List of top Mathematics Questions asked in JEE Main

Let the parabola \( y = x^2 + px + q \) passing through the point \( (1, -1) \) be such that the distance between its vertex and the x-axis is minimum. Then the value of \( p^2 + q^2 \) is:

Let O be the origin, and P and Q be two points on the rectangular hyperbola \( xy = 12 \) such that the midpoint of the line segment PQ is \( \left( \frac{1}{2}, -\frac{1}{2} \right) \). Then the area of the triangle OPQ equals:

Let a circle pass through the origin and its center be the point of intersection of two mutually perpendicular lines \( x + (k-1)y + 3 = 0 \) and \( 2x + k2y - 4 = 0 \). If the line \( x - y + 2 = 0 \) intersects the circle at the points A and B, then \( (AB)^2 \) is equal to:

The mean and variance of \( n \) observations are 8 and 16, respectively. If the sum of the first \( (n-1) \) observations is 48 and the sum of squares of the first \( (n-1) \) observations is 496, then the value of \( n \) is:

A man throws a fair coin repeatedly. He gets 10 points for each head he throws and 5 points for each tail he throws. If the probability that he gets exactly 30 points is \( \frac{m}{n} \), gcd \( (m, n) = 1 \), then \( m + n \) is equal to:

Let \( p_n \) denote the total number of triangles formed by joining the vertices of an \( n \)-side regular polygon. If \( p_{n+1} - p_n = 66 \), then the sum of all distinct prime divisors of \( n \) is:

The sum \( \frac{1^3}{1} + \frac{2^3}{1+3} + \frac{3^3}{1+3+5} + \cdots \) up to 8 terms is:

Let \( a_1, a_2, a_3, \dots \) be an A.P. and \( g_1 = a_1, g_2 = a_2, g_3 = a_3, \dots \) be an increasing G.P. If \( a_1 = a_2 + g_2 = 1 \) and \( a_3 + g_3 = 4 \), then \( a_{10} + g_5 \) is equal to:

If the system of equations} \[ x + 5y + 6z = 4 \] \[ 2x + 3y + 4z = 7 \] \[ x + 6y + az = b \] has infinitely many solutions, then the point \( (a, b) \) lies on the line}

Let the circles \( C_1 : |z| = r \) and \( C_2 : |z - 3 - 4i| = 5, z \in \mathbb{C} \), be such that \( C_2 \) lies within \( C_1 \). If \( z_1 \) moves on \( C_1 \), \( z_2 \) moves on \( C_2 \) and \( \min |z_1 - z_2| = 2 \), then \( \max |z_1 - z_2| \) is equal to:

Let \( \alpha, \beta \) be the roots of the equation \( x^2 - 3x + r = 0 \), and \( \frac{\alpha}{2}, 2\beta \) be the roots of the equation \( x^2 + 3x + r = 0 \). If the roots of the equation \( x^2 + 6x = m \) are \( 2\alpha + \beta + 2r \) and \( \alpha - 2\beta - \frac{r}{2} \), then \( m \) is equal to:

Let \[ f(x)= \begin{cases} x^3+8, & x<0,\\ x^2-4, & x\ge0, \end{cases} \qquad g(x)= \begin{cases} (x-8)^{1/3}, & x<0,\\ (x+4)^{1/2}, & x\ge0. \end{cases} \] Then the number of points where the function \(g\circ f\) is discontinuous is ______.

Let \( R=\{(x,y)\in \mathbb{N}\times\mathbb{N}:\log_e(x+y)\le2\} \). Then the minimum number of elements required to be added in \(R\) to make it a transitive relation is ________.

Let \[ \lim_{x\to2}\frac{\tan(x-2)\,[x^2+(p-2)x-2p]}{(x-2)^2}=5 \] for some \(p,r\in\mathbb{R}\). If the set of all possible values of \(q\), such that the roots of the equation \(rx^2-px+q=0\) lie in \( (0,2) \), be the interval \( (\alpha,\beta) \), then \(4(\alpha+\beta)\) equals :

Let $f(x) = \begin{cases} e^{x-1}, & x<0 \\ x^2 - 5x + 6, & x \ge 0 \end{cases}$ and $g(x) = f(|x|) + |f(x)|$. If the number of points where $g$ is not continuous and is not differentiable are $\alpha$ and $\beta$ respectively, then $\alpha + \beta$ is equal to _______.

From a month of 31 days, 3 different dates are selected at random. If the probability that these dates are in an increasing A.P. is equal to $a/b$, where $a, b \in \mathbb{N}$ and $\gcd(a, b) = 1$, then $a + b$ is equal to _______

Let \( A, B, C \) be three \( 2\times2 \) matrices with real entries such that \[ B=(I+A)^{-1} \quad \text{and} \quad A+C=I. \] If \[ BC=\begin{bmatrix}1 & -5 \\-1 & 2\end{bmatrix} \quad \text{and} \quad B\begin{bmatrix}x_1\\x_2\end{bmatrix} =\begin{bmatrix}12\\-6\end{bmatrix}, \] then \( x_1+x_2 \) is:

Let the image of the point \(P(0,-5,0)\) in the line \[ \frac{x-1}{2}=\frac{y}{1}=\frac{z+1}{-2} \] be the point \(R\) and the image of the point \(Q(0,-\frac12,0)\) in the line \[ \frac{x-1}{-1}=\frac{y+9}{4}=\frac{z+1}{1} \] be the point \(S\). Then the square of the area of the parallelogram \(PQRS\) is ______.

Let \[ A= \begin{bmatrix} 1 & 3 & -1\\ 2 & 1 & \alpha\\ 0 & 1 & -1 \end{bmatrix} \] be a singular matrix. Let \[ f(x)=\int_{0}^{x}(t^2+2t+3)\,dt,\quad x\in[1,\alpha]. \] If \(M\) and \(m\) are respectively the maximum and the minimum values of \(f\) in \([1,\alpha]\), then \(3(M-m)\) is equal to :

Let \( \vec a = 2\hat i + 3\hat j + 3\hat k \) and \( \vec b = 6\hat i + 3\hat j + 3\hat k \). Then the square of the area of the triangle with adjacent sides determined by the vectors \( (2\vec a + 3\vec b) \) and \( (\vec a - \vec b) \) is :

The shortest distance between the lines \[ \frac{x-4}{1}=\frac{y-3}{2}=\frac{z-2}{-3} \] and \[ \frac{x+2}{2}=\frac{y-6}{4}=\frac{z-5}{-5} \] is :

Let \( A = \begin{bmatrix} 1 & 0 & 0\\ 3 & 1 & 0\\ 9 & 3 & 1 \end{bmatrix} \) and \( B = [b_{ij}], 1 \le i,j \le 3 \). If \( B = A^{99} - I \), then the value of \( \dfrac{b_{31}-b_{21}}{b_{32}} \) is:

The sum of all possible values of \( \theta \in [0,2\pi] \), for which the system of equations : \[ x\cos3\theta - 8y - 12z = 0 \] \[ x\cos2\theta + 3y + 3z = 0 \] \[ x + y + 3z = 0 \] has a non-trivial solution, is equal to :

Let \( PQR \) be a triangle such that \[ \vec{PQ}=-2\hat i-\hat j+2\hat k,\quad \vec{PR}=a\hat i+b\hat j-4\hat k,\ a,b\in\mathbb{Z}. \] Let \( S \) be the point on \( QR \) which is equidistant from the lines \( PQ \) and \( PR \). If \[ |\vec{PR}|=9 \quad \text{and} \quad \vec{PS}=\hat i-7\hat j+2\hat k, \] then the value of \( 3a-4b \) is: