People make me, save me, change me, raise me. What am I?

Find the mistake in the below maths equations

A = 2
A(A-1) = 2(A-1)
A2-A = 2A-2
A2-2A = A-2
A(A-2) = A-2
A = 1

There's one "sport" in which neither the spectators nor the participants know the score or the leader until the contest ends.

What is it?

Can you fill the blank with the correct number?

6 30 870 756030 _____

What does the following BrainBat signifies?

ABCDEFGHIJKLMNOPQRSTUVWYZ

QQQQQQQQQQQQQQQQQQQQQQQQQ.

A girl was sitting in her hotel room when she heard a knock on the door. She opened the door and found that a man was standing outside.
The man said, "Oh! I am really sorry, I thought this was my room."
He then walked through the corridor to the elevator. The girl did not know the man. She closed her door and called security asking them to apprehend the man.
What made her suspicious of that man? He might have been genuinely mistaken.

All of you like to eat me. I am a five-lettered word. Remove the first two and I turn into an infamous animal. Remove just the first and I become a heinous crime. Remove the first and the last and you can groove on me.

Can you guess who am I?

If you had an infinite supply of water and 5-quart and 3-quart pails, how would you measure exactly 4 quarts? What is the least number of steps you need?

A bridge is about to collapse. There are four people P, Q, R and S on one of the sides. Before the bridge collapses, they want to cross it. Now since the bridge is too weak, it can only stand the weight of two people at a time. Also, it is night time and nothing is visible. They have just one torch with them.

Now P takes one minute to cross the bridge, Q takes two minutes to cross, R takes five minutes to cross and S takes ten minutes to cross.

The bridge will collapse in seventeen minutes. How will they be able to cross the bridge before it collapses?

How many sides does a circle have?