Solve the given two equations and select the correct answer from the given options.
$I.$ $x - \sqrt{169} = 0$
$II.$ $y^2 - 169 = 0$

The value of $k$ for which the roots $\alpha, \beta$ of the equation $x^{2}-6x+k=0$ satisfy the relation $3\alpha+2\beta=20$ is:

The diagram shows the graph of $y = ax^2 + bx + c$. Then:

If $x+\frac{1}{y}=1$ and $y+\frac{1}{z}=1$,then find the value of $z+\frac{1}{x}$.

$Quantity \, 1$: 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. $Quantity \, 2$: $120 \, \text{liters}$.

If $1 - i$ is a root of the equation ${x^2} - ax + b = 0$, then $b = $