Let $P$ and $Q$ be the points on the line $\dfrac{x+3}{8}=\dfrac{y-4}{2}=\dfrac{z+1}{2}$ which are at a distance of $6$ units from the point $R(1,2,3)$. If the centroid of the triangle $PQR$ is $(\alpha,\beta,\gamma)$, then $\alpha^{2}+\beta^{2}+\gamma^{2}$ is:

1

Let $\alpha$ be a non-zero real number. Suppose $f:\mathbf{R}\to\mathbf{R}$ is a differentiable function such that $f…

Topic: JEE Main 2024 (1 February Evening Shift)

2

Let the locus of the midpoints of the chords of the circle $x^{2}+(y-1)^{2}=1$ drawn from the origin intersect the line…

Topic: JEE Main 2024 (1 February Evening Shift)

3

The number of solutions of the equation $4\sin^{2}x-4\cos^{3}x+9-4\cos x=0,\; x\in[-2\pi,2\pi]$ is:

Topic: JEE Main 2024 (1 February Evening Shift)

4

Let $f(x)=\left|2x^{2}+5|x|-3\right|,\; x\in\mathbb{R}$. If $m$ and $n$ denote the number of points where $f$ is not …

Topic: JEE Main 2024 (1 February Evening Shift)

5

Consider $10$ observations $x_{1},x_{2},\ldots,x_{10}$ such that $\displaystyle \sum_{i=1}^{10}(x_{i}-\alpha)=2$ and $\…

Topic: JEE Main 2024 (1 February Evening Shift)

6

Let $P$ be a point on the ellipse $\dfrac{x^{2}}{9}+\dfrac{y^{2}}{4}=1$. Let the line passing through $P$ and parallel …

Topic: JEE Main 2024 (1 February Evening Shift)

7

Let Ajay will not appear in JEE exam with probability $p=\dfrac{2}{7}$, while both Ajay and Vijay will appear in the e…

Topic: JEE Main 2024 (1 February Evening Shift)

8

Let $\alpha$ and $\beta$ be the roots of the equation $p x^{2}+q x-r=0$, where $p\ne 0$. If $p,q,r$ are consecutive ter…

Topic: JEE Main 2024 (1 February Evening Shift)

9

If $\displaystyle \int_{0}^{\pi/3}\!\cos^{4}x\,dx=a\pi+b\sqrt{3}$, where $a$ and $b$ are rational numbers, then $9a+8b$…

Topic: JEE Main 2024 (1 February Evening Shift)

10

Let the system of equations $x+2y+3z=5,\quad 2x+3y+z=9,\quad 4x+3y+\lambda z=\mu$ have infinite number of solutions. T…

Topic: JEE Main 2024 (1 February Evening Shift)

1

The value of $\int_{0}^{1} (2x^{3} - 3x^{2} - x + 1)^{\frac{1}{3}} \, dx$ is equal to:

Topic: JEE Main 2024 (1 February Evening Shift)

2

If the mirror image of the point $P(3, 4, 9)$ in the line $\dfrac{x-1}{3} = \dfrac{y+1}{2} = \dfrac{z-2}{1}$ is $(\al…

Topic: JEE Main 2024 (1 February Evening Shift)

3

Let $S_n$ denote the sum of the first $n$ terms of an arithmetic progression. If $S_{10} = 390$ and the ratio of the t…

Topic: JEE Main 2024 (1 February Evening Shift)

4

Let $m$ and $n$ be the coefficients of the seventh and thirteenth terms respectively in the expansion of $\left(\dfrac…

Topic: JEE Main 2024 (1 February Evening Shift)

5

Let $f(x)= \begin{cases} x-1, & x \text{ is even},\\ 2x, & x \text{ is odd}, \end{cases}\quad x\in\mathbb N.$ If for …

Topic: JEE Main 2024 (1 February Evening Shift)