If the set of all values of $a$, for which the equation $5 x ^3-15 x - a =0$ has three distinct real roots, is the interval $(\alpha, \beta)$, then $\beta-2 \alpha$ is equal to _____

The real roots of the equation ${x^2} + 5|x| + \,\,4 = 0$ are

The number of real values of $x$ for which the equality $\left| {\,3{x^2} + 12x + 6\,} \right| = 5x + 16$ holds good is

The number of real roots of the polynomial equation $x^4-x^2+2 x-1=0$ is

If $x$ is real, then the value of $\frac{{{x^2} + 34x - 71}}{{{x^2} + 2x - 7}}$ does not lie between

If $\alpha ,\beta$ are the roots of $x^2 -ax + b = 0$ and if $\alpha^n + \beta^n = V_n$, then -