A) \[R\]
B) \[[1,2)\]
C) \[\left\{ \log \frac{\pi }{2} \right\}\]
D) \[\{-sin1\}\]
Correct Answer: C
Solution :
[c] \[{{\sin }^{-1}}(\log [x])\] is defined if \[-1\le \log [x]\le 1\] and \[[x]>0\] \[\Rightarrow \frac{1}{e}\le [x]\le e\Rightarrow [x]=1,2\Rightarrow x\in [1,3)\] Again, \[\log ({{\sin }^{-1}}[x])\]is defined if \[{{\sin }^{-1}}[x]>0\] and \[-1\le [x]\le 1\] \[\Rightarrow [x]>0\,\,and\,\,-1\le [x]\le 1\Rightarrow 0<[x]\le 1\] \[\Rightarrow x\in [1,2)\] \[\therefore \]Domain of \[f(x)=[1,2)\] For \[1\le x<2,[x]=1\] \[\therefore f(x)=si{{n}^{-1}}0+\log \frac{\pi }{2}=\log \frac{\pi }{2},\forall x\in [1,2)\] \[\therefore \]Range of \[f(x)=\left\{ \log \frac{\pi }{2} \right\}\]
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