List of top Mathematics Questions asked in JEE Main

Let \(A\) be the focus of the parabola \(y^2=8x\). Let the line \(y=mx+c\) intersect the parabola at two distinct points \(B\) and \(C\). If the centroid of triangle \(ABC\) is \(\left(\frac{7}{3},\frac{4}{3}\right)\), then \((BC)^2\) is equal to:

Let $z=(1+i)(1+2i)(1+3i)\cdots(1+ni)$, where $i=\sqrt{-1}$. If $|z|^2=44200$, then $n$ is equal to

Let \( a_1 = 1 \) and for \( n \ge 1 \), \[ a_{n+1} = \frac{1}{2}a_n + \frac{n^2 - 2n - 1}{n^2 (n+1)^2}. \] Then \[ \sum_{n=1}^{\infty} \left( a_n - \frac{2}{n^2} \right) \] is equal to

Let \( \vec{a} = 2\hat{i} - \hat{j} + \hat{k} \) and \( \vec{b} = \lambda \hat{j} + 2\hat{k} \), where \( \lambda \in \mathbb{Z} \), be two vectors. Let \( \vec{c} = \vec{a} \times \vec{b} \) and \( \vec{d} \) be a vector of magnitude \(2\) in the \(yz\)-plane. If \( |\vec{c}| = \sqrt{53} \), then the maximum possible value of \( (\vec{c}\cdot\vec{d})^2 \) is equal to

If the mean deviation about the median of the numbers \[ k,\,2k,\,3k,\,\ldots,\,1000k \] is \(500\), then \(k^2\) is equal to

Evaluate: \[ \frac{6}{3^{26}}+\frac{10\cdot1}{3^{25}}+\frac{10\cdot2}{3^{24}}+\frac{10\cdot2^{2}}{3^{23}}+\cdots+\frac{10\cdot2^{24}}{3}. \]

Let \(y=y(x)\) be the solution of the differential equation \[ x\frac{dy}{dx}=y-x^2\cot x,\quad x\in(0,\pi) \] If \(y\!\left(\frac{\pi}{2}\right)=\frac{\pi^2}{2}\), then \[ 6y\!\left(\frac{\pi}{6}\right)-8y\!\left(\frac{\pi}{4}\right) \] is equal to:

Let \( \cos(\alpha + \beta) = -\dfrac{1}{10} \) and \( \sin(\alpha - \beta) = \dfrac{3}{8} \), where \( 0<\alpha<\dfrac{\pi}{3} \) and \( 0<\beta<\dfrac{\pi}{4} \).
If \[ \tan 2\alpha = \frac{3(1 - r\sqrt{5})}{\sqrt{11}(s + \sqrt{5})}, \quad r, s \in \mathbb{N}, \] then the value of \( r + s \) is ______________.

Let \[ A = \begin{bmatrix} 0 & 2 & -3 \\ -2 & 0 & 1 \\ 3 & -1 & 0 \end{bmatrix} \] and \( B \) be a matrix such that \[ B(I - A) = I + A. \] Then the sum of the diagonal elements of \( B^{T}B \) is equal to

The mean and variance of a data of 10 observations are 10 and 2, respectively. If an observation $\alpha$ in this data is replaced by $\beta$, then the mean and variance become $10.1$ and $1.99$, respectively. Then $\alpha+\beta$ equals

Let \( A \) be a \( 3 \times 3 \) matrix such that \( A + A^{T} = O \). If

\[ A \begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix} = \begin{bmatrix} 3 \\ 3 \\ 2 \end{bmatrix}, \quad A^{2} \begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix} = \begin{bmatrix} -3 \\ 19 \\ -24 \end{bmatrix} \] and

\[ \det\!\big(\operatorname{adj}(2\,\operatorname{adj}(A+I))\big) = 2^{\alpha}\,3^{\beta}\,11^{\gamma}, \] where \( \alpha, \beta, \gamma \) are non-negative integers, then the value of
\( \alpha + \beta + \gamma \) is ________.

The positive integer \( n \), for which the solutions of the equation \( x(x+2) + (x+2)(x+4) + \dots + (x+2n-2)(x+2n) = \frac{8n}{3} \) are two consecutive even integers, is :

Let the relation \( R \) on the set \( M = \{1, 2, 3, \ldots, 16\} \) be given by \[ R = \{(x, y) : 4y = 5x - 3,\; x, y \in M\}. \] Then the minimum number of elements required to be added in \( R \), in order to make the relation symmetric, is equal to

Let \[ \sum_{k=1}^{n} a_k = \alpha n^2 + \beta n. \] If \( a_{10} = 59 \) and \( a_6 = 7a_1 \), then \( \alpha + \beta \) is equal to

Let \( \overrightarrow{AB} = 2\hat{i} + 4\hat{j} - 5\hat{k} \) and \( \overrightarrow{AD} = \hat{i} + 2\hat{j} + \lambda \hat{k} \), \( \lambda \in \mathbb{R} \). Let the projection of the vector \( \vec{v} = \hat{i} + \hat{j} + \hat{k} \) on the diagonal \( \overrightarrow{AC} \) of the parallelogram \( ABCD \) be of length one unit. If \( \alpha, \beta \), where \( \alpha>\beta \), be the roots of the equation \( \lambda^2 x^2 - 6\lambda x + 5 = 0 \), then \( 2\alpha - \beta \) is equal to

Let the direction cosines of two lines satisfy the equations : \( 4l + m - n = 0 \) and \( 2mn + 5nl + 3lm = 0 \). Then the cosine of the acute angle between these lines is :

Let the line \( y - x = 1 \) intersect the ellipse \( \frac{x^2}{2} + \frac{y^2}{1} = 1 \) at the points A and B. Then the angle made by the line segment AB at the center of the ellipse is:

Let the mean and variance of 8 numbers -10, -7, -1, x, y, 9, 2, 16 be \( 2 \) and \( \frac{293}{4} \), respectively. Then the mean of 4 numbers x, y, x+y+1, |x-y| is:

Let \( f : [1,\infty) \to \mathbb{R} \) be a differentiable function. If
\[ 6\int_{1}^{x} f(t)\,dt = 3x f(x) + x^3 - 4 \] for all \( x \ge 1 \), then the value of \( f(2) - f(3) \) is

Let a circle of radius $4$ pass through the origin $O$, the points $A(-\sqrt{3}a,0)$ and $B(0,-\sqrt{2}b)$, where $a$ and $b$ are real parameters and $ab\neq0$. Then the locus of the centroid of $\triangle OAB$ is a circle of radius

Let \[ f(x)=\int \frac{dx}{2\left(\frac{3}{2}\right)^x+2x\left(\frac12\right)^x} \] such that \(f(0)=-26+24\log_e(2)\). If \(f(1)=a+b\log_e(3)\), where \(a,b\in\mathbb{Z}\), then \(a+b\) is equal to:

Let
\[ A=\begin{pmatrix} 3 & -4 \\ 1 & -1 \end{pmatrix} \] and \( B \) be two matrices such that
\[ A^{100}-100B+I=0. \]
Then the sum of all the elements of \( B^{100} \) is _______.

\(6 \int_{0}^{\pi} (\sin 3x + \sin 2x + \sin x)dx\) is equal to:

Let \[ f(x)=x^{2025}-x^{2000},\quad x\in[0,1] \] and the minimum value of the function $f(x)$ in the interval $[0,1]$ be \[ (80)^{80}(n)^{-81}. \] Then $n$ is equal to