- Any periodic signal can be given as a sum of sine waves of different frequencies. For example, a 1 kHz square wave consists of a 1 kHz base harmonic and other harmonics at 3, 5, 7, 9, etc. kHz (with decreasing amplitude), ideally infinitely many. If you chose the cut-off frequency of your LPF to let through the base harmonic but attenuate any other harmonics, you will get a signal that resembles a sine wave. As the other components are only attenuated, not eradicated, this won't be a true sine wave, but it might be difficult to tell the difference with your naked eyes.
- A modulation index of 1 means that them amplitude of the carrier (A) equals the amplitude of the modulating signal (M). The sidebands have amplitudes of \$\frac{M}{2}\$, in this case it "simplifies" to \$\frac{A}{2}\$. Half the amplitude means one quarter of power, because power is proportional to the square of amplitude.
a) If your signal is AM-DSB, the original carrier power is \$P\$ and "cutting off" the one sideband means cancelling it out completely, then the calculations are as following: A 20dB amplification equals a power amplification of 100. Both the left and unattenuated sideband and the reduced carrier get amplified.
\$\frac{P}{4}\cdot 100+\frac{P}{100}\cdot 100=200W\$
\$26P=200W\$
\$P\approx7.692W\$
b) However, if it was an AM-SSB signal and you cancelled out the only one sideband, your calculations are correct.