Test Of Logic

Let's look at question 1 first.

We know the answer can't be A or B, because it contradicts itself. Therefore, we're left with C, D or E as the answer.

If we choose C: Then the answer to question 3 would be B, but since in order for this to happen, we have to choose C as the answer for question 1 and not A, this is not an option.

If we choose D: Then the answer to question 4 would be B, but if the answer to question 4 is B then so is question 2, but question 4 is the FIRST question to have B as its answer, therefore D is not an option either.

And so we're left with E as the answer for question 1, and so now we know that the answer to question 5 is B, and the answers to questions 2, 3 and 4 cannot be B.

Now let's look at question 4.

We know the answer to question 1 is E, so A is not correct.

B is not an option either, as mentioned earlier.

If the answer is C, then the answer to question 3 is also C, but we know that the answer to question 5 is B, and not A, so C is not an option either.

We know the answer to question 5 is B, so D isn't an option either.

So we're also left with E as the answer for question 4, and so the answer to question 6 is also E.

Now let's look at question 3.

It can't be A, because the answer to question 4 is E.

It can't be B, as mentioned earlier.

It can't be C, because the answer to question 5 is B.

It can't be E, because the answer to question 6 is E.

And so we're left with D as the answer for this question, and so the answer for question 2 is A.

And so we have our answers: EADEBE

Therefore, $a = 3$ and $b = 2$ , and so $a^2 + b = 3^2 + 2 = 9 + 2 = 11$

Looking at the 1st question, options A and B are immediately showing contradiction. If we go for option C, them that implies A is the answer of 1 and again contradiction. Going for option D means that second question has answer B and it makes B as the answer of 1 which is again a contradiction. So the only option left is option E. This means answer to 5th question is B.

Second question now has options A,C D and E possible.

In the third question, option A says that answer of 4th question is A and that immediately says the answer of 1st question is A. Contradiction. option B can be expelled similarly. Option C says the answer to 5th question is A but it actually is B from the first question. Contradiction. The possibilities D and E stays for now.

The fourth question: A is an immediate contradiction from question1. if B is true, then second question has option B as the answer. Contradiction. Option C says 5th question has answer A. Contradiction. Option D says 5th question has answer D. Contradiction. So E is the answer. So 6th question has answer E.

Fifth question: Answer B from first question

Sixth question: Option E from 4th question.

Third question: from answer to 6th question we can eliminate option E and option D turns out to be it's answer. This immediately implies answer to second question is A.

so it is EADEBE.

$a=3\\ b=2\\ \implies a^2+b=9+2=11$