List of top Mathematics Questions asked in JEE Main

Let a relation \( R \) on \( \mathbb{N} \times \mathbb{N} \) be defined as: $$(x_1, y_1) \, R \, (x_2, y_2) \text{ if and only if } x_1 \leq x_2 \text{ or } y_1 \leq y_2.$$ 
Consider the two statements:
[(I)] \( R \) is reflexive but not symmetric. 
[(II)] \( R \) is transitive.
Then which one of the following is true:

The sum of all rational terms in the expansion of $$ \left( \frac{1}{2^5} + \frac{1}{5^3} \right)^{15} $$ is equal to: 

Let \( f : \mathbb{R} \rightarrow \mathbb{R} \) be a function defined by
$$ f(x) = \frac{x}{(1 + x^4)^{1/4}} $$
and \( g(x) = f(f(f(x))) \). Then
$$ 18 \int_{\sqrt[3]{\frac{8}{3}}}^{\sqrt[3]{\frac{4}{3}}} x^3 g(x) \, dx $$
equals: 

If \[\int \frac{1}{a^2 \sin^2 x + b^2 \cos^2 x} \, dx = \frac{1}{12} \tan^{-1}(3 \tan x) + \text{constant},\]then the maximum value of $a \sin x + b \cos x$ is:

Let \( \alpha, \beta \) be the distinct roots of the equation $$ x^2 - (t^2 - 5t + 6)x + 1 = 0, \, t \in \mathbb{R} \, \text{and} \, a_n = \alpha^n + \beta^n. $$ Then the minimum value of \( \frac{a_{2023} + a_{2025}}{a_{2024}} \) is: 

The integral $$ \int \frac{x^8 - x^2}{(x^{12} + 3x^6 + 1) \tan^{-1}\left( \frac{x^3 + 1}{x^3} \right)} \, dx $$ is equal to: 

Consider the line $L$ passing through the points $(1, 2, 3)$ and $(2, 3, 5)$. The distance of the point $$ \left( \frac{11}{3}, \frac{11}{3}, \frac{19}{3} \right) $$ from the line $L$ along the line $$ \frac{3x - 11}{2} = \frac{3y - 11}{1} = \frac{3z - 19}{2} $$ is equal to: 

Let \( C: x^2 + y^2 = 4 \) and \( C': x^2 + y^2 - 4\lambda x + 9 = 0 \) be two circles. If the set of all values of \( \lambda \) such that the circles \( C \) and \( C' \) intersect at two distinct points is \( R = [a, b] \), then the point \( (8a + 12, 16b - 20) \) lies on the curve:

Let \( P \) be a point on the hyperbola \( H: \frac{x^2}{9} - \frac{y^2}{4} = 1 \), in the first quadrant such that the area of the triangle formed by \( P \) and the two foci of \( H \) is \( 2 \sqrt{13} \). Then, the square of the distance of \( P \) from the origin is

If \[ f(x) = \begin{vmatrix} 2 \cos^4 x & 2 \sin^4 x & 3 + \sin^2 2x \\ 3 + 2 \cos^4 x & 2 \sin^4 x & \sin^2 2x \\ 2 \cos^4 x & 3 + 2 \sin^4 x & \sin^2 2x \end{vmatrix} \] then \( \frac{1}{5} f'(0) \) is equal to

Suppose AB is a focal chord of the parabola \( y^2 = 12x \) of length \( l \) and slope \( m<\sqrt{3} \). If the distance of the chord AB from the origin is \( d \), then \( ld^2 \) is equal to _________.

The 20^th term from the end of the progression \[ 20, 19, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, \ldots, -129 \frac{1}{4} \] is:

Let \( A (a, b), B(3, 4) \) and \( (-6, -8) \) respectively denote the centroid, circumcentre, and orthocentre of a triangle. Then, the distance of the point \( P(2a + 3, 7b + 5) \) from the line \( 2x + 3y - 4 = 0 \) measured parallel to the line \( x - 2y - 1 = 0 \) is

\[ \lim_{x \to \frac{\pi}{2}} \frac{\int_{x^3}^{\left(\frac{\pi}{2}\right)^3} \left( \sin\left(2t^{1/3}\right) + \cos\left(t^{1/3}\right) \right) \, dt}{\left( x - \frac{\pi}{2} \right)^2} \] is equal to:

The square of the distance of the image of the point \( (6, 1, 5) \) in the line \[ \frac{x - 1}{3} = \frac{y}{2} = \frac{z - 2}{4}, \] from the origin is _____ .

The position vectors of the vertices \( A, B \) and \( C \) of a triangle are \[ 2\mathbf{i} - 3\mathbf{j} + 3\mathbf{k}, \quad 2\mathbf{i} + 2\mathbf{j} + 3\mathbf{k} \quad \text{and} \quad -\mathbf{i} + \mathbf{j} + 3\mathbf{k} \] respectively. Let \( l \) denote the length of the angle bisector \( AD \) of \( \angle BAC \) where \( D \) is on the line segment \( BC \). Then \( 2l^2 \) equals:

Let \( f : \mathbb{R} \to \mathbb{R} \) be a function given by
\[f(x) = \begin{cases} \frac{1 - \cos 2x}{x^2}, & x < 0 \\\alpha, & x = 0, \text{ where } \alpha, \beta \in \mathbb{R}. \\\beta \sqrt{1 - \cos x} / x, & x > 0 \end{cases} \]
If \( f \) is continuous at \( x = 0 \), then \( \alpha^2 + \beta^2 \) is equal to:

Let the point, on the line passing through the points \( P(1, -2, 3) \) and \( Q(5, -4, 7) \), farther from the origin and at a distance of 9 units from the point \( P \), be \( (\alpha, \beta, \gamma) \). Then \( \alpha^2 + \beta^2 + \gamma^2 \) is equal to:

Let \( f(x) = \begin{cases} -2, & -2 \leq x \leq 0 \\ x - 2, & 0 < x \leq 2 \end{cases} \) and \( h(x) = f(|x|) + |f(x)| \). Then \(\int_{-2}^{2} h(x) \, dx\) is equal to:

Let a unit vector which makes an angle of \(60^\circ\) with \( 2\hat{i} + 2\hat{j} - \hat{k} \) and an angle of \(45^\circ\) with \( \hat{i} - \hat{k} \) be \( \vec{C} \). Then \( \vec{C} + \left( -\frac{1}{2} \hat{i} + \frac{1}{3\sqrt{2}} \hat{j} - \frac{\sqrt{2}}{3} \hat{k} \right) \) is:

Let the sum of the maximum and the minimum values of the function \( f(x) = \frac{2x^2 - 3x + 8}{2x^2 + 3x + 8} \) be \( \frac{m}{n} \), where \( \text{gcd}(m, n) = 1 \). Then \( m + n \) is equal to:

One of the points of intersection of the curves \( y = 1 + 3x - 2x^2 \) and \( y = \frac{1}{x} \) is \( \left( \frac{1}{2}, 2 \right) \). Let the area of the region enclosed by these curves be \[\frac{1}{24} \left( \ell \sqrt{5} + m \right) - n \log_e \left( 1 + \sqrt{5} \right),\] where \( \ell, m, n \in \mathbb{N} \). Then \( \ell + m + n \) is equal to:

Let \( a, b \in \mathbb{R} \). Let the mean and the variance of 6 observations \(-3, 4, 7, -6, a, b\) be 2 and 23, respectively. The mean deviation about the mean of these 6 observations is:

The vertices of a triangle are A(–1, 3), B(–2, 2) and C(3, –1). A new triangle is formed by shifting the sides of the triangle by one unit inwards. Then the equation of the side of the new triangle nearest to origin is :

Let the first three terms \(2\), \(p\), and \(q\), with \(q \neq 2\), of a G.P. be respectively the \(7^\text{th}\), \(8^\text{th}\), and \(13^\text{th}\) terms of an A.P. If the \(5^\text{th}\) term of the G.P. is the \(n^\text{th}\) term of the A.P., then \(n\) is equal to: