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

Wheels of diameters $7 \,cm$ and $14 \,cm$ start rolling simultaneously from $X$ and $Y$,which are $1990.50 \,cm$ apart,towards each other in opposite directions. Both of them make the same number of revolutions per second. Both of them meet after $10 \,s$.
Quantity $1$: Speed of the smaller wheel.
Quantity $2$: $21 \pi \,cm/s$.

If $\alpha$ and $\beta$ are roots of the equation $x^2 + px + \frac{3p}{4} = 0,$ such that $|\alpha - \beta| = \sqrt{10},$ then $p$ belongs to the set

Let $x = \frac{\sqrt{13} + \sqrt{11}}{\sqrt{13} - \sqrt{11}}$ and $y = \frac{1}{x}$. Then the value of $3x^2 - 5xy + 3y^2$ is:

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

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.