Complex Numbers
Algebra of Complex Numbers
Grade Class 11
Question:
<p>Let \( a = \cos\dfrac{2\pi}{7} + i\sin\dfrac{2\pi}{7} \). The quadratic whose roots are \( a+a^2+a^4 \) and \( a^3+a^5+a^6 \) is:</p>
x^2 + x + 2 = 0
x^2 - x + 2 = 0
x^2 + x - 2 = 0
x^2 - x - 2 = 0
Step-by-Step Solution
Key Concept: Sum of roots = (a+a^2+a^3+a^4+a^5+a^6) = -1 (sum of all 7th roots of unity except 1). Product = (a+a^2+a^4)(a^3+a^5+a^6) = 2. Quadratic: x^2 + x + 2 = 0.
<p>Sum = $\sum_{k=1}^{6} a^k = -1$. Product: $(a+a^2+a^4)(a^3+a^5+a^6) = a^4+a^6+a^7+a^5+a^7+a^8+a^7+a^9+a^{10}$. Reducing mod 7: $= a^4+a^6+1+a^5+1+a+1+a^2+a^3 = 3+\sum_{k=1}^6 a^k = 3-1=2$. Quadratic: $x^2+x+2=0$. ✓</p>
Correct Answer: ABCD