A) \[\left( 0,2\pi \right)\]
B) \[\left( -\pi ,0 \right)\]
C) \[\left( -\frac{\pi }{2},\frac{\pi }{2} \right)\]
D) \[\left( 0,\pi \right)\]
Correct Answer: D
Solution :
| The given equation is \[\left( \cos \,p-1 \right){{x}^{2}}+\left( \cos \,p \right)x+\sin p=0\] |
| For this equation to have real roots \[D\ge 0\]\[\Rightarrow {{\cos }^{2}}p-4\sin p\left( \cos p-1 \right)\ge 0\] |
| \[\Rightarrow {{\cos }^{2}}p-4\sin p\cos p+4{{\sin }^{2}}p+4\sin p-4{{\sin }^{2}}p\ge 0\]\[\Rightarrow {{\left( \cos p-2\sin p \right)}^{2}}+4\sin p\left( 1-\sin \,p \right)\ge 0\] |
| For every real value of \[p{{\left( \cos p-2\sin p \right)}^{2}}\ge 0\] |
| and \[1-\sin p\ge 0\therefore D\ge 0,\,\forall p\in \left( 0,\pi \right)\] |
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