List of top Mathematics Questions asked in JEE Main

If the probability that the random variable \( X \) takes values \( x \) is given by \( P(X = x) = k(x + 1) 3^{-x}, x = 0, 1, 2, \dots \), where \( k \) is a constant, then \( P(X \geq 2) \) is equal to:

Let the vectors \(\mathbf{u}_1 = \hat{i} + \hat{j} + a\hat{k}, \mathbf{u}_2 = \hat{i} + b\hat{j} + \hat{k}\), and \(\mathbf{u}_3 = c\hat{i} + \hat{j} + \hat{k}\) be coplanar. If the vectors \(\mathbf{v}_1 = (a + b)\hat{i} + c\hat{j} + c\hat{k}, \mathbf{v}_2 = a\hat{i} + (b + c)\hat{j} + a\hat{k}, \mathbf{v}_3 = b\hat{i} + b\hat{j} + (c + a)\hat{k}\) are also coplanar, then \(6(a + b + c)\) is equal to:

The area of the quadrilateral having vertices as (1,2), (5,6), (7,6), (-1,-6) is?

For \(a, b \in \mathbb{Z}\) and \(|a - b| \leq 10\), let the angle between the plane \(P: ax + y - z = b\) and the line \(L: x - 1 = a - y = z + 1\) be \(\cos^{-1}\left(\frac{1}{3}\right)\). If the distance of the point \((6, -6, 4)\) from the plane \(P\) is \(3\sqrt{6}\), then \(a^4 + b^2\) is equal to:

Let \( A(0, 1) \), \( B(1, 1) \), and \( C(1, 0) \) be the midpoints of the sides of a triangle with incentre at the point \( D \). If the focus of the parabola \( y^2 = 4ax \) passing through \( D \) is \( (\alpha + \beta \sqrt{3}, 0) \), where \( \alpha \) and \( \beta \) are rational numbers, then \( \frac{\alpha}{\beta^2} \) is equal to:

Let \( A = \{1, 2, 3, 4, 5, 6, 7\} \). Then the relation \( R = \{(x, y) \in A \times A : x + y = 7\} \) is:

Let the area enclosed by the lines \( x + y = 2 \), \( y = 0 \), \( x = 0 \), and the curve \( f(x) = \min \left\{ x^2 + \frac{3}{4}, 1 + [x] \right\} \), where \( [x] \) denotes the greatest integer less than or equal to \( x \), be \( A \). Then the value of \( 12A \) is ____________.

Let the solution curve \( x = x(y) \), \( 0 < y \leq \frac{\pi}{2} \), of the differential equation \[ (\log_e (\cos y))^2 \cos y \, dx - (1 + 3x \log_e (\cos y)) \sin y \, dy = 0 \] satisfy \( x \left( \frac{\pi}{3} \right) = \frac{1}{2 \log_e 2} \). If \( x \left( \frac{\pi}{6} \right) = \frac{1}{\log_e m - \log_e n} \), where \( m \) and \( n \) are co-prime integers, then \( mn \) is equal to ______.

A parabola with focus (3, 0) and directrix x = –3. Points P and Q lie on the parabola and their ordinates are in the ratio 3 : 1. The point of intersection of tangents drawn at points P and Q lies on the parabola

Let P₁ be the plane 3x-y-7z = 11 and P₂ be the plane passing through the points (2,-1,0), (2,0,-1), and (5,1,1). If the foot of the perpendicular drawn from the point (7,4,-1) on the line of intersection of the planes P₁ and P₂ is (α, β, γ), then a + ẞ+ y is equal to

Let A = {1, 2, 3, 4,.....10} and B = {0, 1, 2, 3, 4}. The number of elements in the relation R = {(a, b) ∈ A × A: 2(a – b)2 + 3(a – b) ∈ B} is______.

Let S be the set of all values of θ Ε [-π, π] for which the system of linear equations
x+y+√3z=0
-x+(tanθ)y+ √7z=0
x+y+(tanθ)z = 0 has non-trivial solution.
Then 120/π ∑θ_θ∈s is equal to

The coefficient of \(x^{18}\) in the expansion of \(\left(x^4 - \frac{1}{x^3}\right)^{15}\) is ________

If α > β > 0 are the roots of the equation ax2 + bx + 1 = 0, and \(\lim\limits_{x \to \frac{1}{\alpha}}\left(\frac{1-cos(x^2+bx+a)}{2(1-ax)^2}\right)^{1/2}=\frac{1}{k}\left(\frac{1}{\beta}-\frac{1}{\alpha}\right),\) then k is equal to

Let a_n be the n^th term of the series 5 + 8 + 14 + 23 + 35 + 50 +.... and Sn = \(\displaystyle\sum_{k=1}^{n} a_k.\) Then S³⁰ - a⁴⁰ is equal to

A circle passing through the point P\(( \alpha , \beta )\) in the first quadrant touches the two coordinate axes at the points A and B. The point P is above the line AB. The point Q on the line segment AB is the foot of perpendicular from P on AB. If PQ is equal to 11 units, then value of \(( \alpha \beta )\) is ___

Let the mean and variance of 8 numbers \(x, y, 10, 12, 6, 12, 4, 8\) be \(9\) and \(9.25\) respectively. If \(x > y\), then \(3x - 2y\) is equal to:

Let \(\vec{a} = 6\hat{i} + 9\hat{j} + 12\hat{k}\), \(\vec{b} = a\hat{i} + 11\hat{j} - 2\hat{k}\), and \(\vec{c}\) be vectors such that \(\vec{a} \times \vec{c} = \vec{a} \times \vec{b}\). If \(\vec{a} \cdot \vec{c} = -12\) and \(\vec{c} \cdot (\hat{i} - 2\hat{j} + \hat{k}) = 5\), then \(\vec{c} \cdot (\hat{i} + \hat{j} + \hat{k})\) is equal to:

Let \(\lambda_1, \lambda_2\) be the values of \(\lambda\) for which the points \((1, \lambda, \frac{1}{2})\) and \((-2, 0, 1)\) are at equal distance from the plane \(2x + 3y - 6z + 7 = 0\). If \(\lambda_1 > \lambda_2\), then the distance of the point \((1 - \lambda_2, \lambda_2, \lambda_1)\) from the line \(\frac{x-5}{1} = \frac{y-1}{2} = \frac{z+7}{2}\) is:

Consider a circle \(C_1 : x^2 + y^2 - 4x - 2y = \alpha - 5\). Let its mirror image in the line \(y = 2x + 1\) be another circle \(C_2 : 5x^2 + 5y^2 - 10x - 10y + 36 = 0\). Let \(r\) be the radius of \(C_2\). Then \(\alpha + r\) is equal to:

If the solution curve of the differential equation \( (y - 2 \log x) dx + (x \log x^2) dy = 0 \) passes through the points \( \left( e^{4/3}, \alpha \right) \) and \( \left( e^4, \alpha \right) \), then \( \alpha \) is equal to:

Let [t] denote the greatest integer ≤t. Then \(\frac{2}{π} \int\limits_{\frac{\pi}{6}}^{\frac{5\pi}{6}} (8 [cosec x]-5[cot x]) dx\)is equal to ______.

If a_a is the greatest term in the sequence \(a_n=\frac{n^3}{n^4+147},n=1,2,3,...,\) then a is equal to______________.

Let \((\alpha, \beta, \gamma)\) \(\text{ be the image of the point }\) \(P(3, 3, 5) \text{ in the plane } 2x + y - 3z = 6\). \(\text{ Then } \alpha + \beta + \gamma \text{ is equal to:}\)

Let O be the origin and OP and OQ be the tangents to the circle \( x^2 + y^2 - 6x + 4y + 8 = 0 \) at the points P and Q on it. If the circumcircle of the triangle OPQ passes through the point \( \left( \alpha, \frac{1}{2} \right) \), then a value of \( \alpha \) is.