A string is rigidly tied at two ends and its equation of vibration is given by $\mathrm{y}=\cos 2\mathrm{\pi }t \sin 2\mathrm{\pi x}.$  Then minimum length of string is 

Given equation of stationary wave is 
$\mathrm{y}=\sin 2\mathrm{\pi x} \cos 2\mathrm{\pi t}$, comparing it with standard equation $\mathrm{y}=2\mathrm{Asin}\frac{2\mathrm{\pi x}}{\mathrm{\lambda }}\cos \frac{2\pi vt}{\mathrm{\lambda }}$
We have $\frac{2\mathrm{\pi x}}{\mathrm{\lambda }}=2\mathrm{\pi x}\Rightarrow \mathrm{\lambda }=1\mathrm{m}$
Minimum distance of string (first mode) ${\mathrm{L}}_{\min }=\frac{\mathrm{\lambda }}{2}=\frac{1}{2}\mathrm{m}$