Sequences & Series
Geometric Progression
Grade 11
Question:
<p>In the quadratic equation \(ax^2 + bx + c = 0\), if \(\Delta = b^2 - 4ac\) and \(\alpha + \beta\), \(\alpha^2 + \beta^2\), \(\alpha^3 + \beta^3\) are in G.P., where \(\alpha\), \(\beta\) are the roots of \(ax^2 + bx + c = 0\), then</p>
<p>(A) \(\Delta \neq 0\)</p>
<p>(B) \(b\Delta = 0\)</p>
<p>(C) \(c\Delta = 0\)</p>
<p>(D) \(\Delta = 0\)</p>
Step-by-Step Solution
Key Concept: Express the power sums in terms of Vieta's formulas and use the G.P. condition to derive a constraint on the discriminant and coefficients.
<p>Let \(S_n = \alpha^n + \beta^n\). We have \(S_1 = -\frac{b}{a}\), \(S_2 = S_1^2 - 2\frac{c}{a}\), and \(S_3\) can be computed using recurrence. For these to be in G.P., \(S_2^2 = S_1 S_3\). Solving this constraint yields \(c\Delta = 0\).</p>
Correct Answer: C