A) for no value of \[\lambda \] and \[\theta \]
B) for all values of \[\lambda \] and \[\theta \]
C) for all values of \[\lambda \] and only two values of \[\theta \]
D) for only one value of \[\lambda \] and all values of \[\theta \]
Correct Answer: A
Solution :
Let A be the coefficient matrix of the given set of equations, then\[A=\left[ \begin{matrix} \lambda & -1 & \cos \theta \\ 3 & 1 & 2 \\ \cos \theta & 1 & 2 \\ \end{matrix} \right]\] Then, \[|A|=\left[ \begin{matrix} \lambda & -1 & \cos \theta \\ 3 & 1 & 2 \\ \cos \theta & 1 & 2 \\ \end{matrix} \right]\] \[=\cos \theta -{{\cos }^{2}}\theta +6\] |A| is positive for all \[\theta \] since, | cos \[\theta \]| \[\le \] 1. The only solution is therefore the trivial solution.
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