Question

If $\alpha$ and $\beta$ are the roots of the equation $l{x}^2+mx+n=0$, then the equation, whose roots are ${\alpha }^3\beta$ and $\alpha {\beta }^3$, is

• A. $l^4{x}^2–nl(m^2–2nl)x+n^4=0$
• B. $l^4{x}^2+nl(m^2–2nl)x+n^4=0$
• C. $l^4{x}^2+nl(m^2–2nl)x–n^4=0$
• D. $l^4{x}^2-nl(m^2+2nl)x+n^4=0$

Answer 1

The correct option is A

$l^4{x}^2–nl(m^2–2nl)x+n^4=0$

Explanation for the correct option:

Step1. Expressing the given equation $\mathit{l}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{}\mathit{m}\mathit{x}\mathbf{+}\mathbf{}\mathit{n}\mathbf{}\mathbf{=}\mathbf{}\mathbf{0}$:

Given that $\alpha$ and $\beta$ are the roots of the equation $l{x}^2+mx+n=0$

$\Rightarrow \alpha +\beta =\frac{-m}{l},\alpha .\beta =\frac{n}{l}$

$$\begin{gathered} {\alpha }^2+{\beta }^2={(\alpha +\beta )}^2–2\alpha \beta \\ \Rightarrow {\alpha }^2+{\beta }^2={\left(\frac{-m}{l}\right)}^2–2\left(\frac{n}{l}\right) \\ \Rightarrow {\alpha }^2+{\beta }^2=\frac{m^2}{l^2}-\frac{2n}{l}........\left(i\right) \end{gathered}$$

Step2. Find the equation :

The required equation is

${\mathit{x}}^{\mathbf{2}}\mathbf{-}\mathbf{(}\mathit{s}\mathit{u}\mathit{m}\mathbf{}\mathit{o}\mathit{f}\mathbf{}\mathit{t}\mathit{h}\mathit{e}\mathbf{}\mathit{r}\mathit{o}\mathit{o}\mathit{t}\mathit{s}\mathbf{)}\mathit{x}\mathbf{-}\mathbf{(}\mathit{m}\mathit{u}\mathit{l}\mathit{t}\mathit{i}\mathit{p}\mathit{l}\mathit{i}\mathit{c}\mathit{a}\mathit{t}\mathit{i}\mathit{o}\mathit{n}\mathbf{}\mathit{o}\mathit{f}\mathbf{}\mathit{t}\mathit{h}\mathit{e}\mathbf{}\mathit{r}\mathit{o}\mathit{o}\mathit{t}\mathit{s}\mathbf{)}\mathbf{=}\mathbf{0}$

$$\begin{gathered} x^2–({\alpha }^3\beta +\alpha {\beta }^3)x+{\alpha }^3\beta \left(\alpha {\beta }^3\right)=0 \\ \Rightarrow x^2–\alpha \beta ({\alpha }^2+{\beta }^2)x+{\left(\alpha \beta \right)}^4=0 \\ \Rightarrow x^2–\left(\frac{n}{l}\right)\left(\frac{m^2}{l^2}-\frac{2n}{l}\right)x+{\left(\frac{n}{l}\right)}^4=0\mathbf{}\mathbf{}\mathbf{\left(}\mathit{f}\mathit{r}\mathit{o}\mathit{m}\mathbf{\right(}\mathit{i}\mathbf{\left)}\mathbf{\right)} \\ \Rightarrow l^4{x}^2–nl(m^2–2nl)x+n^4=0 \end{gathered}$$

Hence, the correct option is A.