Complex Numbers
Algebra of Complex Numbers
Grade Class 11
Question:
<p>If \( z + z^2 = 0 \), then which of the following must be true on the complex plane?</p><ul><li>(A) \( \text{Re}(z) = 0 \)</li><li>(B) \( \text{Im}(z) = 0 \)</li><li>(C) Both \( \text{Re}(z) = 0 \) and \( \text{Im}(z) = 0 \)</li><li>(D) \( z^2 = 1 \)</li></ul>
Re(z) = 0
Im(z) = 0
Both Re(z) = 0 and Im(z) = 0
z^2 = 1
Step-by-Step Solution
Key Concept: z + z^2 = 0 \Rightarrow z(1 + z) = 0 \Rightarrow z = 0 or z = -1; both are real, so Im(z) = 0.
<p>\( z + z^2 = 0 \Rightarrow z(1 + z) = 0 \Rightarrow z = 0 \) or \( z = -1 \). Both values are real, so \( \text{Im}(z) = 0 \).</p>
Correct Answer: B