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

Let $a \ne b, c \ne 0$. If the equations $x^2 + ax + bc = 0$ and $x^2 + bx + ac = 0$ have a common root,then:
Statement $-1$: The equation of the other roots is $x^2 + cx + ab = 0$.
Statement $-2$: $a + b + c = 0$.

$P$ 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).

Solve the given two equations and select the correct answer from the given options.
$I.$ $(289)^{\frac{1}{2}} x - \sqrt{324} = 203$
$II.$ $(484)^{\frac{1}{2}} y - \sqrt{225} = 183$

If $\alpha$ and $\beta$ are the roots of $9x^2 + 6x + 1 = 0$,then the equation with the roots $\frac{1}{\alpha}$ and $\frac{1}{\beta}$ is

If $x=a(b-c), y=b(c-a)$ and $z=c(a-b),$ then $\left(\frac{x}{a}\right)^{3}+\left(\frac{y}{b}\right)^{3}+\left(\frac{z}{c}\right)^{3}=?$