List of top Questions asked in JEE Main

Let 

be a continuous function at $x=0$, then the value of $(a^2+b^2)$ is (where $[\ ]$ denotes greatest integer function). 
 

Let $P$ and $Q$ be any two $3\times3$ matrices where $P=[p_{ij}]_{3\times3}$, $Q=[q_{ij}]_{3\times3}$ such that $q_{ij}=2^{\,i+j-1}p_{ij}$. Find $|\operatorname{adj}(\operatorname{adj}P)|$.
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 \(f\) and \(g\) be functions satisfying \[ f(x+y) = f(x)f(y), \quad f(1) = 7 \] \[ g(x+y) = g(xy), \quad g(1) = 1, \] for all \(x,y \in \mathbb{N}\). If \[ \sum_{x=1}^{n} \left(\frac{f(x)}{g(x)}\right) = 19607, \] then \(n\) is equal to
If two lines drawn from a point $P(2,3)$ intersect the line $x+y=6$ at a distance $\sqrt{\dfrac{2}{3}}$, then the angle between the lines is
Let the locus of the mid-point of the chord through the origin \(O\) of the parabola \(y^2 = 4x\) be the curve \(S\). Let \(P\) be any point on \(S\). Then the locus of the point, which internally divides \(OP\) in the ratio \(3:1\), is
Let \(\alpha,\beta,\gamma\ (0<\alpha,\beta,\gamma<\tfrac{\pi}{2})\) be the angles between non–zero vectors \(\vec a\) and \(\vec b\), \(\vec b\) and \(\vec c\), \(\vec c\) and \(\vec a\) respectively. If \(\theta\) is the angle that the vector \(\vec a\) makes with the plane containing \(\vec b\) and \(\vec c\), then:
If shortest distance between the lines \[ \frac{x+1}{\alpha}=\frac{y-2}{-2}=\frac{z-4}{-2\alpha} \quad \text{and} \quad \frac{x}{\alpha}=\frac{y-1}{1}=\frac{z-1}{\alpha} \] is $\sqrt{2}$, then find the sum of all possible values of $\alpha$.
If the function \[ f(x)=\frac{e^x\left(e^{\tan x - x}-1\right)+\log_e(\sec x+\tan x)-x}{\tan x-x} \] is continuous at $x=0$, then the value of $f(0)$ is equal to
The sum of all the real solutions of the equation \[ \log_{(x+3)}\left(6x^2 + 28x + 30\right) = 5 - 2\log_{(6x+10)}\left(x^2 + 6x + 9\right) \] is equal to .
Let the ellipse \[ E=\frac{x^2}{a^2}+\frac{y^2}{b^2}=1 \] have eccentricity equal to the greatest value of the function \[ f(t)=-\frac34+2t-t^2 \] and the length of its latus rectum is $30$. Find the value of $a^2+b^2$.
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
If 3 balls are taken from a box without replacement and found to be all black. If all configurations of red balls and black balls are equally likely, then the probability that the box contained 1 red and 9 black balls is \(\dfrac{p}{q}\) for some coprime natural numbers \(p\) and \(q\). Find \(p+q\).
If the line $ax + 4y = \sqrt{7}$, where $a \in \mathbb{R}$, touches the ellipse $3x^2 + 4y^2 = 1$ at the point $P$ in the first quadrant, then one of the focal distances of $P$ is :
Let \(x=x(t)\) be the solution curve of the differential equation \[ \frac{dx}{dt}=-kx, \] with \[ x(0)=100,\quad x\!\left(\frac{1}{2}\right)=80. \] If \(x(t_\alpha)=5\), then \(t_\alpha\) is equal to:

Let a vector \[ \vec a=\sqrt{2}\,\hat i-\hat j+\lambda \hat k,\ \lambda>0, \] make an obtuse angle with the vector \[ \vec b=-\lambda^2\hat i+4\sqrt{2}\hat j+4\sqrt{2}\hat k \] and an angle \( \theta \), \(\frac{\pi}{6}<\theta<\frac{\pi}{2}\), with the positive \(z\)-axis. If the set of all possible values of \( \lambda \) is \((\alpha,\beta)-\{\gamma\}\), then\(\alpha+\beta+\gamma\) is equal to ____________. 

Let one end of focal chord of the parabola \( y^2 = 16x \) be \( (16,16) \).
If \( P(\alpha,\beta) \) divides this focal chord internally in the ratio \( 5:2 \), then the minimum value of \( \alpha + \beta \) is equal to
A spring of force constant $15\,\text{N/m}$ is cut into two pieces. If the ratio of their lengths is $1:3$, then the force constant of the smaller piece is ___ $\text{N/m}$.
A lift is going up to the \(10^{\text{th}}\) floor. Number of ways in which \(3\) people can exit the lift at {three different floors, if the lift will not stop at I$^{\text{st}}$, II$^{\text{nd}}$ and III$^{\text{rd}}$ floors, is: }
Let \(\triangle ABC\) such that \(A(0,0)\) and vertices \(B\) and \(C\) lie on the parabola \[ y^2=8x. \] If \(\left(\frac{7}{3},\frac{4}{3}\right)\) is the centroid of \(\triangle ABC\), then \((BC)^2\) is equal to:
Let \(X\) have a binomial distribution \(B(6,p)\). If the sum of the mean and the variance of \(X\) is \(\dfrac{21}{8}\), then \[ \frac{P(2\le X<4)}{P(4<X<6)} \] is equal to:
Let \(A=[a_{ij}]\), \(\det(A)\neq 0\), and \(B=[b_{ij}]\) be two \(3\times 3\) matrices. If \[ b_{ij}=3^{\,i-j}\,a_{ij}\quad \text{for all } i,j=1,2,3, \] then:
If \[ a_n=(2n^2-n+2)(n!) , \] then \[ \sum_{n=1}^{20} a_n \] is equal to:
Let \(m\) and \(n\) be non–negative integers such that for \[ x\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right),\qquad \tan x+\sin x=m,\quad \tan x-\sin x=n. \] Then the possible ordered pair \((m,n)\) is:

If the area of the region \[ \{(x, y) : 1 - 2x \le y \le 4 - x^2,\ x \ge 0,\ y \ge 0\} \] is \[ \frac{\alpha}{\beta}, \] \(\alpha, \beta \in \mathbb{N}\), \(\gcd(\alpha, \beta) = 1\), then the value of \[ (\alpha + \beta) \] is :