A network consisting of a finite number of linear resistor (R), inducer (L), and capacitor (C) elements, connected all in series or all in parallel, is excited with a source of the form $$\sum\limits_{k = 1}^3 {{a_x}\,\cos \left( {k{\omega _0}t} \right),{\rm{were}}\,{a_k} \ne 0,} \,{\omega _0} \ne 0.$$ The source has nonzero impedance. Which one of the following is a possible form of the output measured across a resistor in the network?

Two monochromatic waves having frequencies $$\omega $$ and $$\omega + \Delta \omega \left( {\Delta \omega \ll \omega } \right)$$ and corresponding wavelengths $$\lambda $$ and $$\lambda - \Delta \lambda \left( {\Delta \lambda \ll \lambda } \right)$$ of same polarization, travelling along X-axis are superimposed on each other. The phase velocity and group velocity of the resultant wave are respectively given by

A

$$\frac{{\omega \lambda }}{{2\pi }},\,\frac{{\Delta \omega {\lambda ^2}}}{{2\pi \Delta \lambda }}$$

B

$$\omega \lambda ,\,\frac{{\Delta \omega {\lambda ^2}}}{{\Delta \lambda }}$$

C

$$\frac{{\omega \Delta \lambda }}{{2\pi }},\,\frac{{\Delta \omega \Delta \lambda }}{{2\pi }}$$

D

$$\omega \Delta \lambda ,\,\omega \Delta \lambda $$

The parallel combination of a 6.8 kΩ resistor and a 10 kΩ resistor is in series with the parallel combination of a 2.2 kΩ resistor and a 1 kΩ resistor. A 100 V source is connected across the circuit. The resistor(s) with the greatest voltage drop is (are)

A

6.8 kΩ

B

2.2 kΩ

C

6.8 kΩ and 10 kΩ

D

2.2 kΩ and 1 kΩ

For a function g(t), it is given that
$$\int\limits_{ - \infty }^{ + \infty } {g\left( t \right){e^{ - j\omega t}}dt = \omega {e^{ - 2{\omega ^2}}}} $$ for any real value $$\omega $$ . If $$y\left( t \right) = \int\limits_{ - \infty }^t {g\left( \tau \right)} d\tau ,\,{\rm{then}}\,\int\limits_{ - \infty }^{ + \infty } {y\left( t \right)} dt$$ is. . . . . . . .

A

0

B

-j

C

$$ - {j \over 2}$$

D

$${j \over 2}$$

A periodic signal x(t) has a trigonometric Fourier series expansion
$$x\left( t \right) = {a_0} + \sum\limits_{n = 1}^\infty {\left( {{a_n}\,\cos \,n{\omega _0}t + {b_n}\sin \,n{\omega _0}t} \right)} $$
If $$x\left( t \right) = - x\left( { - t} \right) = - x\left( {{{t - \pi } \over {{\omega _0}}}} \right),$$ we can conclude that

A

an are zero for all n and bn are zero for n even

B

an are zero for all n and bn are zero for n odd

C

an are zero for n even and bn are zero for n odd

D

an are zero for n odd and bn are zero for n even

The Fourier transform of a signal
h(t) is $$H\left( {j\omega } \right) = {{\left( {2\cos \omega } \right)\left( {\sin \omega } \right)} \over \omega }$$
The value of h(0) is

A

$${1 \over 4}$$

B

$${1 \over 2}$$

C

1

D

2

A parallel resonant band-pass filter consists of a 90 Ω resistor in series with a parallel network made up of a 60 mH coil and a 0.02 µF capacitor. The output is taken across the capacitor/coil. The coil winding has a resistance of 20 Ω. What is the center frequency of the filter?

A

459 Hz

B

4,591 Hz

C

999 Hz

D

2,176 Hz

The raised cosine pulse p(t) is used for zero ISI in digital communications. The expression for p(t) with unity roll-off factor is given by $$p\left( t \right) = \frac{{\sin 4\pi \omega t}}{{4\pi \omega t\left( {1 - 16{\omega ^2}{t^2}} \right)}}.$$ The value of p(t) at $$t = \frac{1}{{4\omega }}$$ is

A

-0.5

B

0

C

0.5

D

∞

At a 33kV 50 Hz to a bus a load of 35 MW is connected. To improve the power factor to unity a synchronous phase modifier is connected taking 5 MW. If the SPM is represented as delta connected impedance, where in each phase we have capacitor connected parallel to resistance forming RC parallel network, then the value of this resistance is ___________

A

653.4 Ω

B

753.6 Ω

C

280.4 Ω

D

682 Ω

At a 33kV 50 Hz to a bus a load of 35 MW is connected. To improve the power factor to unity a synchronous phase modifier is connected taking 5 MW. If the SPM is represented as delta connected impedance, where in each phase we have capacitor connected parallel to resistance forming RC parallel network, then the value of this capacitance is __________

A

25.57 μF

B

76.2 μF

C

32.2 μF

D

24 μF

The parallel combination of a 470 Ω resistor and a 1.2 kΩ resistor is in series with the parallel combination of three 3 kΩ resistors. A 200 V source is connected across the circuit. The resistor with the most current has a value of

A

470 Ω or 1.2 kΩ

B

3 kΩ

C

470 Ω

D

1.2 kΩ