Solve the given two equations and select the correct answer from the given options.
$I. \quad 4x + 7y = 209$
$II. \quad 12x - 14y = -38$

Solve the given two equations and select the correct option.
$I.$ $x^{2}-8 \sqrt{3} x+45=0$
$II.$ $y^{2}-\sqrt{2} y-24=0$

If $\alpha$ and $\beta$ are the roots of $9x^2 + 6x + 1 = 0$,then the equation with the roots $\frac{1}{\alpha}$ and $\frac{1}{\beta}$ is

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

Let $\lambda \neq 0$ be in $\mathbb{R}$. If $\alpha$ and $\beta$ are the roots of the equation $x^{2}-x+2 \lambda=0$ and $\alpha$ and $\gamma$ are the roots of the equation $3x^{2}-10x+27 \lambda=0$,then $\frac{\beta \gamma}{\lambda}$ is equal to:

$\alpha_r$ and $\beta_r$ $(\alpha_r < \beta_r)$ are the roots of $x^2 - r^2(r + 1)x + r^5 = 0$. The value of $\sum_{r=1}^n (3\alpha_r + 2\beta_r)$ is: