Question

Let S be the set of all real roots of the equation, $3^x\left(3^x-1\right)+2=\left|3^x-1\right|+\left|3^x-2\right|.$ Then S :

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

Answer 1

Answer: (A)

Let $3^x=t;t>0$
$t\left(t-1\right)+2=|t-1|+|t-2|$
$t^2-t+2=|t-1|+|t-2|$
Case-I : $t<1<br/>t^2-t+2=1-t+2-t$
$t^2+2=3-t$
$t^2+t-1=0$
$t=\frac{-1\pm \sqrt{5}}{2}$
$t=\frac{\sqrt{5}-1}{2}$ is only acceptable
Case-II : $1\le t<2<br/>t^2-t+2=t-1+2-t$
$t^2-t+1=0$
$D<0$ no real solution
Case-III : $t\ge 2$
$t^2-t+2=t-1+t-2$
$t^2-3t5=0\Rightarrow D<0$ no real solution