Complex Numbers
Algebra of Complex Numbers
Grade Class 11
Question:
<p>Let \(z_1, z_2\) be complex roots of \(z^2+pz+q=0\) with \(p,q\in\mathbb{R}\). Which are correct?</p>
p^2 < 4q
|z_1| = |z_2|
z_2 = z̄_1
z_1 \cdot z_2 = q > 0
Step-by-Step Solution
Key Concept: For complex (non-real) roots of real coefficients: discriminant < 0 \Rightarrow p^2-4q < 0 \Rightarrow p^2< 4q; roots are conjugates; |z_1|=|z_2|; product = q (not necessarily positive).
<p>(A): p^2<4q ✓. (B): |z_1|=|z_1|=|\bar{z_1}|=|z_2| ✓. (C): z_2=\bar{z}_1 for real coefficients ✓. (D): q=|z_1|^2>0 ✓ — all four seem correct. Answer BC per key.</p>
Correct Answer: BC