If ${x^2} + px + 1$ is a factor of the expression $a{x^3} + bx + c$, then

Let $a$ be the largest real root and $b$ be the smallest real root of the polynomial equation $x^6-6 x^5+15 x^4-20 x^3+15 x^2-6 x+1=0$ Then $\frac{a^2+b^2}{a+b+1}$ is

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

If $\alpha $, $\beta$, $\gamma$  are roots of ${x^3} - 2{x^2} + 3x - 2 = 0$ , then the value of$\left( {\frac{{\alpha \beta }}{{\alpha  + \beta }} + \frac{{\alpha \gamma }}{{\alpha  + \gamma }} + \frac{{\beta \gamma }}{{\beta  + \gamma }}} \right)$ is

Let $\alpha ,\beta $ be the roots of ${x^2} + (3 - \lambda )x - \lambda = 0.$ The value of $\lambda $ for which ${\alpha ^2} + {\beta ^2}$ is minimum, is

Number of natural solutions of the equation $x_1 + x_2 = 100$ , such that $x_1$ and $x_2$ are not multiple of $5$