| \[x+3y+7z=0\] \[-x+4y+7z=0\] \[\left( sin3\theta \right)x+\left( cos2\theta \right)y+2z=0\] |
| has a non-trivial solution, is- |
A) two
B) one
C) four
D) three
Correct Answer: A
Solution :
For non-trivial solution \[\Delta =0\] \[\Rightarrow \,\,\,\left| \begin{matrix} 1 & 3 & 7 \\ -1 & 4 & 7 \\ \sin \,3\theta & \cos \,2\,\theta & 2 \\ \end{matrix} \right|\,\,=\,\,0\] \[\Rightarrow \,\,\,\,4\,si{{n}^{3}}\theta +4\,si{{n}^{2}}\theta -3\,sin\,\theta =0\] \[\Rightarrow \,\,\,sin\,\theta =0\,or\,sin\,\theta =\frac{1}{2}\,\,or\text{ }sin\,\theta =-\frac{3}{2}\] For \[\theta \,\in \,(0,\,\,\pi )\] \[\theta =\frac{\pi }{6}\,\,and\,\,\theta =\frac{5\pi }{6}\,\] are satisfy the equation Number of values of \[\theta =2\]
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