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1
JEE Main 2021 (Online) 17th March Evening Shift
Numerical
+4
-1
Let $${I_n} = \int_1^e {{x^{19}}{{(\log |x|)}^n}} dx$$, where n$$\in$$N. If (20)I10 = $$\alpha$$I9 + $$\beta$$I8, for natural numbers $$\alpha$$ and $$\beta$$, then $$\alpha$$ $$-$$ $$\beta$$ equals to ___________.
2
JEE Main 2021 (Online) 17th March Evening Shift
Numerical
+4
-1
Let f : [$$-$$3, 1] $$\to$$ R be given as

$$f(x) = \left\{ \matrix{ \min \,\{ (x + 6),{x^2}\}, - 3 \le x \le 0 \hfill \cr \max \,\{ \sqrt x ,{x^2}\} ,\,0 \le x \le 1. \hfill \cr} \right.$$

If the area bounded by y = f(x) and x-axis is A, then the value of 6A is equal to ___________.
3
JEE Main 2021 (Online) 17th March Morning Shift
Numerical
+4
-1
If [ . ] represents the greatest integer function, then the value of

$$\left| {\int\limits_0^{\sqrt {{\pi \over 2}} } {\left[ {[{x^2}] - \cos x} \right]dx} } \right|$$ is ____________.
Let the curve y = y(x) be the solution of the differential equation, $${{dy} \over {dx}}$$ = 2(x + 1). If the numerical value of area bounded by the curve y = y(x) and x-axis is $${{4\sqrt 8 } \over 3}$$, then the value of y(1) is equal to _________.