List of top Questions asked in JEE Main

Let the system of equations \[x + 2y + 3z = 5, \quad 2x + 3y + z = 9, \quad 4x + 3y + \lambda z = \mu\]have an infinite number of solutions. Then $\lambda + 2\mu$ is equal to:

The area (in square units) of the region \[ S = \{ z \in \mathbb{C} : |z - 1| \leq 2; (z + \bar{z}) + i (z - \bar{z}) \leq 2, \, \operatorname{Im}(z) \geq 0 \} \] is:

The area (in square units) of the region described by: $$ \{(x, y) : y^2 \leq 2x, \, \text{and} \, y \geq 4x - 1\} $$ is:

If the net electric field at point \( P \) along \( Y \)-axis is zero, then the ratio of \[ \left| \frac{q_2}{q_3} \right| \] is \[\frac{8}{5\sqrt{x}},\] where \( x = \dots \dots \dots \dots \dots \).

Let \[ f(x) = \int_{0}^{x} \left( t + \sin\left(1 - e^t\right) \right) \, dt, \, x \in \mathbb{R}. \] Then \[ \lim_{x \to 0} \frac{f(x)}{x^3} \] is equal to:

Let \( A = \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix} \) and \[B = I + \text{adj}(A) + (\text{adj}(A))^2 + \dots + (\text{adj}(A))^{10}.\]Then, the sum of all the elements of the matrix \( B \) is:

The number of solutions of the equation \(e^{\sin x} - 2e^{-\sin x} = 2\) is

If the function \( f : (-\infty, -1] \rightarrow (a, b) \) defined by \(f(x) = e^{x^3 - 3x + 1}\) is one-one and onto, then the distance of the point \( P(2b + 4, a + 2) \) from the line \( x + e^{-3} y = 4 \) is:

Let three real numbers a,b,c be in arithmetic progression and a + 1, b, c + 3 be in geometric progression. If a>10 and the arithmetic mean of a,b and c is 8, then the cube of the geometric mean of a,b and c is

Consider the function \( f : (0, \infty) \rightarrow \mathbb{R} \) defined by \(f(x) = e^{-| \log_e x |}.\)If \( m \) and \( n \) be respectively the number of points at which \( f \) is not continuous and \( f \) is not differentiable, then \( m + n \) is

Let the mean and the variance of 6 observation a,b, 68, 44, 48, 60 be 55 and 194, respectively if a > b, then a + 3b is

Let a relation \( R \) on \( \mathbb{N} \times \mathbb{N} \) be defined as: $$(x_1, y_1) \, R \, (x_2, y_2) \text{ if and only if } x_1 \leq x_2 \text{ or } y_1 \leq y_2.$$ 
Consider the two statements:
[(I)] \( R \) is reflexive but not symmetric. 
[(II)] \( R \) is transitive.
Then which one of the following is true:

The area of the region enclosed by the parabola \( y = 4x - x^2 \) and \( 3y = (x - 4)^2 \) is equal to

Let \( f : \mathbb{R} \rightarrow (0, \infty) \) be a strictly increasing function such that \(\lim_{x \to \infty} \frac{f(7x)}{f(x)} = 1.\)Then, the value of \(\lim_{x \to \infty} \left[ \frac{f(5x)}{f(x)} - 1 \right]\)is equal to

Let C be a circle with radius \( \sqrt{10} \) units and centre at the origin. Let the line \( x + y = 2 \) intersects the circle C at the points P and Q. Let MN be a chord of C of length 2 unit and slope \(-1\). Then, a distance (in units) between the chord PQ and the chord MN is

Let 2nd, 8th, and 44th terms of a non-constant A.P. be respectively the 1st, 2nd, and 3rd terms of a G.P. If the first term of A.P. is 1, then the sum of the first 20 terms is equal to

The temperature \( T(t) \) of a body at time \( t = 0 \) is \( 160^\circ \)F and it decreases continuously as per the differential equation \[ \frac{dT}{dt} = -K(T - 80), \] **where \( K \) is a positive constant. If \( T(15) = 120^\circ \)F, then \( T(45) \) is equal to

Let \( P \) be a parabola with vertex \( (2, 3) \) and directrix \( 2x + y = 6 \). Let an ellipse \( E : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), \( a > b \), of eccentricity \( \frac{1}{\sqrt{2}} \) pass through the focus of the parabola \( P \). Then the square of the length of the latus rectum of \( E \) is

Let \( f, g : (0, \infty) \rightarrow \mathbb{R} \) be two functions defined by \(f(x) = \int_{-x}^{x} (|t| - t^2) e^{-t^2} \, dt \quad \text{and} \quad g(x) = \int_{0}^{x} t^{1/2} e^{-t} \, dt.\)Then the value of \( f \left( \sqrt{\log_e 9} \right) + g \left( \sqrt{\log_e 9} \right) \) is equal to

Let \( (\alpha, \beta, \gamma) \) be the mirror image of the point \( (2, 3, 5) \) in the line \(\frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4}\). Then \( 2\alpha + 3\beta + 4\gamma \) is equal to

Let a variable line passing through the centre of the circle x² + y² – 16x – 4y = 0, meet the positive co-ordinate axes at the point A and B. Then the minimum value of OA + OB, where O is the origin, is equal to

Let \( z_1 \) and \( z_2 \) be two complex numbers such that \( z_1 + z_2 = 5 \) and \( z_1^3 + z_2^3 = 20 + 15i \). Then \( \left| z_1^4 + z_2^4 \right| \) equals

The number of ways in which 21 identical apples can be distributed among three children such that each child gets at least 2 apples, is

The following data were obtained during the first order thermal decomposition of a gas A at constant volume:\[\text{A(g)} \rightarrow 2\text{B(g)} + \text{C(g)}\]\[\begin{array}{|c|c|c|}\hline\text{S.No} & \text{Time/s} & \text{Total pressure/(atm)} \\\hline1 & 0 & 0.1 \\2 & 115 & 0.28 \\\hline\end{array}\]The rate constant of the reaction is _____ $\times 10^{-2}$ s$^{-1}$ (nearest integer).