, the amplitude of the sinusoidal voltage (in Volts) across the capacitor is ______

Solution: Key = 24 to 26

Q7. In the network shown in the figure, all resistors are identical with R = 300 Ω. The resistance Rab (in Ω) of the network is ________.

Solution: Key = 99.5 to 100.5

Q8. In the given circuit, the values of V1 and V2 respectively are

(A) 5 V, 25 V     (B) 10 V, 30 V               (C) 15 V, 35 V                 (D) 0 V, 20 V

Solution: (A)

Q9. A region of negative differential resistance is observed in the current voltage characteristics of silicon PN junction if

(A) both the P-region and the N-region are heavily doped

(B) the N-region is heavily doped compared to the P-region

(C) the P-region is heavily doped compared to the N-region

(D) an intrinsic silicon region is inserted between the P-region and the N-region

Solution: (A)

Q10. A silicon sample is uniformly doped with donor type impurities with a concentration of 1016/cm3. The electron and hole mobilities int eh sample are 1200 cm2/V-s respectively. Assume complete ionization of impurities. The charge of an electron is 1.6 × 10−19 C. The resistivity of the sample (in Ω –cm) is _______.

Solution: Key = 0.50 to 0.54

Q11. For the circuit with ideal diodes shown in the figure, the shape of the output  (vout) for the given sin wave input (vin) will be

(A)          (B)        (C)       (D)

Solution: (C)

Q12. In the circuit shown below, the Zener diode is ideal and the Zener voltage is 6 V. The output voltage V0(in volts) is _________.

Solution: Key = 5

Q13. In the circuit shown, the switch SW is thrown from position A to position B at time t = 0. The energy (in μJ) taken from the 3 V source to charge the 0.1 μF capacitor from 0V to 3 V is

(A) 0.3                   (B) 0.45                  (C) 0.9                     (D) 3

Solution: (C)

Q14. In an 8085 microprocessor, the shift registers which store the result of an addition and the overflow bit are, respectively

(A) B and F                    (B) A and F                    (C) H and F                   (D) A and C

Solution: (B)

Q15. A 16 Kb (=16,384 bit) memory array is designed as a square with an aspect ratio of one (number of rows is equal to the number of columns). The minimum number of address lines needed for the row decoder is _______.

Solution: Key = 7

Q16. Consider a four bit D to A converter. The analog value corresponding to digital signals of values 0000 and 0001 are 0V and 0.0625 V respectively. The analog value (in Volts) corresponding to the digital signal 1111 is ________.

Solution: Key = 0.93 to 0.94

Q17. The result of the convolution x(–t) ∗ δ(–t – t0) is

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

Solution: (D)

Q18. The waveform of a periodic signal x(t) is shown in the figure.

A signal g(t) is defined by   The average power of g(t) is _______

Solution: Key = 2

Q19. Negative feedback in a closed-loop control system DOES NOT

(A) reduce the overall gain                                 (B) reduce bandwidth

(C) improve disturbance rejection                    (D) reduce sensitivity to parameter variation

Solution: (B)

Q20. A unit negative feedback system as the open-loop transfer function $$G(s)=\frac{K}{s(s+1)(s+3)}$$ The value of the gain K(>0) at which the root locus crosses the imaginary axis is __________.

Solution: Key = 12

Q21. The polar plot of the transfer function $$G(s)=\frac{100(s+1)}{s+10}$$ for 0 ≤ ω < ∞ will be in the

Solution: (A)

Q22. A sinusoidal signal of 2 kHz frequency is applied to a delta modulator. The sampling rate and step-size ∆ of the delta modulator are 20,000 samples per second and 0.1 V, respectively. To prevent slope overload, the maximum amplitude of the sinusoidal signal (in volts) is

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

Solution: (A)

Q23. Consider the signal  denotes the Hilbert transform of m(t) and the bandwidth of m(t) is very small compared of fc. The single s(t) is a

(A) high-pas signal               (B) low-pass signal

(C) band-pass signal            (D) double sideband suppressed carrier signal

Solution: (C)

Q24. Consider a straight, infinitely long, current carrying conductor lying on the z-axis. Which one of the following plots (in linear scale) qualitatively represents the dependence of Hϕ or r, where Hϕ is the magnitude of the azimuthal component of magnetic field outside the conductor and r is the radial distance from the conductor?

(A)            (B)

(C)            (D)

Solution: (C)

Q25. The electric field component of a plane wave traveling in a lossless dielectric medium is given by The wavelength (in m) for the wave is ______

Solution: Key = 8.85 to 8.92

Q26. The solution of differential equation  with y(0) = y′(0) = 1 is

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

Solution: (B)

Q27. A vector  is given by  Which one of the following statements is TRUE?

(A) is solenoidal, but not irrotational

(B) is irrotational, but not solenoidal

(C) is neither solenoidal nor irrotational

(D) is both solenoidal and irrotational

Solution: (A)

Q28. Which one of the following graphs describes the function f(x) = e−x (x2 + x + 1)?

(A)              (B)

(C)              (D)

Solution: (B)

Q29. The maximum area (in square units) of a rectangle whose vertices lie on the ellipse x2 + 4y2 = 1 is _________.

Solution: Key = 0.95 to 1.05

Q30. The damping ratio of a series RLC circuit can be expressed as

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

Solution: (C)

Q31. In the circuit shown, switch SW is closed at t = 0. Assuming zero initial conditions, the value of vc(t)(in volts) at t = 1 sec is ______.

Solution: Key = 2.48 to 2.58

Q32.In the given circuit, the maximum power (in Watts) that can be transferred to the load RL is ________.

Solution: Key = 1.6 to 1.7

Q33. The built-in potential of an abrupt p- n junction is 0.75 V. If its junction capacitance (CJ) at a reverse bias (VR) of 1.25 V is 5 pF, the value of CJ (in pF) when VR = 7.25 V is  ________.

Solution: Key = 2.4 to 2.6

Q34. A MOSFET is saturation has a drain current of mA for VDS = 0.5 V. If the channel length modulation coefficient is 0.05 V−1, the output resistance (in kΩ) of the MOSFET is ________.

Solution: Key = 19 to 21

Q35. For a silicon diode with long P and N regions, the accepter and donor impurity concentrations are 1 × 1017 cm3 and 1 × 1015 cm3, respectively. The lifetimes of electrons in P region and holes in N region are both 100 μs. The electron and hole diffusion coefficients are 49 cm2/s and 36cm2/s, respectively. Assume kT/q = 26 mV, the intrinsic carrier concentration is 1 × 1010 cm3, and q = 1.6 × 1019 C. When a forward voltage of 208 mV is applied across the diode, the hole current density (in nA/cm2) injected from P region to N region is __________

Solution: Key = 28 to 30

Q36. The Boolean expression  converted into the canonical product of sum (POS) form is

(A)

(B)

(C)

(D)

Solution: (A)

Q37. All the logic gates shown in the figure have propagation delay of 20 ns. Let A = C = 0 and B = 1 until time t = 0. At t = 0, al the inputs flip (i.e., A = C = 1 and B = 0) and remain in that state. For t > 0, output Z = 1 for a duration (in ns) of

Solution: Key = 40

Q38. A 3-input majority gate is defined by the logic function M(a, b, c) = ab + bc + ca. Which one of the following gates is represented by the function

(A) 3-input NAND gate                           (B) 3-input XOR gate

(C) 3-input NOR gate                              (D) 3-input XNOR gate

Solution: (B)

Q39. For the NMOSFET in the circuit shown, the threshold voltage is Vth, where Vth > 0. The source voltage VSS is varied from 0 to VDD. Neglecting the channel length modulation, the drain current ID as a function of VSS is represented by

(A)               (B)

(C)                  (D)

Solution: (A)

Q40. In the circuit shown, assume that the opamp is ideal. The bridge output voltage V0(in mV) for δ = 0.05 is _______.

Solution: Key = 249 to 251

Q41. The circuit shown in the figure has an ideal opamp. The oscillation frequency and the condition to sustain the oscillations, respectively, are

(A)               (B)

(C)             (D)

Solution: (D)

Q42. In the circuit shown, I1 = 80 mA and I2 = 4 mA. Transistors, T1 and T­2 are identical. Assume that the thermal voltage VT is 26 mV at 27℃. At 50℃, ,the value of the voltage V12 = V1 – V2(in mV) is _________.

Solution: Key = 83.5 to 84.0

Q43. Two sequences [a, b, c] and [A, B, C] are related as,

The another sequence [p, q, r] is derived as,

then the relationship between the sequences [p, q, r] and [a, b, c] is

(A) [p, q, r] = [b, a, c]             (B) [p, q, r] = [b, c, a]          (C) [p, q, r] = [c, a, b]          (D) [p, q, r] = [c, b, a]

Solution: (C)

Q44. For the discrete-time system shown in the figure, the poles of the system transfer function are located at

(A) 2, 3                 (B) 1/2, 3                 (C) 1/2, 1/3                 (D) 2, 1/3

Solution: (C)

Q45. The pole-zero diagram of a casual and stable discrete-time system is shown in the figure. The zero at the origin has multiplicity 4. The impulse response of the system is h[n]. If h[0] = 1, we can conclude

(A) h[n] is real for all n                         (B) h[n] is purely imaginary for all n

(C) h[n] is real for only even n            (D) h[n] is purely imaginary for only odd n

Solution: (A)

Q46. The open-loop transfer function of a plant in a unity feedback configuration is given as  The value of the gain K(>0) for which −1 + j2 lies on the root locus is _______.

Solution: Key = 25 to 26

Q47. A lead compensator network includes a parallel combination of R and C in the feed-forward path. If the transfer function of the compensator is  the value of RC is ________.

Solution: Key = 0.5

Q48. A plant transfer function is given as  When the plant operates in unity feedback configuration, the condition for the stability of the closed loop system is

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

Solution: (A)

Q49. The input X to the Binary Symmetric Channel (BSC) shown in the figure is ‘1’ with probability 0.8. The cross-over probability is 1/7. If the received bit Y = 0, the conditional probability that ‘1’ was transmitted is ______.

Solution: Key = 0.39 to 0.41

Q50. The transmitted signal in a GSM  system is of 200 khZ bandwidth and 8 users share a common bandwidth using TDMA. If at a given time 12 users are talking in a cell, the total bandwidth of the signal received by the base station of the cell will be at least (in kHz) _________.

Solution: Key = 400

Q51. In the system shown in Figure (a), m(t) is a low-pass signal with bandwidth W Hz. The frequency response of the band-pass filter H(f) is shown in Figure (b). If it is desired that the output signal z(t) = 10x (t), the maximum value of W (in Hz) should be strictly less than ________.

Solution: Key = 349 to 350

Q52. A source emits bit 0 with probability 1/3 and bit 1 with probability 2/3. The emitted bits are communicated to the receiver. The receiver decides for either 0 or 1 based on the received value R. It is given that the conditional density functions of R are as s many as

The minimum decision error probability is

(A) 0                   (B) 1/2                   (C) 1/9                   (D) 1/6

Solution: (D)

Q53.The longitudinal component of the magnetic field inside an air-filled rectangular waveguide made of a perfect electric conductor is given by the following expression

Hz(x, y, z, t) = 0.1 cos(25πx) cos(30.3 πy) cos(12π × 109t – βz ) (A/m)

The cross –sectional dimensions of the waveguide are given as a = 0.08 ma dn b = 0.033 m. The mode of propagation inside the waveguide is

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

Solution: (C)

Q54. The electric field intensity of a plane wave traveling in free space is given by the following expression

E(x, t) =  ay 24π cos(ωt – k0x) (V/m)

In this field, consider a square area 10 cm x 10 cm on a plane x + y = 1. The total time-averaged power (in mW) passing through the square area is

Solution: Key = 53 to 54

Q55. Consider a uniform plane wave with amplitude (E0) of 10 V/m and 1.1 GHz frequency travelling in air, and incident normally on a dielectric medium with complex relative permittivity (εr) and permeability (μr) as shown in the figure.

The magnitude of the transmitted electric field component (in V/m) after it has travelled a distance of 10 cm inside the dielectric region is _______.

Solution: 0.08 to 0.12