The equation $x^4-x^3-6x^2+4x+8=0$ has two equal roots. If $\alpha$ and $\beta$ are the other two roots of this equation,then $\alpha^2+\beta^2=$

If $m, n$ are the roots of the equation ${x^2} - x - 1 = 0$,then the value of $\frac{{\left( {1 + m{{\log }_e}3 + \frac{{{{(m{{\log }_e}3)}^2}}}{{2!}} + ...\infty } \right)\left( {1 + n{{\log }_e}3 + \frac{{{{(n{{\log }_e}3)}^2}}}{{2!}} + ...\infty } \right)}}{{\left( {1 + mn{{\log }_e}3 + \frac{{{{(mn{{\log }_e}3)}^2}}}{{2!}} + ...\infty } \right)}}$ is:

The least positive value of $a$ for which the equation $2x^{2} + (a-10)x + \frac{33}{2} = 2a$ has real roots is

The sum of all the real values of $x$ satisfying the equation $(x^2-7x+11)^{x^2-6x-7}=1$ is

Consider the cubic equation $x^3+ax^2+bx+c=0$ where $a, b, c$ are real numbers. Which of the following statements is correct?

Find the product of all real roots of the equation $|x|^{6/5} - 26|x|^{3/5} - 27 = 0$.