List of top Questions asked in JEE Main

Bag A contains 9 white and 8 black balls, while bag B contains 6 white and 4 black balls. One ball is picked from B and put in A. Then a ball is drawn from A. Probability it is white is $p/q$. Find $p+q$.

If the solution curve $y = f(x)$ of the differential equation $(x^2 - 4) y' - 2xy + 2x(4 - x^2)^2 = 0, x>2$, passes through the point $(3, 15)$, then the local maximum value of $f$ is ___

The system of linear equations
$x + y + z = 6$
$2x + 5y + az = 36$
$x + 2y + 3z = b$
has

Let S denote the set of 4-digit numbers abcd such that $a>b>c>d$ and P denote the set of 5-digit numbers having product of its digits equal to 20. Then $n(S) + n(P)$ is equal to ___

Let \[ I(x) = \int \frac{3\,dx}{(4x+6)\sqrt{4x^2 + 8x + 3}} \] and \[ I(0) = \frac{\sqrt{3}}{4} + 20. \] If \[ I\left(\frac{1}{2}\right) = \frac{a\sqrt{2}}{b} + c, \] where \(a, b, c \in \mathbb{N}\) and \(\gcd(a,b)=1\), then find the value of \[ a + b + c. \]

The number of ways 16 oranges distributed to 4 children, each gets at least one.

Let \[ A = \begin{bmatrix} 0 & 2 & -3 \\ -2 & 0 & 1 \\ 3 & -1 & 0 \end{bmatrix} \] and \( B \) be a matrix such that \[ B(I - A) = I + A. \] Then the sum of the diagonal elements of \( B^{T}B \) is equal to

The area of the region enclosed between the circles $x^2 + y^2 = 4$ and $x^2 + (y - 2)^2 = 4$ is:

Let \( \vec{a}, \vec{b}, \vec{c} \) be three vectors such that \[ \vec{a} \times \vec{b} = 2(\vec{a} \times \vec{c}). \] If \( |\vec{a}| = 1 \), \( |\vec{b}| = 4 \), \( |\vec{c}| = 2 \), and the angle between \( \vec{b} \) and \( \vec{c} \) is \(60^\circ\), then \( |\vec{a} \cdot \vec{c}| \) is equal to.

The least value of $(\cos^2 \theta - 6\sin \theta \cos \theta + 3\sin^2 \theta + 2)$ is

Let \( \vec{a} = \hat{i} - 2\hat{j} + 3\hat{k} \), \( \vec{b} = 2\hat{i} + \hat{j} - \hat{k} \), \( \vec{c} = \lambda \hat{i} + \hat{j} + \hat{k} \) and \( \vec{v} = \vec{a} \times \vec{b} \). If \( \vec{v} \cdot \vec{c} = 11 \) and the length of the projection of \( \vec{b} \) on \( \vec{c} \) is \( p \), then find the value of \( 9p^2 \).

The number of elements in the set \[ S = \left\{ x : x \in [0,100] \text{ and } \int_{0}^{x} t^2 \sin(x - t)\,dt = x^2 \right\} \] is

Points of intersection of ellipses $x^2 + 2y^2 - 6x - 12y + 23 = 0$ and $4x^2 + 2y^2 - 20x - 12y + 35 = 0$ lie on a circle. Value of $ab + 18r^2$ is

An equilateral triangle OAB is inscribed in the parabola $y^2 = 4x$ with the vertex O at the vertex of the parabola. Then the minimum distance of the circle having AB as a diameter from the origin is

The value of the integral \( \int_{\frac{\pi}{24}}^{\frac{5\pi}{24}} \frac{dx}{1 + \sqrt[3]{\tan 2x}} \) is :

The vertices B and C of a triangle ABC lie on the line \( \frac{x}{1} = \frac{1-y}{2} = \frac{z-2}{3} \). The coordinates of A and B are (1, 6, 3) and (4, 9, 6) respectively and C is at a distance of 10 units from B. The area (in sq. units) of \( \triangle ABC \) is:

Let \( S = \{z : 3 \le |2z - 3(1+i)| \le 7\ \) be a set of complex numbers. Then \( \min_{z \in S} \left| z + \frac{1}{2}(5+3i) \right| \) is equal to :}

Number of solutions of \( \sqrt{3} \cos 2\theta + 8 \cos \theta + 3\sqrt{3} = 0, \theta \in [-3\pi, 2\pi] \) is:

Let \( \alpha \) and \( \beta \) respectively be the maximum and the minimum values of the function \( f(\theta) = 4\left(\sin^4\left(\frac{7\pi}{2} - \theta\right) + \sin^4(11\pi + \theta)\right) - 2\left(\sin^6\left(\frac{3\pi}{2} - \theta\right) + \sin^6(9\pi - \theta)\right) \). Then \( \alpha + 2\beta \) is equal to :

Let the mean and variance of 8 numbers -10, -7, -1, x, y, 9, 2, 16 be \( 2 \) and \( \frac{293}{4} \), respectively. Then the mean of 4 numbers x, y, x+y+1, |x-y| is:

Let the line \( y - x = 1 \) intersect the ellipse \( \frac{x^2}{2} + \frac{y^2}{1} = 1 \) at the points A and B. Then the angle made by the line segment AB at the center of the ellipse is:

A building construction work can be completed by two masons A and B together in 22.5 days. Mason A alone can complete the construction work in 24 days less than mason B alone. Then mason A alone will complete the construction work in :

The value of \( \frac{100 C_{50}}{51} + \frac{100 C_{51}}{52} + \dots + \frac{100 C_{100}}{101} \) is :

From the first 100 natural numbers, two numbers first \( a \) and then \( b \) are selected randomly without replacement. If the probability that \( a - b \ge 10 \) is \( m/n \), \( \text{gcd}(m, n) = 1 \), then \( m + n \) is equal to :

The sum of all possible values of \( n \in \mathbb{N} \), so that the coefficients of \(x, x^2\) and \(x^3\) in the expansion of \((1+x^2)^2(1+x)^n\) are in arithmetic progression is :