List of top Mathematics Questions asked in JEE Main

Let the line $L_1$ be parallel to the vector $-3\hat{i }+ 2\hat{j} + 4\hat{k}$ and pass through the point $(2, 6, 7)$, and the line $L_2$ be parallel to the vector $2\hat{i} + \hat{j} + 3\hat{k}$ and pass through the point $(4, 3, 5)$. If the line $L_3$ is parallel to the vector $-3\hat{i} + 5\hat{j} + 16\hat{k}$ and intersects the lines $L_1$ and $L_2$ at the points $C$ and $D$, respectively, then $|\vec{CD}|^2$ is equal to :

$A_1$ is the area bounded by $y=x^2+2$, $x+y=8$, and the $y$-axis in the first quadrant, and $A_2$ is the area bounded by $y=x^2+2$, $y^2=x$, $x=0$ and $x=2$ in the first quadrant. Find $(A_1-A_2)$.

The coefficient of $x^{48}$ in \[ (1+x) + 2(1+x)^2 + 3(1+x)^3 + \cdots + 100(1+x)^{100} \] is equal to

The value of $\dfrac{\sqrt{3}\cosec 20^\circ - \sec 20^\circ}{\cos 20^\circ \cos 40^\circ \cos 60^\circ \cos 80^\circ}$ is equal to

Let $ A = \{0,1,2,\ldots,9\} $. Let $ R $ be a relation on $ A $ defined by $(x,y) \in R$ if and only if $ |x - y| $ is a multiple of $3$. Given below are two statements:
Statement I: $ n(R) = 36 $.
Statement II: $ R $ is an equivalence relation.
In the light of the above statements, choose the correct answer from the options given below.

Let the domain of the function \[ f(x) = \log_3 \log_5 \left( 7 - \log_2 \left( x^2 - 10x + 15 \right) \right) + \sin^{-1} \left( \frac{3x - 7}{17 - x} \right) \] be $ (\alpha, \beta) $, then $ \alpha + \beta $ is equal to:

The number of strictly increasing functions $f$ from the set $\{1, 2, 3, 4, 5, 6\}$ to the set $\{1, 2, 3, ...., 9\}$ such that $f(i)>i$ for $1 \le i \le 6$, is equal to:

Let \[ f(t)=\int \left(\frac{1-\sin(\log_e t)}{1-\cos(\log_e t)}\right)dt,\; t>1. \] If $f(e^{\pi/2})=-e^{\pi/2}$ and $f(e^{\pi/4})=\alpha e^{\pi/4}$, then $\alpha$ equals

If $f(x^2 + 1) = x^4 + 5x^2 + 1$, then find $\int_{0}^{3} f(x) dx$ :

Statement 1 : $2^{513}+2^{013}+8^{13}+3^{13}$ is divisible by $7$.
Statement 2 : The value of integral part of $(7+4\sqrt{3})^{25}$ is an odd number.

Let $A_1$ be the bounded area enclosed by the curves $y=x^2+2$, $x+y=8$ and $y$-axis that lies in the first quadrant. Let $A_2$ be the bounded area enclosed by the curves $y=x^2+2$, $y^2=x$, $x=2$ and $y$-axis that lies in the first quadrant. Then $A_1-A_2$ is equal to

The number of solutions of \[ \tan^{-1}(4x) + \tan^{-1}(6x) = \frac{\pi}{6}, \] where \[ -\frac{1}{2\sqrt{6}}<x<\frac{1}{2\sqrt{6}}, \] is equal to

For a triangle $ ABC $, let $ \vec{p} = \vec{BC} $, $ \vec{q} = \vec{CA} $ and $ \vec{r} = \vec{BA} $. If $ |\vec{p}| = 2\sqrt{3}, |\vec{q}| = 2 $ and $ \cos \theta = -\frac{1}{\sqrt{3}} $, where $ \theta $ is the angle between $ \vec{p} $ and $ \vec{q} $, then $ |\vec{p} \times (\vec{q} - 3\vec{r})|^2 + 3|\vec{r}|^2 $ is equal to :

If the domain of the function
\[ f(x)=\sin^{-1}\!\left(\frac{5-x}{3+2x}\right)+\frac{1}{\log_e(10-x)} \]
is $ (-\infty,\alpha] \cup [\beta,\gamma) - \{\delta\} $, then $ 6(\alpha+\beta+\gamma+\delta) $ is equal to

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

The probability distribution of a random variable $X$ is given below:

If $E(X)=\dfrac{263}{15}$, then $P(X<20)$ 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 \[ f(x)=\int \frac{7x^{10}+9x^8}{(1+x^2+2x^9)^2}\,dx \] and $f(1)=\frac14$. Given that

and $B=\operatorname{adj}(\operatorname{adj}A)$, if $|B|=81$, find the value of $\alpha^2$ (where $\alpha\in\mathbb{R}$).

Let $(\alpha, \beta, \gamma)$ be the co-ordinates of the foot of the perpendicular drawn from the point (5, 4, 2) on the line $\vec{r}=(-\hat{i}+3\hat{j}+\hat{k})+\lambda(2\hat{i}+3\hat{j}-\hat{k})$. Then the length of the projection of the vector $\alpha\hat{i}+\beta\hat{j}+\gamma\hat{k}$ on the vector $6\hat{i}+2\hat{j}+3\hat{k}$ 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 $p+q$.

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 $f$ be a differentiable function satisfying \[ f(x+y)=f(x)+f(y)-xy \quad \text{for all } x,y\in\mathbb{R}. \] If \[ \lim_{h\to 0}\frac{f(h)}{h}=3, \] then the value of \[ \sum_{n=1}^{10} f(n) \] is equal to:

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

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 $ P(10, 2\sqrt{15}) $ be a point on the hyperbola \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1, \] whose foci are $ S $ and $ S' $. If the length of its latus rectum is $8$, then the square of the area of $ \triangle PSS' $ is equal to