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P Pankaj Sanodiya
As we know that the general   term   in the binomial expansion of    is given by  So,  Term in  the expansion of  :  Term in  the expansion of  :  Term in  the expansion of  : Now, As given in the question, From here, we get , Which can be written as  From these equations we get,   

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P Pankaj Sanodiya
Given  By Binomial Theorem It can also be written as  Now, Again By Binomial Theorem, From (1) and (2) we get,  

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P Pankaj Sanodiya
Given the expression, Binomial expansion of this expression is  Now Applying Binomial Theorem again, And  Now, From (1), (2) and (3) we get,

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P Pankaj Sanodiya
Given, the expression  Fifth term from the beginning  is  And Fifth term from the end is, Now, As given in the question, So, From Here , From here, Hence the value of n is 10.      

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P Pankaj Sanodiya
As we can write 0.99 as 1-0.01, Hence the value of     is 0.951 approximately.

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P Pankaj Sanodiya
First, lets simplify the expression  using binomial expansion, And Now, Now, Putting  we get,  

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P Pankaj Sanodiya
First let's simplify the expression  using binomial theorem, So, And  Now, Now, Putting  we get 

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P Pankaj Sanodiya
we need to prove,    where k is some natural number. Now let's add and subtract b from a so that we can prove the above result, Hence, is a factor of .

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P Pankaj Sanodiya
First, lets expand both expressions individually, So, And  Now, Now, for the coefficient of , we multiply and add those terms whose product gives .So, The term which contain are, Hence the coefficient of  is 171.

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P Pankaj Sanodiya
As we know the Binomial expansion of  is given by  Given in the question, Now, dividing (1) by (2) we get, Now, Dividing (2) by (3) we get,  Now, From (4) and (5), we get,

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P Pankaj Sanodiya
As we know that the general   term   in the binomial expansion of    is given by  So, the general   term   in the binomial expansion of    is  will come when . So, The coeficient of   in the binomial expansion of    = 6  Hence the positive value of m for which the coefficient of  in the expansion is 6, is 4.

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P Pankaj Sanodiya
As we know that the general   term   in the binomial expansion of    is given by  So, general   term   in the binomial expansion of   is,  will come when , So, Coefficient of  in the binomial expansion of   is, Now, the general   term   in the binomial expansion of   is, Here also  will come when , So, Coefficient of  in the binomial expansion of   is, Now, As we can see Hence, the...

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P Pankaj Sanodiya
As we know that the general   term   in the binomial expansion of    is given by  So, the general  term   in the binomial expansion of    is given by  Now, as we can see  will come when  and  will come when  So,  Coefficient of  : CoeficientCoefficient of  : As we can see . Hence it is proved that the coefficients of  and  are equal.

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P Pankaj Sanodiya
As we know that the middle term in the expansion of   when n is even is, , Hence the middle term of the expansion        is, Which is  Now,  As we know that the general   term   in the binomial expansion of    is given by  So the  term of the expansion of    is Hence the middle term of the expansion of   is  .

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P Pankaj Sanodiya
As we know that the middle  terms in the expansion of   when n is odd are, Hence the middle term of the expansion       are  Which are  Now,  As we know that the general   term   in the binomial expansion of    is given by  So the  term of the expansion of    is And the  Term of the expansion of     is, Hence the middle terms of the expansion of given expression are    

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P Pankaj Sanodiya
As we know that the general   term   in the binomial expansion of    is given by  So the  term of the expansion of        is

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P Pankaj Sanodiya
As we know that the general   term   in the binomial expansion of    is given by  So the  term of the expansion of  is .

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P Pankaj Sanodiya
As we know that the general   term   in the binomial expansion of    is given by  So the general term of the expansion of  is .

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P Pankaj Sanodiya
As we know that the general   term   in the binomial expansion of    is given by  So the general term of the expansion of   : .  

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P Pankaj Sanodiya
As we know that the  term   in the binomial expansion of    is given by  Now let's assume  happens in the  term of the binomial expansion of  So, On comparing the indices of x we get, Hence the coefficient of the    in  is 
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