$I. 6x^2 + 41x + 63 = 0$
$II. 4y^2 + 8y + 3 = 0$

Find the value of $(a+b+c)^{3}-3(b+c)(c+a)(a+b)$ if $a=5, b=3$ and $c=2$.

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:

The value of $a$ for which one root of the quadratic equation $(a^2 - 5a + 3)x^2 + (3a - 1)x + 2 = 0$ is twice as large as the other,is

$A$ man who swims $48 \, m/min$ in still water,swims $200 \, m$ against the current and $200 \, m$ with the current. The difference between the time taken by him against the stream and with the stream is $10 \, minutes.$
$Quantity \, 1$: Speed of current.
$Quantity \, 2$: Speed of a man who completes $3$ rounds of a circular path of radius $49 \, m$ in $14 \, minutes.$

If $\alpha, \beta$ are the roots of the equation $x^{2}-5x+6=0$,construct a quadratic equation whose roots are $\frac{1}{\alpha}, \frac{1}{\beta}$.