Question

     i.       Find the coefficients of  and  in the expansion of .

   ii.       It is given that, when  is expanded, there is no term in . Find the value of the constant .

Solution

i.
 

Expression for the general term in the Binomial expansion of  is:

First we rewrite the expression in the standard form;

In the given case:

Hence;

Since we are looking for the terms with : we can  equate

Now we can find the two terms with ;

Substituting ;	Substituting ;

Hence the coefficients of  in the expansion of  are   &  respectively.

ii.
 

It is evident that that coefficient of the term containing  in the product of  is zero.

We have found in (i) that in the expansion of   we have terms;

Therefore  can be written for terms containing  as follows;

It can be expanded as;

As per given condition coefficient of the term containing  is zero, therefore;

Hence;