For $f(x)=\sin x+3x-\dfrac{2}{\pi}(x^{2}+x)$, where $x\in\left[0,\tfrac{\pi}{2}\right]$, consider: (I) $f$ is increasing in $\left(0,\tfrac{\pi}{2}\right)$. (II) $f'$ is decreasing in $\left(0,\tfrac{\pi}{2}\right)$.

1

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

Topic: JEE Main 2024 (5 April Morning Shift)

2

The value of $\int_{-\pi}^{\pi}\dfrac{2y(1+\sin y)}{1+\cos^{2}y},dy$ is:

Topic: JEE Main 2024 (5 April Morning Shift)

3

Let the line $2x+3y-k=0,\ k>0$ intersect the $x$-axis and $y$-axis at points $A$ and $B$, respectively. If the circle h…

Topic: JEE Main 2024 (5 April Morning Shift)

4

If the function $f(x)=\dfrac{\sin 3x+\alpha\sin x-\beta\cos 3x}{x^{3}},; x\in\mathbb{R},$ is continuous at $x=0$, then …

Topic: JEE Main 2024 (5 April Morning Shift)

5

Let $d$ be the distance of the point of intersection of the lines $\dfrac{x+6}{3}=\dfrac{y}{2}=\dfrac{z+1}{1}$ and $\df…

Topic: JEE Main 2024 (5 April Morning Shift)

6

If $\dfrac{1}{\sqrt{1}+\sqrt{2}}+\dfrac{1}{\sqrt{2}+\sqrt{3}}+\cdots+\dfrac{1}{\sqrt{99}+\sqrt{100}}=m$ and $\dfrac{1}{…

Topic: JEE Main 2024 (5 April Morning Shift)

7

The integral $\displaystyle \int_{0}^{\pi/4}\frac{136\sin x}{3\sin x+5\cos x},dx$ is equal to:

Topic: JEE Main 2024 (5 April Morning Shift)

8

If the system $11x+y+\lambda z=-5,\quad 2x+3y+5z=3,\quad 8x-19y-39z=\mu$ has infinitely many solutions, then $\lambda^{…

Topic: JEE Main 2024 (5 April Morning Shift)

9

The coefficients $a,b,c$ in the quadratic $ax^{2}+bx+c=0$ are chosen from the set ${1,2,3,4,5,6,7,8}$. The probability …

Topic: JEE Main 2024 (5 April Morning Shift)

10

Let $f(x)=x^{5}+2x^{3}+3x+1,; x\in\mathbb{R}$, and let $g(x)$ be a function such that $g(f(x))=x$ for all $x\in\mathbb{…

Topic: JEE Main 2024 (5 April Morning Shift)

1

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

Topic: JEE Main 2024 (5 April Morning Shift)

2

If $y=y(x)$ solves the differential equation $\dfrac{dy}{dx}+2y=\sin(2x)$ with $y(0)=\dfrac{3}{4}$, then $y!\left(\dfra…

Topic: JEE Main 2024 (5 April Morning Shift)