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

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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