The roots of the equation $ax^2 + bx + c = 0$ will be reciprocal if

Number of equations of the form $ax^2 + bx + 1 = 0$ having real roots,where $a, b \in \{1, 2, 3, 4\}$ is-

If $\alpha$ and $\beta$ are roots of the equation $x^2 - 4\sqrt{2}kx + 2e^{4\ln k} - 1 = 0$ for some $k$, and $\alpha^2 + \beta^2 = 66$, then $\alpha^3 + \beta^3$ is equal to: (in $\sqrt{2}$)

$P$ can complete a piece of work in $16$ days,which $Q$ can complete in $32$ days. If $P$ and $Q$ work on alternate days,find the relationship between Quantity $I$ (Total days taken if $P$ starts) and Quantity $II$ (Total days taken if $Q$ starts).

$I. \quad p^{2}-18 p+77=0$
$II. \quad 3 q^{2}-25 q+28=0$
$Quantity \, 1$: Value of $p$
$Quantity \, 2$: Value of $q$

Let $p, q, r \in \mathbb{R}$ and $r > p > 0.$ If the quadratic equation $px^2 + qx + r = 0$ has two complex roots $\alpha$ and $\beta,$ then $|\alpha| + |\beta|$ is: