1
JEE Main 2023 (Online) 30th January Morning Shift
Numerical
+4
-1 $$\lim_\limits{x \rightarrow 0} \frac{48}{x^{4}} \int_\limits{0}^{x} \frac{t^{3}}{t^{6}+1} \mathrm{~d} t$$ is equal to ___________.

2
JEE Main 2023 (Online) 30th January Morning Shift
Numerical
+4
-1 Let $$\alpha$$ be the area of the larger region bounded by the curve $$y^{2}=8 x$$ and the lines $$y=x$$ and $$x=2$$, which lies in the first quadrant. Then the value of $$3 \alpha$$ is equal to ___________.

3
JEE Main 2023 (Online) 25th January Evening Shift
Numerical
+4
-1 If $$\int\limits_{{1 \over 3}}^3 {|{{\log }_e}x|dx = {m \over n}{{\log }_e}\left( {{{{n^2}} \over e}} \right)}$$, where m and n are coprime natural numbers, then $${m^2} + {n^2} - 5$$ is equal to _____________.

4
JEE Main 2023 (Online) 25th January Morning Shift
Numerical
+4
-1 If the area enclosed by the parabolas $$\mathrm{P_1:2y=5x^2}$$ and $$\mathrm{P_2:x^2-y+6=0}$$ is equal to the area enclosed by $$\mathrm{P_1}$$ and $$\mathrm{y=\alpha x,\alpha > 0}$$, then $$\alpha^3$$ is equal to ____________.

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