List of top Questions asked in JEE Main

Consider two sets \[ A = \{ x \in \mathbb{Z} : |(|x-3|-3)| \le 1 \} \] and \[ B = \left\{ x \in \mathbb{R} - \{1,2\} : \frac{(x-2)(x-4)}{x-1}\,\log_e(|x-2|) = 0 \right\}. \] Then the number of onto functions \( f : A \to B \) 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 $P=[p_{ij}]$ 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

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 $f$ 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 $y=y(x)$ be a differentiable function in the interval $(0,\infty)$ such that $y(1)=2$, and \[ \lim_{t\to x}\left(\frac{t^2y(x)-x^2y(t)}{x-t}\right)=3 \text{ for each } x>0. \] Then $2y(2)$ 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 $(A_1-A_2)$.

Consider an A.P. $a_1,a_2,\ldots,a_n$ with $a_1>0$, $a_2-a_1=-\dfrac{3}{4}$ and $a_n=\dfrac{a_1}{4}$. If \[ \sum_{i=1}^{n} a_i=\frac{525}{2}, \] then find $\sum_{i=1}^{17} a_i$.

Given that \[ \vec a=2\hat i+\hat j-\hat k,\quad \vec b=\hat i+\hat j,\quad \vec c=\vec a\times\vec b, \] \[ |\vec d\times\vec c|=3,\quad \vec d\cdot\vec c=\frac{\pi}{4},\quad |\vec a-\vec d|=\sqrt{11}, \] find $\vec a\cdot\vec d$.

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 \(\lambda\).

If all the letters of the word 'UDAYPUR' are arranged in all possible permutations and these permutations are listed in dictionary order, then the rank of the word 'UDAYPUR' is

Let a function $f(x)$ satisfy \[ 3f(x)+2f\!\left(\frac{m}{19x}\right)=5x \] where $m=\sum_{i=1}^{9} i^2$. Find $f(5)+f(2)$.

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

Let \(x \frac{dy}{dx} - \sin 2y = x^3(2 - x^3) \cos^2 y ; y(2) = 0\), then find \(\tan(y(1))\) :

Consider a circle \(C_1\), passing through origin and lying in region \(0 \le x\) only, with diameter = 10. Consider a chord \(PQ\) of \(C_1\) with equation \(y = x\) and another Circle \(C_2\) which has \(PQ\) as diameter. A chord is drawn to \(C_2\) passing through (2, 3) such that distance of chord from centre of \(C_2\) is maximum has equation \(x + ay + b = 0\) then \(|b - a|\) is equal to :

Product of first 3 terms of a G.P. is 27 and sum is \(R - \{a,b\}\), then \(a^2 + b^2\) is equal to :

If \(g(x) = 3x^2 + 2x - 3, f(0) = -3, 4g(f(x)) = 3x^2 - 32x + 72\) then find \(f(g(2))\) where \(f(x)>0\) for all valid x:

If \( P(10, 2\sqrt{15}) \) lies on the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) and the length of the latus rectum is 8, then the square of the area of \( \Delta PS_1S_2 \) is [where \( S_1 \) and \( S_2 \) are the foci of the hyperbola].

If \( a, b, c \) are in A.P. where \( a + b + c = 1 \) and \( a, 2b, c \) are in G.P., then the value of \( 9(a^2 + b^2 + c^2) \) is equal to:

If \[ \int_0^6 \left( x^3 + \lfloor x^{1/3} \rfloor \right) \, dx = \alpha \] and \[ \int_0^{\frac{\pi}{2}} \frac{\sin^2 x}{\sin^6 x + \cos^6 x} \, dx = \beta, \] then the value of \( ab^2 \) is equal to:

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: