# Electronics and Communication Engg - GATE 2006 Paper Solution

The rank of the matrix

$\left[\begin{array}{ccc}1& 1& 1\\ 1& -1& 0\\ 1& 1& 1\end{array}\right]$ is:

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

$\nabla ×\nabla ×P,$, where P is a vector, is equal to

 (A) $P×\nabla ×P-{\nabla }^{2}P$ (B) ${\nabla }^{2}P+\nabla \left(\nabla .P\right)$ (C) ${\nabla }^{2}P+\nabla ×P$; (D) $\nabla \left(\nabla .P\right)-{\nabla }^{2}P$

$?\left(\nabla ×P\right).ds,$where P is a vector, is equal to

A probability density function is of the form

$p\left(x\right)=K{e}^{-\alpha \left|x\right|},x\in \left(-\infty ,\infty \right)$

The value of K is

 (A) 0.5 (B) 1 (C) 0.5$\alpha$ (D) $\alpha$

A solution for the differential equation

$\stackrel{.}{x}\left(t\right)+2x\left(t\right)=\delta \left(t\right)$

with initial condition x (0 −) = 0 is:

 (A) ${e}^{-2t}u\left(t\right)$ (B) ${e}^{2t}u\left(t\right)$ (C) ${e}^{-t}u\left(t\right)$ (D) ${e}^{t}u\left(t\right)$

A low-pass filter having a frequency response $H\left(j\omega \right)=A\left(\omega \right){e}^{j?\left(\omega \right)}$ does not produce any phase distortion if

 (A) $A\left(\omega \right)=C{\omega }^{2},\varphi \left(\omega \right)=K{\omega }^{3}$ (B) $A\left(\omega \right)=C{\omega }^{2},\varphi \left(\omega \right)=K\omega$ (C) $A\left(\omega \right)=C\omega ,\varphi \left(\omega \right)=K{\omega }^{2}$ (D) $A\left(\omega \right)=C,\varphi \left(\omega \right)=K{\omega }^{-1}$

The values of voltage (VD )  across a tunnel-diode corresponding to peak and valley currents are and VP and VV respectively. The range of tunnel-diode voltage VD for which the slope of its I-VD characteristics is negative would be

 (A) (B) $0\le {V}_{D}<{V}_{P}$ (C) ${V}_{P}\le {V}_{D}<{V}_{V}$ (D) ${V}_{D}>{V}_{V}$

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

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

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.

The input impedance (${Z}_{i}$ )  and the output impedance ( ${Z}_{o}$)  of an ideal transconductance (voltage controlled current source) amplifier are

 (A)${Z}_{i}=0,{Z}_{0}=0$ (B) ${Z}_{i}=0,{Z}_{0}=\infty$ (C) ${Z}_{i}=\infty ,{Z}_{0}=0$ (D) ${Z}_{i}=\infty ,{Z}_{0}=\infty$

An n-channel depletion MOSFET has following two points on its ${I}_{D}-{V}_{GS}$  curve:

(i) ${V}_{GS}=0$ at ${I}_{D}=12MA$ and

(ii) at ${I}_{D}=0$

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

 (A) Volts (A) Volts (C) Volts (D) Volts

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)

 1 0 0 1 0 d 0 0 0 0 d 1 1 0 0 1
 (A) 2 (B) 3 (C) 4 (D) 5

Let be Fourier Transform pair. The Fourier Transform of the signal x (5t − 3) in terms of X ( jw ) is given as

 (A) $\frac{1}{5}{e}^{-\frac{j3\omega }{5}}X\left(\frac{j\omega }{5}\right)$ (B) $\frac{1}{5}{e}^{\frac{j3\omega }{5}}X\left(\frac{j\omega }{5}\right)$ (C) $\frac{1}{5}{e}^{-j3\omega }X\left(\frac{j\omega }{5}\right)$ (D) $\frac{1}{5}{e}^{j3\omega }X\left(\frac{j\omega }{5}\right)$

The Dirac delta function $\delta \left(t\right)$ is defined as

 (A) $\delta \left(t\right)=\left\{\begin{array}{ll}1& t=0\\ 0& otherwise\end{array}\right\$ (B) $\delta \left(t\right)=\left\{\begin{array}{ll}\infty & t=0\\ 0& otherwise\end{array}\right\$ (C) (D)

If the region of convergence of ${x}_{1}\left[n\right]+{x}_{2}\left[n\right]$ is $\frac{1}{3}<\left|z\right|<\frac{2}{3},$ then the region of convergence of ${x}_{n}\left[n\right]-{x}_{2}\left[n\right]$ includes

 (A) $\frac{1}{3}<\left|z\right|<3$ (B) $\frac{2}{3}<\left|z\right|<3$ (C) $\frac{3}{2}<\left|z\right|<3$ (D) $\frac{1}{3}<\left|z\right|<\frac{2}{3}$

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

$G\left(s\right)=\frac{K}{\left(s+1\right)\left(s+2\right)}$

The gain margin of the system in dB is given by

 (A) 0 (B) 1 (C) 20 (D) $\infty$

In the system shown below, x (t ) = (sint )u (t ). In steady-sate, the response y (t )will be: (A) $\frac{1}{\sqrt{2}}\mathrm{sin}\left(t-\frac{\mathrm{\pi }}{4}\right)$ (B) $\frac{1}{\sqrt{2}}\mathrm{sin}\left(t+\frac{\mathrm{\pi }}{4}\right)$ (C) $\frac{1}{\sqrt{2}}{e}^{-t}\mathrm{sin}t$ (D) $\mathrm{sin}t-\mathrm{cos}t$

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

.

The wave is

 (A) linearly polarized in the z-direction (B) elliptically polarized (C) left-hand circularly polarized (D) right-hand circularly polarized