JEE MAIN - Mathematics (2020 - 6th September Morning Slot - No. 1)
The area (in sq. units) of the region
A = {(x, y) : |x| + |y| $$ \le $$ 1, 2y2 $$ \ge $$ |x|}
A = {(x, y) : |x| + |y| $$ \le $$ 1, 2y2 $$ \ge $$ |x|}
$${1 \over 6}$$
$${5 \over 6}$$
$${1 \over 3}$$
$${7 \over 6}$$
Explanation
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For point of intersection
x + y = 1 $$ \Rightarrow $$ x = 1 – y
y2 = $${x \over 2}$$ $$ \Rightarrow $$ 2y2 = x
2y2 = 1 – y $$ \Rightarrow $$ 2y2 + y – 1 = 0
$$ \Rightarrow $$ (2y – 1) (y + 1) = 0
$$ \Rightarrow $$ y = $${1 \over 2}$$ or -1
Total area = $$4\int\limits_0^{{1 \over 2}} {\left[ {\left( {1 - x} \right) - \left( {\sqrt {{x \over 2}} } \right)} \right]} dx$$
= $$4\left[ {x - {{{x^2}} \over 2} - {1 \over {\sqrt 2 }}{{{x^{3/2}}} \over {3/2}}} \right]_0^{{1 \over 2}}$$
= $$4\left[ {{1 \over 2} - {1 \over 8} - {{\sqrt 2 } \over 3}{{\left( {{1 \over 2}} \right)}^{3/2}}} \right]$$
= 4 $$ \times $$ $${5 \over {24}}$$ = $${5 \over 6}$$
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