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 : $t < 1
t^{2} - t + 2 = 1 - t + 2 - t$
$t^{2} + 2 = 3 - t$
$t^{2} + t - 1 = 0$
$t = \frac{-1\pm\sqrt{5}}{2}$
$t = \frac{\sqrt{5}-1}{2}$ is only acceptable
Case-II : $1 \le t < 2
t^{2} - t + 2 = t - 1 + 2 - t$
$t^{2} - t + 1 = 0$
$D < 0$ no real solution
Case-III : $t \ge 2$
$t^{2} - t + 2 = t - 1 + t - 2$
$t^{2} - 3t 5 = 0 \Rightarrow D < 0$ no real solution