Question

Construct an equation whose roots are $n^{th}$ powers of the roots of the equation $x^2-2x\cos \theta +1=0.$

• A. $x^2-2n\cos n\theta x+1=0$
• B. $x^2-2n\cos \theta x+1=0$
• C. $x^2-2\cos n\theta x+1=0$
• D. $x^2-2{\cos }^n\theta x+1=0$

Answer 1

The correct option is C $x^2-2\cos n\theta x+1=0$
We know, $\alpha =\cos \theta +i\sin \theta ,\beta =\cos \theta -i\sin \theta$
${\alpha }^n=\cos n\theta +i\sin n\theta ,$
${\beta }^n=\cos n\theta -i\sin n\theta$
$S=2\cos n\theta ,P=1\therefore x^2-Sx+P=0$
or $x^2-2\cos n\theta x+1=0$ is the required equation.
Ans: C