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Q1. If the ratio of the coefficients of 3rd and 4th terms in the expansion of is 1:2 then find the value of n.

Solution

Given expansion is   

Q2. Find the  term independent of x  in the expansion of    :

Solution

We have:   The term will be independent of x  if the index of x is zero, i.e. 9-3r = 0, Thus, r = 3 Hence, term is independent of x and is given by 489888.

Q3. Show that  is divisible by 64, whenever n is a positive integer:

Solution

Hence,  is divisible by 64.

Q4. Find the general term in the expansion of  : 

Solution

The term of the expansion is given by:

Q5. Find the middle term in the expansion of .

Solution

As n is even, the middle term of the expansion   isterm.

Q6. Find the coefficient of in the expansion of :

Solution

In the expansion of , Let  be the term containing . So, 12- r = 5 and r = 7 Hence, r = 7 Hence the coefficient of = 101376.

Q7. If the coefficients of   terms of are in arithmetic progression, then find the value of r:

Solution

  Since the terms of given expansion are in arithmetic progression, we have,             or (15-2r) (r +1) = (13-2r) (15 –r) or 15r + 15 - - 2r = 195 – 30r -13r + or - 56r + 180 = 0 or - 4r + 45 = 0 or (r - 5) (r- 9) = 0 r = 5, 9.

Q8. In the binomial expansion of ,the coefficients of the 5^th, 6^th and7^h terms are in AP. Find all the values of n for which this can happen.

Solution

The coefficients of the 5^th, 6^th and 7^th terms are Since the coefficients of the 5^th, 6^th and 7^th terms are in AP, we have,

Q9. Using binomial theorem, evaluate .

Solution

We know that: 104 = 100 + 4