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

On a certain day, John celebrated his birthday. Two days later, his older twin brother Jacob celebrated his birthday.

Is this even possible? How can it be?

A and B have a certain number of chocolates with them. If B gives one chocolate to A, they will have an equal number of chocolates. But if A gives one chocolate to B, then A will be left with half the number of chocolates that B has.

Can you find out the number of chocolates they have right now?

John has played 50 ODI's and his average is 50. How many runs should he score in his 51st ODI, so that his average score jumps to 51?

It comprises roots which nobody sees,
Much taller than those trees,
Up and up it goes,
Still it never grows?

If you drop a 15 kg iron bar and a 5 kg bag of cotton from a height of 50 meters which will reach the ground first?

A king sentenced a man to the death sentence for some crime he had committed. Known for his kindness, the king told the culprit that he had a choice to die in a way he decides.

The culprit was clever and said something that saved him from death. What method do you think he must have chosen for his death?

John is found dead in a bare field. Nobody killed John,
John did not commit suicide.

Inspector Jacob found a bag just a few meters away from John which could save his life. What was in the bag?

During which month do people sleep the least?

I am the largest in my family of fifty.
The youngest of our family lives separately from the rest of the family and so do I.
I am regarded as harsh and tough.
I am so large, you cut me in two equal halves,
each half would still be larger than the second largest member of the family.
Who am I?