List of top Questions asked in JEE Main

A long straight wire of radius \(a\) carries a steady current \(I\). The current is uniformly distributed across its cross section. The ratio of the magnetic field at \(\frac{a}{2}\) and \(2a\) from axis of the wire is:

Given below are two statements :
Statement (I) : The mean free path of gas molecules is inversely proportional to square of molecular diameter.
Statement (II) : Average kinetic energy of gas molecules is directly proportional to absolute temperature of gas.
In the light of the above statements, choose the correct answer from the option given below:

A cube of ice floats partly in water and partly in kerosene oil. The radio of volume of ice immersed in water to that in kerosene oil (specific gravity of Kerosene oil = 0.8, specific gravity of ice = 0.9)

A thin circular disc of mass \(M\) and radius \(R\) is rotating in a horizontal plane about an axis passing through its centre and perpendicular to its plane with angular velocity \(\omega\). If another disc of same dimensions but of mass \(\frac{M}{2}\) is placed gently on the first disc co-axially, then the new angular velocity of the system is:

If \( \epsilon_0 \) is the permittivity of free space and \( E \) is the electric field, then \( \epsilon_0 E^2 \) has the dimensions:

If \[ \int \frac{1}{\sqrt[5]{(x - 1)^4}(x + 3)^6} \, dx = A \left( \frac{\alpha x - 1}{\beta x + 3} \right)^B + C, \] where \(C\) is the constant of integration, then the value of \(\alpha + \beta + 20AB\) is _______.

Let \(a, b, c \in \mathbb{N}\) and \(a<b<c\). Let the mean, the mean deviation about the mean and the variance of the 5 observations \(9, 25, a, b, c\) be \(18, 4\) and \(\frac{136}{5}\), respectively. Then \(2a + b - c\) is equal to _______.

The number of distinct real roots of the equation \[ |x + 1| |x + 3| - 4|x + 2| + 5 = 0, \] is _______.

Let \(P(\alpha, \beta, \gamma)\) be the image of the point \(Q(1, 6, 4)\) in the line \[ \frac{x}{1} = \frac{y - 1}{2} = \frac{z - 2}{3}. \] Then \(2\alpha + \beta + \gamma\) is equal to _______.

Let \(A\) be the region enclosed by the parabola \(y^2 = 2x\) and the line \(x = 24\). Then the maximum area of the rectangle inscribed in the region \(A\) is ________.

For $a, b > 0$, let $ f(x) = \begin{cases} \frac{\tan((a+1)x) + b \tan x}{x}, & x < 0, \\ \frac{x}{3}, & x = 0, \\ \frac{\sqrt{ax + b^2x^2} - \sqrt{ax}}{b\sqrt{a x \sqrt{x}}}, & x > 0 \end{cases} $ be a continuous function at $x = 0$. Then $\frac{b}{a}$ is equal to: 

Let A= {2, 3, 6, 8, 9, 11} and B = {1, 4, 5, 10, 15} Let R be a relation on A × B define by (a, b)R(c, d) if and only if 3ad – 7bc is an even integer. Then the relation R is

Let \[ \int_{\log_e a}^{4} \frac{dx}{\sqrt{e^x - 1}} = \frac{\pi}{6}. \] Then \(e^\alpha\) and \(e^{-\alpha}\) are the roots of the equation:

If the function \(f(x) = 2x^3 - 9ax^2 + 12a^2x + 1, \, a>0\) has a local maximum at \(x = \alpha\) and a local minimum at \(x = \alpha^2\), then \(\alpha\) and \(\alpha^2\) are the roots of the equation:

If the line segment joining the points \((5, 2)\) and \((2, a)\) subtends an angle \(\frac{\pi}{4}\) at the origin, then the absolute value of the product of all possible values of \(a\) is:

If the value of \[ \frac{3 \cos 36^\circ + 5 \sin 18^\circ}{5 \cos 36^\circ - 3 \sin 18^\circ} = \frac{a\sqrt{5} - b}{c}, \] where \(a, b, c\) are natural numbers and \(\text{gcd}(a, c) = 1\), then \(a + b + c\) is equal to:

If the system of equations \(x + 4y - z = \lambda\), \(7x + 9y + \mu z = -3\), \(5x + y + 2z = -1\) has infinitely many solutions, then \((2\mu + 3\lambda)\) is equal to:

In an increasing geometric progression of positive terms, the sum of the second and sixth terms is \[ \frac{70}{3} \] and the product of the third and fifth terms is 49. Then the sum of the \(4^\text{th}, 6^\text{th}\), and \(8^\text{th}\) terms is:

If \(\alpha \neq a\), \(\beta \neq b\), \(\gamma \neq c\) and \[ \begin{vmatrix} \alpha & b & c \\ a & \beta & c \\ a & b & \gamma \end{vmatrix} = 0,\] then \[ \frac{a}{\alpha - a} + \frac{b}{\beta - b} + \frac{\gamma}{\gamma - c} \] is equal to:

Let \(\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}\), \(\vec{b} = 2\hat{i} + 3\hat{j} - 5\hat{k}\), and \(\vec{c} = 3\hat{i} - \hat{j} + \lambda\hat{k}\) be three vectors. Let \(\vec{r}\) be a unit vector along \(\vec{b} + \vec{c}\). If \(\vec{r} \cdot \vec{a} = 3\), then \(3\lambda\) is equal to:

If the image of the point \((-4, 5)\) in the line \(x + 2y = 2\) lies on the circle \((x + 4)^2 + (y - 3)^2 = r^2\), then \(r\) is equal to:

Number of colourless lanthanoid ions among the following is ____.
\[ \text{Eu}^{3+}, \, \text{Lu}^{3+}, \, \text{Nd}^{3+}, \, \text{La}^{3+}, \, \text{Sm}^{3+} \]

Number of ambidentate ligands among the following is ____.
\[ \text{NO}_2^{-}, \text{SCN}^{-}, \text{C}_2\text{O}_4^{2-}, \text{NH}_3, \text{CN}^{-}, \text{SO}_4^{2-}, \text{H}_2\text{O} \]

When equal volume of 1M HCl and 1M H$_2$SO$_4$ are separately neutralized by excess volume of 1M NaOH solution, X and Y kJ of heat is liberated respectively. The value of Y/X is ____.

How many compounds among the following compounds show inductive, mesomeric as well as hyperconjugation effects?