Solve the given two equations and select the correct answer from the given options.
$I.$ $15 x^{2}-41 x+14=0$
$II.$ $2 y^{2}-13 y+20=0$

If $\alpha, \beta, \gamma$ are roots of $x^3 - 2x^2 + 3x - 2 = 0$,then the value of $\left( \frac{\alpha\beta}{\alpha + \beta} + \frac{\alpha\gamma}{\alpha + \gamma} + \frac{\beta\gamma}{\beta + \gamma} \right)$ is

If $x = a^{1/2} + a^{-1/2}$ and $y = a^{1/2} - a^{-1/2}$,then find the value of $(x^4 - x^2 y^2 - 1) + (y^4 - x^2 y^2 + 1)$.

If $\alpha$ and $\beta$ are the roots of the equation $x^2 - 6x + a = 0$ and satisfy the relation $3\alpha + 2\beta = 16$,then the value of $a$ is

Solve the given two equations and provide the correct answer from the given options.
$I.$ $4x^2 - 29x + 45 = 0$
$II.$ $3y^2 - 19y + 28 = 0$

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 $a$.