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 $\mathrm{S}$
is an empty set.
contains at least four elements.
contains exactly two elements
is a singleton
If $a < 0$ then the inequality $a{x^2} - 2x + 4 > 0$ has the solution represented by
Equation $\frac{3}{{x - {a^3}}} + \frac{5}{{x - {a^5}}} + \frac{7}{{x - {a^7}}} = 0,a > 1$ has
Consider a three-digit number with the following properties:
$I$. If its digits in units place and tens place are interchanged, the number increases by $36$ ;
$II.$ If its digits in units place and hundreds place are interchanged, the number decreases by $198 .$
Now, suppose that the digits in tens place and hundreds place are interchanged. Then, the number
The number of positive integers $x$ satisfying the equation $\frac{1}{x}+\frac{1}{x+1}+\frac{1}{x+2}=\frac{13}{2}$ is.
The sum of all the roots of the equation $\left|x^2-8 x+15\right|-2 x+7=0$ is: