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

If $$y(x) = {\left( {{x^x}} \right)^x},\,x > 0$$, then $${{{d^2}x} \over {d{y^2}}} + 20$$ at x = 1 is equal to ____________.

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2
JEE Main 2022 (Online) 27th June Evening Shift
Numerical
+4
-1
Change Language

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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3
JEE Main 2022 (Online) 27th June Evening Shift
Numerical
+4
-1
Change Language

Let $$y = y(x)$$ be the solution of the differential equation $$(1 - {x^2})dy = \left( {xy + ({x^3} + 2)\sqrt {1 - {x^2}} } \right)dx, - 1 < x < 1$$, and $$y(0) = 0$$. If $$\int_{{{ - 1} \over 2}}^{{1 \over 2}} {\sqrt {1 - {x^2}} y(x)dx = k} $$, then k$$-$$1 is equal to _____________.

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4
JEE Main 2022 (Online) 27th June Evening Shift
Numerical
+4
-1
Out of Syllabus
Change Language

Let a circle C of radius 5 lie below the x-axis. The line L1 : 4x + 3y + 2 = 0 passes through the centre P of the circle C and intersects the line L2 = 3x $$-$$ 4y $$-$$ 11 = 0 at Q. The line L2 touches C at the point Q. Then the distance of P from the line 5x $$-$$ 12y + 51 = 0 is ______________.

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