Complex Numbers
Quadratic Equations with Complex Roots
Grade Class 11
Question:
<p>If \( \alpha, \beta \) are the roots of \( x^2 - 2x + 2 = 0 \), and \( n \) is the least positive integer such that \( \left(\dfrac{\alpha}{\beta}\right)^n = 1 \), then \( n \) is:</p>
Step-by-Step Solution
Key Concept: Roots: \alpha = 1+i, \beta = 1-i. \alpha/\beta = (1+i)/(1-i) = i. The least n with iⁿ = 1 is n = 4.
<p>$ x = \dfrac{2 \pm \sqrt{4-8}}{2} = 1 \pm i $. So $ \alpha/\beta = \dfrac{1+i}{1-i} = \dfrac{(1+i)^2}{2} = i $. Least $n$ with $i^n = 1$ is $n = 4$.</p>
Correct Answer: D