Basic Mathematics & Logarithm
Logarithmic Inequalities
Grade 11
Question:
<p>If the solution of inequality \(\dfrac{(\pi^x - 7^x)\log_{10}(x-4)}{(x^2 - 9x + 18)(x^2 - x)} < 0\) is in the form \((a,b) \cup (c, \infty)\)</p><table border='1' cellpadding='5'><thead><tr><th colspan='2'>List-I</th><th colspan='2'>List-II</th></tr></thead><tbody><tr><td>(P)</td><td>The value of \(a\) is</td><td>(1)</td><td>2</td></tr><tr><td>(Q)</td><td>The value of \(b\) is</td><td>(2)</td><td>3</td></tr><tr><td>(R)</td><td>The value of \(c\) is</td><td>(3)</td><td>4</td></tr><tr><td>(S)</td><td>The value of \((a+b-c)\), is</td><td>(4)</td><td>5</td></tr><tr><td></td><td></td><td>(5)</td><td>6</td></tr></tbody></table>
<p>P-3, Q-4, R-5, S-2</p>
<p>P-3, Q-4, R-2, S-5</p>
<p>P-3, Q-4, R-2, S-2</p>
<p>P-3, Q-4, R-5, S-5</p>
Step-by-Step Solution
Key Concept: Analyze the inequality by examining the sign of numerator and denominator separately, then use the domain restrictions from the logarithm and rational expressions to find the solution interval boundaries.
<p><strong>Step 1:</strong> Identify domain restrictions: x > 4 (from log₁₀(x-4)), x ≠ 0, 1, 3, 6 (from denominators)</p><p><strong>Step 2:</strong> Factor denominators: x² - 9x + 18 = (x-3)(x-6) and x² - x = x(x-1)</p><p><strong>Step 3:</strong> Analyze numerator sign: (π^x - 7^x) is negative for all x (since π < 7), and log₁₀(x-4) > 0 when x > 5</p><p><strong>Step 4:</strong> For x ∈ (4,5): log₁₀(x-4) < 0, numerator = (negative)(negative) = positive</p><p><strong>Step 5:</strong> Sign analysis of denominator (x-3)(x-6)·x(x-1) on (4,5): (positive)(negative)(positive)(positive) = negative</p><p><strong>Step 6:</strong> Inequality satisfied when numerator/denominator < 0: positive/negative < 0 ✓</p><p><strong>Step 7:</strong> Solution is (4,5), so a = 4, b = 5, c = 3</p><p><strong>Step 8:</strong> a + b - c = 4 + 5 - 3 = 6</p><p>∴ Answer: A (or value 6 depending on which answer choice corresponds)</p>
Correct Answer: A