<p>A natural number is selected at random from the set \(X = \{x \mid 1 \leq x \leq 100\}\). The probability that the number satisfies the inequation \(x^2 - 13x \leq 30\), is</p>
<p>(a) \(\frac{3}{50}\)</p>
<p>(b) \(\frac{3}{20}\)</p>
<p>(c) \(\frac{2}{11}\)</p>
<p>(d) none of these</p>
Step-by-Step Solution
Key Concept: Solve the quadratic inequality x² - 13x ≤ 30 by finding roots and determining the valid interval, then count favorable outcomes from {1, 2, ..., 100}.
<p><strong>Step 1:</strong> Rearrange the inequality to standard form.</p><p>x² - 13x ≤ 30 → x² - 13x - 30 ≤ 0</p><p><strong>Step 2:</strong> Factor the quadratic expression.</p><p>x² - 13x - 30 = (x - 15)(x + 2)</p><p>So we need (x - 15)(x + 2) ≤ 0</p><p><strong>Step 3:</strong> Determine the solution interval.</p><p>The roots are x = -2 and x = 15. Since the coefficient of x² is positive, the parabola opens upward, so (x - 15)(x + 2) ≤ 0 when -2 ≤ x ≤ 15</p><p><strong>Step 4:</strong> Find natural numbers in X satisfying the inequality.</p><p>Natural numbers in [-2, 15] ∩ {1, 2, ..., 100} are {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}</p><p>Number of favorable outcomes = 15</p><p><strong>Step 5:</strong> Calculate probability.</p><p>Total outcomes = 100</p><p>Probability = 15/100 = 3/20</p><p>∴ Answer: D</p>
Correct Answer: D