If $x$ is real, then the value of ${x^2} - 6x + 13$ will not be less than
$4$
$6$
$7$
$8$
If $\alpha ,\beta ,\gamma $are the roots of the equation ${x^3} + x + 1 = 0$, then the value of ${\alpha ^3}{\beta ^3}{\gamma ^3}$
If $\alpha, \beta$ are the roots of the equation, $x^2-x-1=0$ and $S_n=2023 \alpha^n+2024 \beta^n$, then
The solution set of the equation $pq{x^2} - {(p + q)^2}x + {(p + q)^2} = 0$ is
The number of real solutions of the equation $|{x^2} + 4x + 3| + 2x + 5 = 0 $are
If $a$ and $b$ are the roots of equation $x^2-7 x-1=0$, then the value of $\frac{a^{21}+b^{21}+a^{17}+b^{17}}{a^{19}+b^{19}}$ is equal to $........$.