Matrices & Determinants
Determinant of Order 3
Grade 12
Question:
<p>If <span style='display:none'>f(θ) = </span><math><mrow><mrow><mo>|</mo><mtable columnalign='center'><mtr><mtd><mn>1</mn></mtd><mtd><mi>tan</mi><mi>θ</mi></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>−</mo><mi>tan</mi><mi>θ</mi></mtd><mtd><mn>1</mn></mtd><mtd><mi>tan</mi><mi>θ</mi></mtd></mtr><mtr><mtd><mo>−</mo><mn>1</mn></mtd><mtd><mo>−</mo><mi>tan</mi><mi>θ</mi></mtd><mtd><mn>1</mn></mtd></mtr></mtable><mo>|</mo></mrow></mrow></math>, then the set <math><mrow><mrow><mo>{</mo><mi>θ</mi><mo>:</mo><mi>f</mi><mo>(</mo><mi>θ</mi><mo>)</mo><mo>=</mo><mn>0</mn><mo>,</mo><mi>θ</mi><mo>∈</mo><mo>[</mo><mo>−</mo><mi>π</mi><mo>,</mo><mi>π</mi><mo>]</mo><mo>}</mo></mrow></mrow></math> is</p>
<p>(a) \((-\pi, -1) \cup (1, \pi)\)</p>
<p>(b) \([2, \pi)\)</p>
<p>(c) \((-\pi, 0] \cup [2, \pi)\)</p>
<p>(d) \((-\pi, -1] \cup [1, \pi)\)</p>
Step-by-Step Solution
Key Concept: Expand the determinant and solve the resulting trigonometric equation for the given interval.
<p><strong>Solution:</strong> Expand the determinant and find when it equals zero. The solution involves analyzing the trigonometric expression and finding the valid range for $\theta$ in $[-\pi, \pi]$.</p>
Correct Answer: D