Question

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

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

Answer 1

The correct option is C is a singleton.

$3^x(3^x-1)+2=|3^x-1|+|3^x-2|$

Let $3^x=t$

Then $t(t-1)+2=|t-1|+|t-2|$
$\Rightarrow t^2-t+2=|t-1|+|t-2|$

We plot $t^2-t+2$ and $|t-1|+|t-2|$
As $t=3^x$ is always positive, therefore only positive values of $t$ will be the solution.



The graphs intersect for only positive value of $t$.

So, there is single value of $x$ given by $x={\log }_{3}t$

Therefore, we have only one solution.