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

Detective Rockford was jogging near the beach at 4:30 am.
He hears a sound near the shack "No Michael, Please Do not shoot me".
Next instance he heard the sound of gunfire. Rockford rushes to the shack where he finds women lying dead and a gun in close proximity of three "Doctore Lawyer and a Teacher".

Rockford immediately knew that the Lawyer has committed the crime. How?

If in a code language, DISTANCE is written as IDTUBECN and DOCUMENT is written as ODDVNTNE, the how will THURSDAY be coded as in the same language?

Replace the question mark with the correct number, given the pair of numbers exhibits a similar relationship?

? : 3839 :: 11 : 1209

I am an odd number. Take away a letter and I become even. What number am I?

After a heavy Thanksgiving meal, the night watchman went to work. In the morning, he told his boss he had dreamed that a saboteur planted a bomb in the factory and that he felt it was a warning. The boss promptly fired him. Worker confused, Why boss fire him?

How can you tell a raw egg from a hard-boiled egg?

Outside a massage parlour, there is a board that reads
"I only massage those who do not massage themselves."

Reading the above statement, does the masseur massage himself?

Okay before we begin with the puzzle, do keep in mind that you have to answer it quickly.

You are participating in a car race. Seeking the perfect moment, you overtake the second person in the race. What positions are you in now?

Two in a corner, one in a room, zero in a house, but one in a shelter. What am I?