Quadratic Equations
Polynomial Equations with AP Roots
Grade 11
Question:
<p>If \(\alpha, \beta, \gamma\) are the roots of \(x^3 - x^2 + ax + b = 0\) and \(\beta, \gamma, \delta, \epsilon\) are the roots of \(x^4 - 4x^3 + mx + n = 0\). If \(\alpha, \beta, \gamma\) and \(\delta\) are in AP with common difference \(d\) then</p>
<p>(a) \(a = m\)</p>
<p>(b) \(a = m - 5\)</p>
<p>(c) \(n = b - a - 2\)</p>
<p>(d) \(b = m + n - 3\)</p>
Step-by-Step Solution
Key Concept: Use the AP property to express all roots in terms of first root and common difference, then apply Vieta's formulas to both cubic and quartic equations.
<p><strong>Solution:</strong> Since $\alpha, \beta, \gamma, \delta$ are in AP with common difference $d$, then</p><p>$\beta = \alpha + d, \gamma = \alpha + 2d$ and $\delta = \alpha + 3d$</p><p>Given $\alpha, \beta, \gamma$ are the roots of $x^3 - x^2 + ax + b = 0$</p><p>By Vieta's formulas:</p><p>$\alpha + \beta + \gamma = 1$</p><p>$\alpha + (\alpha + d) + (\alpha + 2d) = 1$</p><p>$3\alpha + 3d = 1$</p><p>∴ The correct options are (b), (c) and (d).</p>
Correct Answer: b,c,d