Solve the given two equations and select the correct answer from the given options.
$I.$ $x = \sqrt[3]{2197}$
$II.$ $y^2 = 169$

Let $\alpha, \alpha^2$ be the roots of $x^2 + x + 1 = 0$. Then the equation whose roots are $\alpha^{31}, \alpha^{62}$ 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 $14 \, minutes.$

The number of solutions of the equation $2^x = x^2$ is

If $x-\sqrt{3}-\sqrt{2}=0$ and $y-\sqrt{3}+\sqrt{2}=0,$ then the value of $(x^{3}-20\sqrt{2})-(y^{3}+2\sqrt{2})$ is:

Let $p(x)$ be a quadratic polynomial such that $p(0) = 1$. If $p(x)$ leaves a remainder of $4$ when divided by $x - 1$ and a remainder of $6$ when divided by $x + 1$,then: