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?

The correct answer is $\boxed{\text{B}}$.

The output measured across a resistor in the network is a sum of sine waves with different frequencies and phases. The frequencies of the sine waves are multiples of the frequency of the source, and the phases are arbitrary. The amplitudes of the sine waves are determined by the impedance of the resistor and the source.

Option $\text{A}$ is incorrect because the amplitudes of the sine waves are not equal to the amplitudes of the source. Option $\text{C}$ is incorrect because the output does not contain all of the sine waves in the source. Option $\text{D}$ is incorrect because the output contains more sine waves than the source.

Here is a more detailed explanation of each option:

• Option $\text{A}$ is incorrect because the amplitudes of the sine waves are not equal to the amplitudes of the source. The output measured across a resistor in the network is a sum of sine waves with different frequencies and phases. The frequencies of the sine waves are multiples of the frequency of the source, and the phases are arbitrary. The amplitudes of the sine waves are determined by the impedance of the resistor and the source. In Option $\text{A}$, the amplitudes of the sine waves are not equal to the amplitudes of the source. This is because the impedance of the resistor is not taken into account.
• Option $\text{C}$ is incorrect because the output does not contain all of the sine waves in the source. The output measured across a resistor in the network is a sum of sine waves with different frequencies and phases. The frequencies of the sine waves are multiples of the frequency of the source, and the phases are arbitrary. The amplitudes of the sine waves are determined by the impedance of the resistor and the source. In Option $\text{C}$, the output does not contain all of the sine waves in the source. This is because the phases of the sine waves are not arbitrary.
• Option $\text{D}$ is incorrect because the output contains more sine waves than the source. The output measured across a resistor in the network is a sum of sine waves with different frequencies and phases. The frequencies of the sine waves are multiples of the frequency of the source, and the phases are arbitrary. The amplitudes of the sine waves are determined by the impedance of the resistor and the source. In Option $\text{D}$, the output contains more sine waves than the source. This is because the frequencies of the sine waves are not multiples of the frequency of the source.