List of top Mathematics Questions asked in JEE Main

In a G.P., if the product of the first three terms is \(27\) and the set of all possible values for the sum of its first three terms is \( \mathbb{R} - (a,b) \), then \( a^2+b^2 \) is equal to:

Let \( S = \{x^3 + ax^2 + bx + c : a, b, c \in \mathbb{N} \text{ and } a, b, c \le 20\} \) be a set of polynomials. Then the number of polynomials in \( S \), which are divisible by \( x^2 + 2 \), is:

Let \( y = y(x) \) be the solution of the differential equation \[ x\frac{dy}{dx} - \sin 2y = x^3(2 - x^3)\cos^2 y,\; x \ne 0. \] If \( y(2) = 0 \), then \( \tan(y(1)) \) is equal to:

The value of \[ \lim_{x\to 0}\frac{\log_e\!\big(\sec(ex)\cdot \sec(e^2x)\cdots \sec(e^{10}x)\big)} {e^2-e^{2\cos x}} \] is equal to:

If \( g(x) = 3x^2 + 2x - 3 \), \( f(0) = -3 \) and \( 4g(f(x)) = 3x^2 - 32x + 72 \), then \( f(g(2)) \) is equal to:

Let \( y = x \) be the equation of a chord of the circle \( C_1 \) (in the closed half-plane \( x \ge 0 \)) of diameter 10 passing through the origin. Let \( C_2 \) be another circle described on the given chord as diameter. If the equation of the chord of the circle \( C_2 \), which passes through the point \( (2, 3) \) and is farthest from the center of \( C_2 \), is \( x + ay + b = 0 \), then \( b \) is equal to:

Let \( ABC \) be an equilateral triangle with orthocenter at the origin and the side \( BC \) lying on the line \( x+2\sqrt{2}\,y=4 \). If the coordinates of the vertex \( A \) are \( (\alpha,\beta) \), then the greatest integer less than or equal to \( |\alpha+\sqrt{2}\beta| \) is:

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 $\vec{a}=2\hat{i}-\hat{j}-\hat{k}$, $\vec{b}=\hat{i}+3\hat{j}-\hat{k}$ and $\vec{c}=2\hat{i}+\hat{j}+3\hat{k}$. Let $\vec{v}$ be the vector in the plane of $\vec{a}$ and $\vec{b}$, such that the length of its projection on the vector $\vec{c}$ is $\dfrac{1}{\sqrt{14}}$. Then $|\vec{v}|$ 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 number of elements in the set $\{x\in[0,180^\circ]: \tan(x+100^\circ)=\tan(x+50^\circ)\tan x\tan(x-50^\circ)\}$ 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

If $f(x)$ satisfies the relation \[ f(x)=e^x+\int_0^1 (y+x e^x)f(y)\,dy, \] then $e+f(0)$ is equal to

Let $(h,k)$ lie on the circle $C:x^2+y^2=4$ and the point $(2h+1,\,3k+2)$ lie on an ellipse with eccentricity $e$. Then the value of $\dfrac{5}{e^2}$ 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

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,

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

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 $f(\alpha)$ 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

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

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

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

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

Let the image of parabola $x^2=4y$ in the line $x-y=1$ be $(y+a)^2=b(x-c)$, where $a,b,c\in\mathbb{N}$. Then $a+b+c$ is equal to