Solution:

Since each one of them gave one answer wrong, this means that each one of them gave two answers right.

Let us assume that Student A gave a wrong answer to the first question. This will mean that Student B also gave a wrong answer for the first. This will conclude that the rest of the two answers given by them are correct. However, the answers are different and thus it is not possible.

Thus both Student A and Student B must be right with the first question and the answer to the first is two.

If you keep applying the same logic, you will come to a conclusion that following are the correct answers:
Q1. Two
Q2. Three
Q3. Two

Solution:

The Stars

Solution:

The check was for dollars 31.63.

The bank teller gave her dollars 63.31
She spent .05, and then she had dollars 63.26, which is twice the check.

Let x be the dollars of the check, and y be the cent.

The check was for 100x + y cent
He was given 100y + x cent

Also
100y + x - 5 = 2(100x + y)
Expanding this out and rearranging, we find:
98y = 199x + 5

or 199x ≡ -5 (mod 98)
or 98*2*x + 3x ≡ -5 (mod 98)
3x ≡ -5 ≡ 93 (mod 98)
this quickly leads to x = 31

Solution:

Ganpat had a black suit and a brown suit.
Harish had a brown suit and a pink suit.
Manoj had a white suit and a black suit.
Tarun had a pink suit and a white suit.

Solution:

NONE, holes are empty.

Solution:

It is best if the person sits on the back of the train. After the stop, the train will be accelerating as it goes into the tunnel. Thus the train will be much master when the back of the train enters the tunnel. In this way, he will have to spend much less time in the tunnel.

Solution:

On the back, since bullet train is accelerating, therefore,the person at the back will spend less time than one in front.

Solution:

Ms. Britney did it.

We will take up every person and analyze the statements over all of them one by one.

Case 1: Ms. Britney did it, then
Statement 1 => true
Statement 2 => false (as Rihanna did not do it)
Statement 3 => false (as Ms. Britney is lying)
Statement 4 => false (as Ms. Gwyneth is lying)
Our first guess itself was right but let us confirm our analysis by analyzing the statements over others.

Case 2: Ms. Rihanna did it, then
Statement 1 => false (as Ms. Britney did not do it)
Statement 2 => true
Statement 3 => true (as Ms. Britney is not lying)
As per the conditions, three statements had to be wrong but here, we have two statements right which clearly contradict the condition.

Case 3: Ms. Gwyneth did it, then
Statement 1 => false (as Ms. Britney did not do it)
Statement 2 => false (as Ms. Rihanna did not do it)
Statement 3 => false (as Ms. Britney is lying)
Statement 4 => false (as Ms. Gwyneth is lying)
She did not do it as all the statements are false.

Case 4: Ms. Yoon did it, then
Statement 1 => false (as Ms. Britney did not do it)
Statement 2 => false (as Ms. Rihanna did not do it)
Statement 3 => false (as Ms. Britney is lying)
Statement 4 => false (as Ms. Gwyneth is lying)
All four statements are false in her case as well thus she did not do it.

Therefore we have proved that Ms. Britney did it.

Solution:

The ball will take infinite bounces before it is stopped. As per the question it will keep on bouncing half way up every time it hits the ground. But gravity will play its part at certain point of time which will make it stop.

Solution:

Align the toothpicks to form TEN. Use one toothpick for each of the lines.

Solution:

You must put labels on the tablets as A and B before using. In that case, if you pour tablets together, you will get 3A, 2A 1B, 1A 2B or 3B. If they are from the same bottles you can take one from another bottle and save the remaining two for another day. If you get two from same and one from other, you can draw one from another bottle and you will have two pairs of which you can eat one and save the other.