1
JEE Main 2024 (Online) 5th April Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

The value of $$\int_\limits{-\pi}^\pi \frac{2 y(1+\sin y)}{1+\cos ^2 y} d y$$ is :

A
$$\frac{\pi}{2}$$
B
$$\pi^2$$
C
$$\frac{\pi^2}{2}$$
D
$$2 \pi^2$$
2
JEE Main 2024 (Online) 5th April Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Suppose $$\theta \in\left[0, \frac{\pi}{4}\right]$$ is a solution of $$4 \cos \theta-3 \sin \theta=1$$. Then $$\cos \theta$$ is equal to :

A
$$\frac{6-\sqrt{6}}{(3 \sqrt{6}-2)}$$
B
$$\frac{4}{(3 \sqrt{6}+2)}$$
C
$$\frac{6+\sqrt{6}}{(3 \sqrt{6}+2)}$$
D
$$\frac{4}{(3 \sqrt{6}-2)}$$
3
JEE Main 2024 (Online) 5th April Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

For the function

$$f(x)=\sin x+3 x-\frac{2}{\pi}\left(x^2+x\right), \text { where } x \in\left[0, \frac{\pi}{2}\right],$$

consider the following two statements :

(I) $$f$$ is increasing in $$\left(0, \frac{\pi}{2}\right)$$.

(II) $$f^{\prime}$$ is decreasing in $$\left(0, \frac{\pi}{2}\right)$$.

Between the above two statements,

A
only (I) is true.
B
both (I) and (II) are true.
C
only (II) is true.
D
neither (I) nor (II) is true.
4
JEE Main 2024 (Online) 5th April Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

If $$\mathrm{A}(1,-1,2), \mathrm{B}(5,7,-6), \mathrm{C}(3,4,-10)$$ and $$\mathrm{D}(-1,-4,-2)$$ are the vertices of a quadrilateral ABCD, then its area is :

A
$$24 \sqrt{7}$$
B
$$48 \sqrt{7}$$
C
$$24 \sqrt{29}$$
D
$$12 \sqrt{29}$$
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