For the equation $|{x^2}| + |x| - 6 = 0$, the roots are
One and only one real number
Real with sum one
Real with sum zero
Real with product zero
The sum of the roots of the equation, ${x^2}\, + \,\left| {2x - 3} \right|\, - \,4\, = \,0,$ is
The set of all real numbers $x$ for which ${x^2} - |x + 2| + x > 0,$ is
If $\alpha ,\beta,\gamma$ are the roots of equation $x^3 + 2x -5 = 0$ and if equation $x^3 + bx^2 + cx + d = 0$ has roots $2 \alpha + 1, 2 \beta + 1, 2 \gamma + 1$ , then value of $|b + c + d|$ is (where $b,c,d$ are coprime)-
Suppose $a$ is a positive real number such that $a^5-a^3+a=2$. Then,
If $\alpha, \beta$ are the roots of the equation, $x^2-x-1=0$ and $S_n=2023 \alpha^n+2024 \beta^n$, then