If a= cos (2 π/7)+i sin (2 π/7) then the quadratic equation whose roots are α=a+a2+a4 and β=a3+a5+ a6 is

Question

If a= cos (2 π/7)+i sin (2 π/7) then the quadratic equation whose roots are α=a+a2+a4 and β=a3+a5+ a6 is (A) x2+x-2=0 (B) x2-x+2=0 (C) x2-x-2=0 (D) x2+x+2=0

Tardigrade Admin · Accepted Answer

Sum of roots =a+a2+a3+a4+a5+a6=-1 product of root =3 a7+(a+a2+a3+a4+a5+a6)=3-1=2 ⇒ quadratic equation is x2+x+2=0

Q. If $a = cos \frac{2 π}{7} + i sin \frac{2 π}{7}$ then the quadratic equation whose roots are $α = a + a^{2} + a^{4}$ and $β = a^{3} + a^{5} +$ $a^{6}$ is

$x^{2} + x - 2 = 0$

$x^{2} - x + 2 = 0$

$x^{2} - x - 2 = 0$

$x^{2} + x + 2 = 0$

Sum of roots $= a + a^{2} + a^{3} + a^{4} + a^{5} + a^{6} = - 1$
product of root $= 3 a^{7} + (a + a^{2} + a^{3} + a^{4} + a^{5} + a^{6}) = 3 - 1 = 2$
$\Rightarrow$ quadratic equation is $x^{2} + x + 2 = 0$

Q. If a=cos72π​+isin72π​ then the quadratic equation whose roots are α=a+a2+a4 and β=a3+a5+ a6 is

x2+x−2=0

x2−x+2=0

x2−x−2=0

x2+x+2=0