Question

If $\alpha =\cos \frac{2\pi }{7}+i\sin \frac{2\pi }{7}$ and $p=\alpha +{\alpha }^2+{\alpha }^4$ and $q={\alpha }^3+{\alpha }^5+{\alpha }^6,$ then find the equation whose roots are $p$ and $q$.

• A. $x^2-x+2=0$ is the required equation.
• B. $x^2+x-2=0$ is the required equation.
• C. $x^2+x+2=0$ is the required equation.
• D. $x^2-x-2=0$ is the required equation.

Answer 1

The correct option is C $x^2+x+2=0$ is the required equation.

${\alpha }^7=\cos 2\pi +i\sin 2\pi =1$
$S=\alpha +{\alpha }^2+{\alpha }^3+\cdots {\alpha }^6=\frac{\alpha (1-{\alpha }^5)}{1-\alpha }$
$S=\frac{\alpha -{\alpha }^7}{1-\alpha }=\frac{\alpha -1}{1-\alpha }=-1$ ...(1)

Product, $pq=$ Nine terms

$=({\alpha }^4+{\alpha }^6+{\alpha }^7)+({\alpha }^5+{\alpha }^7+{\alpha }^8)+({\alpha }^7+{\alpha }^9+{\alpha }^{10})$

$=({\alpha }^4+{\alpha }^5+1)+({\alpha }^5+1+\alpha )+(1+{\alpha }^2+{\alpha }^3)$

$=3+(\alpha +{\alpha }^2+{\alpha }^3+\cdots {\alpha }^6)=3+S$

$=3-1=2$ ------ from (1)

$\therefore x^2-Sx+P=0$
$x^2+x+2=0$ is the required equation.
Ans: C