| Which of the following statement is not possible? |
| I. \[2{{\sin }^{2}}\theta -\cos \theta +4=0\] |
| II. \[{{\sin }^{2}}\theta +\cos e{{c}^{2}}\theta \ge 2\] |
| III. \[4{{\sin }^{2}}\theta +4\sin \theta -15=0\] |
A) Only II
B) I and II
C) I and III
D) None of these
Correct Answer: A
Solution :
| I. \[2{{\sin }^{2}}\theta -\cos \theta +4=0\] |
| \[\Rightarrow \] \[2\,\,(1-{{\cos }^{2}}\theta )-\cos \theta +4=0\] |
| \[\Rightarrow \] \[2{{\cos }^{2}}\theta +\cos \theta -6=0\] |
| \[\therefore \] \[\cos \theta =\frac{-1\pm \sqrt{49}}{4}=\frac{-1\pm 7}{4}\] |
| \[\Rightarrow \] \[\cos \theta =\frac{3}{2}\]or \[-\,\,2\] |
| But \[-1\le \cos \theta \le 1\]so |
| \[\cos \theta \ne \frac{3}{2}\]or\[-\,\,2\]hence not possible. |
| II. \[{{\sin }^{2}}\theta +\text{cose}{{\text{c}}^{\text{2}}}\theta \] |
| \[={{\sin }^{2}}\theta +\text{cose}{{\text{c}}^{2}}\theta -2\sin \theta \,\,\text{cosec}\theta +\text{2}\,\,\text{sin}\theta \,\,\text{cosec}\theta \]\[=(\sin \theta -\text{cosec}{{\theta }^{2}}+2)\] |
| \[\Rightarrow \] \[{{(\sin \theta -\text{cosec}\theta )}^{2}}\ge 0\] |
| \[\therefore \] \[{{(\sin \theta -\text{cosec}\theta )}^{2}}+2\ge 2\] |
| \[\Rightarrow \] \[{{\sin }^{2}}\theta +\text{cose}{{\text{c}}^{2}}\theta \ge 2\] |
| The least value of \[{{\sin }^{2}}\theta +\text{cose}{{\text{c}}^{2}}\theta \]is 2. |
| Here, statement is true. |
| III. Not possible like statement I |
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