List of top Questions asked in JEE Main

The positive integer \( n \), for which the solutions of the equation \( x(x+2) + (x+2)(x+4) + \dots + (x+2n-2)(x+2n) = \frac{8n}{3} \) are two consecutive even integers, is :
Let the relation \( R \) on the set \( M = \{1, 2, 3, \ldots, 16\} \) be given by \[ R = \{(x, y) : 4y = 5x - 3,\; x, y \in M\}. \] Then the minimum number of elements required to be added in \( R \), in order to make the relation symmetric, is equal to

Match List-I with List-II.

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10 mole of oxygen is heated at constant volume from 30 \(^{\circ}\)C to 40 \(^{\circ}\)C. The change in the internal energy of the gas is_____ cal.
(The molecular specific heat of oxygen at constant pressure, \(C_p=7\) cal/mol \(^{\circ}\)C and \(R=2\) cal/mol \(^{\circ}\)C.)

Consider an A.P. $a_1,a_2,\ldots,a_n$; $a_1>0$. If $a_2-a_1=-\dfrac{3}{4}$, $a_n=\dfrac{1}{4}a_1$, and \[ \sum_{i=1}^{n} a_i=\frac{525}{2}, \] then $\sum_{i=1}^{17} a_i$ is equal to

Let \[ \sum_{k=1}^{n} a_k = \alpha n^2 + \beta n. \] If \( a_{10} = 59 \) and \( a_6 = 7a_1 \), then \( \alpha + \beta \) is equal to

An object is projected with kinetic energy K from point A at an angle 60° with the horizontal. The ratio of the difference in kinetic energies at points B and C to that at point A (see figure), in the absence of air friction 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 \( \overrightarrow{AB} = 2\hat{i} + 4\hat{j} - 5\hat{k} \) and \( \overrightarrow{AD} = \hat{i} + 2\hat{j} + \lambda \hat{k} \), \( \lambda \in \mathbb{R} \). Let the projection of the vector \( \vec{v} = \hat{i} + \hat{j} + \hat{k} \) on the diagonal \( \overrightarrow{AC} \) of the parallelogram \( ABCD \) be of length one unit. If \( \alpha, \beta \), where \( \alpha>\beta \), be the roots of the equation \( \lambda^2 x^2 - 6\lambda x + 5 = 0 \), then \( 2\alpha - \beta \) is equal to
Let the direction cosines of two lines satisfy the equations : \( 4l + m - n = 0 \) and \( 2mn + 5nl + 3lm = 0 \). Then the cosine of the acute angle between these lines 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:
The final product [B] is:
If \(x^2+x+1=0\), then the value of \(\left(x + \frac{1}{x}\right)^2 + \left(x^2 + \frac{1}{x^2}\right)^2 + \left(x^3 + \frac{1}{x^3}\right)^2 + \dots + \left(x^{25} + \frac{1}{x^{25}}\right)^2\) is:
Two cars $A$ and $B$ each of mass $10^3$ kg are moving on parallel tracks separated by a distance of $10$ m, in same direction with speeds $72$ km/h and $36$ km/h. The magnitude of angular momentum of car $A$ with respect to car $B$ is _________ J.s.
The pH and conductance of a weak acid (HX) was found to be 5 and \(4\times10^{-5}\) S, respectively. The conductance was measured under standard condition using a cell where the electrode plates having a surface area of 1 cm\(^2\) were at a distance of 15 cm apart. The value of the limiting molar conductivity (\(\Lambda_m^\circ\)) is___ S cm\(^2\) mol\(^{-1}\) (nearest integer) (Given: degree of dissociation \(\alpha \ll 1\))
  
As shown in the diagram, when the incident ray is parallel to base of the prism, the emergent ray grazes along the second surface. If refractive index of the material of prism is $\sqrt{2}$, the angle $\theta$ of prism is :
As compared with chlorocyclohexane, which of the following statements correctly apply to chlorobenzene?
The magnitude of negative charge is more on chlorine atom.
The C--Cl bond has partial double bond character.
C--Cl bond is less polar.
C--Cl bond is longer due to repulsion between delocalised electrons of the aromatic ring and lone pairs of electrons of chlorine.
The C--Cl bond is formed using \(sp^2\) hybridised orbital of carbon.
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Electric field in a region is given by \[ \vec{E} = A x\,\hat{i} + B y\,\hat{j}, \] where \( A = 10 \,\text{V/m}^2 \) and \( B = 5 \,\text{V/m}^2 \). If the electric potential at a point \( (10, 20) \) is \(500\ \text{V}\), then the electric potential at origin is __________ V.

Match the LIST-I with LIST-II 

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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:
A \(7.9\ \text{MeV}\) \(\alpha\)-particle scatters from a target material of atomic number \(79\). From the given data, the estimated diameter of the nuclei of the target material is (approximately) _________ m. \[ \left[\frac{1}{4\pi\varepsilon_0}=9\times10^9\ \text{Nm}^2\text{/C}^2 \text{ and electron charge }=1.6\times10^{-19}\ \text{C}\right] \]
The letters of the word ``UDAYPUR'' are written in all possible ways with or without meaning and these words are arranged as in a dictionary. The rank of the word ``UDAYPUR'' is
Let $\vec{a}=2\hat{i}-\hat{j}-\hat{k}$, $\vec{b}=\hat{i}+3\hat{j}-\hat{k}$ and $\vec{c}=2\hat{i}+\hat{j}+3\hat{k}$. Let $\vec{v}$ be the vector in the plane of $\vec{a}$ and $\vec{b}$, such that the length of its projection on the vector $\vec{c}$ is $\dfrac{1}{\sqrt{14}}$. Then $|\vec{v}|$ is equal to
A point charge of \(10^{-8}\) C is placed at origin. The work done in moving a point charge 2 \(\mu\)C from point A(4, 4, 2) m to B(2, 2, 1) m is_______ J. (\(\frac{1}{4\pi\epsilon_0} = 9 \times 10^9\) in SI units)
Let \( f : [1,\infty) \to \mathbb{R} \) be a differentiable function. If
\[ 6\int_{1}^{x} f(t)\,dt = 3x f(x) + x^3 - 4 \] for all \( x \ge 1 \), then the value of \( f(2) - f(3) \) is