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The value of x for which the sixth term ...

The value of `x` for which the sixth term in the expansion of `[2^(log)2sqrt(9^(x-1)+7)+1/(2^(1/5)(log)_2(3^((x-1)+1)))]^7` is 84 is a. `4` b. `1or2` c. `0or1` d. `3`

A

4

B

3

C

2

D

1

Text Solution

Verified by Experts

The correct Answer is:
c,d
We have , 6th term in the expansion of
`{2^(log_(2) sqrt((9^(x-1) + 7) ))+ 2 (1)/(2^((1//5) log_(2) (3^(x -1)+1)))}`
or ` { sqrt((9^(x-1) + 7)) + (1)/(3^(x-1) + 1)^(1//5)}^(7) " is " T_(6) = T_(5+1)`
` = ""^(7)C_(2) . (9^(x -1) + 7)^(2){(1)/((3^(x-1) + 1)^(1//5))}^(5)`
` =""^(7)C_(2)((9^(x-1) + 7))/((3^(x-1) +1) )= 21. ((9^(x -1) + 7))/((3^(x-1) + 1)) = 84 ` [ given]
`rArr (9^(x-1) + 7) = 4(3^(x-1) +1)`
Let ` 3^(x-1) = lambda ` , then
` lambda^(2) - 4lambda + 3 = 0 `
or ` (lambda - 3 ) (lambda - 1) = 0 `
` therefore lambda = 3, 1`
` rArr 3^(x -1) = 3^(1) , 3^(0) `
or ` x - 1 = 1 , 0 or x = 2, 1`
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