List of top Mathematics Questions asked in JEE Main

Let \( f: \mathbb{R} \to \mathbb{R} \) be defined as: \(f(x) =  \begin{cases}  \frac{a - b \cos 2x}{x^2}, & x < 0, \\  x^2 + cx + 2, & 0 \leq x \leq 1, \\  2x + 1, & x > 1.  \end{cases}\)
If \( f \) is continuous everywhere in \( \mathbb{R} \) and \( m \) is the number of points where \( f \) is NOT differentiable, then \( m + a + b + c \) equals:  
If one of the diameters of the circle \(x^2+y^2-10x+4y+13=0\) is a chord of another circle and whose center is the point of intersection of the lines \(2x + 3y = 12\) and \(3x - 2y = 5 \) then the radius of the circle is

Let \( 0 \le r \le n \). If \( ^{n+1}C_{r+1} : ^nC_r : ^{n-1}C_{r-1} = 55 : 35 : 21 \), then \( 2n + 5r \) is equal to:

The sum of the coefficients of \( x^{2/3} \) and \( x^{-2/5} \) in the binomial expansion of $$ \left( x^{2/3} + \frac{1}{2} x^{-2/5} \right)^9 $$ is: 
Let \( A \) and \( B \) be two finite sets with \( m \) and \( n \) elements respectively. The total number of subsets of the set \( A \) is 56 more than the total number of subsets of \( B \). Then the distance of the point \( P(m, n) \) from the point \( Q(-2, -3) \) is:
Let \( \alpha = \sum_{k=0}^{n} \left( \frac{\binom{n}{k}}{k+1} \right)^2 \) and \( \beta = \sum_{k=0}^{n-1} \left( \frac{\binom{n}{k} \binom{n}{k+1}}{k+2} \right) \).
If \( 5\alpha = 6\beta \), then \( n \) equals __________.
Let \( A \) be a square matrix of order 2 such that \( |A| = 2 \) and the sum of its diagonal elements is \(-3\). If the points \((x, y)\) satisfying \( A^2 + xA + yI = 0 \) lie on a hyperbola, whose transverse axis is parallel to the x-axis, eccentricity is \( e \) and the length of the latus rectum is \( \ell \), then \( e^4 + \ell^4 \) is equal to _____
If the system of equations \(x + 4y - z = \lambda\), \(7x + 9y + \mu z = -3\), \(5x + y + 2z = -1\) has infinitely many solutions, then \((2\mu + 3\lambda)\) is equal to:
If the foot is perpendicular from (1, 2, 3) to the line \(\frac{x+1}{2} = \frac{y-2}{5} = \frac{z-1}{1}\) is \(( a, \beta, \gamma)\), then find \(a + \beta + \gamma\)

Let \( \vec{a} = 3\hat{i} + \hat{j} - 2\hat{k} \), \( \vec{b} = 4\hat{i} + \hat{j} + 7\hat{k} \), and \( \vec{c} = \hat{i} - 3\hat{j} + 4\hat{k} \) be three vectors.
If a vector \( \vec{p} \) satisfies \( \vec{p} \times \vec{b} = \vec{c} \times \vec{b} \) and \( \vec{p} \cdot \vec{a} = 0 \), then \( \vec{p} \cdot (\hat{i} - \hat{j} - \hat{k}) \) is equal to

The value of the integral \[ \int_{-1}^{2} \log_e \left( x + \sqrt{x^2 + 1} \right) \, dx \] is:
Let \(\overrightarrow{a},\overrightarrow{b}\) and \(\overrightarrow{c}\) be three non-zero vectors such that \(\overrightarrow{b}\) and \(\overrightarrow{c}\) are non-collinear. If \(\overrightarrow{a}+5\overrightarrow{b}\) is collinear with \(\overrightarrow{c},\overrightarrow{b}+6\overrightarrow{c}\) is collinear with \(\overrightarrow{a}\) and \(\overrightarrow{a}+ α\overrightarrow{b} + β\overrightarrow{c} = 0\), then α + β is equal to
Let $A = \begin{bmatrix} 2 & a & 0 \\ 1 & 3 & 1 \\ 0 & 5 & b \end{bmatrix}$. If $A^3 = 4A^2 - A - 21I$, where I is the identity matrix of order $3 \times 3$, then $2a + 3b$ is equal to:
If the system of equations
\( x + \left( \sqrt{2} \sin \alpha \right) y + \left( \sqrt{2} \cos \alpha \right) z = 0 \)
\( x + \left( \cos \alpha \right) y + \left( \sin \alpha \right) z = 0 \)
\( x + \left( \sin \alpha \right) y - \left( \cos \alpha \right) z = 0 \)
has a non-trivial solution, then \( \alpha \in \left( 0, \frac{\pi}{2} \right) \) is equal to:
The set of all α, for which the vectors $\vec{a} = \alpha \hat{ti} + 6\hat{j} - 3\hat{k}$ and $\vec{b} = \hat{ti} - 2\hat{j} - 2\alpha t\hat{k}$ are inclined at an obtuse angle for all $t \in \mathbb{R}$ is:
The integral $$ \int \frac{x^8 - x^2}{(x^{12} + 3x^6 + 1) \tan^{-1}\left( \frac{x^3 + 1}{x^3} \right)} \, dx $$ is equal to: 
The value of \(\frac {120}{\pi^3}|∫_0^\pi\frac {x^2sinx.cosx}{(sinx)^4+(cosx)^4}dx|\) is
Let \( f : \mathbb{R} \to \mathbb{R} \) be a function given by
\[f(x) = \begin{cases} \frac{1 - \cos 2x}{x^2}, & x < 0 \\\alpha, & x = 0, \text{ where } \alpha, \beta \in \mathbb{R}. \\\beta \sqrt{1 - \cos x} / x, & x > 0 \end{cases} \]
If \( f \) is continuous at \( x = 0 \), then \( \alpha^2 + \beta^2 \) is equal to:
\((α, β)\) lie on the parabola \(y^2 = 4x\) and \((α, β)\) also lie on chord with midpoint \((1,\frac 54)\) of another parabola \(x^2 = 8y\), then value of \(|(8 – β)(α – 28)|\) is
The area (in square units) of the region described by: $$ \{(x, y) : y^2 \leq 2x, \, \text{and} \, y \geq 4x - 1\} $$ is:
Consider a hyperbola \( H \) having its centre at the origin and foci on the \( x \)-axis. Let \( C_1 \) be the circle touching the hyperbola \( H \) and having its centre at the origin. Let \( C_2 \) be the circle touching the hyperbola \( H \) at its vertex and having its centre at one of its foci. If the areas (in square units) of \( C_1 \) and \( C_2 \) are \( 36\pi \) and \( 4\pi \), respectively, then the length (in units) of the latus rectum of \( H \) is:
Let \( g(x) \) be a linear function and \[ f(x) = \begin{cases} g(x), & x \leq 0 \\ \left( \frac{1 + x}{2 + x} \right)^{\frac{1}{x}}, & x > 0 \end{cases} \] is continuous at \( x = 0 \). If \( f'(1) = f(-1) \), then the value of \( g(3) \) is
If the system of equations \[2x + 7y + \lambda z = 3,\]\[3x + 2y + 5z = 4,\]\[x + \mu y + 32z = -1\]has infinitely many solutions, then $(\lambda - \mu)$ is equal to ________.
The frequency distribution of the age of students in a class of 40 students is given below:
\(Age\)151617181920
No. of Students58512xy

If the mean deviation about the median is 1.25, then \(4x + 5y\) is equal to:
There are 4 men and 5 women in Group A, and 5 men and 4 women in Group B. If 4 persons are selected from each group, then the number of ways of selecting 4 men and 4 women is