List of top Mathematics Questions asked in JEE Main

Let a tangent be drawn to the ellipse x²/27 + y² = 1 at (3$\sqrt{3}$ cosθ, sinθ) where θ ∈ (0, π/2). Then the value of θ such that the sum of intercepts on axes made by this tangent is minimum is equal to :
In a triangle ABC, if |BC| = 8, |CA| = 7, |AB| = 10, then the projection of the vector AB on AC is equal to :
If 15 sin⁴ α + 10 cos⁴ α = 6, for some α ∈ R, then the value of 27 sec⁶ α + 8 cosec⁶ α is equal to :
If P and Q are two statements, then which of the following compound statement is a tautology ?
Let in a series of 2n observations, half of them are equal to a and remaining half are equal to -a. Also by adding a constant b in each of these observations, the mean and standard deviation of new set become 5 and 20, respectively. Then the value of a² + b² is equal to :
A pole stands vertically inside a triangular park ABC. Let the angle of elevation of the top of the pole from each corner of the park be π/3. If the radius of the circumcircle of ΔABC is 2, then the height of the pole is equal to :
Let in a Binomial distribution, consisting of 5 independent trials, probabilities of exactly 1 and 2 successes be 0.4096 and 0.2048 respectively. Then the probability of getting exactly 3 successes is equal to :
Let the centroid of an equilateral triangle ABC be at the origin. Let one of the sides of the equilateral triangle be along the straight line x+y=3. If R and r be the radius of circumcircle and incircle respectively of ΔABC, then (R + r) is equal to :
Let \( P = \begin{bmatrix} 3 & -1 & -2 \\ 2 & 0 & \alpha \\ 3 & -5 & 0 \end{bmatrix} \), where \( \alpha \in \mathbb{R} \). Suppose \( Q = [q_{ij}] \) is a matrix satisfying \( PQ = k I_3 \) for some non-zero \( k \in \mathbb{R} \). If \( q_{23} = -\dfrac{k}{8} \) and \( |Q| = \dfrac{k^3}{2} \), then \( \alpha^2 + k^2 \) is equal to __________.
The equation of the plane passing through the point (1, 2, -3) and perpendicular to the planes $3x + y - 2z = 5$ and $2x - 5y - z = 7$, is :
The system of linear equations 3x - 2y - kz = 10, 2x - 4y - 2z = 6, x + 2y - z = 5m is inconsistent if :
If \( f(x) = [x - 1]\cos\!\left( \frac{2x - 1}{2}\pi \right) \), then \( f \) is :
The function \( f(x) = \dfrac{4x^3 - 3x^2}{6} - 2 \sin x + (2x - 1)\cos x \) :
\( \displaystyle \lim_{x \to 0} \frac{\int_{0}^{x^2} \sin(\sqrt{t}) \, dt}{x^3} \) is equal to :
The value of \( 2^{15}C_1 + 2^{15}C_2 - 3^{15}C_3 + \cdots - 15^{15}C_{15} + \binom{14}{1} + \binom{14}{3} + \binom{14}{5} + \cdots + \binom{14}{11} \) is :
Let \( p \) and \( q \) be two positive numbers such that \( p + q = 2 \) and \( p^4 + q^4 = 272 \). Then \( p \) and \( q \) are roots of the equation :
Let \( f : \mathbb{R} \to \mathbb{R} \) be defined as \( f(x) = 2x - 1 \) and \( g : \mathbb{R} \setminus \{1\} \to \mathbb{R} \) be defined as \( g(x) = \dfrac{x - \frac{1}{2}}{x - 1} \). Then the composition function \( f(g(x)) \) is :
\(\lim_{n \to \infty} \tan \left\{ \sum_{r=1}^{n} \tan^{-1} \left( \frac{1}{1 + r + r^2} \right) \right\}\) is equal to __________.
Let three vectors \(\vec{a}, \vec{b}\) and \(\vec{c}\) be such that \(\vec{c}\) is coplanar with \(\vec{a}\) and \(\vec{b}, \vec{a} \cdot \vec{c} = 7\) and \(\vec{b}\) is perpendicular to \(\vec{c}\), where \(\vec{a} = -\hat{i} + \hat{j} + \hat{k}\) and \(\vec{b} = 2\hat{i} + \hat{k}\), then the value of \(|2\vec{a} + \vec{b} + \vec{c}|^2\) is __________
If one of the diameters of the circle \(x^2 + y^2 - 2x - 6y + 6 = 0\) is a chord of another circle 'C', whose center is at (2, 1), then its radius is ________
If \(\int_{-a}^{a} (|x| + |x-2|) dx = 22, (a>2)\) and \([x]\) denotes the greatest integer \(\le x\), then \(\int_{a}^{-a} (x + [x]) dx\) is equal to ______
Let \(M\) be any \(3\times3\) matrix with entries from the set {0, 1, 2}. The maximum number of such matrices, for which the sum of diagonal elements of \(M^T M\) is seven, is __________.
Let \(A = \{n \in N : n \text{ is a 3-digit number}\}\), \(B = \{9k + 2 : k \in N\}\) and \(C = \{9k + l : k \in N\}\) for some \(l\) (0<l<9) If the sum of all the elements of the set \(A \cap (B \cup C)\) is \(274 \times 400\), then \(l\) is equal to __________
The statement among the following that is a tautology is :
Let \(B_i\) (\(i=1, 2, 3\)) be three independent events in a sample space. The probability that only \(B_1\) occurs is \(\alpha\), only \(B_2\) occurs is \(\beta\) and only \(B_3\) occurs is \(\gamma\). Let \(p\) be the probability that none of the events \(B_i\) occurs and these 4 probabilities satisfy the equations \((\alpha - 2\beta) p = \alpha\beta\) and \((\beta - 3\gamma) p = 2\beta\gamma\) (All the probabilities are assumed to lie in the interval (0, 1)). Then \(\frac{P(B_1)}{P(B_3)}\) is equal to __________.