First, let's review some ideas of argumentation.
With deduction, we can talk about arguments about being sound and valid. Valid means the structure of the argument leads to the correct conclusion independent of the premises, whereas soundness implies the argument is not only valid, but has true premises. For instance, "If Socrates is in the kitchen, he is in the house, therefore Socrates is in the house" is a valid argument, however it's sound only if it's actually true "Socrates is in the kitchen".
Remember, a deduction is a deterministic form of inference (things MUST follow), and induction is a form of inference that is probabilistic (things PROBABLY follow).
Strength and cogency for our purposes here will mirror validity and soundness in induction. Hence a strong inductive argument is one that relies on many good techniques to establish a certain probability exists, but ultimately, if those techniques are faulty because they make bad assumptions, then argument ultimately isn't cogent.
Now let's examine each statement and see where the analysis leads.
A) One and the same argument cannot be both inductive and cogent
FALSE. (Reworded:"An inductive argument cannot be cogent".) Any inductive argument that has strong form and is based on true premises by definition is cogent.
B) One and the same argument can be both sound and logically weak
FALSE. By definition, soundness applies to deduction and weakness applies to induction.
C) An inductive argument can be cogent
TRUE. This is the opposite assertion of A. Again, it bears reminding that a cogent argument is an inductive argument that is both logically strong and based on true premises.
D) A deductive argument can be invalid
TRUE. Deductive arguments can be valid or invalid depending on their logical form. An example of an invalid deductive argument would be "All men are mortal, Socrates is a man, therefore Socrates is not mortal." Note both premises are true, however, the conclusion will always be wrong.
Maybe I missed something, but it does appear to me that both A and B are false. Presuming you haven't transcribed the problem wrong, I would endorse the view that the problem isn't with your answer, but someone else's question, although I'm amenable to criticism. As someone who has written questions, it's quite easy to forget to add a negation or to switch between vocabulary for one thing when it applies to the other. I would politely bring it to the attention of your instructor, who if they practice what they teach will have to concede your deduction that the question is invalid is both valid and sound!