Question

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

• A. $x^2+x-1=0$
• B. $x^2+x-2=0$
• C. $x^2+x+1=0$
• D. $x^2+x+2=0$

Answer 1

Answer: (D)

$x^2+x+2=0$

$a=cos\left(2\pi /7\right)+isin\left(2\pi /7\right)$
$⟹a^7=\left[cos\right(2\pi /7)+isin(2\pi /7){]}^7$
$=cos2\pi +isin2\pi =1$ (1)
$S=\alpha +\beta =(a+a^2+a^4)+(a^3+a^5+a^6)$
$=a+a^2+a^3+a^4+a^5+a^6=\frac{a(1-a^6)}{1-a}$
$=\frac{a-a^7}{1-a}=\frac{a-1}{1-a}=-1$ (2)
$P=\alpha \beta =(a+a^2+a^4)(a^3+a^5+a^6)$
$=a^4+a^6+a^7+a^5+a^7+a^8+a^7+a^9+a^{10}$
$=a^4+a^6+1+a^5+1+a+1+a^2+a^3$ [From Eq. (1)]
$=3+(a+a^2+a^3+a^4+a^5+a^6)$
$=3+S=3-1=2$ [From Eq. (2)]
Therefore, the required equation is
$x^2-Sx+P=0$
$⟹x^2+x+2=0$
Ans: D