Complex Numbers
Modulus and Argument
Grade Class 11
Question:
<p>Which of the following hold for all complex numbers \(z, z_1, z_2\)?</p><ul><li>(A) \( |\text{Re}(z)| + |\text{Im}(z)| \leq \sqrt{2}|z| \)</li><li>(B) \( |z_1+z_2|^2 + |z_1-z_2|^2 = 2(|z_1|^2+|z_2|^2) \)</li><li>(C) \( |z_1+z_2| \leq |z_1|+|z_2| \)</li><li>(D) \( \text{Re}(iz) = -\text{Im}(z) \)</li></ul>
|Re(z)| + |Im(z)| \leq \sqrt{2}|z|
Parallelogram law
Triangle inequality
Re(iz) = -Im(z)
Step-by-Step Solution
Key Concept: All four are standard identities/inequalities. (A) follows from AM-GM; (B) is parallelogram law; (C) triangle inequality; (D) Re(i(x+iy))=Re(ix-y)=-y=-Im(z).
<p>(A): $(|x|+|y|)^2 \leq 2(x^2+y^2) = 2|z|^2$ by AM-GM. ✓<br>(B): Direct expansion. ✓<br>(C): Triangle inequality. ✓<br>(D): $iz = i(x+iy)=ix-y\Rightarrow\text{Re}(iz)=-y=-\text{Im}(z)$. ✓</p>
Correct Answer: ABCD