JEE MAIN 2024 Previous Year Questions (PYQs) – Page 37 of 39

Consider $10$ observations $x_{1},x_{2},\ldots,x_{10}$ such that $\displaystyle \sum_{i=1}^{10}(x_{i}-\alpha)=2$ and $\displaystyle \sum_{i=1}^{10}(x_{i}-\beta)^{2}=40$, where $\alpha,\beta$ are positive integers. Let the mean and the variance of the observations be $\dfrac{6}{5}$ and $\dfrac{84}{25}$ respectively. Then $\dfrac{\beta}{\alpha}$ is equal to:

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 continuous and not differentiable respectively, then $m+n$ is equal to:

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:

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 $x+y=1$ at $P$ and $Q$. Then, the length of $PQ$ is:

Let $\alpha$ be a non-zero real number. Suppose $f:\mathbf{R}\to\mathbf{R}$ is a differentiable function such that $f(0)=2$ and $\displaystyle \lim_{x\to -\infty} f(x)=1$. If $f'(x)=\alpha f(x)+3$, for all $x\in\mathbf{R}$, then $f(-\log_{e}2)$ is equal to:

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:

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

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 $(\alpha, \beta, \gamma)$, then $14(\alpha + \beta + \gamma)$ is:

Let $S_n$ denote the sum of the first $n$ terms of an arithmetic progression. If $S_{10} = 390$ and the ratio of the tenth and the fifth terms is $15 : 7$, then $S_{15} - S_{5}$ is equal to:

Let $m$ and $n$ be the coefficients of the seventh and thirteenth terms respectively in the expansion of $\left(\dfrac{1}{3}x^{\tfrac13}+\dfrac{1}{2x^{\tfrac23}}\right)^{18}$. Then $\left(\dfrac{n}{m}\right)^{\tfrac13}$ is: