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. $x^2-x+2$
• B. $x^2+x-2$
• C. $x^2-x-2$
• D. $x^2+x+2$

Answer 1

The correct option is D

$x^2+x+2$

Explanation for correct option:

Step 1. Find the required quadratic equation:

Given, $a=\cos \frac{2\pi }{7}+i\sin \frac{2\pi }{7}$

$\Rightarrow$ $a^7=1$

Step 2. Find the sum of roots:

$\begin{array}{rcl}\alpha +\beta & =& a+a^2+a^4+a^3+a^5+a^6\\ & =& \frac{a(a^6-1)}{(a-1)}\\ & =& \frac{a^7-a}{a-1}\\ & =& \frac{1-a}{a-1}\\ & =& -1\end{array}$

Step 3. Find the product of roots:

$\begin{array}{rcl}\alpha \beta & =& (a+a^2+a^4)(a^3+a^5+a^6)\\ & =& a^4+a^6+a^7+a^5+a^7+a^8+a^7+a^9+a^{10}\\ & =& 2a^7+a^4+a^5+a^6+a^7+a^8+a^9+a^{10}\\ & =& 2+a^4(1-1)\\ & =& 2\end{array}$

Required equation is $x^2-(\alpha +\beta )x+\alpha \beta =0$

$\mathbf{\therefore }{\mathit{x}}^{\mathbf{2}}\mathbf{}\mathbf{+}\mathbf{}\mathit{x}\mathbf{}\mathbf{+}\mathbf{}\mathbf{2}\mathbf{}\mathbf{=}\mathbf{}\mathbf{0}$

Hence, Option ‘D’ is Correct.