Let $f(\theta)=3\big(\sin^{4}\!\left(\tfrac{3\pi}{2}-\theta\right)+\sin^{4}\!(3\pi+\theta)\big)-2\big(1-\sin^{2}2\theta\big)$ and $S=\left\{\theta\in[0,\pi]:\, f'(\theta)=-\dfrac{\sqrt{3}}{2}\right\}$. If $4\beta=\displaystyle\sum_{\theta\in S}\theta$, then $f(\beta)$ is equal to:

1

Three rotten apples are mixed accidentally with seven good apples and four apples are drawn one by one without replacem…

Topic: JEE Main 2023 (29 January Morning Shift)

2

Let $B$ and $C$ be the two points on the line $y+x=0$ such that $B$ and $C$ are symmetric with respect to the origin. …

Topic: JEE Main 2023 (29 January Morning Shift)

3

Let $\lambda\ne 0$ be a real number. Let $\alpha,\beta$ be the roots of the equation $14x^{2}-31x+3\lambda=0$ and $\alp…

Topic: JEE Main 2023 (29 January Morning Shift)

1

Fifteen football players of a club-team are given 15 T-shirts with their names written on the backside. If the player…

Topic: JEE Main 2023 (29 January Morning Shift)

2

Let the tangents at the points $A(4,-11)$ and $B(8,-5)$ on the circle $x^{2}+y^{2}-3x+10y-15=0$, intersect at the point…

Topic: JEE Main 2023 (29 January Morning Shift)

3

Let $f(x)=x+\dfrac{a}{\pi^{2}-4}\sin x+\dfrac{b}{\pi^{2}-4}\cos x,\ x\in\mathbb{R}$ be a function which satisfies $\d…

Topic: JEE Main 2023 (29 January Morning Shift)

4

Let $\alpha$ and $\beta$ be real numbers. Consider a $3\times 3$ matrix $A$ such that $A^{2}=3A+\alpha I$. If $A^{4}=…

Topic: JEE Main 2023 (29 January Morning Shift)

5

A light ray emits from the origin making an angle $30^\circ$ with the positive $x$-axis. After getting reflected by t…

Topic: JEE Main 2023 (29 January Morning Shift)

6

For two non-zero complex numbers $z_{1}$ and $z_{2}$, if $\operatorname{Re}(z_{1}z_{2})=0$ and $\operatorname{Re}(z_{1}…

Topic: JEE Main 2023 (29 January Morning Shift)

7

Let $\Delta$ be the area of the region $\{(x,y)\in\mathbb{R}^{2}:\ x^{2}+y^{2}\le 21,\ y^{2}\le 4x,\ x\ge 1\}$. Then …

Topic: JEE Main 2023 (29 January Morning Shift)

8

The domain of $$f(x)=\frac{\log_{(x+1)}(x-2)}{e^{2\log_e x}-(2x+3)},\quad x\in\mathbb{R}$$ is:

Topic: JEE Main 2023 (29 January Morning Shift)

9

Let $f:\mathbb{R}\to\mathbb{R}$ be a function such that $$f(x)=\frac{x^{2}+2x+1}{x^{2}+1}.$$ Then:

Topic: JEE Main 2023 (29 January Morning Shift)

10

Let $[x]$ denote the greatest integer $\le x$. Consider the function $$f(x)=\max\{x^{2},\,1+[x]\}.$$ Then the value o…

Topic: JEE Main 2023 (29 January Morning Shift)