| STATEMENT-1: The system of linear equations\[\begin{align} & x+(\sin \theta )y+(\cos \theta )z=0 \\ & x+(\cos \theta )y+(\sin \theta )z=0 \\ & x-(\sin \theta )y-(\cos \theta )z=0 \\ \end{align}\]has a non-trivial solution for only one value of \[\theta \] lying in the interval\[\left( 0,\frac{\pi }{2} \right)\]. |
| STATEMENT-2: The equation in \[\theta \] \[\left| \begin{matrix} \cos \theta & \sin \theta & \cos \theta \\ \sin \theta & \cos \theta & \sin \theta \\ \cos \theta & -\sin \theta & -\cos \theta \\ \end{matrix} \right|=0\] has only one solution lying in the interval\[\left( 0,\frac{\pi }{2} \right)\]. |
A) Statement-1 is true, Statement-2 is true and Statement-2 is correct explanation for statement- 1.
B) Statement-1 is true, Statement-2 is true and Statement-2 is NOT the correct explanation for statement-1.
C) Statement-1 is true, Statement-2 is false.
D) Statement-1 is false, Statement-2 is true.
Correct Answer: A
Solution :
Statement-1: For non-trivial solution, put \[\Delta =0\] \[\Rightarrow \left| \begin{matrix} 1 & \sin \theta & \cos \theta \\ 1 & \cos \theta & \sin \theta \\ 1 & -\sin \theta & -\cos \theta \\ \end{matrix} \right|=0\] \[\Rightarrow 1(-{{\cos }^{2}}\theta +{{\sin }^{2}}\theta )-\sin \theta (-\cos \theta -\sin \theta )\]\[+\cos \theta (-\sin \theta -\cos \theta )=0\] \[\Rightarrow \]\[(\sin \theta +\cos \theta )(\sin \theta -\cos \theta +\sin \theta -\cos \theta )=0\] \[\Rightarrow \]\[2(\sin \theta +\cos \theta ).(\sin \theta -\cos \theta )=0\] \[\therefore \tan \theta =-1,\,\,1\Rightarrow \theta =\frac{\pi }{4}\in \left( 0,\,\,\frac{\pi }{2} \right)\] Statement-2: We have, \[\left| \begin{matrix} \cos \theta & \sin \theta & \cos \theta \\ \sin \theta & \cos \theta & \sin \theta \\ \cos \theta & -\sin \theta & -\cos \theta \\ \end{matrix} \right|=0\] \[\Rightarrow \cos \theta (-{{\cos }^{2}}\theta +{{\sin }^{2}}\theta )-\sin \theta (-2\sin \theta .\cos \theta )\]\[+\cos \theta (-1)=0\] \[\Rightarrow -\cos \theta ,\,\,\cos 2\theta +\sin \theta .\sin 2\theta =\cos \theta \] \[\Rightarrow \cos \theta +\cos 3\theta =0\] \[\Rightarrow \]\[2\cos 2\theta .\cos \theta =0\] \[\therefore \]\[\cos \theta =0\]or\[\cos 2\theta =0\] \[\Rightarrow \]\[\theta =\frac{\pi }{4}\in \left( 0,\,\,\frac{\pi }{2} \right)\]
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