If one root of the equation $ax^2 + bx + c = 0$ is the square of the other,then $a(c - b)^3 = cX$,where $X$ is

Construct a quadratic equation whose roots are $\sqrt{2}$ and $2 \sqrt{2}$.

If $m = -4$ and $n = -2$,then the value of $m^{3} - 3m^{2} + 3m + 3n + 3n^{2} + n^{3}$ is

Let $r_1, r_2, r_3$ be the roots of the equation $x^3 - 2x^2 + 4x + 5074 = 0$. Then the value of $(r_1 + 2)(r_2 + 2)(r_3 + 2)$ is:

The largest possible right circular cylinder is cut out from a wooden cube of edge $7 \, cm$.
$Quantity \, I$: Volume of the cube left over after cutting out the cylinder.
$Quantity \, II$: Surface area of the cube remaining after cutting out the cylinder.
$Note$: Compare the magnitudes of both quantities.

If $\alpha, \beta$ are the roots of the equation $x^2 - px + q = 0$,then the quadratic equation whose roots are $(\alpha^2 - \beta^2)(\alpha^3 - \beta^3)$ and $\alpha^3\beta^2 + \alpha^2\beta^3$ is (where $S = p[p^4 - 5p^2q + 5q^2]$ and $P = p^2q^2(p^4 - 5p^2q + 4q^2)$).