<p>If <span class="math">z^2 + z + 1 = 0</span>, where <span class="math">z</span> is a complex number, then find the value of <span class="math">\left(z + \frac{1}{z}\right)^2 + \left(z^2 + \frac{1}{z^2}\right)^2 + \left(z^3 + \frac{1}{z^3}\right)^2 + \cdots + \left(z^6 + \frac{1}{z^6}\right)^2</span></p>
Step-by-Step Solution
Key Concept: Recognize that z satisfies the cube roots of unity equation, use the property that z³ = 1 and 1/z = z² to simplify the expression.
<p><strong>Step 1:</strong> From <span class="math">z^2 + z + 1 = 0</span>, we get</p><p><span class="math">z = \frac{-1 \pm \sqrt{1-4}}{2} = \frac{-1 \pm \sqrt{3}i}{2}</span></p><p><strong>Step 2:</strong> These are the complex cube roots of unity, so <span class="math">z = w</span> or <span class="math">z = w^2</span>, where <span class="math">w = \frac{-1 + \sqrt{3}i}{2}</span> and <span class="math">w^2 = \frac{-1 - \sqrt{3}i}{2}</span></p><p><strong>Step 3:</strong> Since <span class="math">z^2 + z + 1 = 0</span>, we have <span class="math">z^3 = 1</span>, and <span class="math">\frac{1}{z} = z^2</span></p><p><strong>Step 4:</strong> The series simplifies using the periodic nature of powers of <span class="math">z</span> with period 3, and evaluating each pair of terms yields the sum of 12.</p><p>∴ Answer is (b) 12.</p>
Correct Answer: B