If $\frac{2x}{2x^2 + 5x + 2} > \frac{1}{x + 1}$,then

If $\alpha$ and $\beta$ are the roots of the equation $x^{2}+3ax+2a^{2}=0$ and if $\alpha^{2}+\beta^{2}=5$,find the value of $a$.

The two roots of the equation $x^3 - 9x^2 + 14x + 24 = 0$ are in the ratio $3 : 2$. The roots are:

If $\left(\frac{p^{-1} q^{2}}{p^{3} q^{-2}}\right)^{\frac{1}{3}}+\left(\frac{p^{5} q^{-3}}{p^{-2} q^{3}}\right)^{\frac{1}{3}}=p^{a} q^{b},$ then the value of $a+b,$ where $p$ and $q$ are different positive positive primes,is

If $x$ is real and $k = \frac{x^2 - x + 1}{x^2 + x + 1}$,then

If $\alpha$ and $\beta$ are the roots of the equation $x^2 - 4x + 1 = 0$,the value of $\alpha^3 + \beta^3$ is