A) \[[1-\sqrt{2},\,\,1+\sqrt{2}]\]
B) \[[2-\sqrt{3},\,\,2+\sqrt{3}]\]
C) \[[0,\,\,2+\sqrt{3}]\]
D) \[\left[ \frac{1-\sqrt{2}}{2},\,\,\frac{1+\sqrt{2}}{2} \right]\]
Correct Answer: D
Solution :
We have, \[\sin x(\sin x+\cos x)=a\] \[\Rightarrow \] \[2{{\sin }^{2}}x+2\sin x\cos x=2a\] \[\Rightarrow \] \[1-\cos 2x+\sin 2x=2a\] \[\Rightarrow \] \[\sin 2x-\cos 2x=2a-1\] This equation will have real solutions, if \[|2a-1|\,\,\le \sqrt{1+1}\] \[\Rightarrow \] \[1-\sqrt{2}\le 2a\le 1+\sqrt{2}\] \[\Rightarrow \] \[a\in \left[ \frac{1-\sqrt{2}}{2},\,\,\frac{1+\sqrt{2}}{2} \right]\]
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