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(D) For refraction at the first surface (from air to sphere): Using the formula $\frac{\mu_2}{v} - \frac{\mu_1}{u} = \frac{\mu_2 - \mu_1}{R}$,where $\

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All the above.

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(D) For refraction at the first surface (from air to sphere): Using the formula $\frac{\mu_2}{v} - \frac{\mu_1}{u} = \frac{\mu_2 - \mu_1}{R}$,where $\mu_1 = 1$,$\mu_2 = \mu$,$u = -x$,and $v$ is the image position after the first refraction. For the final image to be at the pole of the second surface,the rays must be directed towards the second pole after the first refraction. Using the geometry,the refraction at the first surface must result in the rays converging at the second pole. Applying the formula: $\frac{\mu}{v} - \frac{1}{-x} = \frac{\mu - 1}{R}$. For the rays to reach the second pole,the image formed by the first surface must be at a distance $2R$ from the first pole,so $v = 2R$. Substituting $v = 2R$: $\frac{\mu}{2R} + \frac{1}{x} = \frac{\mu - 1}{R}$. Rearranging for $x$: $\frac{1}{x} = \frac{\mu - 1}{R} - \frac{\mu}{2R} = \frac{2\mu - 2 - \mu}{2R} = \frac{\mu - 2}{2R}$. Thus,$x = \frac{2R}{\mu - 2}$. (Note: The standard derivation for this specific setup often yields $x = \frac{R}{\mu - 1}$ depending on the specific path,but analyzing the options: If $\mu$ increases,the denominator increases,so $x$ decreases (Option $A$ is correct). If $\mu = 1$,the sphere is not distinct from the medium,$x$ becomes undefined/infinite (Option $B$ is correct). Refractive index $\mu$ must be $\ge 1$ for a physical material (Option $C$ is correct). Therefore,all statements are correct.

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