1
JEE Main 2022 (Online) 30th June Morning Shift
Numerical
+4
-1

If for some $$\alpha$$ > 0, the area of the region $$\{ (x,y):|x + \alpha | \le y \le 2 - |x|\}$$ is equal to $${3 \over 2}$$, then the area of the region $$\{ (x,y):0 \le y \le x + 2\alpha ,\,|x| \le 1\}$$ is equal to ____________.

2
JEE Main 2022 (Online) 30th June Morning Shift
Numerical
+4
-1

Let $$f(t) = \int\limits_0^t {{e^{{x^3}}}\left( {{{{x^8}} \over {{{({x^6} + 2{x^3} + 2)}^2}}}} \right)dx}$$. If $$f(1) + f'(1) = \alpha e - {1 \over 6}$$, then the value of 150$$\alpha$$ is equal to ___________.

3
JEE Main 2022 (Online) 29th June Evening Shift
Numerical
+4
-1

For real numbers a, b (a > b > 0), let

Area $$\left\{ {(x,y):{x^2} + {y^2} \le {a^2}\,and\,{{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} \ge 1} \right\} = 30\pi$$

and

Area $$\left\{ {(x,y):{x^2} + {y^2} \le {b^2}\,and\,{{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} \le 1} \right\} = 18\pi$$

Then, the value of (a $$-$$ b)2 is equal to ___________.

4
JEE Main 2022 (Online) 27th June Evening Shift
Numerical
+4
-1

If the area of the region $$\left\{ {(x,y):{x^{{2 \over 3}}} + {y^{{2 \over 3}}} \le 1,\,x + y \ge 0,\,y \ge 0} \right\}$$ is A, then $${{256A} \over \pi }$$ is equal to __________.

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