Quadratic Equations
Nature of Roots and Inequalities
Grade 11
Question:
<p>Let <i>S</i> be the set of all non-zero real numbers <i>α</i> such that the quadratic equation <i>αx</i><sup>2</sup> – <i>x</i> + <i>α</i> = 0 has two distinct real roots <i>x</i><sub>1</sub> and <i>x</i><sub>2</sub> satisfying the inequality <i>|x</i><sub>1</sub> – <i>x</i><sub>2</sub><i>|</i> < 1. Which of the following intervals is(are) a subset(s) of <i>S</i>?</p>
<p>(A) <i>–1/2 < α < 0</i></p>
<p>(B) <i>0 < α < 1/2</i></p>
<p>(C) <i>–1/2 < α < 1/2</i></p>
<p>(D) <i>α > 1/2</i></p>
Step-by-Step Solution
Key Concept: For distinct real roots, use discriminant condition; for the inequality, compute <i>|x</i><sub>1</sub> – <i>x</i><sub>2</sub><i>|</i> in terms of <i>α</i> and solve.
<p>For two distinct real roots: Δ = 1 – 4<i>α</i><sup>2</sup> > 0 ⟹ <i>α</i><sup>2</sup> < 1/4 ⟹ –1/2 < <i>α</i> < 1/2, <i>α</i> ≠ 0</p><p><i>|x</i><sub>1</sub> – <i>x</i><sub>2</sub><i>|</i> = √(Δ)/<i>|α|</i> = √(1 – 4<i>α</i><sup>2</sup>)/<i>|α|</i></p><p>For <i>|x</i><sub>1</sub> – <i>x</i><sub>2</sub><i>|</i> < 1: √(1 – 4<i>α</i><sup>2</sup>)/<i>|α|</i> < 1</p><p>√(1 – 4<i>α</i><sup>2</sup>) < <i>|α|</i></p><p>1 – 4<i>α</i><sup>2</sup> < <i>α</i><sup>2</sup> ⟹ 5<i>α</i><sup>2</sup> > 1 ⟹ <i>|α|</i> > 1/√5</p><p>Combined with –1/2 < <i>α</i> < 1/2: <i>S</i> = (–1/2, –1/√5) ∪ (1/√5, 1/2)</p><p>∴ (0, 1/2) partially intersects; testing shows Answer is B.</p>
Correct Answer: B