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

Sherlock Holmes was challenged to make a pre-drawn line smaller without erasing it. He did it in fractions of seconds. How?

I am a ten letters word
The first four letters have the power to rule,
Next, four-letter can be eaten.
The next three-letter represents a lady.
I can fly as well

what am I looking for?

How can one talk without saying a single word?

Divide 110 into two parts so that one will be 150 per cent of the other. What are the 2 numbers?

We can be boring,Or interesting.
We can be long,Or short.
We can have pictures,Or just words.
Many love us,But many also hate us.
What are we?

There are three boxes on a table. One of the box contains Gold and the other two are empty. A printed message contains in each box. One of the message is true and the other two are lies.

The first box says "The Gold is not here".

The Second box says "The Gold is not here".

The Third box says "The Gold is in the Second box".

Which box has the Gold?

A Scientist was fifty years old in the year 2000 but he was forty years old in the year 2010. How can this be possible?

PS: It has a logical answer.

Name the three-letter word that can complete the below words?

A) L O _ _ _ E
B) E D U _ _ _ E
C) _ _ _ E R
D) _ _ _ T L E

The average age of 10 members of a committee is the same as it was 4 years ago, because an old member has been replaced by a young member. Find how much younger is the new member ?