List of top Questions asked in JEE Main

(X) $\xrightarrow[\text{FeBr}_3]{\text{Br}_2}$ (i) $\text{NaOH}$, 623 K, 300 atm; (ii) dil. HCl $\xrightarrow{\text{HNO}_3}$ A + B. The organic product (A) and (B) can be separated by :
In the given situation, the force acting at the center on a \(1\,\text{kg}\) mass is \(F_1\). Now, if the masses \(4m\) and \(3m\) are interchanged, the force becomes \(F_2\). Given that \[ \frac{F_1}{F_2}=\frac{2}{\sqrt{\alpha}}, \] find the value of \(\alpha\).
Consider a set \( S = \{a, b, c, d\} \). Then the number of reflexive as well as symmetric relations from \( S \to S \) is:
A rectangle is formed by lines \( x = 0, y = 0, x = 3 \) and \( y = 4 \). A line perpendicular to \( 3x + 4y + 6 = 0 \) divides the rectangle into two equal parts. Then the distance of the line from the point \( \left( -1, \frac{3}{2} \right) \) is:
Statement--1: Liquid pressure is only exerted on solid surfaces in contact and is exerted in between the layers of liquid. Statement--2: Surface tension arises due to difference in potential energy of molecules in the bulk of liquid and at the surface.
Consider a geometric sequence $729,\,81,\,9,\,1,\ldots$ If $P_n$ denotes the product of first $n$ terms of the G.P. such that \[ \sum_{n=1}^{40} (P_n)^{\frac{1}{n}}=\frac{3^{\alpha}-1}{2\times 3^{\beta}}, \] then find the value of $(\alpha+\beta)$.
The function \[ f(x)=\sin 2x+2\cos x,\qquad x\in\left(-\frac{3\pi}{4},\frac{3\pi}{4}\right) \] has:
Consider the 10 observations 2, 3, 5, 10, 11, 13, 15, 21, a and b such that mean of observation is a and variance is 34.2. Then the mean deviation about median, is :
Let \(f:[-2a,2a]\to\mathbb{R}\) be a thrice differentiable function and define \[ g(x)=f(a+x)+f(a-x). \] If \(m\) is the minimum number of roots of \(g'(x)=0\) in the interval \((-a,a)\) and \(n\) is the minimum number of roots of \(g''(x)=0\) in the interval \((-a,a)\), then \(m+n\) is equal to:
If the image of a point \(P(3,2,1)\) in the line \[ \frac{x-1}{1}=\frac{y}{2}=\frac{z-1}{1} \] is \(Q\), then the distance of \(Q\) from the line \[ \frac{x-9}{3}=\frac{y-9}{2}=\frac{z-5}{-2} \] is:
If dataset $A=\{1,2,3,\ldots,19\}$ and dataset $B=\{ax+b;\,x\in A\}$. If mean of $B$ is $30$ and variance of $B$ is $750$, then sum of possible values of $b$ is
The sum of coefficients of \(x^{499}\) and \(x^{500}\) in the expression: \[ (1+x)^{1000} + x(1+x)^{999} + x^2(1+x)^{998} + \cdots + x^{1000} \] is:
If \( y = \operatorname{sgn}(\sin x) + \operatorname{sgn}(\cos x) + \operatorname{sgn}(\tan x) + \operatorname{sgn}(\cot x) \), where \(\operatorname{sgn}(p)\) denotes the signum function of \(p\), then the sum of elements in the range of \(y\) is:
The value of \(S = \sum_{r=1}^{20} \sqrt{\pi \int_{0}^{r} x |\sin \pi x| dx}\) is :
Let \(\tan \left( \frac{\pi}{4} + \frac{1}{2} \cos^{-1} \frac{2}{3} \right) + \tan \left( \frac{\pi}{4} - \frac{1}{2} \sin^{-1} \frac{2}{3} \right) = k\). Then number of solution of the equation \(\sin^{-1}(kx - 1) = \sin x - \cos^{-1} x\) is/are :
If the domain of \[ f(x)=\log_{(10x^2-17x+7)}\,(18x^2-11x+1) \] is $(-\infty,a)\cup(b,c)\cup(d,\infty)-\{e\}$, then find $90(a+b+c+d+e)$.
If \[ \sum_{r=1}^{25}\frac{r}{r^4+r^2+1}=\frac{p}{q}, \] where \(p\) and \(q\) are coprime positive integers, then \(p+q\) is equal to:
Evaluate \[ \left(\frac{4}{7}+\frac{1}{3}\right)+\left(\frac{4}{7}+\frac{4}{3}\right)\left(\frac{1}{3}\right) +\left(\frac{4}{7}+\frac{4}{3}\right)^2\left(\frac{1}{3}\right)^2+\cdots \]
Let \(\int \frac{1 - 5 \cos^2 x}{\sin^5 x \cos^2 x} dx = f(x) + c\) then find \(f\left(\frac{\pi}{4}\right) - f\left(\frac{\pi}{6}\right)\) :
Statement 1 : The function \(f:\mathbb{R}\to\mathbb{R}\) defined by \[ f(x)=\frac{x}{1+|x|} \] is one–one.
Statement 2 : The function \(f:\mathbb{R}\to\mathbb{R}\) defined by \[ f(x)=\frac{x^2+4x-30}{x^2-8x+18} \] is many–one.
Which of the following is correct?
If \(\alpha, \beta\) are roots of the quadratic equation \[ \lambda x^2 - (\lambda+3)x + 3 = 0 \] and \(\alpha<\beta\) such that \[ \frac{1}{\alpha} - \frac{1}{\beta} = \frac{1}{3}, \] then find the sum of all possible values of \(\lambda\).
If \(g(x) = 3x^2 + 2x - 3, f(0) = -3, 4g(f(x)) = 3x^2 - 32x + 72\) then find \(f(g(2))\) where \(f(x)>0\) for all valid x:
Let \(x \frac{dy}{dx} - \sin 2y = x^3(2 - x^3) \cos^2 y ; y(2) = 0\), then find \(\tan(y(1))\) :
Consider the 10 observations \(2, 3, 5, 10, 11, 13, 15, 21, a\) and \(b\) such that the mean of observations is \(9\) and variance is \(34.2\). Then the mean deviation about median is:
If \(f(x^2 + 1) = x^4 + 5x^2 + 1\), then find \(\int_{0}^{3} f(x) dx\) :