Question

$S_1:[X+\frac{1}{2}]=\left[X\right]$, if $X<\frac{1}{2}$
$S_2:\left[nx\right]=n\left[x\right]$, if $x<\frac{1}{n}$
where [ ], { } stands for greatest integer and fraction part functions respectively and n is a natural number

• A. $S_1$ and $S_2$ both are correct
• B. $S_1$ is correct and $S_2$ is INCORRECT
• C. $S_1$ is INCORRECT and $S_2$ is correct
• D. $S_1$ and $S_2$ both are INCORRECT

Answer 1

The correct option is A $S_1$ and $S_2$ both are correct

$S_1:[X+\frac{1}{2}]=\left[X\right]$ if $X<\frac{1}{2}$-(i)

$S_2:\left[nx\right]=n\left[x\right]$ if $x<\frac{1}{n}$-(ii)

from $\left(i\right)and\left(ii\right)$

$\left[2x\right]=2n$

$\Rightarrow 2n\le 2x<2n+1$

$n\le x<n+\frac{1}{2}$

$\therefore n\le x+\frac{1}{2}<n+1$

$\Rightarrow [x+\frac{1}{2}]=n$

$\therefore S1andS2$ both are correct.