If $x+y=3$ and $xy=2$,then find the value of $x^3-y^3$.

The roots of the equation $(a^2 + b^2)t^2 - 2(ac + bd)t + (c^2 + d^2) = 0$ are equal,then

Find the value of $x^{3}-\frac{1}{x^{3}}$ when $x-\frac{1}{x}=a$.

Solve the given two equations and select the correct answer from the given options.
$I.$ $x^{2}-24x+144=0$
$II.$ $y^{2}-26y+169=0$

The equation of the smallest degree with real coefficients having $1 + i$ as one of the roots is

If $\alpha$ and $\beta$ are the roots of the equation $375x^2 - 25x - 2 = 0$,then $\lim_{n \to \infty} \sum_{r=1}^n \alpha^r + \lim_{n \to \infty} \sum_{r=1}^n \beta^r$ is equal to