Matrices & Determinants
System of Linear Equations
Grade 12
Question:
<p>If the system of linear equations</p><p>\((\cos\theta) x + (\sin\theta) y + \cos\theta = 0\)</p><p>\((\sin\theta) x + (\cos\theta) y + \sin\theta = 0\)</p><p>\((\cos\theta) x + (\sin\theta) y - \cos\theta = 0\)</p><p>is consistent, then the number of possible values of \(\theta\), \(\theta \in [0, 2\pi]\) is:</p>
Step-by-Step Solution
Key Concept: For a system to be consistent, analyze when contradictory equations do not arise. Look for immediate contradictions between equations.
<p><strong>Solution:</strong> For the system to be consistent, the determinant of the coefficient matrix must equal zero, or the system must satisfy consistency conditions.</p><p>Notice that equations 1 and 3 are: \((\cos\theta) x + (\sin\theta) y = -\cos\theta\) and \((\cos\theta) x + (\sin\theta) y = \cos\theta\)</p><p>For these to both hold, we need: \(-\cos\theta = \cos\theta\), which gives \(\cos\theta = 0\)</p><p>This occurs when \(\theta = \frac{\pi}{2}\) or \(\theta = \frac{3\pi}{2}\) in \([0, 2\pi]\)</p><p>We can verify these values satisfy all three equations consistently.</p><p>Therefore, there are <strong>2</strong> possible values of \(\theta\).</p>
Correct Answer: 2