When $a = -5, b = -6$ and $c = 10$,find the value of $\frac{a^{3} + b^{3} + c^{3} - 3abc}{ab + bc + ca - a^{2} - b^{2} - c^{2}}$.

Let $p, q, r \in \mathbb{R}$ and $r > p > 0.$ If the quadratic equation $px^2 + qx + r = 0$ has two complex roots $\alpha$ and $\beta,$ then $|\alpha| + |\beta|$ is:

The equation $x^{2}-p x+q=0, p, q \in R$ has no real root if:

If $4a - \frac{4}{a} + 3 = 0$,then the value of $a^{3} - \frac{1}{a^{3}} + 3$ is equal to?

Quantity $I$: Overall profit percentage if the cost prices of two shirts are equal. One shirt is sold for $20 \%$ profit and the other is sold for $10 \%$ loss.
Quantity $II$: Profit percentage made in selling each meter if the profit made in selling $20 \, m$ of a cloth equals the cost price of $5 \, m$ of that cloth.

If $\alpha$ and $\beta$ are the roots of the equation $x^{2}+3ax+2a^{2}=0$ and if $\alpha^{2}+\beta^{2}=5$,find the value of $a$.