. The value of a for which roots of this equation are real and distinct, is
A)
B)
C)
D)
Correct Answer: A
Solution :
\[\therefore \] The roots of equation \[(sin\,a) {{x}^{2}}+ (sin\,a) x+ (1 - cos\,\,a)= 0\] are real and distinct. \[\therefore \,\,si{{n}^{2}}a -\,\,4 sin a \left( 1 - cos a \right) > 0\] \[\Rightarrow \,\,\,\sin \,a >0 or (sin\,a -4 + 4 cos\,a) >0\] \[\Rightarrow \,\,\,\operatorname{a}\in (0,\,\,\pi )\,\,or\,\,\frac{1-\cos \,a}{\sin \,a}<\frac{1}{4}\] \[\Rightarrow \,\,\,\operatorname{a}\in (0,\,\,\pi )\,\,or\,\,a\in \left( 0,\,\,2{{\tan }^{-1}}\left( \frac{1}{4} \right) \right]\] \[\Rightarrow \,\,\,\operatorname{a}\in \left( 0,\,\,2\,\,{{\tan }^{-1}}\left( \frac{1}{4} \right) \right]\]
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