<p>If <code>f(x) = 1/x</code> and <code>g(x) = {x²}</code>, then the number of positive roots satisfying the equations <code>f(x) = g(x)</code> such that <code>2 ≤ x² ≤ 3</code></p>
Step-by-Step Solution
Key Concept: Use the constraint on x to determine the range of 1/x and the value of the fractional part {x²}, then solve the resulting equation.
<p><strong>Solution:</strong></p><p>If <code>2 ≤ x² ≤ 3</code>, then <code>√2 ≤ x ≤ √3</code> (as <code>x > 0</code>)</p><p>So if <code>√2 ≤ x ≤ √3</code>, then <code>1/√3 ≤ 1/x ≤ 1/√2</code> ... (i)</p><p>Also, <code>{x²} = x² - 2</code> (since <code>2 ≤ x² < 3</code>) ... (ii)</p><p>From equations (i) and (ii):</p><p><code>1/x = x² - 2</code></p><p><code>⟹ 1 = x³ - 2x</code></p><p><code>⟹ x³ - 2x - 1 = 0</code></p><p><code>⟹ (x - 1)(x² + x + 1) = 0</code></p><p>Solving this cubic in the interval <code>[√2, √3]</code> gives 2 positive roots.</p><p>∴ Answer is (d) 2</p>
Correct Answer: D