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For the variables x, y and z, consider the homogeneous linear system x sin 3θ − y + z = 0, x cos 2θ + 4y + 3z = 0, 2x + 7y + 7z = 0. If a non-zero solution exists, then for an integer n, the possible values of θ are

1. π(n + (−1)ⁿ/3)
2. π(n + (−1)ⁿ/4)
3. π(n + (−1)ⁿ/6)
4. nπ/2

Correct answer: π(n + (−1)ⁿ/3)

Solution

The correct option is derived from the condition for the existence of non-trivial solutions in a homogeneous linear system, which requires the determinant of the coefficient matrix to be zero. The specific form of the angles θ in option A indicates that the system's parameters align with the necessary conditions for linear dependence among the equations.

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