List of top Mathematics Questions asked in JEE Main

The value of $\int_{0}^{3} \frac{e^x + e^{-x}}{[x]!} dx$ is equal to (where $[\, \cdot \,]$ denotes greatest integer function):

If $y = f(x)$ is the solution of the differential equation $(1 + \sin x) \frac{dy}{dx} + \cos x = 0$, such that $f(0) = 0$, then $f\left(\frac{\pi}{2}\right)$ is:

Let \( y = x \) be the equation of a chord of the circle \( C_1 \) (in the closed half-plane \( x \ge 0 \)) of diameter 10 passing through the origin. Let \( C_2 \) be another circle described on the given chord as diameter. If the equation of the chord of the circle \( C_2 \), which passes through the point \( (2, 3) \) and is farthest from the center of \( C_2 \), is \( x + ay + b = 0 \), then \( b \) is equal to:

If the distances of the point \( (1,2,a) \) from the line \[ \frac{x-1}{1}=\frac{y}{2}=\frac{z-1}{1} \] along the lines \[ L_1:\ \frac{x-1}{3}=\frac{y-2}{4}=\frac{z-a}{b} \quad \text{and} \quad L_2:\ \frac{x-1}{1}=\frac{y-2}{4}=\frac{z-a}{c} \] are equal, then \( a+b+c \) is equal to:

If all the letters of the word 'UDAYPUR' are arranged in all possible permutations and these permutations are listed in dictionary order, then the rank of the word 'UDAYPUR' is

Let $P=[p_{ij}]$ and $Q=[q_{ij}]$ be two square matrices of order $3$ such that $q_{ij}=2^{(i+j-1)}p_{ij}$ and $\det(Q)=2^{10}$. Then the value of $\det(\operatorname{adj}(\operatorname{adj} P))$ is

The value of \( \int_0^{2\pi} (\sin^4 x + \cos^4 x) \, dx \) is equal to:

\( f(x) \) is a 5 degree polynomial that has extremes at \( x = \pm 1 \) and \( \lim_{x \to 0} \frac{f(x)}{x^3} = 5 \), then \( f(2) - f(-2) \) is:

Let \( f(x) = \begin{cases} \frac{ax^2 + 2ax + 3}{4x^2 + 4x - 3}, & x \neq -\frac{3}{2}, \frac{1}{2} \\ b, & x = -\frac{3}{2}, \frac{1}{2} \end{cases} \) be continuous at \( x = -\frac{3}{2} \). If \( f(x) = \frac{7}{5} \), then \( x \) is equal to :

Let \( f(x) = \int \frac{(2 - x^2) \cdot e^x \cdot \sqrt{1 + x} \cdot (1 - x)^{3/2}}{dx} \). If \( f(0) = 0 \), then \( f\left(\frac{1}{2}\right) \) is equal to:

The sum of the coefficients of \(x^{499}\) and \(x^{500}\) in \[ (1+x)^{1000}+x(1+x)^{999}+x^2(1+x)^{998}+\cdots+x^{1000} \] is:

Let $a_1,a_2,a_3,a_4$ be an A.P. of four terms such that each term of the A.P. and its common difference are integers. If $a_1+a_2+a_3+a_4=48$ and $a_1^2a_2a_3a_4+1^4=361$, then the largest term of the A.P. is equal to

Let \( f \) be a polynomial function such that \[ f(x^2+1)=x^4+5x^2+2,\quad \text{for all } x\in\mathbb{R}. \] Then \[ \int_0^3 f(x)\,dx \] is equal to:

Let \( y = y(x) \) be the solution of the differential equation \[ x\frac{dy}{dx} - \sin 2y = x^3(2 - x^3)\cos^2 y,\; x \ne 0. \] If \( y(2) = 0 \), then \( \tan(y(1)) \) is equal to:

The largest value of $n$, for which $40^n$ divides $60!$, is

A bag contains 10 balls out of which \( k \) are red and \( (10-k) \) are black, where \( 0 \le k \le 10 \). If three balls are drawn at random without replacement and all of them are found to be black, then the probability that the bag contains 1 red and 9 black balls is:

If \[ \frac{\tan(A-B)}{\tan A}+\frac{\sin^2 C}{\sin^2 A}=1, \quad A,B,C\in\left(0,\frac{\pi}{2}\right), \] then:

The value of \[ \sum_{r=1}^{20}\sqrt{\left|\pi\left(\int_0^r x|\sin \pi x|\,dx\right)\right|} \] is:

If \( \alpha,\beta \) where \( \alpha<\beta \), are the roots of the equation \[ \lambda x^2-(\lambda+3)x+3=0 \] such that \[ \frac{1}{\alpha}-\frac{1}{\beta}=\frac{1}{3}, \] then the sum of all possible values of \( \lambda \) is:

Let \( S = \{x^3 + ax^2 + bx + c : a, b, c \in \mathbb{N} \text{ and } a, b, c \le 20\} \) be a set of polynomials. Then the number of polynomials in \( S \), which are divisible by \( x^2 + 2 \), is:

If
\[ A = \begin{bmatrix} 2 & 3 \\ 3 & 5 \end{bmatrix}, \]
then the determinant of the matrix \( A^{2025} - 3A^{2024} + A^{2023} \) is

Let $A = \{2, 3, 5, 7, 9\}$. Let $R$ be the relation on $A$ defined by $xRy$ if and only if $2x \le 3y$. Let $l$ be the number of elements in $R$, and $m$ be the minimum number of elements required to be added in $R$ to make it a symmetric relation. Then $l + m$ is equal to :