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Question

The function $$\displaystyle f\left ( x \right )=\frac{\sec ^{-1}x}{\sqrt{x-\left [ x \right ]}}$$, where $$\displaystyle \left [ x \right ] $$ denotes greatest integer function less than or equal to $$x$$ is defined for $$x$$ belongs to


A
R
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B
R{(1,1)n:nI}
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C
R+(0,1)
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D
R+(n:nN)
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Solution

The correct option is A $$\displaystyle R-\left \{ \left ( -1,1 \right )\cup n:n\in I \right \}$$
Note that $${\sec ^{ - 1}}x$$ is defined for $$x\in \left(-\infty,-1\right] \cup \left[1,\infty\right)$$.
Also the function is not defined when $$\displaystyle \sqrt {x - \left[ x \right]}=0$$. 
Hence the function is not defined when $$x=\left[x\right]$$ or when $$x\in I$$
$$ \therefore$$ The function is defined for $$x \ \ where \ \ x\in \displaystyle R-\left \{ \left ( -1,1 \right )\cup n:n\in I \right \}$$

Mathematics

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