Solve the given two equations and select the correct answer from the given options.
$I.$ $2x^2 - 3x - 35 = 0$
$II.$ $y^2 - 7y + 6 = 0$

Solve the given two equations and select the correct answer from the given options.
$I.$ $13x - 21 = 200 - 4x$
$II.$ $y = \sqrt[3]{2197}$

Consider the equation $x^2+x-n = 0$,where $n \in N$ and $n \in [5, 100]$. The total number of different values of $n$ such that the given equation has integral roots is:

Solve the given two equations and select the correct answer from the given options.
$I.$ $x = \sqrt{172}$
$II.$ $y^{2} - 29y + 210 = 0$

In a cubic equation,the coefficient of $x^2$ is zero and the remaining coefficients are real. If one root is $\alpha = 3 + 4i$ and the remaining roots are $\beta$ and $\gamma$,then find the value of $\alpha \beta \gamma$.

If $\alpha, \beta$ are the roots of the equation $x^2 - px + q = 0$,then the quadratic equation whose roots are $(\alpha^2 - \beta^2)(\alpha^3 - \beta^3)$ and $\alpha^3\beta^2 + \alpha^2\beta^3$ is (where $S = p[p^4 - 5p^2q + 5q^2]$ and $P = p^2q^2(p^4 - 5p^2q + 4q^2)$).