A) \[2\le k\le 6\]
B) \[k>2\]
C) \[k>6\]
D) \[k<2\]
Correct Answer: A
Solution :
Given,\[k\sin \theta +\cos 2\theta =2k-7\] \[\Rightarrow \]\[k\sin \theta +1-2{{\sin }^{2}}\theta =2k-7\] \[\Rightarrow \]\[2{{\sin }^{2}}\theta -k\sin \theta +(2k-8)=0\] For the existence of real roots, discriminant\[\ge 0.\] \[\Rightarrow \]\[{{k}^{2}}-4\times 2(2k-8)\ge 0\Rightarrow {{(k-8)}^{2}}\ge 0\], which is always true. Roots of the quadratic equation are \[\frac{k-4}{2},\,\,2\]but\[\sin \theta \ne 2\]. \[\therefore \] \[\sin \theta =\frac{k-4}{2}\]but\[-1\le \sin \theta \le 1\]. \[\Rightarrow \]\[-1\le \frac{k-4}{2}\le 1\Rightarrow -2\le k-4\le 2\Rightarrow 2\le k\le 6\]
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