A) \[\frac{1}{2}\]
B) \[\frac{2}{3}\]
C) \[\frac{3}{4}\]
D) \[\frac{1}{4}\]
Correct Answer: B
Solution :
Picking two points \[x\] and \[y\] randomly from the intervals [0, 2] and [0, 1] is equivalent to picking a single point (x, y) randomly from the rectangle S shown in the adjacent figure, which has vertices at (0, 0), (2, 0), (2, 1) and (0, 1). |
So, we take S as our sample space. |
Now, the condition \[y\le {{x}^{2}}\] is satisfied if and only if the point (x, y) lies in the shaded region. It is the portion of the rectangle lying |
Below the parabola \[y={{x}^{2}}\] |
\[\therefore \]Required probability |
\[a=\frac{Area of the shaded region}{Area of the rec\tan gle S}\] |
Area of rectangle \[S=2\times 1=2\] |
Area of shaded region |
\[=\int\limits_{0}^{1}{{{x}^{2}}dx+1\times 1=\frac{1}{3}+1=\frac{4}{3}}\] |
\[\therefore \]Required probability\[\frac{\frac{4}{3}}{2}=\frac{2}{3}\] |
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