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Study Stuff for MCA Examinations - NIMCET
Study Stuff for MCA Examinations - NIMCET
Previous Year Question (PYQs)
Which of these statements about the floor and ceiling functions are correct?
Statement I: $\lfloor 2x\rfloor=\lfloor x\rfloor+\left\lfloor x+\frac{1}{2}\right\rfloor$ for all real number $x$
Statement II: $\lceil x+y\rceil=\lceil x\rceil+\lceil y\rceil$ for all real numbers $x$ and $y$
Statement I: $\lfloor 2x\rfloor=\lfloor x\rfloor+\left\lfloor x+\frac{1}{2}\right\rfloor$ for all real number $x$
Statement II: $\lceil x+y\rceil=\lceil x\rceil+\lceil y\rceil$ for all real numbers $x$ and $y$
Solution
Statement I is a standard identity of floor function:$\lfloor 2x\rfloor=\lfloor x\rfloor+\left\lfloor x+\frac{1}{2}\right\rfloor$
So, Statement I is true.
Statement II is not always true.
Take $x=0.2$ and $y=0.2$.
$\lceil x+y\rceil=\lceil 0.4\rceil=1$
But,
$\lceil x\rceil+\lceil y\rceil=\lceil 0.2\rceil+\lceil 0.2\rceil=1+1=2$
So, Statement II is false.
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