Question

If $\alpha ,\beta \in R-\left\{0\right\}$, are the roots of $x^2+px+q=0$ and $x^{2n}+p^n{x}^n+q^n=0$ and if $\frac{\alpha }{\beta },\frac{\beta }{\alpha }$ are the roots of $x^n+1+(x+1{)}^n=0$, then $n$

• A. must be an odd integer
• B. may be any integer
• C. must be an even integer
• D. cannot say anything

Answer 1

The correct option is C must be an even integer

Since, $\alpha$ and $\beta$ are the roots of $x^2+px+q=0$
$\Rightarrow \alpha +\beta =-p$ ........(1)
Since, $\alpha$ and $\beta$ are the roots of $x^{2n}+p^n{x}^n+q^n=0$
${\alpha }^{2n}+p^n{\alpha }^n+q^n=0$ .....(2)
${\beta }^{2n}+p^n{\beta }^n+q^n=0$ .....(3)
Subtracting (3) from (2), we get
${\alpha }^{2n}-{\beta }^{2n}+p^n({\alpha }^n-{\beta }^n)=0$
$\Rightarrow {\alpha }^n+{\beta }^n=-p^n$ ......(4)
Now, since, $\left(\frac{\alpha }{\beta }\right)$ is the root of $x^n+1+(x+1{)}^n=0$
${\left(\frac{\alpha }{\beta }\right)}^n+1+{(\frac{\alpha }{\beta }+1)}^n=0$
$\Rightarrow {\alpha }^n+{\beta }^n+(\alpha +\beta {)}^n=0$ .....(5)
Also, since, $\left(\frac{\beta }{\alpha }\right)$ is the root of $x^n+1+(x+1{)}^n=0$
${\left(\frac{\beta }{\alpha }\right)}^n+1+{(\frac{\beta }{\alpha }+1)}^n=0$
$\Rightarrow {\alpha }^n+{\beta }^n+(\alpha +\beta {)}^n=0$ ....(6)
So, (5) and (6) can be written as
$-p^n+(-p{)}^n=0$ .... (by (1) and (4))
which is possible only when $n$ is an even integer.