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

Options

A. is an empty set
B. contains at least four elements
C. conatains exactly exactly two elements
D. is a singleton

Answer

D

Explanation

Let `4^(x)= t ,t gt 0`
`t(t-1)+2=|t-1|+|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^(2)+2=3-t`
`t^(2)+t-1=0, " "t=(-1pmsqrt5)/2`
`t = (sqrt5-1)/2` is only acceptable
Case II : `1 lt t lt 2`
`t^(2)-t+2=t-1=0, " "t^(2)-t+1=0`
`D lt 0` no real solution
`t^(2)-t+2 = t-1+t-2`
`t^(2)-3t 5=0 implies D lt 0` no real solution