In a scenario where the user has to select simple true/false or active/inactive states then I would really suggest that you use a checkbox control rather than a control which would feature redundant input, eg:
In a case such as this there is no reason to add additional input controls which provides the same result as a more simple checkbox control. Either the sync is on or not, the sync being not ON is the same as the sync being OFF.
However, there are of course cases where two alternative answers can't be provided with a simple Yes, I agree with what the checkbox label says.
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This would be very confusing to try and convey using a checkbox control. So it's really all about what type of question you're asking.
But regarding whether you should use a radio button control or a dropdown for these types of questions, for a two alternative question definitely go for the radio button since that provides a better scanability of a form etc.