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

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

A function $$y=f(x)$$ satisfies $$f(x) \sin 2 x+\sin x-\left(1+\cos ^2 x\right) f^{\prime}(x)=0$$ with condition $$f(0)=0$$. Then, $$f\left(\frac{\pi}{2}\right)$$ is equal to

A
2
B
1
C
$$-$$1
D
0
3
JEE Main 2024 (Online) 29th January Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

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

The area (in sq. units) of the part of the circle $$x^2+y^2=169$$ which is below the line $$5 x-y=13$$ is $$\frac{\pi \alpha}{2 \beta}-\frac{65}{2}+\frac{\alpha}{\beta} \sin ^{-1}\left(\frac{12}{13}\right)$$, where $$\alpha, \beta$$ are coprime numbers. Then $$\alpha+\beta$$ is equal to __________.

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