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Solve this: 3. The values ofO, for which the following equations Sinox cos0y+ O; cosex + sin0y — = O; + l)y+ z = O have non u-ivial solution, is (A) 0 = nit, R — {0} (B) 0 = 2M', is any rational number E R, nel

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$$\begin{gathered} \mathrm{We} \mathrm{have} \\ \mathrm{sin\theta x}-\mathrm{cos\theta y}+\left(\mathrm{\lambda }+1\right)\mathrm{z}=0....\left(\mathrm{i}\right) \\ \mathrm{cos\theta x}+\mathrm{sin\theta y}-\mathrm{\lambda z}=0...\left(\mathrm{ii}\right) \\ \mathrm{\lambda x}+\left(\mathrm{\lambda }+1\right)\mathrm{y}+6+\mathrm{cos\theta z}=0...\left(\mathrm{iii}\right) \\ \mathrm{For} \mathrm{non}-\mathrm{trivial} \mathrm{solution}, \mathrm{determinant} \mathrm{formed} \mathrm{by} \mathrm{coefficients} \mathrm{of} \left(\mathrm{i}\right),\left(\mathrm{ii}\right) \\ \mathrm{and} \left(\mathrm{iii}\right) \mathrm{will} \mathrm{be} 0 \\ \Rightarrow \left|\begin{array}{ccc}\mathrm{sin\theta }& -\mathrm{cos\theta }& \mathrm{\lambda }+1\\ \mathrm{cos\theta }& \mathrm{sin\theta }& -\mathrm{\lambda }\\ \mathrm{\lambda }& \mathrm{\lambda }+1& \mathrm{cos\theta }\end{array}\right|=0 \\ \mathrm{Expand} \mathrm{along} {\mathrm{R}}_1 \\ \mathrm{sin\theta }\left[\mathrm{sin\theta cos\theta }+\mathrm{\lambda }\left(\mathrm{\lambda }+1\right)\right]+\mathrm{cos\theta }\left[{\cos }^2\mathrm{\theta }+{\mathrm{\lambda }}^2\right]+\left(\mathrm{\lambda }+1\right)\left[\left(\mathrm{\lambda }+1\right)\mathrm{cos\theta }-\mathrm{\lambda sin\theta }\right]=0 \\ \Rightarrow {\sin }^2\mathrm{\theta cos\theta }+\mathrm{sin\theta \lambda }\left(\mathrm{\lambda }+1\right)+{\cos }^3\mathrm{\theta }+{\mathrm{\lambda }}^2\mathrm{cos\theta }+{\left(\mathrm{\lambda }+1\right)}^2\mathrm{cos\theta }-\mathrm{sin\theta \lambda }\left(\mathrm{\lambda }+1\right)=0 \\ \Rightarrow {\sin }^2\mathrm{\theta cos\theta }+{\cos }^3\mathrm{\theta }+{\mathrm{\lambda }}^2\mathrm{cos\theta }+{\left(\mathrm{\lambda }+1\right)}^2\mathrm{cos\theta }=0 \\ \Rightarrow \mathrm{cos\theta }\left[{\sin }^2\mathrm{\theta }+{\cos }^2\mathrm{\theta }+{\mathrm{\lambda }}^2+\mathrm{cos\theta }+{\left(\mathrm{\lambda }+1\right)}^2\right]=0 \\ \mathrm{Here} \mathrm{if} \mathrm{cos\theta }=0 \mathrm{then} \mathrm{determinant} \mathrm{will} \mathrm{always} \mathrm{be} 0, \mathrm{irrespective} \mathrm{of} \mathrm{value} \\ \mathrm{of} \mathrm{\lambda }. \\ \mathrm{Hence} \\ \mathrm{cos\theta }=0 \mathrm{and} \mathrm{\lambda }\in \mathrm{R} \\ \mathbf{\Rightarrow }\mathbf{\theta }\mathbf{=}\left(\mathbf{2}\mathbf{n}\mathbf{+}\mathbf{1}\right)\frac{\mathbf{\pi }}{\mathbf{2}}\mathbf{,}\mathbf{ }\left(\mathbf{n}\mathbf{ }\mathbf{is}\mathbf{ }\mathbf{integer}\right)\mathbf{ }\mathbf{and}\mathbf{ }\mathbf{\lambda }\mathbf{\in }\mathbf{R} \end{gathered}$$

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