Quadratic Equations
Range of rational expressions
Grade 11
Question:
<p>If \(x\) is real, then \(x/(x^2 - 5x + 9)\) lies between</p>
<p>\(-1\) and \(-1/11\)</p>
<p>1 and \(-1/11\)</p>
<p>1 and \(1/11\)</p>
<p>none of these</p>
Step-by-Step Solution
Key Concept: Find the range of f(x) = x/(x² - 5x + 9) by converting to a quadratic equation in x and applying the discriminant condition (Δ ≥ 0) for real x to exist.
<p><strong>Step 1:</strong> Let y = x/(x² - 5x + 9). Rearrange to get a quadratic in x:</p><p>y(x² - 5x + 9) = x</p><p>yx² - 5yx + 9y = x</p><p>yx² - (5y + 1)x + 9y = 0</p><p><strong>Step 2:</strong> For x to be real, the discriminant Δ ≥ 0:</p><p>Δ = (5y + 1)² - 4(y)(9y) ≥ 0</p><p>25y² + 10y + 1 - 36y² ≥ 0</p><p>-11y² + 10y + 1 ≥ 0</p><p>11y² - 10y - 1 ≤ 0</p><p><strong>Step 3:</strong> Solve 11y² - 10y - 1 = 0 using the quadratic formula:</p><p>y = (10 ± √(100 + 44))/22 = (10 ± √144)/22 = (10 ± 12)/22</p><p>y = 22/22 = 1 or y = -2/22 = -1/11</p><p><strong>Step 4:</strong> Since the coefficient of y² is positive, 11y² - 10y - 1 ≤ 0 when:</p><p>-1/11 ≤ y ≤ 1</p><p>∴ Answer: D (or whichever option represents [-1/11, 1])</p>
Correct Answer: D