A) \[p\in (-\pi ,0)\]
B) \[p\in \left( -\frac{\pi }{2},\frac{\pi }{2} \right)\]
C) \[p\in (0,\pi )\]
D) \[p\in (0,\,2\pi )\]
Correct Answer: C
Solution :
Given equation \[(\cos \,p-1){{x}^{2}}+(\cos p)x+\sin p=0\] Its discriminant \[D\ge 0\] [\[\because \] roots are real] \[\Rightarrow \,{{\cos }^{2}}p-4\,(\cos \,p-1)\,\sin p\ge 0\] \[\Rightarrow \,{{\cos }^{2}}p-4\,\,\cos p\sin p+4\sin p\ge 0\] \[\Rightarrow \,\,{{(\cos p-2\sin p)}^{2}}-4{{\sin }^{2}}p+4\sin p\ge 0\] \[\Rightarrow \,\,{{(\cos p-2\sin p)}^{2}}+4\sin p(1-\sin p)\ge 0\] ?.(i) Now \[(1-\sin p)\ge 0\]for all real p, \[\sin p>0\] for \[0<p<\pi .\] Therefore, \[4\sin p(1-\sin p)\ge 0\] when \[0<p<\pi \]or \[p\in (0,\pi ).\]
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