NOTE: The word 'her child' in the second sentence indicates that the sentence is about the lady's possession.
Option A - 'Whom' is used as a relative pronoun for people. It is only used as an objective pronoun/noun. As the person in the sentence is the subject and not an object, we can't use 'whom' here. Hence, option A is incorrect.
Option C - 'Which' is used as a relative pronoun for the things that have been previously mentioned. Here, the sentence is about the lady's possession.Hence, option C is incorrect.
Option D - 'That' is a relative pronoun, which is used to demonstrate things or people. As the given sentence doesn't demonstrate anything or anyone, we can't use 'that' here. Hence, option D is incorrect.
Option B - 'Whose' is used as a relative pronoun to show possession of a noun or a pronoun. As the given sentence indicates that the lady possesses her child, we can say 'whose child was beaten up'. Hence, option B is the correct answer.
'This is the lady whose child was beaten up.'