<p>Find the complete set of values of <span style="font-style: italic;">a</span> for which <span style="font-style: italic;">f</span>(θ) < 0 for all θ ∈ [0, π/2), where <span style="font-style: italic;">f</span>(θ) = tan²θ + (<span style="font-style: italic;">a</span> + 1)tanθ – (<span style="font-style: italic;">a</span> – 3)</p>
Step-by-Step Solution
Key Concept: A quadratic with positive leading coefficient must eventually become positive as the variable approaches infinity.
<p><strong>Solution:</strong> Let <span style="font-style: italic;">t</span> = tanθ, so <span style="font-style: italic;">t</span> ∈ [0, ∞). We need <span style="font-style: italic;">g</span>(<span style="font-style: italic;">t</span>) = <span style="font-style: italic;">t</span>² + (<span style="font-style: italic;">a</span> + 1)<span style="font-style: italic;">t</span> – (<span style="font-style: italic;">a</span> – 3) < 0 for all <span style="font-style: italic;">t</span> ∈ [0, ∞).</p><p>However, as <span style="font-style: italic;">t</span> → ∞, <span style="font-style: italic;">g</span>(<span style="font-style: italic;">t</span>) → ∞ (since the coefficient of <span style="font-style: italic;">t</span>² is positive).</p><p>Therefore, <span style="font-style: italic;">g</span>(<span style="font-style: italic;">t</span>) cannot be negative for all <span style="font-style: italic;">t</span> ∈ [0, ∞).</p><p>∴ The answer is (D): Null set</p>
Correct Answer: D