Question

If $a=\cos \frac{2\pi }{7}+i\sin \frac{2\pi }{7}$, then the quadratic equation whose roots are $\alpha =a+a^2+a^4$ and $\beta =a^3+a^5+a^6$, is

• A. $x^2-x+2=0$
• B. $x^2+2x+2=0$
• C. $x^2+x+2=0$
• D. $x^2+x-2=0$

Answer 1

The correct option is C $x^2+x+2=0$

Given, $a=\cos \frac{2\pi }{7}+i\sin \frac{2\pi }{7}$
$\therefore a^7=\cos 2\pi +i\sin 2\pi$
$[\because e^{i\theta }=\cos \theta +i\sin \theta ]$
$=1$
Also, $\alpha =a+a^2+a^4,\beta =a^3+a^5+a^6$
then the sum of roots,
$S=\alpha +\beta =a+a^2+a^3+a^4+a^5+a^6$
$\Rightarrow S=\frac{a(1-a^6)}{1-a}=\frac{a-a^7}{1-a}$
$=\frac{a-1}{1-a}=-1[\because a^7=1]$
Product of the roots,
$P=\alpha \beta =(a+a^2+a^4)(a^3+a^5+a^6)$
$=a^4+a^5+1+a^6+1+a^2+1+a+a^3[\because a^7=1]$
$=3+(a+a^2+a^3+a^4+a^5+a^6)=3-1=2$
Hence, the required quadratic equation is
$x^2+x+2=0$.