To solve this problem, we need to understand the concept of resolving power of an optical microscope.$\\$The resolving power $R$ of an optical microscope is given by the formula:$\\ R = \frac{1}{\lambda} \\$where $\lambda$ is the wavelength of light used.$\\$Given two wavelengths $\lambda_1 = 4000$ Å and $\lambda_2 = 6000$ Å, we need to find the ratio of their resolving powers.$\\$The resolving power for $\lambda_1$ is:$\\ R_1 = \frac{1}{\lambda_1} = \frac{1}{4000} \\$The resolving power for $\lambda_2$ is:$\\ R_2 = \frac{1}{\lambda_2} = \frac{1}{6000} \\$The ratio of the resolving powers $\left(\frac{R_1}{R_2}\right)$ is:$\\ \frac{R_1}{R_2} = \frac{\frac{1}{4000}}{\frac{1}{6000}} = \frac{6000}{4000} \\$Simplify the ratio:$\\ \frac{6000}{4000} = \frac{3}{2} \\$Therefore, the ratio of the resolving powers is $3:2. \\$This corresponds to Option 2.