Let $S$ be the set of all possible integral values of $\lambda$ in the interval $(-3,7)$ for which the roots of the quadratic equation $\lambda x^2+13 x+7=0$ are all rational numbers. Then the sum of the elements in $S$ is

$\alpha$ is the maximum value of $1-2 x-5 x^2$ and $\beta$ is the minimum value of $x^2-2 x+r$. If $5 \alpha x^2+\beta x+6>0$ for all real values $x$, then the interval in which $r$ lies is

For the equation $x^4+x^3-4 x^2+x-1=0$ the ratio of the sum of the squares of all the roots to the product of the distinct roots is

If $\alpha_1, \beta_1, \gamma_1, \delta_1$ are the roots of the equation $a x^4+b x^3+c x^2+d x+e=0$ and $\alpha_2, \beta_2, \gamma_2, \delta_2$ are the roots of the equation $e x^4+d x^3+c x^2+b x+a=0$ such that $0<\alpha_1<\beta_1<\gamma_1<\delta_1, 0<\alpha_2<\beta_2<\gamma_2<\delta_2$, $\alpha_1-\delta_2=2=\beta_1-\gamma_2 ; \gamma_1-\beta_2=\delta_1-\alpha_2=4$, then $a+b+c+d+e=$