Quadratic Equations
Roots and Coefficients
Grade 11
Question:
<p>If \(\alpha\) and \(\beta\) are the roots of the equation \(x^2 - mx + 2 = 0\) and \(\alpha + \dfrac{1}{\beta}\), \(\beta + \dfrac{1}{\alpha}\) are the roots of the equation \(x^2 - px + q = 0\), then the value of \(2q\) equals:</p>
<p>1</p>
<p>3</p>
<p>6</p>
<p>9</p>
Step-by-Step Solution
Key Concept: Use Vieta's formulas on the original equation to express α + β and αβ, then compute the sum and product of the new roots (α + 1/β) and (β + 1/α) in terms of m and the constraint that their product equals q.
<p><strong>Step 1:</strong> From x² - mx + 2 = 0, by Vieta's formulas: α + β = m and αβ = 2</p><p><strong>Step 2:</strong> Find the sum of new roots: (α + 1/β) + (β + 1/α) = α + β + 1/β + 1/α = α + β + (α + β)/(αβ) = m + m/2 = 3m/2</p><p>So p = 3m/2</p><p><strong>Step 3:</strong> Find the product of new roots: (α + 1/β)(β + 1/α) = αβ + α·(1/α) + (1/β)·β + (1/β)·(1/α) = αβ + 1 + 1 + 1/(αβ) = 2 + 2 + 1/2 = 9/2</p><p>So q = 9/2</p><p><strong>Step 4:</strong> Therefore, 2q = 2 · (9/2) = 9</p><p>∴ Answer: D</p>
Correct Answer: D