Can you solve below mathematical equation?

2^1234 - 2^1233

I am taken from a mine and shut up in a wooden case, from which I am never released, and yet I am used by almost every person. What am I?

Some friends went on a vacation to a resort. It was raining heavily and it kept raining for thirteen days.

When it rained in the morning, the afternoon was beautiful and when it rained in the afternoon, the day was blessed with a clear morning.

Overall, the friends experienced eleven nice mornings and twelve nice afternoons. Can you find out the number of days they spent on vacation?

2, 3, 5, 9, 17, _ What is the next number in the sequence?

Irene killed hundreds of people mercilessly and yet the police were not able to put her behind the bars. Why?

A word I know,
Six letters contain,
Subtract just one,
And twelve is what remains.

You want to boil a two-minute egg. If you only have a three-minute timer (hourglass), a four-minute timer and a five-minute timer, how can you boil the egg for only two minutes?

Intelligence Agency need to break the vault but the problem is that they are not able to break the vault.

On the vault, a text is written as:

31 11 33 33 43 45
21 25
31 24 11 31 51 41

To Break the code Intelligence Agency brings a mathematician named Joseph.
Minutes later he deciphers the code.

Whats the code?

Name a key which is hardest to turn.

A Man gave one of his sons 10 cents and another son was given 15 cents. What time is it?
Hint: Figure out the relation between time and number, is not so easy..:)