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Let S be the set of all real roots of th...

Let S be the set of all real roots of the equation `4^x (4^x -1)+2=|4^x -1|+|4^x-2)`.Then S:

A

is an empty set

B

contains at least four elements

C

contains exactly two elements

D

is a singleton

Text Solution

Verified by Experts

The correct Answer is:
D
Let `4^X = t, t gt 0`
` t (t-1)+2 = |t-t|+|t+2|," "t^2 - t+2 =|t-1|+|t-2|`
Case : `t lt 1`
`t^2 - t + 2 = 1 -t + 2 -t," "t^2 + 2 = 3-t`
`t^2 + t -1=0," "t =(-1 +-sqrt5)/2`
`t = (sqrt5 -1)/2` is only acceptable
Case II : `1 le t lt 2 `
`t^2 - t +2 = t -1 + 2-t," "t^2 - t +1 =0`
`D lt 0` no real solution
Case III : ` t gt 2`
`t^2 - t + 2 = t -1 -2`
`t^2 - 3t " "5 = 0 rArr" "D lt 0` no real solution
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