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For real parameters \(\alpha\) and \(\beta\), consider the homogeneous system \[ \lambda x + (\sin\alpha)y + (\cos\alpha)z = 0, \] \[ x + (\cos\alpha)y + (\sin\alpha)z = 0, \] \[ -x + (\sin\alpha)y - (\cos\alpha)z = 0. \] The collection of all \(x\)-values for which this system admits a non-zero solution is:

1. \([0,\sqrt{2}]\)
2. \([-\sqrt{2},0]\)
3. \([-\sqrt{2},\sqrt{2}]\)
4. None of these

Correct answer: None of these

Solution

The system of equations represents a homogeneous linear system, and for it to have a non-trivial solution, the determinant of the coefficient matrix must be zero. Upon calculating the determinant, we find that it does not yield a valid range of x-values that fits any of the provided options, confirming that 'None of these' is the correct choice.

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