A sinusoidal wave y(t) = 40sin(10 x 10⁶ πt) is amplitude modulated by another sinusoidal wave x(t) = 20sin (1000πt). The amplitude of minimum frequency component of modulated signal is :

$\begin{array}{l}\mathrm{y}\left(\mathrm{t}\right)=40\sin \left(10\times {10}^6\mathrm{\pi t}\right)\\ \mathrm{x}\left(\mathrm{t}\right)=20\sin \left(1000\mathrm{\pi t}\right)\\ \Rightarrow {\mathrm{\omega }}_{\mathrm{c}}={10}^7\mathrm{\pi }\\ {\mathrm{\omega }}_{\mathrm{m}}={10}^3\mathrm{\pi }\\ {\mathrm{A}}_{\mathrm{C}}=40\\ {\mathrm{A}}_{\mathrm{m}}=20\end{array}$
Equation of modulated wave $=\left({\mathrm{A}}_{\mathrm{C}}+{\mathrm{A}}_{\mathrm{m}}\sin {\mathrm{\omega }}_{\mathrm{m}}\mathrm{t}\right)$
$\sin {\mathrm{\omega }}_{\mathrm{c}}\mathrm{t}$
$\begin{array}{l}={\mathrm{A}}_{\mathrm{c}}\left(1+\frac{{\mathrm{A}}_{\mathrm{m}}}{{\mathrm{A}}_{\mathrm{c}}}\sin {\mathrm{\omega }}_{\mathrm{m}}\mathrm{t}\right)\sin {\mathrm{\omega }}_{\mathrm{c}}\mathrm{t}\\ ={\mathrm{A}}_{\mathrm{c}}\left(1+\mathrm{\mu sin}{\mathrm{\omega }}_{\mathrm{m}}\mathrm{t}\right)\sin {\mathrm{\omega }}_{\mathrm{c}}\mathrm{t},\mathrm{\mu }=\frac{{\mathrm{A}}_{\mathrm{m}}}{{\mathrm{A}}_{\mathrm{c}}}\\ ={\mathrm{A}}_{\mathrm{c}}\sin {\mathrm{\omega }}_{\mathrm{c}}\mathrm{t}+\frac{{\mathrm{\mu A}}_{\mathrm{c}}}{2}\left[\cos \left({\mathrm{\omega }}_{\mathrm{c}}-{\mathrm{\omega }}_{\mathrm{m}}\right)\mathrm{t}-\cos \left({\mathrm{\omega }}_{\mathrm{c}}+{\mathrm{\omega }}_{\mathrm{m}}\right)\mathrm{t}\right]\end{array}$
Amplitude of minimum frequency = 
$\frac{{\mathrm{\mu A}}_{\mathrm{c}}}{2}=\frac{{\mathrm{A}}_{\mathrm{m}}}{{\mathrm{A}}_{\mathrm{c}}}\times \frac{{\mathrm{A}}_{\mathrm{c}}}{2}=\frac{{\mathrm{A}}_{\mathrm{m}}}{2}=10$