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Let $c\left ( t \right )=A_{c} \cos \left ( 2\pi f_{c}t\right )$ and $m\left ( t \right )=\cos \left ( 2\pi f_{m}t\right )$. It is given that $f_{c}>> 5f_{m}.$ The signal $c\left ( t \right )+m\left ( t \right )$ is applied to the input of a non-linear device, whose output $v_{o}\left ( t \right )$ is related to the input $v_{i}\left ( t \right )$ as $v_{o}\left ( t \right )=av_{i}\left ( t \right )+bv_{i}^{2}\left ( t \right ),$ where $a$ and $b$ are positive constants. The output of the non-linear device is passed through an ideal band-pass filter with center frequency $f_{c}$ and bandwidth $3f_{m},$ to produce an amplitude modulated (AM) wave. If it is desired to have the sideband power of the AM wave to be half of the carrier power, then $a/b$ is

1. $0.25$
2. $0.5$
3. $1$
4. $2$