# (Paper) GATE | Electronics and Communication Question Paper 2006

### GATE | Electronics and Communication Question Paper 2006

Q.1 — Q.20 Carry One Mark Each.

**1. The rank of the matrix**

1 1 1

1 -1 0 is:

1 1 1

(A) 0

(B) 1

(C) 2

(D) 3

**2. VxVxP, wherePisa vector, is equal to**

(A) PxVxP-V2P

(B) V2P+V(V.P)

(C) V2P+VxP

(D) v(v.P)-v2P

**3. [[(vxP).ds, where P is a vector, is equal to**

(A) 4P.dI

(B) VxVxP.dI

(C) 4VxP.dI

(D) [[[V.Pdv

**4. A probability density function is of the form**

p(x) = Ke aix1 XE

The value of K is

(A) 0.5

(B) 1

(C) O.5cL

(D) x

**5. A solution for the differential equation k(t)+2x(t) = 5(t) with initial condition x(0 —) = 0 is:**

(A) e 2u(t)

(B) e2u(t)

(C) e u(t)

(D) &u(t)

**6. A low-pass filter having a frequency response H(ja) = A(ai)edoes not produce any phase distortion if**

(A) A(a) = Cai2,Ø(ai) = kai3

(B) A(a) = Cai2,Ø(ai) = ko.

(C) A(a)=Ca,Ø(a)=ka2

(D) A(a)=C, )=ka1

**7. The values of voltage (V3) across a tunnel-diode corresponding to peak and valley currents are V, and V1 respectively-:- The range of tunnel-diode voltage VD for which the slope of its IVD characteristics is negative would be**

(A) V3<O

(B) O<VJ<V?

(C) V,V<V

(D) VdVV

**8. The concentration of minority carriers in an extrinsic semiconductor under equilibrium is:**

(A) directly proportional to the doping concentration

(B) inversely proportional to the doping concentration

(C) directly proportional to the intrinsic concentration

(D) inversely proportional to the intrinsic concentration

**9. Under low level injection assumption, the injected minority carrier current for an extrinsic semiconductor is essentially the**

(A) diffusion current

(B) drift current

(C) recombination current

(D) induced current

**10. The phenomenon known as “Early Effect” in a bipolar transistor refers to a reduction of the effective base-width caused by**

(A) electron-hole recombination at the base

(B) the reverse biasing of the base-collector junction

(C) the forward biasing of emitter-base junction

(D) the early removal of stored base charge during saturation-to-cutoff switching.

**11. The input impedance (Z,)and the output impedance (Z0)of an ideal transconductance (voltage controlled current source) amplifier are**

(A) Z,=0,Z0=0

(B) Z,=O,Z0=oo

(C) Z,=oo,Z0=O

(D) Z = 00, =

**12. An n-channel depletion MOSFET has following two points on its ID VGS curve:**

(i) VGS = 0 at ID = l2mA and

(ii) VGS = —6 Volts at ID =

Which of the following Q-points will give the highest trans-conductance gain for small signals?

(A) VGS = —6 Volts

(B) VGS = —3 Volts

(C) VGS = 0 Volts

(D) VGS = 3 Volts

**13. The number of product terms in the minimized sum-of-product expression obtained through the following K-map is (where “d” denotes don’t care states)**

1001

0d00

00d 1

1001

(A) 2

(B) 3

(C) 4

(D) 5

**14. The Dirac delta function 8(t) is defined as**

(A) 8(t)=1 t=O.

L° otherwise

100 t=O

(B) 8(t)=.

LO otherwise

(C) 8(t) = = . and [8(t)dt = 1

L otherwise

(D) 8(t)=4°° t=O and [8(t)dt=1

L otherwise

**15. The open-loop transfer function of a unity-gain feedback control system is given by The gain margin of the system in dB is given by**

(A) 0

(B) 1

(C) 20

(D)

**16. The electric field of an electromagnetic wave propagating in the positive zdirection is given by**

E=asin(at—flz)+asinot—flz+. The wave is

(A) linearly polarized in the z-direction

(B) elliptically polarized

(C) left-hand circularly polarized

(D) right-hand circularly polarized

**17. A transmission line is feeding 1 Watt of power to a horn antenna having a gain of 10 dB. The antenna is matched to the transmission line. The total power radiated by the horn antenna into the free-space is:**

(A) 10 Watts

(B) 1 Watt

(C) 0.1 Watt

(D) 0.01 Watt

**18. The eigenvalues and the corresponding eigenvectors of a 2 x 2 matrix are given by Eigenvalue Eigenvector**

A1=8

A2=4 v2=[hil

The matrix is:

(A) 6 2 L2 6

(B) 6 L6 4

(C) r L 2

(D) 8 L8 4

**19. For the function of a complex variable W = lnZ (where, W=u+jv and Z=x+jy),the u=constant lines get mapped in Z-plane as**

(A) set of radial straight lines

(B) set of concentric circles

(C) set of confocal hyperbolas

(D) set of confocal ellipses

**20. Three companies, X, Y and Z supply computers to a university. The percentage of computers supplied by them and the probability of those being defective are tabulated below. Given that a computer is defective, the probability that it was supplied by Y is:**

(A) 0.1

(B) 0.2

(C) 0.3

(D) 0.4

**21. For the matrix [ jthe eigenvalue corresponding to the eigenvector is:**

(A) 2

(B) 4

(C) 6

(D) 8

**22. For the differential equation + k2y = 0 the boundary conditions are Company °h of computers supplied Probability of being defective**

X 60°h 0.01

Y 30°h 0.02

Z 10°h 0.03

(i) y=Oforx=Oand

(ii) y=Oforx=a

The form of non-zero solutions of y (where mvaries over all integers) are m,rx

(A) y=AmsIn

m a

m,rx

(B) y= AmCO5

m a

(C) Y=4m

mrx

(D) y=Amea

**23. Consider the function f (t) having Laplace transform**

2 2 Re[sl>O

S +a)0

The final value of f(t)would be:

(A) 0

(B) 1

(C) —1 f(oo) 1

(D)

**24. As x is increased from — to 00, the function ex f(x)= 1 + ex**

(A) monotonically increases

(B) monotonically decreases

(C) increases to a maximum value and then decreases

(D) decreases to a minimum value and then increases

**25. The first and the last critical frequencies (singularities) of a driving point impedance function of a passive network having two kinds of elements, are a pole and a zero respectively. The above property will be satisfied by**

(A) RL network only

(B) RC network only

(C) LC network only

(D) RC as well as RL networks

**26. In the figure shown below, assume that all the capacitors are initially uncharged. If v,(t)=lOu(t) Volts,v0(t) is given by**

(A) Se °°° Volts

(B) 8(1 — e °°°) Volts

(C) 8u(t) Volts

(D) 8 Volts

**27. Consider two transfer functions G1(S)=s2+as+b and G2(S)=s2+as+b• Vo(t) The 3-dB bandwidths of their frequency responses are, respectively**

(A) %Ja2 _4b,%Ja2 +4b

(B) %Ja2 + 4b, %Ja2 — 4b

**28. A negative resistance Rneg is connected to a passive network N having driving point impedance Z1 (s)as shown below. For Z2 (s)to be positive real,**

Z2(s)

(A) R I<ReZ1(ja),Va neg —

(B) R I —

(C) R 1< ImZ1(ja),va neg —

(D) R <zz (jai),Vai neg— 1

**29. In the circuit shown below, the switch was connected to position 1 at t < 0 and at t = 0, it is changed to position 2. Assume that the diode has zero voltage drop and a storage time t5. For 0 < t < t5,v is given by (all in Volts)**

(A) v=—5

(B) v=–5

(D) —5 <v <0

**30. The majority carriers in an n-type semiconductor have an average drift velocity v in a direction perpendicular to a uniform magnetic field B. the electric field E induced due to Hall effect acts in the direction**

(A) v x B

(B) Bxv

(C) along v

(D) opposite to v

**31. A heavily doped n—type semiconductor has the following data: Hole-electron mobility ratio : 0.4**

Doping concentration : 4.2x 108 atoms/m3

Intrinsic concentration : 1.5x 10 atoms/m3

The ratio of conductance of the n —type semiconductor to that of the intrinsic semiconductor of same material and at the same temperature is given by

(A) 0.00005

(B) 2,000

(C) 10,000

(D) 20,000

**32. For the circuit shown below; assume that the zener diode is ideal with a breakdown voltage of 6 Volts. The waveform observed across R is:**

(A)6V

(B)l2sinot R VR

(C)12V

** 33. A new Binary Coded Pentary (BCP) number system is proposed in which every digit of a base-5 number is represented by its corresponding 3-bit binary code.**

For example, the base-5 number 24 will be represented by its BCP code 010100. In this numbering system, the BCP code 100010011001 corresponds to the

following number in base-5 system

(A) 423

(B) 1324

(C) 2201

(D) 4231

**34. An I/O peripheral device shown in figure (b) below is to be interfaced to an 8085 microprocessor. To select the I/O device in the I/O address range D4 H — D7 H, its chip-select**

(Cs) should be connected to the output of the decoder shown in figure (a) below:

I/O

Peripheral

(B) output 5

(C) output 2

(D) output 0

** 35. A 4-bit D/A converter is connected to a free-running 3-bit UP counter, as shown in the following figure. Which of the following waveforms will be observed at V3? the flip-flops are in clear state. After applying two clock pulses, the outputs of the full-adder should be Clock**

(A) S=0 C0

(B) S=0 C0

(C) S=1 C0

(D) S=1 C0

**36. Two D-flip-flops, as shown below, are to be connected as a synchronous counter that goes through the following Q1Q0 sequence
00 — 01 — 11 — 10 — 00 —**

The inputs D0 and D1 respectively should be connected as

(A) Q1 and Q0

(B) Q0 and Q1

(C) Q1Q0 and Q1Q0

(D) and Q1Q0

**37. Following is the segment of a 8085 assembly language program:**

LXI SP, EFFF H

CALL 3000 H

3000 H : LXI H, 3CF4 H

PUSH PSW

SPHL

POP PSW

RET

On completion of RET execution, the contents of SP is:

(A) 3CFO H

(B) 3CF8 H

(C) 3FFD H

(D) EFFF H

**38. The point P in the following figure is stuck-at-i. The output f will be**

(A) ABC

(B) A

(C) ABC

(D) A

**39. A signal m(t)with bandwidth 500 Hz is first multiplied by a signal g(t)where g(t) = (_i)’<**

5(t_O.5xiO 4k)

The resulting signal is then passed through an ideal lowpass filter with bandwidth

i kHz. The output of the lowpass filter would be:

(A) 5(t)

(B) m(t)

(C) 0

(D) m(t)(t)

**40. The minimum sampling frequency (in samples/sec) required to reconstruct the following signal from its samples without distortion.**

(sin2,r1OOOt3 (sin2,r1OOOt2

x(t)=5i I i would be:

Tt }

(A) 2×103

(B) 4×103

(C) 6×103

(D) 8×103

**41. A uniformly distributed random variable X with probability density function f (x) =±(u(x+5)-u(x-5))**

Where u(.)is the unit step function is passed through a transformation given in the figure below. The probability density function of the transformed random variable Y would be

(A) (y)=(u(y+2.5)-u(y-2.5))

(B) i,(y)=O.5Y(y)+O.5Y(y—l)

(C) f(y)= O.255(y+2.5)+O.255(y —2.5)+O.55(y)

(D) f(y) = O.258(y+2.5)+O.258(y —2.5)+(u(y+2.5)—u(y —2.5))

**42. A system with input x[nl and output y[n is given as y[n=sinrnx(n). The system is:**

(A) linear, stable and invertible

(B) non-linear, stable and non-invertible

(C) linear, stable and non-invertible

(D) linear, unstable and invertible

**43. The unit-step response of a system starting from rest is given by c(t)=1—e2 for t!=0**

The transfer function of the system is:

(A)1 + 2s

(B) 2+s

(C) 2+s

(D) 2s 1 + 2s

**44. The Nyquist plot of G(ja)H(ja)for a closed loop control system, passes through (—1, j0) point in the GH plane. The gain margin of the system in dB is equal to**

(A) infinite

(B) greater than zero

(C) less than zero

(D) zero

**45. The positive values of “K” and “a” so that the system oscillates at a frequency of 2 rad/sec respectively are shown in the figure below**

(A) 1, 0.75

(B) 2, 0.75

(C) 1, 1

(D) 2, 2

**46. The unit impulse response of a system is: For this system, the steady-state value of the output for unit step input is equal to**

- -1

(C) 1

(D)

**47. A linear system is described by the following state equation k(t)=Ax(t)+Bu(t), A=[°
The state-transition matrix of the system is:**

(A) rcost sint

L—sint cost

(B) r_cost sint

L—sint —cost

(C) r_cost —sint

L—sint cost

(D) rcost —sint

Lcost sint

**48. The minimum step-size required for a Delta-Modulator operating at 32 K samples/sec to track the signal (here u (t) is the unit-step function)**

x(t)=125t(u(t)_u(t_1))+(250_125t)(u(t_1)_u(t_2))

So that slope-overload is avoided, would be

(A) 2 10

(B) 2 8

(C) 2 6

(D) 2

**49. A zero-mean white Gaussian noise is passed through an ideal lowpass filter of bandwidth 10 kHz. The output is then uniformly sampled with sampling period t = 0.03 msec. The samples so obtained would be**

(A) correlated

(B) statistically independent

(C) uncorrelated

(D) orthogonal

**50. A source generates three symbols with probabilities 0.25, 0.25, 0.50 at a rate of 3000 symbols per second. Assuming independent generation of symbols, the most efficient source encoder would have average bit rate as**

(A) 6000 bits/sec

(B) 4500 bits/sec

(C) 3000 bits/sec

(D) 1500 bits/sec

**51. The diagonal clipping in Amplitude Demodulation (using envelope detector) can be avoided if RC time-constant of the envelope detector satisfies the following condition, (here W is message bandwidth and a is carrier frequency both in rad/sec)**

(A) RC<–w

(B) RC>–w

(C) RC<—–o)c

(D) RC>—-o)c

**52. A message signal with bandwidth 10 kHz is Lower-Side Band SSB modulated with carrier frequency t = 1O Hz: The resulting signal is then passed through a Narrow-Band Frequency Modulator with carrier frequency 2 = iO Hz. The bandwidth of the output would be:**

(A) 4×104 Hz

(B) 2×106 Hz

(C) 2×109 Hz

(D) 2×101° Hz

**53. A medium of relative permittivity 2 = 2 forms an interface with free-space. A point source of electromagnetic energy is located in the medium at a depth of 1 meter from the interface. Due to the total internal reflection, the transmitted beam has a circular cross-section over the interface. The area of the beam cross- section at the interface is given by**

(A) 2,rm2

(B) r2m2

(C) -m2

(D) rm2

**54. A medium is divided into regions I and II about x = 0 plane, as shown in the figure below. An electromagnetic wave with electric field E1 = 4a + 3a + 5a is incident normally on the interface form region-I. The electric field E2 in region-Il at the interface is:
Region I Region II**

= °P1 = = °P2 =

E1 E

x<O x=O x>O

(A) E2=E1

(B) 4a +0.75a —1.25a

(C) 3a+3a+5a

(D) —3a + 3a + 5a

**54. A rectangular waveguide having TE10 mode as dominant mode is having a cutoff frequency of 18-GHz for the TE30 mode. The inner broad-wall dimension of the rectangular waveguide is:**

(A) 5— cms

(B) 5 cms

(C) 8 cms

(D) 10 cms

**55. A mast antenna consisting of a 50 meter long vertical conductor operates over a perfectly conducting ground plane. It is base-fed at a frequency of 600 kHz. The radiation resistance of the antenna in Ohms is:**

(A)

(B)

(C)

(D) 20,r2

Common Data Questions:

Common Data for Questions 71, 72, 73:

In the transistor amplifier circuit shown in the following parameters:

I3DC = 60, “BE = 0. 7V, hie — 00, hfe °° The capacitance Ccan be assumed to be infinite.

5.3K figure below, the transistor has the

In the figure above, the ground has been shown by the symbol V

**71. Under the DC conditions, the collector-to-emitter voltage drop is:**

(A) 4.8 Volts

(B) 5.3 Volts

(C) 6.0 Volts

(D) 6.6 Volts

**72. If I3DC is increased by 10°h, the collector-to-emitter voltage drop **

(A) increases by less than or equal to 10%

(B) decreases by less than or equal to 10%

(C) increases by more than 10%

(D) decreases by more than 10°h

**73. The small-signal gain of the amplifier v/v is:**

(A) -10

(B) -5.3

(C) 5.3

(D) 10

Common Data for Questions 74, 75: Let g(t) = p(t) * p(t), where * denotes convolution and p(t) = u(t)— u(t —1) with u (t) being the unit step function

**74. The impulse response of filter matched to the signal s (t) = g (t) — —2) * g (t) is
given as:**

(A) s(1—t)

(B) —s (1 — t) T EForum

(C) —s(t)

(D) s(t)

**75. An Amplitude Modulated signal is given as**

XAM (t) = 100(p(t)+0.5g(t))cosat in the interval 0 < t < 1. One set of possible values of the modulating signal and modulation index would be

(A) t,O.5

(B) t,1.O

(C) t,2.O

(D) t2,0.5

Linked Answer Questions: Q.76 to Q.85 Carry Two Marks Each.

Statement for Linked Answer Questions 76 & 77:

A regulated power supply, shown in figure below, has an unregulated input (UR) of 15 Volts and generates a regulated output Use the component values shown in the figure.

In the figure above, the ground has been shown by the symbol V

**76. The power dissipation across the transistor Qi shown in the figure is:**

(A) 4.8 Watts

(B) 5.0 Watts

(C) 5.4 Watts

(D) 6.0 Watts

**77. If the unregulated voltage increases by 20%, the power dissipation across the transistor Qi**

(A) increases by 20°h

(B) increases by 50°h

(C) remains unchanged

(D) decreases by 20°h

Statement for Linked Answer Questions 78 & 79:

The following two questions refer to wide sense stationary stochastic processes

**78. It is desired to generate a stochastic process (as voltage process) with power spectral density By driving a Linear-Time-Invariant system by zero mean white noise (as voltage process) with power spectral density being constant equal to 1. The system which can perform the desired task could be:**

(A) first order lowpass R-L filter

(B)

(C)

(D)

Statement for Linked Answer Questions 80 & 81:

Consider the following Amplitude Modulated (AM) signal, where m <B:

XAM (t) = 10(1 + 0.5sin2rfmt)cos2rft average side band power fo tie AM signal given above is:

**81. The AM signal gets added to a noise with Power Spectral Density S (f)given in the figure below. The ratio of average sideband power to mean noise power would be:**

(A) 25

(B) 12.5

(C) 6.25

(D) 3.125

Statement for Linked Answer Questions 82 & 83:

Consider a unity-gain feedback control system whose open-loop transfer function is: first order highpass R-c filter tuned L-C filter series R-L-C filter

**82. The value of “a” so that the system has a phase margin equal to -is approximately equal to**

(A) 2.40

(B) 1.40

(C) 0.84

(D) 0.74

**83. With the value of “a” set for a phase-margin of the value of unit-impulse response of the open-loop system at t = 1 second is equal to**

(A) 3.40

(B) 2.40

(C) 1.84

(D) 1.74

Statement for Linked Answer Questions 84 & 85:

A 30-Volts battery with zero source resistance is connected to a coaxial line of characteristic impedance of 50 Ohms at t = 0 second terminated in an unknown resistive load. The line length is that it takes 400 ps for an electromagnetic wave to travel from source end to load end and vice-versa. At t = 400ps, the voltage at the load end is found to be 40 Volts.

**84. The load resistance is**

(A) 25 Ohms

(B) 50 Ohms

(C) 75 Ohms

(D) 100 Ohms

**85. The steady-state current through the load resistance is:**

(A) 1.2 Amps

(B) 0.3 Amps

(C) 0.6 Amps

(D) 0.4 Amps