Question

If $\alpha ,\beta$ are the roots of $x^2+px+q=0$ ans also of $x^{2n}+p^n{x}^n+q^n=0$, and if $\frac{\alpha }{\beta },\frac{\beta }{\alpha }$ are the roots of $x^n+1+{(x+1)}^n=0,$ then n must be an integer. True or false . True=1 False=0

Answer 1

Since $\alpha ,\beta$ are the roots of $x^2+px+q=0$
$\therefore \alpha +\beta =-p,\alpha \beta =q$ ...........(1)
Also $\alpha ,\beta$ are roots of $x^{2n}+p^n{x}^n+q^n=0$
$\therefore {\alpha }^n+{\beta }^n=-p^{n}and{\alpha }^n{\beta }^n=q^n$ ........(2)
Now, $\alpha /\beta$ is a root of $x^n+1+{(x+1)}^n=0,$
$\Rightarrow \frac{{\alpha }^n}{{\beta }^n}+1{(\frac{\alpha }{\beta }+1)}^2=0$
$\Rightarrow \frac{{\alpha }^n+{\beta }^n}{{\beta }^n}+\frac{{(\alpha +\beta )}^n}{{\beta }^n}=0$
$\Rightarrow ({\alpha }^n+{\beta }^n)+{(\alpha +\beta )}^n=0$
$\Rightarrow -p^n+{(-p)}^n=0$ {from (1) and(2) }
This is true only if n is an integer.
Ans: 1