If $\sin\!\left(\dfrac{y}{x}\right)=\log_e|x|+\dfrac{\alpha}{2}$ is the solution of the differential equation $x\cos\!\left(\dfrac{y}{x}\right)\dfrac{dy}{dx}=y\cos\!\left(\dfrac{y}{x}\right)+x$ and $y(1)=\dfrac{\pi}{3}$, then $\alpha^{2}$ is equal to:

1

If $\log_e a,\ \log_e b,\ \log_e c$ are in an A.P. and $\log_e a-\log_e 2b,\ \log_e 2b-\log_e 3c,\ \log_e 3c-\log_e a$ …

Topic: JEE Main 2024 (29 January Evening Shift)

2

Let $y=\log_{e}\!\left(\dfrac{1-x^{2}}{1+x^{2}}\right),\ -1

Topic: JEE Main 2024 (29 January Evening Shift)

3

Let $P(3,2,3)$, $Q(4,6,2)$ and $R(7,3,2)$ be the vertices of $\triangle PQR$. Then, the angle $\angle QPR$ is:

Topic: JEE Main 2024 (29 January Evening Shift)

4

If the mean and variance of five observations are $\dfrac{24}{5}$ and $\dfrac{104}{25}$ respectively, and the mean of t…

Topic: JEE Main 2024 (29 January Evening Shift)

5

If $R$ is the smallest equivalence relation on the set $\{1,2,3,4\}$ such that $\{(1,2),(1,3)\}\subset R$, then the num…

Topic: JEE Main 2024 (29 January Evening Shift)

6

The function $f(x)=\dfrac{x}{x^{2}-6x-16}$, $x\in\mathbb{R}\setminus\{-2,8\}$:

Topic: JEE Main 2024 (29 January Evening Shift)

7

Let a unit vector $\hat{\mathbf u}=x\hat i+y\hat j+z\hat k$ make angles $\dfrac{\pi}{2},\ \dfrac{\pi}{3}$ and $\dfrac{2…

Topic: JEE Main 2024 (29 January Evening Shift)

8

The distance of the point $(2,3)$ from the line $2x-3y+28=0$, measured parallel to the line $\sqrt{3}\,x-y+1=0$, is equ…

Topic: JEE Main 2024 (29 January Evening Shift)

9

If $\displaystyle \int \frac{\sin^{2}x+\cos^{2}x}{\sqrt{\sin^{2}x\,\cos^{2}x}\;\sin(x-\theta)}\,dx = A\sqrt{\cos\theta…

Topic: JEE Main 2024 (29 January Evening Shift)

10

Let $A$ be the point of intersection of the lines $3x+2y=14$ and $5x-y=6$, and $B$ be the point of intersection of the …

Topic: JEE Main 2024 (29 January Evening Shift)

1

An integer is chosen at random from the integers $1,2,3,\ldots,50$. The probability that the chosen integer is a mult…

Topic: JEE Main 2024 (29 January Evening Shift)

2

Let $\overrightarrow{OA}=\vec a,\ \overrightarrow{OB}=12\vec a+4\vec b$ and $\overrightarrow{OC}=\vec b$, where $O$ is …

Topic: JEE Main 2024 (29 January Evening Shift)

3

The function $f(x)=2x+3\,x^{1/3},\ x\in\mathbb{R},$ has

Topic: JEE Main 2024 (29 January Evening Shift)

4

If each term of a geometric progression $a_1,a_2,a_3,\ldots$ with $a_1=\dfrac{1}{8}$ and $a_2\ne a_1$ is the arithmetic…

Topic: JEE Main 2024 (29 January Evening Shift)

5

The sum of the solutions $x \in \mathbb{R}$ of the equation $\frac{3 \cos 2 x+\cos ^3 2 x}{\cos ^6 x-\sin ^6 x}=x^3-x^2…

Topic: JEE Main 2024 (29 January Evening Shift)

6

Let $r$ and $\theta$ respectively be the modulus and amplitude of the complex number $z=2-i\!\left(2\tan\frac{5\pi}{8}\…

Topic: JEE Main 2024 (29 January Evening Shift)

7

Let $x=\dfrac{m}{n}$ ($m,n$ are co-prime natural numbers) be a solution of the equation $\cos\!\left(2\sin^{-1}x\right)…

Topic: JEE Main 2024 (29 January Evening Shift)

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Let $A=\begin{bmatrix} 2&1&2\\ 6&2&11\\ 3&3&2 \end{bmatrix} \quad\text{and}\quad P=\begin{bmatrix} 1&2&0\\ 5&0&2\\ 7&1…

Topic: JEE Main 2024 (29 January Evening Shift)