find n if the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of [{2^(1/4)}+{3^(-1/4)}]^n is (sqrt6):1

 ncert cbse chapter 8 binomial theorem miscellaneous exercise

 8.

find n if the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of [(fourth root of 2) + {1/(fourth root of 3)}]^n is (sqrt6):1

 

 in [{2^(1/4)}+{3^(-1/4)}]^n 

T(r+1)  = C(n.r)[ {2^(1/4)}^(n-r) ][{3^(-1/4)}^(r)]

for fifth term from the beginning

r=4

T(5) = C(n,4)[2^((n-4)/4)][3^(-(4)/4)] =C(n,4)[2^((n-4)/4)] / [3]

For finding the fifth term from the end in the given expansion, we interchange the terms and find the fifth term from the beginning of 

[{3^(-1/4)} + {2^(1/4)} ]^n

t ' (5) =  C(n,4) [3^( -(n-4)/4)][ 2^((4)/4)]=C(n,4) [3^( -(n-4)/4)] [2]

T(5) / t'(5) = [ 6^((n-4)/4) ] / [6^1] 

(a^m)/(a^n)  =a^(m-n)

[(n-4)/4] -[1] = [(n-8)/4]

T(5) / t'(5) = [6^( (n-8)/4 ) ]

using given ratio

[ sqrt(6) / 1] = [6^( (n-8)/4 ) ]

6^(1/2) = [6^( (n-8)/4 ) ]

 (1/2) = ( (n-8)/4 ) 

4/2 = n-8

2=n-8

n=10

exercise 8.2

 5. find the 4th term in the expansion of (x-2y)^12

n=12

T(r+1)=C(12,r) [x^(12-r)][ (-2y)^r]

for 4th term

put r=3

T(3+1) = C(12,3) [x^(12-3)][ (-2y)^3]

T(4) = 220*[x^9][-8y^3]

T(4)=(-1760)[x^9][y^3]

chapter 8 binomial theorem miscellaneous exercise

 1.Find a , b and n in the expansion of (a+b)^n if the first three terms in the expansion are 729, 7290, 30375

solution

 

2.  Find a if the coefficients of (x^2)  & (x^3) in the expansion of {(3+ax)^9} are equal 

solution

  3.find the coefficient of {x^5} in the expansion of{(1+2x)^6}{(1-x)^7}

solution

 

5.evaluate { (sqrt(3) + sqrt(2))^6 } - { (sqrt(3) - sqrt(2))^6 }

solution 

 6.find the value of [(a^2)+sqrt{(a^2)-1}]^4 + [(a^2)-sqrt{(a^2)-1}]^4 

solution

 

7.find an approximate value of (0.99^5) using the first three terms of its expansion

solution  

 

8.find n if the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of [(fourth root of 2) + {1/(fourth root of 3)}]^n is (sqrt6):1 

solution

 

exercise 8.2

Q8) Find the  middle terms in the expansion of [(x/3)+9y)]^10

solution  

 5. find the 4th term in the expansion of (x-2y)^12

solution

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