Complex Numbers
Quadratic Equations with Complex Roots
Grade Class 11
Question:
<p>If \(\alpha, \beta\) are complex roots of \(x^2-px+q=0\) with \(p,q\in\mathbb{R}\), which are always true?</p>
\alpha + \beta = p
\alpha\beta = q
ᾱ = \beta
|\alpha| = |\beta|
Step-by-Step Solution
Key Concept: For real coefficients, complex roots come in conjugate pairs: \alpha = \betā. Sum = p, product = q, |\alpha|=|\beta| (since conjugates have equal modulus).
<p>(A): $\alpha+\beta=p$ by Vieta's — TRUE. (B): $\alpha\beta=q$ — TRUE. (C): $\bar{\alpha}=\beta$ for real coefficients — TRUE. (D): $|\alpha|=|\bar{\alpha}|=|\beta|$ — TRUE. All four are correct.</p>
Correct Answer: BCD