1
JEE Main 2024 (Online) 29th January Morning Shift
+4
-1

If the value of the integral $$\int_\limits{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(\frac{x^2 \cos x}{1+\pi^x}+\frac{1+\sin ^2 x}{1+e^{\sin x^{2123}}}\right) d x=\frac{\pi}{4}(\pi+a)-2$$, then the value of $$a$$ is

A
$$-\frac{3}{2}$$
B
3
C
$$\frac{3}{2}$$
D
2
2
JEE Main 2024 (Online) 27th January Evening Shift
+4
-1

For $$0 < \mathrm{a} < 1$$, the value of the integral $$\int_\limits0^\pi \frac{\mathrm{d} x}{1-2 \mathrm{a} \cos x+\mathrm{a}^2}$$ is :

A
$$\frac{\pi^2}{\pi+a^2}$$
B
$$\frac{\pi^2}{\pi-a^2}$$
C
$$\frac{\pi}{1-\mathrm{a}^2}$$
D
$$\frac{\pi}{1+\mathrm{a}^2}$$
3
JEE Main 2024 (Online) 27th January Morning Shift
+4
-1
If $\int\limits_0^1 \frac{1}{\sqrt{3+x}+\sqrt{1+x}} \mathrm{~d} x=\mathrm{a}+\mathrm{b} \sqrt{2}+\mathrm{c} \sqrt{3}$, where $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are rational numbers, then $2 \mathrm{a}+3 \mathrm{~b}-4 \mathrm{c}$ is equal to :
A
10
B
7
C
4
D
8
4
JEE Main 2024 (Online) 27th January Morning Shift
+4
-1
If $(a, b)$ be the orthocentre of the triangle whose vertices are $(1,2),(2,3)$ and $(3,1)$, and $\mathrm{I}_1=\int\limits_{\mathrm{a}}^{\mathrm{b}} x \sin \left(4 x-x^2\right) \mathrm{d} x, \mathrm{I}_2=\int\limits_{\mathrm{a}}^{\mathrm{b}} \sin \left(4 x-x^2\right) \mathrm{d} x$, then $36 \frac{\mathrm{I}_1}{\mathrm{I}_2}$ is equal to :
A
80
B
72
C
66
D
88
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