The quadratic equations $x^2 - 6x + a = 0$ and $x^2 - cx + 6 = 0$ have one root in common. The other roots of the first and second equations are integers in the ratio $4 : 3$. Then the common root is:

If $3$ is a root of the equation ${x^2} + kx - 24 = 0,$ then $3$ is also a root of which of the following equations?

If $\alpha$ and $\beta$ are the roots of the equation ${x^2} - (1 + {n^2})x + \frac{1}{2}(1 + {n^2} + {n^4}) = 0$,then the value of ${\alpha ^2} + {\beta ^2}$ is:

Solve the given two equations and select the correct answer from the given options.
$I.$ $x = \sqrt{625}$
$II.$ $y = \sqrt{676}$

The number of solutions of $\log_4(x - 1) = \log_2(x - 3)$ is:

$Quantity \, 1:$ Original duration of flight. In a flight of $3000 \, km$,an aircraft was slowed down by bad weather. Its average speed for the trip was reduced by $100 \, km/hr$ and the time increased by $1 \, hour$.
$Quantity \, 2:$ Usual time of a man who,when walking at $\frac{3}{4}$ of his usual pace,reaches his office $20 \, minutes$ late.