When the nucleus of a fully differentiated carrot cell is injected into a frog egg whose nucleus has been removed, the injected donor nucleus is capable of programming the recipient egg to produce a normal carrot?

In this case, it is given that the nucleus of the carrot cell is injected into the frog's egg lacking nucleus. In this case, the recipient egg (frog egg) would not be able to produce the carrot. Every species differ from from each other, sometimes, mating between closely related species can produce hybrid offspring, but frog and carrot cannot be combined at all in any aspect.

The carrot is a plant and the frog is an animal. All the physiological processes vary a lot in between plant and the animal. Hence, this growth would not be supported. So, it is impossible to generate a carrot by inserting it's nucleus in the frog's cell.