Solution:

This is what Birbal said after bowing in front of the real king:
"Your majesty, the other kings were watching you closely in order to imitate you accurately while you were sitting with your calm and composed demeanor. This is how I found out that you are the real king."

His Quick wit flabbergasted the king and he was rewarded with luxurious gifts and jewels in return.

Solution:

Just One!

example
If u get Blue rock with label as "blue+red" => this is red as all labels are wrong.
In this way u can logically solve this puzzle.

Solution:

The woman told him to take out one nut each from the rest of the wheels and use them.

Solution:

Birbal drew another line that was larger than the pre-drawn line. Now the pre drawn line was smaller.

Solution:

Old password : altered

New Password : distinct

Solution:

13 + 11 + 6 = 30

Explanation
The trick is that you just need to rotate '9' to form '6'.

Solution:

It can be down as shown below.

Solution:

1653

Explanation:
For an ideal case, the batsman will hit a six on each ball. But if he hits six on the last ball of the over, the strike will change in the next over. Thus, the best he can do in the last ball is run 3 runs so that he retains the strike even in the next over. Thus the total runs that he can score in each over:
6 * 5 + 3 = 33
But this will have to go like it only till the 49th over. In the last over, he can hit a six in the last ball as well as that will be the last ball of the match.
Thus runs for the last over will be 6 * 6 = 36.
Hence the maximum runs = 33 * 49 + 36 = 1653

Solution:

They use the bricks and dirt from the tunnel to stand on it.

Solution:

The mules is moving in circular path and hence outside legs travels more distance than inner legs.

Solution:

For this answer is 3^0, 3^1, 3^2... That is 1,3,9,27,81,243 and 729.