1
JEE Main 2023 (Online) 24th January Evening Shift
Numerical
+4
-1 Let $$f$$ be $$a$$ differentiable function defined on $$\left[ {0,{\pi \over 2}} \right]$$ such that $$f(x) > 0$$ and $$f(x) + \int_0^x {f(t)\sqrt {1 - {{({{\log }_e}f(t))}^2}} dt = e,\forall x \in \left[ {0,{\pi \over 2}} \right]}$$. Then $$\left( {6{{\log }_e}f\left( {{\pi \over 6}} \right)} \right)^2$$ is equal to __________.

2
JEE Main 2023 (Online) 24th January Evening Shift
Numerical
+4
-1 If the area of the region bounded by the curves $$y^2-2y=-x,x+y=0$$ is A, then 8 A is equal to __________

3
JEE Main 2023 (Online) 24th January Morning Shift
Numerical
+4
-1 The value of $$12\int\limits_0^3 {\left| {{x^2} - 3x + 2} \right|dx}$$ is ____________

4
JEE Main 2023 (Online) 24th January Morning Shift
Numerical
+4
-1 The value of $${8 \over \pi }\int\limits_0^{{\pi \over 2}} {{{{{(\cos x)}^{2023}}} \over {{{(\sin x)}^{2023}} + {{(\cos x)}^{2023}}}}dx}$$ is ___________

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