List of top Questions asked in JEE Main

$$ \lim_{n \to \infty} \frac{(1^2 - 1)(n-1) + (2^2 - 2)(n-2) + \ldots + ((n-1)^2 - (n-1))}{(1^3 + 2^3 + \ldots + n^3) - (1^2 + 2^2 + \ldots + n^2)} $$ is equal to: 

Thiosulphate reacts differently with iodine and bromine in the reaction given below:  
$$ 2S_2O_3^{2-} + I_2 \rightarrow S_4O_6^{2-} + 2I^- $$ $$ S_2O_3^{2-} + 5Br_2 + 5H_2O \rightarrow 2SO_4^{2-} + 4Br^- + 10H^+ $$ 
Which of the following statements justifies the above dual behaviour of thiosulphate?

If the sum of the series $$ \frac{1}{1 \cdot (1 + d)} + \frac{1}{(1 + d)(1 + 2d)} + \cdots + \frac{1}{(1 + 9d)(1 + 10d)} $$ is equal to 5, then \(50d\) is equal to: 

If $y = y(x)$ is the solution curve of the differential equation $$ (x^2 - 4) \, dy - (y^2 - 3y) \, dx = 0, $$ with $x > 2$, $y(4) = \frac{3}{2}$ and the slope of the curve is never zero, then the value of $y(10)$ equals: 

In an experiment to measure the focal length (f) of a convex lens, the magnitude of object distance (x) and the image distance (y) are measured with reference to the focal point of the lens. The y-x plot is shown in figure.
The focal length of the lens is_____cm.

Two identical conducting spheres P and S with charge Q on each, repel each other with a force 16N. A third identical uncharged conducting sphere R is successively brought in contact with the two spheres. The new force of repulsion between P and S is :

 Let \( R \) be a relation on \( \mathbb{Z} \times \mathbb{Z} \) defined by \((a, b) R (c, d)\) if and only if \(ad - bc\) is divisible by 5.
Then \( R \) is:

Let \( e_1 \) be the eccentricity of the hyperbola $$ \frac{x^2}{16} - \frac{y^2}{9} = 1 $$ and \( e_2 \) be the eccentricity of the ellipse $$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \quad a > b, $$ which passes through the foci of the hyperbola. If \( e_1 e_2 = 1 \), then the length of the chord of the ellipse parallel to the x-axis and passing through (0, 2) is: 

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: 

The area (in square units) of the region enclosed by the ellipse $$ x^2 + 3y^2 = 18 $$ in the first quadrant below the line \( y = x \) is: 

Consider the relations $R_1$ and $R_2$ defined as \[a R_1 b \iff a^2 + b^2 = 1 \quad \text{for all } a, b \in \mathbb{R},\]and \[(a, b) R_2 (c, d) \iff a + d = b + c \quad \text{for all } (a, b), (c, d) \in \mathbb{N} \times \mathbb{N}.\]Then:

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 sum of all rational terms in the expansion of $$ \left( \frac{1}{2^5} + \frac{1}{5^3} \right)^{15} $$ is equal to: 

Let \( f : \mathbb{R} \rightarrow \mathbb{R} \) be a function defined by
$$ f(x) = \frac{x}{(1 + x^4)^{1/4}} $$
and \( g(x) = f(f(f(x))) \). Then
$$ 18 \int_{\sqrt[3]{\frac{8}{3}}}^{\sqrt[3]{\frac{4}{3}}} x^3 g(x) \, dx $$
equals: 

If \[\int \frac{1}{a^2 \sin^2 x + b^2 \cos^2 x} \, dx = \frac{1}{12} \tan^{-1}(3 \tan x) + \text{constant},\]then the maximum value of $a \sin x + b \cos x$ is:

Given below are two statements:  
Statement I: When the speed of liquid is zero everywhere, the pressure difference at any two points depends on the equation $$ P_1 - P_2 = \rho g (h_2 - h_1). $$ Statement II: In the ventury tube shown, $$ 2gh = v_1^2 - v_2^2. $$

In the light of the above statements, choose the most appropriate answer from the options given below.

Let \( \alpha, \beta \) be the distinct roots of the equation $$ x^2 - (t^2 - 5t + 6)x + 1 = 0, \, t \in \mathbb{R} \, \text{and} \, a_n = \alpha^n + \beta^n. $$ Then the minimum value of \( \frac{a_{2023} + a_{2025}}{a_{2024}} \) 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: 

Consider the line $L$ passing through the points $(1, 2, 3)$ and $(2, 3, 5)$. The distance of the point $$ \left( \frac{11}{3}, \frac{11}{3}, \frac{19}{3} \right) $$ from the line $L$ along the line $$ \frac{3x - 11}{2} = \frac{3y - 11}{1} = \frac{3z - 19}{2} $$ is equal to: 

Let \( C: x^2 + y^2 = 4 \) and \( C': x^2 + y^2 - 4\lambda x + 9 = 0 \) be two circles. If the set of all values of \( \lambda \) such that the circles \( C \) and \( C' \) intersect at two distinct points is \( R = [a, b] \), then the point \( (8a + 12, 16b - 20) \) lies on the curve:

Let \( P \) be a point on the hyperbola \( H: \frac{x^2}{9} - \frac{y^2}{4} = 1 \), in the first quadrant such that the area of the triangle formed by \( P \) and the two foci of \( H \) is \( 2 \sqrt{13} \). Then, the square of the distance of \( P \) from the origin is

How is load transferred to earth?

Given below are two statements:
Statement-I: The orbitals having same energy are called as degenerate orbitals.
Statement-II: In hydrogen atom, 3p and 3d orbitals are not degenerate orbitals.
In the light of the above statements, choose the most appropriate answer from the options given

During an adiabatic process, the pressure of a gas is found to be proportional to the cube of its absolute temperature. The ratio of \(\frac{C_p}{C_v}\) for the gas is :

Number of metal ions characterized by flame test among the following is _______.
$\text{Sr}^{2+}, \text{Ba}^{2+}, \text{Ca}^{2+}, \text{Cu}^{2+}, \text{Zn}^{2+}, \text{Co}^{2+}, \text{Fe}^{2+}$