A) a parabola whose axis is horizontal
B) a parabola whose axis is vertical
C) Integer point of the parabola \[y={{x}^{2}}\]
D) none of the these
Correct Answer: D
Solution :
| \[\because \left[ {{x}^{2}} \right]=\left[ y \right]\] |
| If \[0\le y<1,\]then,\[\left[ y \right]=0\] |
| \[\therefore \left[ {{x}^{2}} \right]=0\Rightarrow 0\le {{x}^{2}}<1\Rightarrow x\in \left( -1,1 \right)\] |
| For \[1\le y<2,then\left[ y \right]=1\therefore \left[ {{x}^{2}} \right]=1\] |
| \[\Rightarrow 1\le {{x}^{2}}<2\Rightarrow x\in \left( -\sqrt{2,-1} \right)\cup \left( 1,\sqrt{2} \right)\] |
| For \[2\le y<3,\]then\[\left[ y \right]=2\]then\[\left[ {{x}^{2}} \right]=2\] |
| \[\Rightarrow 2\le {{x}^{2}}<3\] |
| \[\therefore x\in \left( -\sqrt{3,}-\sqrt{2} \right)\cup \left( \sqrt{2,}\sqrt{3} \right)\] |
| The graph of the region will not only contain of the parabola \[y={{x}^{2}}\]but \[\left[ {{x}^{2}} \right]\]=\[\left[ y \right]\]contain points within the rectangles of side 1, 2; \[1\sqrt{2}-1;1,\sqrt{3}-\sqrt{2}\]etc. |
| Hence, [a] [b] and [c] are incorrect options. |
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