A) \[\frac{7\pi }{24}\] or \[\frac{11\pi }{24}\]
B) \[\frac{5\pi }{24}\]
C) \[\frac{\pi }{24}\]
D) None of these
Correct Answer: A
Solution :
The given determinant (Applying \[{{R}_{1}}\to {{R}_{1}}-{{R}_{3}}\] and \[{{R}_{2}}\to {{R}_{2}}-{{R}_{3}}\]) reduces to \[\left| \,\begin{matrix} 1 & 0 & -1 \\ 0 & 1 & -1 \\ {{\sin }^{2}}\theta & {{\cos }^{2}}\theta & 1+4\sin 4\theta \\ \end{matrix}\, \right|\,=0\] \[\Rightarrow \] \[1+4\sin 4\theta +{{\cos }^{2}}\theta +{{\sin }^{2}}\theta =0\](By expanding along \[{{R}_{1}})\] Þ \[4\sin 4\theta =-2\] Þ \[\sin 4\theta =\frac{-1}{2}\] Þ \[4\theta =\frac{7\pi }{6}\] or \[\frac{11\pi }{6}\], (\[0<4\theta <2\pi \]) Since, \[0<\theta <\frac{\pi }{2}\] Þ \[0<4\theta <2\pi \] Þ \[\theta =\frac{7\pi }{24},\,\,\frac{11\pi }{24}\]
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