If $x+\frac{1}{x}=3,$ then the value of $x^{8}+\frac{1}{x^{8}}$ will be

If $\alpha$ is a root of the quadratic equation $x^2 + 6x - 2 = 0$,then another root $\beta$ is

$Quantity \, 1$: Value of $y$. $A$ vessel contains $2.5 \, \text{liters}$ of water and $10 \, \text{liters}$ of milk. $20 \%$ of the contents of the vessel are removed. To the remaining contents,$x$ liters of water are added to reverse the ratio of water and milk. Then $y$ liters of milk are added again to reverse the ratio of water and milk. $Quantity \, 2$: $120 \, \text{liters}$.

Solve the given two equations and select the correct option.
$I.$ $77 x^{2}+58 x+8=0$
$II.$ $42 y^{2}+59 y+20=0$

The roots of the quadratic equation $2x^2 + 3x + 1 = 0$ 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 $II$ $(\angle ADB)$.