Question

If $x\ge 0$ and $y\ge 0$, then the area bounded by the graph of $\left[x\right]+\left[y\right]=2$ is (where $[.]$ denotes greatest integer function)

• A. $4sq.unit$
• B. $1sq.unit$
• C. $2sq.unit$
• D. $3sq.unit$

Answer 1

The correct option is D $3sq.unit$

According to question,

$\begin{array}{c}\left[x\right]+\left[y\right]=2\\ then,\left[x\right],\left[y\right]=0,1,2\\ case\left(i\right)\left[y\right]=0,\Rightarrow 0\le y<1\\ 2\le x<3\\ A_1=1unit\\ case\left(ii\right)\left[y\right]=1,\Rightarrow 1\le y<2,1\le x<2\\ A_2=1unit\end{array}$

$\begin{array}{c}case\left(iii\right)\left[y\right]=2\Rightarrow 2\le y<3\\ \Rightarrow 0\le x<1\\ A_3=1unit\\ Area=A_1+A_2+A_3\\ =1+1+1=3sq.unit\\ \end{array}$

So, that the correct option is D.