List of top Mathematics Questions asked in JEE Main

Let \( H_n : \frac{x^2}{1 + n} + \frac{y^2}{3 + n} = 1, n \in \mathbb{N} \). Let \( k \) be the smallest even value of \( n \) such that the eccentricity of \( H_n \) is a rational number. If \( l \) is the length of the latus rectum of \( H_k \), then 21 \( l \) is equal to:
For \( m, n>0 \), let \( \alpha(m,n) = \int_{0}^{1} (1 + 3t)^{n} \, dt \). If \( \alpha(10,6) = \int_{0}^{1} (1 + 3t)^{6} \, dt \) and \( \alpha(11,5) = p(14)^{5} \), then \( p \) 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:}
If equation of the plane that contains the point \((-2,3,5)\) and is perpendicular to each of the planes \( 2x + 4y + 5z = 8 \) and \( 3x - 2y + 3z = 5 \), is \( \alpha x + \beta y + \gamma z = 97 \), then \( \alpha + \beta + \gamma \) is:
\( Let f(x) = | x^2 - x | + |x|, \text{ where } x \in \mathbb{R} \text{ and } | t | \text{ denotes the greatest integer less than or equal to } t. \text{ Then, } f \text{ is:} \)
Consider ellipse \( E_k : \frac{x^2}{k} + \frac{y^2}{k} = 1 \), for \( k = 1, 2, \dots, 20 \). Let \( C_k \) be the circle which touches the four chords joining the end points (one on the minor axis and another on the major axis) of the ellipse \( E_k \). If \( r_k \) is the radius of the circle \( C_k \), then the value of \( \sum_{k=1}^{20} r_k^2 \) is:
For any vector \( \mathbf{a} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k} \), with \( 10 | \mathbf{a} |<1 \), \( i = 1, 2, 3 \), consider the following statements:
The number of triplets \( (x, y, z) \), where \( x, y, z \) are distinct non-negative integers satisfying \( x + y + z = 15 \), is:
Let \( w_1 \) be the point obtained by the rotation of \( z_1 = 5 + 4i \) about the origin through a right angle in the anticlockwise direction, and \( w_2 \) be the point obtained by the rotation of \( z_2 = 3 + 5i \) about the origin through a right angle in the clockwise direction. Then the principal argument of \( w_1 - w_2 \) is equal to:
Let \( \mathbf{a} \) be a non-zero vector parallel to the line of intersection of the two planes described by \( i + j + k \) and \( -i - j - k \). If \( \theta \) is the angle between the vector \( \mathbf{a} \) and the vector \( \mathbf{b} = -2i - 2j + 2k \), and \( \left| \mathbf{a} \right| = 6 \), then ordered pair \( (\mathbf{a} \cdot \mathbf{b}) \) is equal to:
Let sets A and B have 5 elements each. Let mean of the elements in sets A and B be 5 and 8 respectively and the variance of the elements in sets A and B be 12 and 20 respectively. A new set C of 10 elements is formed by subtracting 3 from each element of A and adding 2 to each element of B. Then the sum of the mean and variance of the elements of C is:
Area of the region \((x, y) : x^2 + (y - 2)^2 \leq 4, \, x^2 \geq 2y\) is:
The number of elements in the set \(S = \{ \theta \in [0, 2\pi] : 3 \cos^4 \theta - 5 \cos^2 \theta - 2 \sin^2 \theta + 2 = 0 \}\) is:
Let A be a \( 2 \times 2 \) matrix with real entries such that \( A^T = \alpha A + I \), where \( \alpha \in \mathbb{R} \setminus \{-1, 1\} \). If \( \text{det}(A^2 - A) = 4 \), then the sum of all possible values of \( \alpha \) is equal to:
The value of the integral \[ \int_{\log_2}^{-\log_2} e^x \left( \log \left( e^x + \sqrt{1 + e^{2x}} \right) \right) dx \] is equal to:
Let \( f: [2, 4] \to \mathbb{R} \) be a differentiable function such that \( (x \log x) f'(x) + (\log x) f(x) \geq 1 \), \( x \in [2, 4] \) with \( f(2) = \frac{1}{2} \) and \( f(4) = \frac{1}{4} \). Consider the following two statements:
\( (A) \quad f(x) \geq 1 \quad \text{for all} \quad x \in [2, 4] \)
\( (B) \quad f(x) \leq \frac{1}{8} \quad \text{for all} \quad x \in [2, 4] \) Then,
The number of integral solutions of \( \log_2 \left( \frac{x - 7}{2x - 3} \right) \geq 0 \) is:
Let \(x_1, x_2, \dots, x_{100}\) be in an arithmetic progression, with \(x_1 = 2\) and their mean equal to 200. If \(y_i = (i \cdot x_i)\), then the mean of \(y_1, y_2, \dots, y_{100}\) is:

Let [t] denote the greatest integer function. If \(\int\limits_0^{2.4}[x^2]dx=α+β√2+γ√3+δ√5,\) then α+β+γ +δ is equal to _____.

Let \( k \) and \( m \) be positive real numbers such that the function} \[ f(x) = \begin{cases} 3x^2 + \frac{k}{\sqrt{x} + 1}, & 0 < x < 1, \\ mx^2 + k^2, & x \geq 1 \end{cases} \] is differentiable for all \( x > 0 \). Then \( 8f'(8) \left(\frac{1}{f(8)}\right) \) is equal to __________

Let \( 0 < z < y < x \) be three real numbers such that \( \frac{1}{x}, \frac{1}{y}, \frac{1}{z} \) are in an arithmetic progression and \( x, \sqrt{2}y, z \) are in a geometric progression. If \( xy + yz + zx = \frac{3}{\sqrt{2}} xyz \), then \( 3(x + y + z)^2 \) is equal to ____________.

Let \( m \) and \( n \) be the numbers of real roots of the quadratic equations \( x^2 - 12x + [x] + 31 = 0 \) and \( x^2 - 5|x+2| - 4 = 0 \), respectively, where \( [x] \) denotes the greatest integer less than or equal to \( x \). Then \( m^2 + mn + n^2 \) is equal to ___________.

The negation of \( (p \land (\sim q)) \lor (\sim p) \) is equivalent to:

The value of \( 36 \big(4 \cos^2 9^\circ - 1\big) \big(4 \cos^2 27^\circ - 1\big) \big(4 \cos^2 81^\circ - 1\big) \big(4 \cos^2 243^\circ - 1\big) \) is:

Let the mean and variance of 12 observations be \( \frac{9}{2} \) and 4, respectively. Later on, it was observed that two observations were considered as 9 and 10 instead of 7 and 14 respectively. If the correct variance is \( \frac{m}{n} \), where \( m \) and \( n \) are coprime, then \( m + n \) is equal to: