A) \[\frac{1}{2}<\left| a \right|<\frac{1}{\sqrt{2}}\]
B) All real values of a
C) \[\left| a \right|<1/2\]
D) \[\left| a \right|\ge \frac{1}{\sqrt{2}}\]
Correct Answer: C
Solution :
[c] \[{{\sin }^{-1}}x=2{{\sin }^{-1}}a\] \[\therefore -\frac{\pi }{2}\le {{\sin }^{-1}}x\le \frac{\pi }{2},\] \[\frac{-\pi }{2}\le 2{{\sin }^{-1}}a\le \frac{\pi }{2}\] \[\Rightarrow \frac{-\pi }{4}\le {{\sin }^{-1}}a\le \frac{\pi }{4}\] \[\therefore \frac{-1}{\sqrt{2}}\le a\le \frac{1}{\sqrt{2}}\] \[\therefore \left| a \right|\le \frac{1}{\sqrt{2}}\,\,\,\,(as\frac{1}{\sqrt{2}}>\frac{1}{2})\] Out of the given four options no one is absolutely correct, but option (3) could be take into consideration. If \[\left| a \right|<1/2\] is taken as correct, then its domain is not satisfied for \[a=1/\sqrt{3}\], but the equation is satisfied.
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