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:
Previous 10 Questions — JEE Main 2024 (1 February Evening Shift)
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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)
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)
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)
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)
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)
Consider the relations $R_1$ and $R_2$ defined as
$a\,R_1\,b \iff a^2 + b^2 = 1$ for all $a,b\in\mathbb{R}$,
and …
Topic: JEE Main 2024 (1 February Evening Shift)
Consider a $\triangle ABC$ where $A(1,3,2)$, $B(-2,8,0)$ and $C(3,6,7)$.
If the angle bisector of $\angle BAC$ meets …
Topic: JEE Main 2024 (1 February Evening Shift)
If $z$ is a complex number such that $|z|\le1$, then the minimum value of
$\left|z+\dfrac{1}{2}(3+4i)\right|$ is:
Topic: JEE Main 2024 (1 February Evening Shift)
If the domain of the function
$f(x)=\sqrt{\dfrac{x^{2}-25}{4-x^{2}}}+\log_{10}(x^{2}+2x-15)$
is $(-\infty,\alpha)\c…
Topic: JEE Main 2024 (1 February Evening Shift)
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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)
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)
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)
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)
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 $6$…
Topic: JEE Main 2024 (1 February Evening Shift)
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)
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)
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)
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)
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)
