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The value of x in the expression (x+x^((...

The value of `x` in the expression `(x+x^((log)_(10)x))^5` if third term in the expansion is 10,00,000 is/are
a. 10 b. 100 c. `10^(-5//2)` d. `10^(-3//2)`

A

10

B

100

C

`10^(-5//2)`

D

`10^(-3//2)`

Text Solution

Verified by Experts

The correct Answer is:
A, C
Inclusion of log x implies `x gt 0`
Now, `3^(nd)` term in expension is
`T_(2+1) = .^(5)C_(2)x^(5-2)(x^(log_(10)x))^(2) = 10000000` (given)
or `x^(3+2log_(10)x) = 10^(5)`
Taking logarithm of both sides, we get
`(3+2log_(10)x)log_(10)x = 5`
or `2y^(2) + 3 y - 5 = 0`, where `log_(10) x = y`
or `(y-1) (2y+5) = 0 y = 1` or `-5//2`
or `log_(10)x=1` or `-5//2`
`:. x = 10^(1) = 10` or `10^(-5//2)`
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