<p>Let <i>w = (√3 + i)/2</i> and <i>P = {wⁿ : n = 1, 2, 3, ...}</i>. Further, <i>H₁ = {z ∈ ℂ : Re(z) > -1/2}</i> and <i>H₂ = {z ∈ ℂ : Re(z) < 1/2}</i>. Let <i>S = P ∩ H₁ ∩ H₂</i>. Then the number of elements in <i>S</i></p>
<p>(a) is strictly greater than 5</p>
<p>(b) is equal to 5</p>
<p>(c) is strictly greater than 3/2 but less than 5/2</p>
<p>(d) lies in the interval (1, 2)</p>
Step-by-Step Solution
Key Concept: Convert w to exponential form, identify the periodic nature of powers, and find which elements satisfy both half-plane constraints.
<p><strong>Solution:</strong> First, <i>w = (√3 + i)/2 = e^(iπ/6)</i> with <i>|w| = 1</i>.</p><p>The set <i>P = {e^(inπ/6) : n = 1, 2, 3, ...}</i> consists of 12 distinct points on the unit circle (period 12).</p><p>For <i>z = e^(inπ/6)</i>, we have <i>Re(z) = cos(nπ/6)</i>.</p><p>Condition <i>Re(z) > -1/2</i> means <i>cos(nπ/6) > -1/2</i>, satisfied for <i>n ∈ {1, 2, 3, 4, 5, 7, 8, 9, 10, 11}</i>.</p><p>Condition <i>Re(z) < 1/2</i> means <i>cos(nπ/6) < 1/2</i>, satisfied for <i>n ∈ {2, 3, 4, 5, 6, 7, 8, 10, 11, 12}</i>.</p><p>The intersection gives 5 values of <i>n</i> in a period: <i>n ∈ {2, 3, 4, 5, 7}</i>.</p><p>∴ Number of elements in <i>S</i> = 5</p>
Correct Answer: B