Binomial Theorem

Terms greater than 5! i.e., is divisible by 100 For terms ..., remainder is 0. Now consider = 1 + 4+ 36+ 576 = 617 When 617 is divided by 100, its remainder is 17. Required remainder is 17.

The general term of the expansion is
.
We have
Here,
Nineth term

$ 3^{4}=3^{8}=3^{12}=\ldots .3^{100} 2^{4}=2^{8}=2^{12}=\ldots .2^{500}$.So, unit digit is 6 . So, the last digit of the product will be 6 . Dividing by 5 we will have 1 as a remainder.

Given,
Now,
and
According to question
coefficient of coefficient of











and gives imaginary roots.
Alternate Method

Given, and is an
odd positive integer.


If we take ,


On comparing both sides,
...(i)
If we take ,








On comparing the coefficient of on both sides;

...(ii)
From E (i), we see that,
when
From E (ii), we see that,

when
Similarly, for each odd terms:

Given,

and
From question,

Given,
We know that,


On differentiating w.r.t. , we get


On putting , we get

(given)

On comparing, we get

Given expansion is .
According to the question,

Since, (even), so the middle term will be th term, i.e. 6 th term.
Now,

Total number of terms in

Total number of terms in the expansion of
is odd

The correct answer is (A) :12 .