Two real numbers $\alpha$ and $\beta$ are such that $\alpha + \beta = 3$ and $|\alpha - \beta| = 4$. Then $\alpha$ and $\beta$ are the roots of the quadratic equation:

Roots of $ax^2 + b = 0$ are real and distinct if

If the roots of the equation $5x^2 - 7x + k = 0$ are reciprocals of each other,then the value of $k$ is:

If $x = \sqrt{5} + 2$,then the value of $\frac{2x^2 - 3x - 2}{3x^2 - 4x - 3}$ is equal to

$A$ 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 $14 \, minutes.$

Let $\alpha$ and $\beta$ be the roots of $x^2 - \sqrt{2}x + 1 = 0$. Then the value of $\alpha^{50} + \beta^{50}$ is: