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

The correct option is A

is a singleton

Explanation for the correct option :

Determining $S$:

Given, $3^x\left(3^x-1\right)+2=\left|3^x-1\right|+\left|3^x-2\right|$

Solving the equation,

Let $3^x=t$

Then, $t(t-1)+2=\left|t-1\right|+\left|t-2\right|$

$\Rightarrow t^2-t+2=\left|t-1\right|+\left|t-2\right|$

Now plot, $t^2-t+2$ and $\left|t-1\right|+\left|t-2\right|$

As $3^x$ is always positive, therefore only positive values of $t$ will be the solution.

Therefore, we have only one solution.

Hence, the correct option is (A).