A) \[\frac{3}{5}\]or 1
B) \[\frac{2}{3}\] or \[-\frac{2}{3}\]
C) \[\frac{4}{5}\] or \[\frac{3}{4}\]
D) \[\pm \frac{1}{2}\]
Correct Answer: C
Solution :
\[12{{\cot }^{2}}\theta -31\text{ }\cos ec\,\theta +32=0\] \[\Rightarrow \] \[12{{\cos }^{2}}\theta -31\sin \theta +32{{\sin }^{2}}\theta =0\] \[\Rightarrow \] \[12\,(1-si{{n}^{2}}\theta )-31\sin \theta +32{{\sin }^{2}}\theta =0\] \[\Rightarrow \] \[20{{\sin }^{2}}\theta -31\sin \theta +12=0\] This is a quadratic equation in sin 9. \[\therefore \] \[\sin \theta =\frac{3\pm \sqrt{{{31}^{2}}-4.\,20\,.12}}{2.20}\] \[=\frac{31\pm \sqrt{961-960}}{40}=\frac{31\pm 1}{40}\] \[\Rightarrow \] \[\sin \theta =\frac{4}{5},\frac{3}{4}\]
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