1
JEE Main 2024 (Online) 29th January Morning Shift
MCQ (Single Correct Answer)
+4
-1

$$\mathop {\lim }\limits_{x \to {\pi \over 2}} \left( {{1 \over {{{\left( {x - {\pi \over 2}} \right)}^2}}}\int\limits_{{x^3}}^{{{\left( {{\pi \over 2}} \right)}^3}} {\cos \left( {{t^{{1 \over 3}}}} \right)dt} } \right)$$ is equal to

A
$$\frac{3 \pi^2}{4}$$
B
$$\frac{3 \pi^2}{8}$$
C
$$\frac{3 \pi}{4}$$
D
$$\frac{3 \pi}{8}$$
2
JEE Main 2024 (Online) 29th January Morning Shift
MCQ (Single Correct Answer)
+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
3
JEE Main 2024 (Online) 27th January Evening Shift
MCQ (Single Correct Answer)
+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
$$\frac{\pi}{1+\mathrm{a}^2}$$
4
JEE Main 2024 (Online) 27th January Morning Shift
MCQ (Single Correct Answer)
+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
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