List of top Questions asked in JEE Main

Let the centre of a circle, passing through the point \((0, 0)\), \((1, 0)\) and touching the circle \(x^2 + y^2 = 9\), be \((h, k)\). Then for all possible values of the coordinates of the centre \((h, k)\), \(4(h^2 + k^2)\) is equal to __________.

If a function \( f \) satisfies \( f(m + n) = f(m) + f(n) \) for all \( m, n \in \mathbb{N} \) and \( f(1) = 1 \), then the largest natural number \( \lambda \) such that \[ \sum_{k=1}^{2022} f(\lambda + k) \leq (2022)^2 \] is equal to __________.

Let \(\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}\), \(\vec{b} = 2\hat{i} + 3\hat{j} - 5\hat{k}\), and \(\vec{c} = 3\hat{i} - \hat{j} + \lambda\hat{k}\) be three vectors. Let \(\vec{r}\) be a unit vector along \(\vec{b} + \vec{c}\). If \(\vec{r} \cdot \vec{a} = 3\), then \(3\lambda\) is equal to:

If \(\alpha \neq a\), \(\beta \neq b\), \(\gamma \neq c\) and \[ \begin{vmatrix} \alpha & b & c \\ a & \beta & c \\ a & b & \gamma \end{vmatrix} = 0,\] then \[ \frac{a}{\alpha - a} + \frac{b}{\beta - b} + \frac{\gamma}{\gamma - c} \] is equal to:

In an increasing geometric progression of positive terms, the sum of the second and sixth terms is \[ \frac{70}{3} \] and the product of the third and fifth terms is 49. Then the sum of the \(4^\text{th}, 6^\text{th}\), and \(8^\text{th}\) terms is:

The number of ways five alphabets can be chosen from the alphabets of the word MATHEMATICS, where the chosen alphabets are not necessarily distinct, is equal to :

If the system of equations \(x + 4y - z = \lambda\), \(7x + 9y + \mu z = -3\), \(5x + y + 2z = -1\) has infinitely many solutions, then \((2\mu + 3\lambda)\) is equal to:

If the value of \[ \frac{3 \cos 36^\circ + 5 \sin 18^\circ}{5 \cos 36^\circ - 3 \sin 18^\circ} = \frac{a\sqrt{5} - b}{c}, \] where \(a, b, c\) are natural numbers and \(\text{gcd}(a, c) = 1\), then \(a + b + c\) is equal to:

Let \(y = y(x)\) be the solution curve of the differential equation \[ \sec y \frac{dy}{dx} + 2x \sin y = x^3 \cos y, \] \(y(1) = 0\). Then \(y\left(\sqrt{3}\right)\) is equal to:

Let \(\vec{a} = 4\hat{i} - \hat{j} + \hat{k}\), \(\vec{b} = 11\hat{i} - \hat{j} + \hat{k}\), and \(\vec{c}\) be a vector such that \[ (\vec{a} + \vec{b}) \times \vec{c} = \vec{c} \times (-2\vec{a} + 3\vec{b}). \] If \((2\vec{a} + 3\vec{b}) \cdot \vec{c} = 1670\), then \(|\vec{c}|^2\) is equal to:

There are three bags \(X\), \(Y\), and \(Z\). Bag \(X\) contains 5 one-rupee coins and 4 five-rupee coins; Bag \(Y\) contains 4 one-rupee coins and 5 five-rupee coins, and Bag \(Z\) contains 3 one-rupee coins and 6 five-rupee coins. A bag is selected at random and a coin drawn from it at random is found to be a one-rupee coin. Then the probability, that it came from bag \(Y\), is:

Let \[ \int_{\log_e a}^{4} \frac{dx}{\sqrt{e^x - 1}} = \frac{\pi}{6}. \] Then \(e^\alpha\) and \(e^{-\alpha}\) are the roots of the equation:

Let \(f(x) = \begin{cases} -a & \text{if } -a \leq x \leq 0, \\ x + a & \text{if } 0<x \leq a \end{cases} \) where \(a>0\) and \(g(x) = (f(|x|) - |f(x)|)/2\). Then the function \(g : [-a, a] \to [-a, a]\) is:

Let \(A\) be the region enclosed by the parabola \(y^2 = 2x\) and the line \(x = 24\). Then the maximum area of the rectangle inscribed in the region \(A\) is ________.

Let \(S\) be the focus of the hyperbola \(\frac{x^2}{3} - \frac{y^2}{5} = 1\), on the positive x-axis. Let \(C\) be the circle with its centre at \(A\left(\sqrt{6}, \sqrt{5}\right)\) and passing through the point \(S\). If \(O\) is the origin and \(SAB\) is a diameter of \(C\), then the square of the area of the triangle \(OSB\) is equal to -

Let \(P(\alpha, \beta, \gamma)\) be the image of the point \(Q(1, 6, 4)\) in the line \[ \frac{x}{1} = \frac{y - 1}{2} = \frac{z - 2}{3}. \] Then \(2\alpha + \beta + \gamma\) is equal to _______.

The number of distinct real roots of the equation \[ |x + 1| |x + 3| - 4|x + 2| + 5 = 0, \] is _______.

Let \(a, b, c \in \mathbb{N}\) and \(a<b<c\). Let the mean, the mean deviation about the mean and the variance of the 5 observations \(9, 25, a, b, c\) be \(18, 4\) and \(\frac{136}{5}\), respectively. Then \(2a + b - c\) is equal to _______.

Let \(\alpha |x| = |y| e^{xy - \beta}\), \(\alpha, \beta \in \mathbb{N}\) be the solution of the differential equation \[ xdy - ydx + xy(xdy + ydx) = 0, \quad y(1) = 2. \] Then \(\alpha + \beta\) is equal to _.

If \[ \int \frac{1}{\sqrt[5]{(x - 1)^4}(x + 3)^6} \, dx = A \left( \frac{\alpha x - 1}{\beta x + 3} \right)^B + C, \] where \(C\) is the constant of integration, then the value of \(\alpha + \beta + 20AB\) is _______.

The value of $k \in \mathbb{N}$ for which the integral \[ I_n = \int_0^1 (1 - x^k)^n \, dx, \, n \in \mathbb{N}, \] satisfies $147 \, I_{20} = 148 \, I_{21}$ is:

The sum of all the solutions of the equation \[(8)^{2x} - 16 \cdot (8)^x + 48 = 0\]is:

Let the circles $C_1 : (x - \alpha)^2 + (y - \beta)^2 = r_1^2$ and $C_2 : (x - 8)^2 + \left( y - \frac{15}{2} \right)^2 = r_2^2$ touch each other externally at the point $(6, 6)$. If the point $(6, 6)$ divides the line segment joining the centres of the circles $C_1$ and $C_2$ internally in the ratio $2 : 1$, then $(\alpha + \beta) + 4\left(r_1^2 + r_2^2\right)$ equals _____.

The number of critical points of the function $f(x) = (x - 2)^{2/3}(2x + 1)$ is:

Let \( f(x) \) be a positive function such that the area bounded by \( y = f(x) \), \( y = 0 \), from \( x = 0 \) to \( x = a>0 \) is \[ \int_0^a f(x) \, dx = e^{-a} + 4a^2 + a - 1. \] Then the differential equation, whose general solution is \[ y = c_1 f(x) + c_2, \] where \( c_1 \) and \( c_2 \) are arbitrary constants, is: