List of top Mathematics Questions asked in JEE Main

Let \( L_1 : \frac{x - 1}{3} = \frac{y - 1}{-1} = \frac{z + 1}{0} \) and \( L_2 : \frac{x - 2}{2} = \frac{y}{0} = \frac{z + 4}{\alpha} \), where \( \alpha \in \mathbb{R} \), be two lines which intersect at the point \( B \). If \( P \) is the foot of the perpendicular from the point \( A(1, 1, -1) \) on \( L_2 \), then the value of \( 26 \alpha (PB)^2 \) is:

Marks obtained by all the students of class 12 are presented in a frequency distribution with classes of equal width. Let the median of this grouped data be 14 with median class interval 12-18 and median class frequency 12. If the number of students whose marks are less than 12 is 18, then the total number of students is:

Let the area of a triangle \( \triangle PQR \) with vertices \( P(5, 4) \), \( Q(-2, 4) \), and \( R(a, b) \) be 35 square units. If its orthocenter and centroid are \( O(2, \frac{14}{5}) \) and \( C(c, d) \) respectively, then \( c + 2d \) is equal to:

If the first term of an A.P. is 3 and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first 20 terms is equal to:

Let R = {(1, 2), (2, 3), (3, 3)} be a relation defined on the set \( \{1, 2, 3, 4\} \). Then the minimum number of elements needed to be added in \( R \) so that \( R \) becomes an equivalence relation, is:}

Evaluate the following limit: \[ \lim_{x \to \infty} \frac{(2x^2 - 3x + 5) \left( 3x - 1 \right)^{x/2}}{(3x^2 + 5x + 4) \sqrt{(3x + 2)^x}}. \] The value of the limit is:

If \[ I = \int_0^{\frac{\pi}{2}} \frac{\sin^2 \frac{3}{2}x}{\sin^2 x + \cos^2 x} \, dx, \] then \[ \int_0^{\frac{\pi}{2}} \frac{x \sin x \cos x}{\sin^4 x + \cos^4 x} \, dx \] equals:

If the square of the shortest distance between the lines \[ \frac{x-2}{1} = \frac{y-1}{2} = \frac{z+3}{-3} \] and \[ \frac{x+1}{2} = \frac{y+3}{4} = \frac{z+5}{-5} \] is \( \frac{m}{n} \), where \( m \) and \( n \) are co-prime numbers, then \( m+n \) is equal to:

Let \( \alpha, \beta \) be the roots of the equation \( x^2 - ax - b = 0 \) with \( \text{Im}(\alpha) <\text{Im(\beta) \). Let \( P_n = \alpha^n - \beta^n \). If} \[ P_3 = -5\sqrt{7}, \, P_4 = -3\sqrt{7}, \, P_5 = 11\sqrt{7}, \, P_6 = 45\sqrt{7}, \] then \( |\alpha^4 + \beta^4| \) is equal to:

Let \( A = [a_{ij}] \) be a \( 3 \times 3 \) matrix such that: \[ A = \begin{bmatrix} 0 & 0 & 4
1 & 0 & 0
0 & 1 & 3 \end{bmatrix}, A^{-1} = \begin{bmatrix} 0 & 1 & 0
1 & 3 & 0
2 & 1 & 0 \end{bmatrix}. \] Then \( a_{23} \) equals:

The number of complex numbers \( z \), satisfying \( |z| = 1 \) and \[ \left| \frac{z}{\overline{z}} + \frac{\overline{z}}{z} \right| = 1, \] is:

The system of equations \[ x + y + z = 6, \] \[ x + 2y + 5z = 9, \] \[ x + 5y + \lambda z = \mu, \] has no solution if:

Let the shortest distance from \( (a, 0) \), where \( a > 0 \), to the parabola \( y^2 = 4x \) be 4. Then the equation of the circle passing through the point \( (a, 0) \) and the focus of the parabola, and having its center on the axis of the parabola is:

The number of ways, 5 boys and 4 girls can sit in a row so that either all the boys sit together or no two boys sit together is:

If the area of the region \[ \{(x, y) : -1 \leq x \leq 1, 0 \leq y \leq a + e^{|x|} - e^{-x}, a> 0\} \] is \[ \frac{e^2 + 8e + 1}{e}, \] then the value of \(a\) is:

Let the range of the function \[ f(x) = 6 + 16 \cos x \cdot \cos\left(\frac{\pi}{3} - x\right) \cdot \cos\left(\frac{\pi}{3} + x\right) \cdot \sin 3x \cdot \cos 6x, \quad x \in R \text{ be } [\alpha, \beta]. \] Then the distance of the point \((\alpha, \beta)\) from the line \(3x + 4y + 12 = 0\) is:

Let \( X = {R} \times {R} \). Define a relation \( R \) on \( X \) as: \[ (a_1, b_1) \, R \, (a_2, b_2) \iff b_1 = b_2. \] Statement-I: \( R \) is an equivalence relation. Statement-II: For some \( (a, b) \in X \), the set \( S = \{(x, y) \in X : (x, y) R (a, b)\ \) represents a line parallel to \( y = x \).}

Let \( x = x(y) \) be the solution of the differential equation: \[ y = \left( x - y \frac{dx}{dy} \right) \sin\left( \frac{x}{y} \right), \, y > 0 \, \text{and} \, x(1) = \frac{\pi}{2}. \] Then \( \cos(x(2)) \) is equal to:

A board has 16 squares as shown in the figure. Out of these 16 squares, two squares are chosen at random. The probability that they have no side in common is: \includegraphics[]{10.PNG}

The roots of the quadratic equation \( 3x^2 - px + q = 0 \) are the 10th and 11th terms of an arithmetic progression with common difference \( \frac{3}{2} \). If the sum of the first 11 terms of this arithmetic progression is 88, then \( q - 2q \) is equal to:

The length of the chord of the ellipse: \[ \frac{x^2}{4} + \frac{y^2}{2} = 1, \] whose mid-point is \( \left( 1, \frac{1}{2} \right) \), is:

Let the point \( A \) divide the line segment joining the points \( P(-1, -1, 2) \) and \( Q(5, 5, 10) \) internally in the ratio \( r : 1 \) (\( r > 0 \)). If \( O \) is the origin and \[ \left( \frac{|\overrightarrow{OQ} \cdot \overrightarrow{OA}|}{5} \right) - \frac{1}{5} |\overrightarrow{OP} \times \overrightarrow{OA}|^2 = 10, \] then the value of \( r \) is:

The distance of the line \( \frac{x - 2}{2} = \frac{y - 6}{3} = \frac{z - 3}{4} \) from the point \( (1, 4, 0) \) along the line \( \frac{x}{1} = \frac{y - 2}{2} = \frac{z + 3}{3} \) is:

Let A = (x, y) \(\in\) {R \(\times\) {R} : |x + y| \(\geq\) 3} and} B = (x, y) \(\in\) {R \(\times\) {R} : |x| + |y| \(\leq 3\)}. If C = (x, y) \(\in\) A \(\cap\) B : x = 0 \(\text{ or y = 0}\), then \(\sum_{(x, y) \in C} |x| + |y|\) is:

The focus of the parabola \( y^2 = 4x + 16 \) is the center of the circle \( C \) with radius 5. If the values of \( \lambda \), for which \( C \) passes through the point of intersection of the lines \( 3x - y = 0 \) and \( x + \lambda y = 4 \), are \( \lambda_1 \) and \( \lambda_2 \), \( \lambda_1 <\lambda_2 \), then \( 12\lambda_1 + 29\lambda_2 \) is equal to: