Solve the given two equations and select the correct answer from the given options.
$I.$ $x^{2}+13x+40=0$
$II.$ $y^{2}+7y+10=0$

Babu starts from his house at a certain time with a certain speed to pick up his girlfriend from the office at $5:00 \, PM$. One day,his girlfriend left the office at $3:00 \, PM$ and started walking home at a speed of $40 \, km/h$. She met Babu on the way,who had left his home at his usual time. They reached home $40 \, min$ earlier than their usual time. If Quantity $I$ is the speed of Babu and Quantity $II$ is $120 \, km/h$,compare the two quantities.

Given that $\tan \alpha$ and $\tan \beta$ are the roots of the equation $x^2 - px + q = 0$,then the value of $\sin^2(\alpha + \beta)$ is:

Let $\alpha$ and $\beta$ be the roots of the equation $x^2 - 6x - 2 = 0$. If $a_n = \alpha^n - \beta^n$ for $n \ge 1$,then the value of $\frac{a_{10} - 2a_8}{2a_9}$ is equal to:

Solve the given two equations and select the correct option.

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: