List of top Mathematics Questions asked in JEE Main

Let \[ \int_{0}^{x} \sqrt{1 - (y'(t))^2} \, dt = \int_{0}^{x} y(t) \, dt, \quad 0 \leq x \leq 3, \, y \geq 0, \, y(0) = 0. \] Then, at \( x = 2 \), \( y'' + y + 1 \) is equal to:

Let \( S \) be the set of positive integral values of \( a \) for which \[ \frac{a x^2 + 2(a + 1)x + 9a + 4}{x^2 - 8x + 32} < 0, \quad \forall x \in \mathbb{R}. \] Then, the number of elements in \( S \) is:

Two marbles are drawn in succession from a box containing 10 red, 30 white, 20 blue and 15 orange marbles, with replacement being made after each drawing. Then the probability, that first drawn marble is red and second drawn marble is white, is

Let \( \vec{a} \) and \( \vec{b} \) be two vectors such that \( |\vec{a}| = 1 \), \( |\vec{b}| = 4 \) and \( \vec{a} \cdot \vec{b} = 2 \).If \( \vec{c} = (2 \vec{a} \times \vec{b}) - 3 \vec{b} \) and the angle between \( \vec{b} \) and \( \vec{c} \) is \( \alpha \), then \( 192 \sin^2 \alpha \) is equal to _____

If the integral \[ 525 \int_0^{\frac{\pi}{2}} \sin 2x \cos^{\frac{11}{2}} x \left( 1 + \cos^{\frac{5}{2}} x \right)^{\frac{1}{2}} \, dx \] is equal to \[ \left( n \sqrt{2} - 64 \right), \] then \( n \) is equal to ______

Let Q and R be the feet of perpendiculars from the point P(a, a, a) on the lines x = y, z = 1 and x = –y, z = –1 respectively. If ∠QPR is a right angle, then 12a² is equal to _____

The total number of words (with or without meaning) that can be formed out of the letters of the word ‘DISTRIBUTION’ taken four at a time, is equal to _____

The number of integers, between 100 and 1000 having the sum of their digits equals to 14, is______ .

The value of the integral \[ \int_{-1}^{2} \log_e \left( x + \sqrt{x^2 + 1} \right) \, dx \] is:

Let the inverse trigonometric functions take principal values. The number of real solutions of the equation \[ 2 \sin^{-1} x + 3 \cos^{-1} x = \frac{2\pi}{5}, \] is ______ .

If \( \log_e y = 3 \sin^{-1}x \), then \( (1 - x)^2 y'' - xy' \) at \( x = \frac{1}{2} \) is equal to:

For a differentiable function \( f : \mathbb{R} \to \mathbb{R} \), suppose \[ f'(x) = 3f(x) + \alpha, \] where \( \alpha \in \mathbb{R} \), \( f(0) = 1 \), and \[ \lim_{x \to -\infty} f(x) = 7. \] Then \( 9f(-\log_2 3) \) is equal to __________ .

Let \( z_1 \) and \( z_2 \) be two complex numbers such that \( z_1 + z_2 = 5 \) and \( z_1^3 + z_2^3 = 20 + 15i \). Then \( \left| z_1^4 + z_2^4 \right| \) equals

Let a variable line passing through the centre of the circle x² + y² – 16x – 4y = 0, meet the positive co-ordinate axes at the point A and B. Then the minimum value of OA + OB, where O is the origin, is equal to

Let \[ A = \{(x, y) : 2x + 3y = 23, \, x, y \in \mathbb{N}\} \] and \[ B = \{x : (x, y) \in A\}. \] Then the number of one-one functions from \( A \) to \( B \) is equal to _________ .

The integral \[ \int_{1/4}^{3/4} \cos\left( 2 \cot^{-1} \sqrt{\frac{1 - x}{1 + x}} \right) \, dx \] is equal to:

If an unbiased dice is rolled thrice, then the probability of getting a greater number in the \( i \)-th roll than the number obtained in the \( (i-1) \)-th roll, \( i = 2, 3 \), is equal to:

Consider the circle \( C : x^2 + y^2 = 4 \) and the parabola \( P : y^2 = 8x \). If the set of all values of \( \alpha \), for which three chords of the circle \( C \) on three distinct lines passing through the point \( (\alpha, 0) \) are bisected by the parabola \( P \), is the interval \( (p, q) \), then \( (2q - p)^2 \) is equal to ________ .

Consider the matrices: \[ A = \begin{bmatrix} 2 & -5 \\ 3 & m \end{bmatrix}, \quad B = \begin{bmatrix} 20 \\ m \end{bmatrix}, \quad \text{and} \quad X = \begin{bmatrix} x \\ y \end{bmatrix}. \] Let the set of all \( m \), for which the system of equations \( AX = B \) has a negative solution (i.e., \( x < 0 \) and \( y <0 \)), be the interval \( (a, b) \). Then \[ 8 \int_a^b |\det(A)| \, dm \] is equal to _____ .

Let \( f, g : (0, \infty) \rightarrow \mathbb{R} \) be two functions defined by \(f(x) = \int_{-x}^{x} (|t| - t^2) e^{-t^2} \, dt \quad \text{and} \quad g(x) = \int_{0}^{x} t^{1/2} e^{-t} \, dt.\)Then the value of \( f \left( \sqrt{\log_e 9} \right) + g \left( \sqrt{\log_e 9} \right) \) is equal to

The area of the region \[\left\{ (x, y) : y^2 \leq 4x, \, x<4, \, \frac{xy(x - 1)(x - 2)}{(x - 3)(x - 4)}>0, \, x \neq 3 \right\}\]is

Let \( a \) be the sum of all coefficients in the expansion of \( (1 - 2x + 2x^2)^{2023} (3 - 4x^2 + 2x^3)^{2024} \). and \( b = \lim_{x \to 0} \frac{\int_0^x \frac{\log(1 + t)}{t^{2024} + 1} \, dt}{x^2} \).If the equations \( cx^2 + dx + e = 0 \) and \( 2bx^2 + ax + 4 = 0 \) have a common root, where \( c, d, e \in \mathbb{R} \), then \( d : c : e \) equals

If the system of linear equations
\( x - 2y + z = -4 \)
\( 2x + \alpha y + 3z = 5 \)
\( 3x - y + \beta z = 3 \)
has infinitely many solutions, then \( 12\alpha + 13\beta \) is equal to

Let the line \( L : \sqrt{2}x + y = \alpha \) pass through the point of intersection \( P \) (in the first quadrant) of the circle \( x^2 + y^2 = 3 \) and the parabola \( x^2 = 2y \). Let the line \( L \) touch two circles \( C_1 \) and \( C_2 \) of equal radius \( 2\sqrt{3} \). If the centers \( Q_1 \) and \( Q_2 \) of the circles \( C_1 \) and \( C_2 \) lie on the y-axis, then the square of the area of the triangle \( PQ_1Q_2 \) is equal to ____.

The number of elements in the set S = {(x, y, z) : x, y, z ∈ Z, x + 2y + 3z = 42, x, y, z ≥ 0} equals ____