The roots of the equation $a(x^2 + 1) - (a^2 + 1)x = 0$ are

$PBA$ and $PDC$ are two secants. $AD$ is the diameter of the circle with center at $O$. $\angle A = 40^{\circ}$,$\angle P = 20^{\circ}$. Find the relationship between Quantity $I$ $(\angle DBC)$ and Quantity $II$ $(\angle ADB)$.

If $\alpha, \beta$ are the roots of the equation $a x^{2}+b x+b=0,$ then $\sqrt{\frac{\alpha}{\beta}}+\sqrt{\frac{\beta}{\alpha}}+\sqrt{\frac{b}{a}}=$

If one of the roots of the equations $x^2 + ax + b = 0$ and $x^2 + bx + a = 0$ is coincident,then the numerical value of $(a + b)$ is

If $\lambda \in R$ is such that the sum of the cubes of the roots of the equation $x^2 + (2 - \lambda) x + (10 - \lambda) = 0$ is minimum,then the magnitude of the difference of the roots of this equation is

Find the value of $k$ so that the sum of the roots of the equation $3 x^{2}+(2 k+1) x-k-5=0$ is equal to the product of the roots: