Let A, B, C, D, and E be five boolean variables.

Let P1, P2, P3, P4, and P5 be the five propositions listed in the question.

To find all possible answers, we can construct a truth table. Since the validity of each proposition must match the truth value of its matching variable, we can mark rows as contradictions if the two sets of columns are unequal. All rows without contradictions are valid answers to the question. I don't have to convince you of any given value of any given variable, because this is exhaustively examining every possible combination.

A B C D E | P1 P2 P3 P4 P5 | Contradiction?
----------+----------------+---------------
F F F F F | T  T  F  T  F  | Yes
F F F F T | F  T  F  T  F  | Yes
F F F T F | F  F  F  T  F  | No
F F F T T | F  F  F  T  F  | Yes
F F T F F | F  F  F  F  F  | Yes
F F T F T | F  F  F  F  F  | Yes
F F T T F | F  F  F  F  F  | Yes
F F T T T | F  F  F  F  F  | Yes
F T F F F | F  T  F  F  F  | No
F T F F T | F  T  F  F  F  | Yes
F T F T F | F  F  F  F  F  | Yes
F T F T T | F  F  F  F  F  | Yes
F T T F F | F  F  T  F  F  | Yes
F T T F T | F  F  T  F  F  | Yes
F T T T F | F  F  T  F  F  | Yes
F T T T T | F  F  T  F  F  | Yes
T F F F F | T  T  F  F  F  | Yes
T F F F T | F  T  F  F  F  | Yes
T F F T F | F  F  F  F  F  | Yes
T F F T T | F  F  F  F  F  | Yes
T F T F F | F  F  F  F  F  | Yes
T F T F T | F  F  F  F  F  | Yes
T F T T F | F  F  F  F  F  | Yes
T F T T T | F  F  F  F  F  | Yes
T T F F F | F  T  F  F  F  | Yes
T T F F T | F  T  F  F  F  | Yes
T T F T F | F  F  F  F  F  | Yes
T T F T T | F  F  F  F  F  | Yes
T T T F F | F  F  T  F  F  | Yes
T T T F T | F  F  T  F  F  | Yes
T T T T F | F  F  T  F  F  | Yes
T T T T T | F  F  T  F  T  | Yes

In conclusion, there are two possible answers:

(1) Only B is true.
(2) Only D is true.

They cannot be true at the same time.