Complex Numbers
Algebra of Complex Numbers
Grade Class 11
Question:
<p>Let \( z = \left(\dfrac{\sqrt{3}}{2}+\dfrac{i}{2}\right)^5 + \left(\dfrac{\sqrt{3}}{2}-\dfrac{i}{2}\right)^5 \). If \(\text{Re}(z)=a\) and \(\text{Im}(z)=b\), then \((a,b)\) equals:</p>
(0, \sqrt{3})
(0, -\sqrt{3})
(-\sqrt{3}, 0)
(\sqrt{3}, 0)
Step-by-Step Solution
Key Concept: Note (\sqrt{3}/2 \pm i/2) = e^(\pmi\pi/6). So z = e^(5i\pi/6) + e^(-5i\pi/6) = 2cos(5\pi/6) = -\sqrt{3.} Thus (a,b) = (-\sqrt{3}, 0).
<p>$ \dfrac{\sqrt{3}}{2}+\dfrac{i}{2} = e^{i\pi/6} $. So $ z = e^{5i\pi/6}+e^{-5i\pi/6} = 2\cos\dfrac{5\pi}{6} = 2\left(-\dfrac{\sqrt{3}}{2}\right) = -\sqrt{3} $. Hence $(a,b)=(-\sqrt{3},0)$.</p>
Correct Answer: C