question_answer

Area of the square formed by\[|x|+|y|\,\,=1\]is:

A) \[0\,\,sq\,\,unit\]                           

B) \[1\,\,sq\,\,unit\]

C) \[2\,\,sq\,\,unit\]                           

D) \[4\,\,sq\,\,unit\]

Correct Answer: C

Solution :

Given equation of curve is                 \[|x|+|y|\,\,=1\] \[\therefore \]Required area of curve                 \[=\]Area of curve\[ABCD\]                 \[=4\]area of curve\[OAB\]                 \[=4\int_{0}^{1}{(1-x)}\,dx\]                 \[=4\left[ x-\frac{{{x}^{2}}}{2} \right]_{0}^{1}\]                 \[=4\left[ 1-\frac{1}{2} \right]=2\,\,sq\,\,unit\] Alternative Solution: It is clear from the figure that given curve is a square whose length is\[\sqrt{2}\]. \[\therefore \]Area of square\[={{(\sqrt{2})}^{2}}\]                                  \[=2\,\,sq\,\,unit\]