List of top Questions asked in JEE Main

The value of
\[ \left(\frac{1}{3}+\frac{4}{7}\right) +\left(\frac{1}{3^2}+\frac{1}{3}\times\frac{4}{7}+\frac{4}{7^2}\right) +\left(\frac{1}{3^3}+\frac{1}{3^2}\times\frac{4}{7}+\frac{1}{3}\times\frac{4}{7^2}+\frac{4}{7^3}\right) +\cdots \text{ up to infinite terms is} \]

Let $[t]$ denote the greatest integer less than or equal to $t$.
If the function
\[ f(x)= \begin{cases} b^2 \sin\!\left[\dfrac{\pi}{2}\left[\dfrac{\pi}{2}(\cos x+\sin x)\cos x\right]\right], & x < 0 \\ \dfrac{\sin x-\dfrac{1}{2}\sin 2x}{x^3}, & x > 0 \\ a, & x = 0 \end{cases} \] is continuous at $x=0$, then $a^2+b^2$ is equal to

Let $\vec{a}=2\hat{i}-5\hat{j}+5\hat{k}$ and $\vec{b}=\hat{i}-\hat{j}+3\hat{k}$. If $\vec{c}$ is a vector such that $2(\vec{a}\times\vec{c})+3(\vec{b}\times\vec{c})=\vec{0}$ and $(\vec{a}-\vec{b})\cdot\vec{c}=-97$, then $|\vec{c}\times\hat{k}|^2$ is equal to

The smallest positive integral value of $a$, for which all the roots of $x^4-ax^2+9=0$ are real and distinct, is equal to

Let $f(\alpha)$ denote the area of the region in the first quadrant bounded by $x=0$, $x=1$, $y^2=x$ and $y=|\alpha x-5|-|1-\alpha x|+\alpha^2$. Then $(f(0)+f(1))$ is equal to

Let $f$ be a function such that $3f(x)+2f\!\left(\dfrac{m}{19x}\right)=5x$, $x\ne0$, where $m=\displaystyle\sum_{i=1}^{9} i^2$. Then $f(5)-f(2)$ is equal to

Let the length of the latus rectum of an ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ $(a>b)$ be $30$. If its eccentricity is the maximum value of the function $f(t)=-\dfrac{3}{4}+2t-t^2$, then $(a^2+b^2)$ is equal to

If the domain of the function $f(x)=\sin^{-1}\!\left(\dfrac{1}{x^2-2x-2}\right)$ is $(-\infty,\alpha)\cup[\beta,\gamma]\cup[\delta,\infty)$, then $\alpha+\beta+\gamma+\delta$ is equal to

The sum of all values of $\alpha$, for which the shortest distance between the lines $\dfrac{x+1}{\alpha}=\dfrac{y-2}{-1}=\dfrac{z-4}{-\alpha}$ and $\dfrac{x}{\alpha}=\dfrac{y-1}{2}=\dfrac{z-1}{2\alpha}$ is $\sqrt{2}$, is

The letters of the word ``UDAYPUR'' are written in all possible ways with or without meaning and these words are arranged as in a dictionary. The rank of the word ``UDAYPUR'' is

Let the angles made with the positive $x$-axis by two straight lines drawn from the point $P(2,3)$ and meeting the line $x+y=6$ at a distance $\sqrt{\frac{2}{3}}$ from the point $P$ be $\theta_1$ and $\theta_2$. Then the value of $(\theta_1+\theta_2)$ is

Let \( y^2 = 16x \), from point \( (16, 16) \) a focal chord is passing. Point \( (\alpha, \beta) \) divides the focal chord in the ratio 2:3, then the minimum value of \( \alpha + \beta \) is:

Ellipse \( E: \frac{x^2}{36} + \frac{y^2}{25} = 1 \), A hyperbola confocal with ellipse \( E \) and eccentricity of hyperbola is equal to 5. The length of latus rectum of hyperbola is, if principle axis of hyperbola is x-axis?

Consider an ellipse \[ E_1:\ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 \ (a>b) \quad \text{and} \quad E_2:\ \frac{x^2}{A^2}+\frac{y^2}{B^2}=1 \ (B>A), \] where $e=\dfrac{4}{5}$ for both the curves and $\ell_1$ is the length of latus rectum of $E_1$ and $\ell_2$ is the length of latus rectum of $E_2$. Let the distance between the foci of the first curve be $8$. Find the distance between the foci of the second curve. (Given $2\ell_1^2=9\ell_2$).

Let mirror image of parabola $x^2 = 4y$ in the line $x-y=1$ be $(y+a)^2 = b(x-c)$. Then the value of $(a+b+c)$ is

Let $X=\{x\in\mathbb{N}:1\le x\le19\}$ and for some $a,b\in\mathbb{R}$, $Y=\{ax+b:x\in X\}$. If the mean and variance of the elements of $Y$ are $30$ and $750$ respectively, then the sum of all possible values of $b$ 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

Consider the following three statements for the function $f : (0,\infty) \rightarrow \mathbb{R}$ defined by
\[ f(x) = \left| \log_e x \right| - |x - 1| : \]
(I) $f$ is differentiable at all $x>0$.
(II) $f$ is increasing in $(0,1)$.
(III) $f$ is decreasing in $(1,\infty)$.
Then,

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

If \((9+7\alpha-7\beta)^{20} + (9\alpha+7\beta-7)^{20} + (9\beta+7-7\alpha)^{20} + (14+7\alpha+7\beta)^{20}\) is \(m^{10}\) then the value of m is : (where \(\alpha = \frac{-1+i\sqrt{3}}{2}\) \& \(\beta = \frac{-1-i\sqrt{3}}{2}\))

Let the curve $z(1+i) + z(1-i) = 4$, $z \in \mathbb{C}$, divide the region $|z-3| \le 1$ into two parts of areas $\alpha$ and $\beta$. Then $|\alpha - \beta|$ equals:

If \( z = \dfrac{\sqrt{3}}{2} + \dfrac{i}{2} \), then the value of \[ \left(z^{201} - i\right)^8 \] is: