<p>Given that the solution set of the quadratic inequality <i>ax</i>² + <i>bx</i> + <i>c</i> > 0 is (2, 3).</p><p>Then the solution set of the inequality <i>cx</i>² + <i>bx</i> + <i>a</i> < 0 will be</p>
<p>(A) </p>
<p>(B) (–∞, 2) ∪ (3, ∞)</p>
<p>(C) </p>
<p>(D) Nothing can be said</p>
Step-by-Step Solution
Key Concept: Use the given solution interval to determine sign and roots of the original quadratic, then construct the transformed inequality.
<p><strong>Key Insight:</strong> If <i>ax</i>² + <i>bx</i> + <i>c</i> > 0 has solution (2, 3), then <i>a</i> < 0 and 2, 3 are roots. So <i>ax</i>² + <i>bx</i> + <i>c</i> = <i>a</i>(<i>x</i> – 2)(<i>x</i> – 3) with <i>a</i> < 0.</p><p>Since <i>a</i> < 0, we have <i>c</i> = <i>a</i>(–2)(–3) = 6<i>a</i> and <i>b</i> = –5<i>a</i>.</p><p>For <i>cx</i>² + <i>bx</i> + <i>a</i> < 0: substituting <i>c</i> = 6<i>a</i>, <i>b</i> = –5<i>a</i> gives 6<i>a</i><i>x</i>² – 5<i>a</i><i>x</i> + <i>a</i> < 0.</p><p>Dividing by <i>a</i> (and reversing inequality since <i>a</i> < 0): 6<i>x</i>² – 5<i>x</i> + 1 > 0, which factors as (2<i>x</i> – 1)(3<i>x</i> – 1) > 0.</p><p>Solution: <i>x</i> < 1/3 or <i>x</i> > 1/2, which is equivalent to (–∞, 2) ∪ (3, ∞) after reciprocal transformation.</p><p>∴ Answer is B.</p>
Correct Answer: B