Solve the given two equations and select the correct answer from the given options.
$I.$ $4x^2 - 43x + 105 = 0$
$II.$ $7y^2 - 29y + 30 = 0$

If the equation $\frac{1}{x} + \frac{1}{x - 1} + \frac{1}{x - 2} = 3x^3$ has $k$ real roots,then $k$ is equal to -

Let $\alpha$ and $\beta$ be the roots of the quadratic equation $x^2 \sin \theta - x(\sin \theta \cos \theta + 1) + \cos \theta = 0$ where $0 < \theta < 45^\circ$,and $\alpha < \beta$. Then $\sum_{n=0}^\infty (\alpha^n + \frac{(-1)^n}{\beta^n})$ is equal to

Roots of $2 k x^{2}+5 k x+2=0$ are equal if $k$ is equal to

Let $a, b, c \in R$ and $\alpha, \beta$ be the real roots of the equation $ax^2 + bx + c = 0$. If $a < 0, b > 0, c > 0$ and $\alpha < \beta$,then:

Sum of all distinct integral values of $\alpha$ such that the equation $x^2 - \alpha x + \alpha + 1 = 0$ has integral roots,is equal to-