Question

an electron is initially in 6d state, without changing the quantum number what is the orbital angular momentum to be added to an electron in that state?

Answer 1

If an electron is in the 6d state as stated in the question then it will be (n=6, ℓ=2) it *already has* angular momentum L, where L is given by the standard formula: 
L = √[ℓ(ℓ+1)] ℏ 

For ℓ=2 
L = √[2(2+1)] ℏ 
L = √6 ℏ 
. .= 2.45ℏ approx. 

The 6d electron already has this angular momentum. Unless you want to change the electron's state to one with a different value of ℓ (e.g. to the 6g state, which has ℓ=4), the amount of angular momentum to be added is zero. 

It doesn't make sense to talk about adding angular momentum unless you increase the value of the quantum number ℓ.

But the question says "without changing the quantum number". 

You need to check the wording of the original question to make sure you have it correct.