# GATE 2014 EE – SET 1 – Complete Solutions Q1.  Given a system of equations:

x + 2y + 2z = b1

5x + y + 3z = b2

Which of the following regarding its solutions

(A) The system has a unique solution for any given $$b_{1}$$ and $$b_{2}$$.

(B) The system will have infinitely many solutions for any given $$b_{1}$$ and $$b_{2}$$.

(C) Whether or not a solution exists depends on the given $$b_{1}$$ and $$b_{2}$$.

(D) The system would have no solution for any values of $$b_{1}$$ and $$b_{2}$$.

Solution: (B)

Q2. Let f(x) = x e−x. The maximum value of the function in the interval (0, ∞) is

(A) $$e^{-1}$$        (B) $$e$$           (C) $$1-e^{-1}$$          (D) $$1+e^{-1}$$

Solution: (A)

Q3. The solution for the differential equation $$\frac{\mathrm{d} ^{2}x}{\mathrm{d} t^{2}}=-9x$$ with initial conditions x(0) = 1 and $$\frac{\mathrm{d} x}{\mathrm{d} t}\mid _{t=0}=1$$, is

(A) $$t^{2}+t+1$$                                             (B) $$\sin 3t+\frac{1}{3}\cos 3t+\frac{2}{3}$$

(C) $$\frac{1}{3}\sin 3t+\cos 3t$$                (D) $$\cos 3t+t$$

Solution: (C)

Q4. Let $$X(s)=\frac{3s+5}{s^{2}+10s+21}$$ be the Laplace Transform of a signal x(t). Then, x(0++) is

(A) 0             (B) 3               (C) 5               (D) 21

Solution: (B)

Q5. Let S be the set of points in the complex plane corresponding to the unit circle. (That is, S ={z : |z| = 1}). Consider the function f(z) = z z* where z* denotes the complex conjugate of z. The f(z) maps S to which one of the following in the complex plane

(A) unit circle

(B) horizontal axis line segment from origin to (1, 0)

(C) the point (1, 0)

(D) the entire horizontal axis

Solution: (C)

Q6. The three circuit elements shown in the figure are part of an electric circuit. The total power absorbed by the three circuit elements in watts is Solution: Key = 330

Q7. C0 is the capacitance of a parallel plate capacitor with air as dielectric (as in figure (a)). If, half of the entire gap as shown in figure (b) is filled with a dielectric of permittivity ϵr, the expression for the modified capacitance is (A) $$\frac{C_{0}}{2}\left ( 1+\epsilon _{r} \right )$$           (B) $$\left ( C_{0}+\epsilon _{r} \right )$$

(C) $$\frac{C_{0}}{2}\epsilon _{r}$$                                         (D) $$C_{0}\left ( 1+\epsilon _{r} \right )$$

Solution: (A)

Q8. A combination of 1 μF capacitor with an initial voltage vc(0) = −2 V in series with a 100 Ω resistor is connected to a 20 mA ideal dc current source by operating both switches at t = 0 s as shown. Which of the following graphs shown in the options approximates the voltage vs across the current source over the next few seconds? (A) (B) (C) (D) Solution: (C)

Q9. x(t) is nonzero only for Tx < t < T’x, and similarly y, y(t) is nonzero for Ty < t < T’y. Let z(t) be convolution of x(t) and y(t). Which one of the following statements is TRUE?

(A) z(t) can be nonzero over a n unbounded interval

(B) z(t) is nonzero for $$t< T_{x}+T_{y}$$

(C) z(t) is zero outside of $$T_{x}+T_{y}< t< {T_{x}}’+{T_{y}}’$$

(D) z(t) is nonzero for $$t> {T_{x}}’+{T_{y}}’$$

Solution: (C)

Q10. For a periodic square wave, which one of the following statements is TRUE?

(A) The Fourier series coefficients do not exist.

(B) The Fourier series coefficients exist but the reconstruction converges at no point.

(C) The Fourier series coefficients exist and the reconstruction converges at most points.

(D) The Fourier series coefficients exist and the reconstruction converges at every point.

Solution: (C)

Q11. An 8-pole, 3-phase, 50 Hz induction motor is operating at a speed of 700 rpm. The frequency of the rotor current of the motor in Hz is ________.

Solution: Key = 3.2 to 3.5

Q12. For a specified input voltage and frequency, if the equivalent radius of the core of a transformer is reduced by half, the factor by which the number of turns in the primary should change to maintain the same no load current is

(A) 1/4                (B) 1/2               (C) 2                 (D) 4

Solution: (C)

Q13. A star connected 400 V, 50 Hz, 4 pole synchronous machine gave the following open circuit and short circuit test results:

Open circuit test: Voc = 400 V (rms, line-to-line) at field current, If = 2.3 A

Short circuit test: Isc = 10 A (rms, phase) at field current, If = 1.5 A

The value of per phase synchronous impedance in Ω at rated voltage is __________.

Solution: Key = 14.5 to 15.5

Q14. The undesirable property of an electrical insulating material is

(A) high dielectric strength                 (B) high relative permittivity

(C) high relative permittivity              (D) high insulation resistivity

Solution: (B)

Q15. Three-phase to ground fault takes place at locations F1 and F2 in the system shown in the figure If the fault takes place at location F1, then the voltage and the current at bus A are VF1 and IF1 respectively. If the fault takes place at location F2, then the voltage and the current at bus A are VF2 and IF2 respectively. The correct statement about voltages and currents during faults at F1 and F2 is

(A) $$V_{F1}$$ leads $$I_{F1}$$ and $$V_{F2}$$ leads $$I_{F2}$$

(B) $$V_{F1}$$ leads $$I_{F1}$$ and $$V_{F2}$$ lags $$I_{F2}$$

(C) $$V_{F1}$$ lags $$I_{F1}$$ and $$V_{F2}$$ leads $$I_{F2}$$

(D) $$V_{F1}$$ lags $$I_{F1}$$ and $$V_{F2}$$ lags $$I_{F2}$$

Solution: (C)

Q16. A 2-bus system and corresponding zero sequence network are shown in the figure. (A) (B) (C) (D) Solution: (B)

Q17. In the formation of Routh-Hurwitz array for a polynomial, all the elements of a row have zero values. This premature termination of the array indicates the presence of

(A) only one root at the origin           (B) imaginary roots

(C) only positive real roots                 (D) only negative real roots

Solution: (B)

Q18. The root locus of a unity feedback system is shown in the figure The closed loop transfer function of the system is

(A) $$\frac{C(s)}{R(s)}=\frac{K}{(s+1)(s+2)}$$

(B) $$\frac{C(s)}{R(s)}=\frac{-K}{(s+1)(s+2)+K}$$

(C) $$\frac{C(s)}{R(s)}=\frac{K}{(s+1)(s+2)-K}$$

(D) $$\frac{C(s)}{R(s)}=\frac{K}{(s+1)(s+2)+K}$$

Solution: (C)

Q19. Power consumed by a balanced 3-phase, 3-wire load is measured by the two wattmeter method. The first wattmeter reads twice that of the second. Then the load impedance angle in radians is

(A) π/12            (B) π/8              (C) π/6                 (D) π/3

Solution: (C)

Q20. In an oscilloscope screen, linear sweep is applied at the

(A) vertical axis                     (B) horizontal axis

(C) origin                                (D) both horizontal and vertical axis

Solution: (B)

Q21. A cascade of three identical modulo-5 counters has an overall modulus of

(A) 5                 (B) 25                 (C) 125                 (D) 625

Solution: (C)

Q22. In the Wien Bridge oscillator circuit shown in figure, the bridge is balanced when (A) $$\frac{R_{3}}{R_{4}}=\frac{R_{1}}{R_{2}}, \omega =\frac{1}{\sqrt{R_{1}C_{1}R_{2}C_{2}}}$$

(B) $$\frac{R_{2}}{R_{1}}=\frac{C_{2}}{C_{1}}, \omega =\frac{1}{R_{1}C_{1}R_{2}C_{2}}$$

(C) $$\frac{R_{3}}{R_{4}}=\frac{R_{1}}{R_{2}}+\frac{C_{2}}{C_{1}}, \omega =\frac{1}{\sqrt{R_{1}C_{1}R_{2}C_{2}}}$$

(D) $$\frac{R_{3}}{R_{4}}=\frac{R_{1}}{R_{2}}=\frac{C_{2}}{C_{1}}, \omega =\frac{1}{R_{1}C_{1}R_{2}C_{2}}$$

Solution: (C)

Q23. The magnitude of the mid-band voltage gain of the circuit shown in figure is (assuming hfe of the transistor to be 100) (A) 1               (B) 10                  (C) 20               (D) 100

Solution: (D)

Q24. The figure shows the circuit of a rectifier fed from a 230-V (rms), 50-Hz sinusoidal voltage source. If we want to replace the current source with a resistor so that the rms value of the current supplied by the voltage source remains unchanged, the value of the resistance (in ohms) is __________ (Assume diodes to be ideal.) Solution: Key = 23

Q25. Figure shows four electronic switches (i), (ii), (iii) and (iv). Which of the switches can block voltages of either polarity (applied between terminals ‘a’ and ‘b’) when the active device is in the OFF state? (A) (i), (ii) and (iii)                             (B) (ii), (iii) and (iv)

(C) (ii) and (iii)                                    (D) (i) and (iv)

Solution: (C)

Q26. Let g: [0, ∞) → [0, ∞) be a function defined by g(x) = x – [x], where [x] represents the integer part of x. (That is, it is a largest integer which is less than or equal to x).

The value of the constant term in the Fourier series expansion of g(x) is _______

Solution: Key = 0.5

Q27. A fair coin is tossed n times. The probability that the difference between the number of heads and tails is (n − 3) is

(A) $$2^{-n}$$               (B) 0               (C) $$_{n-3}^{n}\textrm{C}2^{-n}$$                 (D) $$2^{-n+3}$$

Solution: (B)

Q28. The line integral of function F = yzi, in the counterclockwise direction, along the circle x2 + y2 = 1 at z = 1 is

(A) −2π                   (B) −π                     (C) π                      (D) 2π

Solution: (B)

Q29. An incandescent lamp is marked 40 W, 240V. If resistance at room temperature (26°C) is 120 Ω, and temperature coefficient of resistance is 4.5 × 10−3/°C, then its ‘ON’ state filament temperature in °C is approximately _______

Solution: Key = 2470 to 2471

Q30. In the figure, the value of resistor R is (25 + I/2) ohms, where I is the current in amperes. The current I is ______ Solution: Key = 10

Q31. In an unbalanced three phase system, phase current Ia = 1 ∠(−90°) pu, negative sequence current Ib2 = 4∠(−150°) pu, zero sequence current Ic0 = 3∠90° pu. The magnitude of phase current Ib in pu is

(A) 1.00                   (B) 7.81                   (C) 11.53                (D) 13.00

Solution: (C)

Q32. The following four vector fields are given in Cartesian co-ordinate system. The vector field which does not satisfy the property of magnetic flux density is

(A) $$y^{2}a_{x}+z^{2}a_{y}+x^{2}a_{z}$$

(B) $$z^{2}a_{x}+x^{2}a_{y}+y^{2}a_{z}$$

(C) $$x^{2}a_{x}+y^{2}a_{y}+z^{2}a_{z}$$

(D) $$y^{2}z^{2}a_{x}+x^{2}z^{2}a_{y}+x^{2}y^{2}a_{z}$$

Solution: (C)

Q33. The function shown in the figure can be represented as (A) $$u(t)-u(t-T)+\frac{(t-T)}{T}u(t-T)-\frac{(t-2T)}{T}u(t-2T)$$

(B) $$u(t)+\frac{t}{T}u(t-T)-\frac{t}{T}u(t-2T)$$

(C) $$u(t)-u(t-T)+\frac{(t-T)}{T}u(t)-\frac{(t-2T)}{T}u(t)$$

(D) $$u(t)+\frac{(t-T)}{T}u(t-T)-2\frac{(t-2T)}{T}u(t-2T)$$

Solution: (A)

Q34. Let $$X(z)=\frac{1}{1-z^{-3}}$$ be the Z-transform of a causal signal x[n]. Then, the values of x and x are

(A) 0 and 0           (B) 0 and 1           (C) 1 and 0           (D) 1 and 1

Solution: (B)

Q35. Let f(t) be continuous time signal and let F(ω) be its Fourier Transform defined by $$F(\omega )=\int_{-\infty }^{\infty }f(t)e^{j\omega t}dt$$

Define g(t) by  $$g(t)=\int_{-\infty }^{\infty }F(u)e^{-jut}du$$

What is the relationship between f(t) and g(t)

(A) g(t) would always be proportional to f(t).

(B) g(t) would be proportional to f(t) if f(t) is an even function.

(C) g(t) would be proportional to f(t) only if f(t) is a sinusoidal function.

(D) g(t) would never be proportional to f(t).

Solution: (B)

Q36. The core loss of a single phase, 230/115 V, 50 Hz power transformer is measured from 230 V side by feeding the primary (230 V side) from a variable voltage variable frequency source while keeping the secondary open circuited. The core loss is measured to be 1050 W for 230 V, 50 Hz input. The core loss is again measured to be 500 W for 138 V, 30 Hz input. The hysteresis and eddy current losses of the transformer for 230 V, 50 Hz input are respectively,

(A) 508W and 542W            (B) 468W and 582W

(C) 498W and 552W            (D) 488W and 562W

Solution: (A)

Q37. A 15 kW, 230 V dc shunt motor has armature circuit resistance of 0.4 Ω and field circuit resistance of 230 Ω. At no load and rated voltage, the motor runs at 1400 rpm and the line current drawn by the motor is 5 A. At full load, the motor draws a line current of 70 A. Neglect armature reaction. The full load speed of the motor in rpm is _________.

Solution: Key = 1239 to 1242

Q38. A 3 phase, 50 Hz, six pole induction motor has a rotor resistance of 0.1 Ω and reactance of 0.92 Ω. Neglect the voltage drop in stator and assume that the rotor resistance is constant. Given that the full load slip is 3%, the ratio of maximum torque to full load torque is

(A) 1.567           (B) 1.712             (C) 1.948             (D) 2.134

Solution: (C)

Q39. A three phase synchronous generator is to be connected to the infinite bus. The lamps are connected as shown in the figure for the synchronization. The phase sequence of bus voltage is R-Y-B and that of incoming generator voltage is R’-Y’-B’. It was found that the lamps are becoming dark in the sequence La-Lb-Lc. It means that the phase sequence of incoming generator is

(A) opposite to infinite bus and its frequency is more than infinite bus

(B) opposite to infinite bus but its frequency is less than infinite bus

(C) same as infinite bus and its frequency is more than infinite bus

(D) same as infinite bus and its frequency is less than infinite bus

Solution: (A)

Q40. A distribution feeder of 1 km length having resistance, but negligible reactance, is fed from both the ends by 400V, 50 Hz balanced sources. Both voltage sources S1 and S2 are in phase. The feeder supplies concentrated loads of unity power factor as shown in the figure. The contributions of S1 and S2 in 100 A current supplied at location P respectively, are

(A) 75 A and 25 A                    (B) 50 A and 50 A

(C) 25 A and 75 A                    (D) 0 A and 100 A

Solution: (D)

Q41. A two bus power system shown in the figure supplies load of 1.0+j0.5 p.u. The values of V1 in p.u. and δ2 respectively are

(A) 0.95 and 6.00°                      (B) 1.05 and −5.44°

(C) 1.1 and −6.00°                       (D) 1.1 and −27.12°

Solution: (B)

Q42. The fuel cost functions of two power plants are

Plant $$P_{1}: C_{1}=0.05P_{g1}^{2}+AP_{g1}+B$$
Plant $$P_{2}: C_{2}=0.10P_{g2}^{2}+AP_{g2}+2B$$

where, Pg1 and pg2 are the generated powers of two plants, and A and B are the constants. If the two plants optimally share 1000 MW load at incremental fuel cost of 100 Rs/MWh, the ratio of load shared by plants P1 and P2 is

(A) 1:4            (B) 2:3               (C) 3:2               (D) 4:1

Solution: (D)

Q43. The over current relays for the line protection and loads connected at the buses are shown in the figure. The relays IDMT in nature having the characteristic The maximum and minimum fault currents at bus B are 2000 A and 500 A respectively. Assuming the time multiplier setting and plug setting for relay RB to be 0.1 and 5A respectively, the operating time of RB (in seconds) is __________

Solution: Key = 0.21 to 0.23

Q44. For the given system, it is desired that the system be stable. The minimum value of α for this condition is ___________. Solution: Key =  0.61 to 0.63

Q45. The Bode magnitude plot of the transfer function Note that −6 dB/octave = −20 dB/decade. The value of $$\frac{a}{bK}$$ is Solution: Key = 0.7 to 0.8

Q46. A system matrix is given as follows. The absolute value of the ratio of the maximum eigenvalue to the minimum eigenvalue is _________

Solution: Key = 2.9 to 3.1

Q47. The reading of the voltmeter (rms) in volts, for the circuit shown in the figure is __________ Solution: Key = 140 to 142

Q48. The dc current flowing in a circuit is measured by two ammeters, one PMMC and another electrodynamometer type, connected in series. The PMMC meter contains 100 turns in the coil, the flux density in the air gap is 0.2 Wb/m2, and the area of the coil is 80 mm2. The electrodynamometer ammeter has a change in mutual inductance with respect to deflection of 0.5 mH/deg. The spring constants of both the meters are equal. The value of current, at which the deflections of the two meters are same, is ________

Solution: Key = 3.0 to 3.4

Q49. Given that the op-amps in the figure are ideal, the output voltage V0 is (A) $$\left ( V_{1}-V_{2} \right )$$

(B) $$2\left ( V_{1}-V_{2} \right )$$

(C) $$\frac{\left ( V_{1}-V_{2} \right )}{2}$$

(D) $$\left ( V_{1}+V_{2} \right )$$

Solution: (B)

Q50. Which of the following logic circuits is a realization of the function F whose Karnaugh map is shown in figure (A)

(C) (D) Solution: (C)

Q51. In the figure shown, assume the op-amp to be ideal. Which of the alternatives gives the correct Bode plots for the transfer function  (A) (B) (C) (D) Solution: (A)

Q52. An output device is interfaced with 8-bit microprocessor 8085A. The interfacing circuit is shown in figure The interfacing circuit makes use of 3 Line to 8 Line decoder having 3 enable lines The address of the device is

(A) $$50_{H}$$                (B) $$5000_{H}$$

(C) $$A0_{H}$$                (D) $$A000_{H}$$

Solution: (B)

Q53. The figure shows the circuit diagram of a rectifier. The load consists of a resistance 10 Ω and an inductance 0.05 H connected in series. Assuming ideal thyristor and ideal diode, the thyristor firing angle (in degree) needed to obtain an average load voltage of 70 V is ______ Solution: Key = 69 to 70

Q54. Figure (i) shows the circuit diagram of a chopper. The switch S in the circuit in figure (i) is switched such that the voltage vD across the diode has the wave shape as shown in figure (ii). The capacitance C is large so that the voltage across it is constant. If switch S and the diode are ideal, the peak to peak ripple (in A) in the inductor current is ______  Solution: Key = 2.49 to 2.51

Q55.  The figure shows one period of the output voltage of an inverter. α should be chosen such that 60° < α < 90°. If rms value of the fundamental component is 50 V, then α in degree is _________ Solution: Key = 76.5 to 78.0