If $\alpha, \beta$ are the roots of the quadratic equation $x^{2}-8x+k=0$,find the value of $k$ such that $\alpha^{2}+\beta^{2}=40$.

Solve the given two equations and select the correct answer from the given options.
$I. \quad 6x^{2} - 49x + 99 = 0$
$II. \quad 5y^{2} + 17y + 14 = 0$

If $a \in R$ and the equation $-3(x - [x])^2 + 2(x - [x]) + a^2 = 0$ (where $[x]$ denotes the greatest integer $\leq x$) has no integral solution,then all possible values of $a$ lie in the interval

If $\alpha$ and $\beta$ are the roots of the equation ${x^2} - a(x + 1) - b = 0$,then $(\alpha + 1)(\beta + 1) = $

Let $p(x)$ be a quadratic polynomial such that $p(0) = 1$. If $p(x)$ leaves a remainder of $4$ when divided by $x - 1$ and a remainder of $6$ when divided by $x + 1$,then:

If $\alpha$ and $\beta$ are the roots of the equation $4x^2 + 3x + 7 = 0$,then $\frac{1}{\alpha} + \frac{1}{\beta} =$