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Question

Consider the system of linear equation in x, y, z.
                              $$(sin 3 \theta)  x - y  + z = 0$$
                            $$(cos 2 \theta) x + 4y + 3z = 0$$
                                       $$2x + 7y + 7z = 0$$ 
Find the values of $$\theta$$ for which this system has non-trivial solution


A
θ=nπ+(1)nπ6nI
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B
θ=(m)π,mI
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C
θ=(m+1/2)π,mI
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D
θ=nπ+(1)nπ3nI
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Solution

The correct options are
A $$\theta = n \pi + (-1)^n \frac{\pi}{6} \forall n \in I$$
D $$\theta = (m) \pi, m \in I$$
The system of the equation has a non-trivial solution if $$\triangle =0$$
$$\displaystyle \Rightarrow \begin{vmatrix} \sin { 3\theta \quad  }  & -1\quad  & 1 \\ \cos { 2\theta  }  & 4 & 3 \\ 2 & 7 & 7 \end{vmatrix}=0$$
Expanding along $${ C }_{ 1 }$$, we get
$$\sin { 3\theta  } .\left( 28-21 \right) -\cos { 2\theta  } \left( -7-7 \right) +2\left( -3-4 \right) =0\\$$

$$ \Rightarrow 7\sin { 3\theta  } +14\cos { 2\theta  } -14=0\\$$

$$ \Rightarrow \sin { 3\theta  } +2\cos { 2\theta  } -1=0\\$$

$$ \Rightarrow 3\sin { \theta  } -4\sin ^{ 3 }{ \theta  } +2\left( 1-2\sin ^{ 2 }{ \theta  }  \right) -2=0\\$$

$$ \Rightarrow \sin { \theta  } \left( \sin ^{ 2 }{ \theta  } +4\sin { \theta -3 }  \right) =0\\$$

$$ \Rightarrow \sin { \theta  } \left( 2\sin { \theta  } -1 \right) \left( 2\sin { \theta  } +3 \right) =0$$
$$\displaystyle \Rightarrow \sin { \theta  } =0,\sin { \theta =\frac { 1 }{ 2 }  } $$   (neglecting $$\displaystyle \sin { \theta  } =-\frac { 3 }{ 2 } $$)
$$\displaystyle \Rightarrow \theta =n\pi ,n\pi +{ \left( -1 \right)  }^{ n }\frac { \pi  }{ 6 } n\in Z$$

Mathematics

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