Let $\alpha$ and $\beta$ be the roots of the equation $x^{2}-x-1=0$. If $p_{k}=(\alpha)^{k}+(\beta)^{k}, k \geq 1$,then which one of the following statements is not true?

$Quantity$ $1$: Height of the tank if the volume of a cylindrical tank is $12320 \, cm^3$. Its radius and height are in the ratio of $7:10$ respectively. $Quantity$ $2$: Height of the kerosene level in a jar if a cone of radius $2 \, cm$ and height $3 \, cm$ is melted and recast into a cylinder of radius $2 \, cm$.

The value of $p$ so that both the roots of the equation $(p - 5)x^2 - 2px + (p - 4) = 0$ are positive,one is less than $2$ and other is lying between $2$ and $3$,lies in the interval:

If the equation $x^2 + \lambda x + \mu = 0$ has equal roots and one root of the equation $x^2 + \lambda x - 12 = 0$ is $2$,then $(\lambda, \mu) = $

If $\alpha, \beta, \gamma$ are roots of $x^3 - 2x^2 + 3x - 2 = 0$,then the value of $\left( \frac{\alpha\beta}{\alpha + \beta} + \frac{\alpha\gamma}{\alpha + \gamma} + \frac{\beta\gamma}{\beta + \gamma} \right)$ is

Solve the given system of three equations and select the correct relationship between $x, y,$ and $z$ from the given options.
$I. x + 3y + 4z = 96$
$II. 2x + 8z = 80$
$III. 2x + 6y = 120$