List of top Mathematics Questions asked in JEE Main

If $\int e^x \left( \frac{x \sin^{-1} x}{\sqrt{1-x^2}} + \frac{\sin^{-1} x}{(1-x^2)^{3/2}} + \frac{x}{1-x^2} \right) dx = g(x) + C$, where C is the constant of integration, then $g\left( \frac{1}{2} \right)$equals:

Let $ P(4, 4\sqrt{3}) $ be a point on the parabola $ y^2 = 4ax $ and PQ be a focal chord of the parabola. If M and N are the foot of the perpendiculars drawn from P and Q respectively on the directrix of the parabola, then the area of the quadrilateral PQMN is equal to:

Let $ \mathbf{a} $ and $ \mathbf{b} $ be two unit vectors such that the angle between them is $ \frac{\pi}{3} $. If $ \lambda \mathbf{a} + 2 \mathbf{b} $ and $ 3 \mathbf{a} - \lambda \mathbf{b} $ are perpendicular to each other, then the number of values of $ \lambda $ in $ [-1, 3] $ is:

The perpendicular distance of the line $ \frac{x - 1}{2} = \frac{y + 2}{-1} = \frac{z + 3}{2} $ from the point $ P(2, -10, 1) $ is:

Suppose that the number of terms in an A.P. is $ 2k, k \in \mathbb{N} $. If the sum of all odd terms of the A.P. is 40, the sum of all even terms is 55, and the last term of the A.P. exceeds the first term by 27, then $ k $ is equal to:

In the expansion of \[ \left( \sqrt[3]{2} + \frac{1}{\sqrt[3]{3}} \right)^n , \, n \in \mathbb{N}, \] if the ratio of the 15^th term from the beginning to the 15^th term from the end is \[ \frac{1}{6}, \] then the value of \[ {}^nC_3 \] is:

1 + 3 + $5^2$ + 7 + $9^2$ + $\ldots$ upto 40 terms is equal to

Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous function satisfying $f(0) = 1$ and $f(2x) - f(x) = x$ for all $x \in \mathbb{R}$. If $\lim_{n \to \infty} \left\{ f(x) - f\left( \frac{x}{2^n} \right) \right\} = G(x)$, then $\sum_{r=1}^{10} G(r^2)$ is equal to

Let $A$ and $B$ be two distinct points on the line $L: \frac{x-6}{3} = \frac{y-7}{2} = \frac{z-7}{-2}$. Both $A$ and $B$ are at a distance $2\sqrt{17}$ from the foot of perpendicular drawn from the point $(1, 2, 3)$ on the line $L$. If $O$ is the origin, then $\overrightarrow{OA} \cdot \overrightarrow{OB}$ is equal to:

Let $f: [0, \infty) \to \mathbb{R}$ be a differentiable function such that $f(x) = 1 - 2x + \int_0^x e^{x-t} f(t) \, dt$ for all $x \in [0, \infty)$. Then the area of the region bounded by $y = f(x)$ and the coordinate axes is

If $\lim_{x \to 1} \frac{(x-1)(6+\lambda \cos(x-1)) + \mu \sin(1-x)}{(x-1)^3} = -1$, where $\lambda, \mu \in \mathbb{R}$, then $\lambda + \mu$ is equal to

Let the shortest distance between the lines $\frac{x-3}{3} = \frac{y-\alpha}{-1} = \frac{z-3}{1}$ and $\frac{x+3}{-3} = \frac{y+7}{2} = \frac{z-\beta}{4}$ be $3\sqrt{30}$. Then the positive value of $5\alpha + \beta$ is

The probability of forming a 12 persons committee from 4 engineers, 2 doctors, and 10 professors containing at least 3 engineers and at least 1 doctor is:

For an integer $n \geq 2$, if the arithmetic mean of all coefficients in the binomial expansion of $(x + y)^{2n-3}$ is 16, then the distance of the point $P(2n-1, n^2-4n)$ from the line $x + y = 8$ is:

Let $A = \{1, 6, 11, 16, \ldots\}$ and $B = \{9, 16, 23, 30, \ldots\}$ be the sets consisting of the first 2025 terms of two arithmetic progressions. Then $n(A \cup B)$ is

Consider the sets $A = \{(x, y) \in \mathbb{R} \times \mathbb{R} : x^2 + y^2 = 25\}$, $B = \{(x, y) \in \mathbb{R} \times \mathbb{R} : x^2 + 9y^2 = 144\}$, $C = \{(x, y) \in \mathbb{Z} \times \mathbb{Z} : x^2 + y^2 \leq 4\}$, and $D = A \cap B$. The total number of one-one functions from the set $D$ to the set $C$ is:

Let $f, g: (1, \infty) \rightarrow \mathbb{R}$ be defined as $f(x) = \frac{2x + 3}{5x + 2}$ and $g(x) = \frac{2 - 3x}{1 - x}$. If the range of the function $fog: [2, 4] \rightarrow \mathbb{R}$ is $[\alpha, \beta]$, then $\frac{1}{\beta - \alpha}$ is equal to

For $ n \geq 2 $, let $ S_n $ denote the set of all subsets of $ \{1, 2, 3, \ldots, n\} $ with no two consecutive numbers. For example, $ \{1, 3, 5\} \in S_6 $, but $ \{1, 2, 4\} \notin S_6 $. Then, find $ n(S_5) $.

The number of singular matrices of order 2, whose elements are from the set $ \{2, 3, 6, 9\} $ is:

Consider the hyperbola $ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1, $ having one of its foci at $ P(-3, 0) $. If the latus rectum through its other focus subtends a right angle at $ P $, and $ a^2b^2 = \alpha\sqrt{2} - \beta, \quad \alpha, \beta \in \mathbb{N}, $ then find $ \alpha $ and $ \beta $.

The number of relations on the set $ A = \{1, 2, 3\} $ containing at most 6 elements including $ (1, 2) $, which are reflexive and transitive but not symmetric, is:

The number of points of discontinuity of the function $ f(x) = \left\lfloor \frac{x^2}{2} \right\rfloor - \left\lfloor \sqrt{x} \right\rfloor, \quad x \in [0, 4], $ where $ \left\lfloor \cdot \right\rfloor $ denotes the greatest integer function, is:

Let $ ABC $ be the triangle such that the equations of lines $ AB $ and $ AC $ are: $ 3y - x = 2 \quad \text{and} \quad x + y = 2, $ respectively, and the points $ B $ and $ C $ lie on the x-axis. If $ P $ is the orthocentre of the triangle $ ABC $, then the area of the triangle $ PBC $ is equal to:

Let the angle $ \theta, 0<\theta<\frac{\pi}{2} $ between two unit vectors $ \hat{a} $ and $ \hat{b} $ be $ \sin^{-1} \left( \frac{\sqrt{65}}{9} \right) $. If the vector $ \vec{c} = 3\hat{a} + 6\hat{b} + 9(\hat{a} \times \hat{b}) $, then the value of $ 9(\vec{c} \cdot \hat{a}) - 3(\vec{c} \cdot \hat{b}) $ is: