List of top Questions asked in JEE Main

Two immiscible liquids of refractive indices \( \frac{8}{5} \) and \( \frac{3}{2} \) respectively are put in a beaker as shown in the figure. The height of each column is 6 cm. A coin is placed at the bottom of the beaker. For near-normal vision, the apparent depth of the coin is \( \frac{\alpha}{4} \) cm. The value of \( \alpha \) is ______.

A thin metallic wire having a cross-sectional area of \( 10^{-4} \, \text{m}^2 \) is used to make a ring of radius 30 cm. A positive charge of \( 2\pi \, C \) is uniformly distributed over the ring, while another positive charge of 30 pC is kept at the center of the ring. The tension in the ring is ______ N; provided that the ring does not deform (neglect the influence of gravity).
(Given, \(\frac{1}{4 \pi \varepsilon_0} = 9 \times 10^9 \, \text{SI units}\))

A particle starts from origin at \( t = 0 \) with a velocity \( 5\hat{i} \, \text{m/s} \) and moves in the \( x\)-\( y \) plane under the action of a force that produces a constant acceleration of \( (3\hat{i} + 2\hat{j}) \, \text{m/s}^2 \). If the \( x \)-coordinate of the particle at that instant is 84 m, then the speed of the particle at this time is \( \sqrt{\alpha} \, \text{m/s} \). The value of \( \alpha \) is ______.

The average kinetic energy of a monatomic molecule is 0.414 eV at temperature: (Use \( k_B = 1.38 \times 10^{-23} \, \text{J/mol-K} \))

A convex lens of focal length 40 cm forms an image of an extended source of light on a photo-electric cell. A current \( I \) is produced. The lens is replaced by another convex lens having the same diameter but focal length 20 cm. The photoelectric current now is:

The radius of the third stationary orbit of an electron in Bohr's atom is \( R \). The radius of the fourth stationary orbit will be:

Identify the physical quantity that cannot be measured using a spherometer:

Which of the following circuits is reverse-biased?

A train is moving with a speed of 12 m/s on rails which are 1.5 m apart. To negotiate a curve radius of 400 m, the height by which the outer rail should be raised with respect to the inner rail is (Given, \( g = 10 \, \text{m/s}^2 \)):

A proton moving with a constant velocity passes through a region of space without any change in its velocity. If \( \vec{E} \) and \( \vec{B} \) represent the electric and magnetic fields respectively, then the region of space may have:
(A) \( \vec{E} = 0, \vec{B} = 0 \) 
(B) \( \vec{E} = 0, \vec{B} \neq 0 \) 
(C) \( \vec{E} \neq 0, \vec{B} = 0 \) 
(D) \( \vec{E} \neq 0, \vec{B} \neq 0 \) 
Choose the most appropriate answer from the options given below:

Position of an ant \( S \) (in meters) moving in the Y-Z plane is given by \( S = 2t \hat{j} + 5t \hat{k} \) (where \( t \) is in seconds). The magnitude and direction of velocity of the ant at \( t = 1 \, s \) will be:

Let \( f(x) = x^3 + x^2 f'(1) + x f''(2) + f'''(3) \), \( x \in \mathbb{R} \). Then \( f'(10) \) is equal to ______.

Let the set of all \( a \in \mathbb{R} \) such that the equation \(\cos 2x + a \sin x = 2a - 7\) has a solution be \([p, q]\) and \( r = \tan 9^\circ - \tan 27^\circ - \frac{1}{\cot 63^\circ + \tan 81^\circ} \), then \( pqr \) is equal to ______.

If \( 8 = 3 + \frac{1}{4}(3 + p) + \frac{1}{4^2}(3 + 2p) + \frac{1}{4^3}(3 + 3p) + \ldots \infty \), then the value of \( p \) is ______.

Let the area of the region \(\{(x, y) : x - 2y + 4 \geq 0, x + 2y^2 \geq 0, x + 4y^2 \leq 8, y \geq 0\}\) be \(\frac{m}{n}\), where \( m \) and \( n \) are coprime numbers. Then \( m + n \) is equal to ______.

If the solution of the differential equation \((2x + 3y - 2) \, dx + (4x + 6y - 7) \, dy = 0, \quad y(0) = 3,\) is  \(\alpha x + \beta y + 3 \log_e |2x + 3y - \gamma| = 6,\) then \(\alpha + 2\beta + 3\gamma\) is equal to ____.

Let for a differentiable function \(f : (0, \infty) \rightarrow \mathbb{R}\), \(f(x) - f(y) \geq \log_e \left( \frac{x}{y} \right) + x - y, \quad \forall \; x, y \in (0, \infty).\) 
Then \(\sum_{n=1}^{20} f'\left(\frac{1}{n^2}\right)\) is equal to ____.

The least positive integral value of \( \alpha \), for which the angle between the vectors \( \alpha \hat{i} - 2\hat{j} + 2\hat{k} \) and \( \alpha \hat{i} + 2\alpha \hat{j} - 2\hat{k} \) is acute, is ______.

Consider the matrix \( f(x) = \begin{bmatrix} \cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1 \end{bmatrix} \).
Given below are two statements:
Statement I: \( f(-x) \) is the inverse of the matrix \( f(x) \).
Statement II: \( f(x) f(y) = f(x + y) \).
In the light of the above statements, choose the correct answer from the options given below:"

If \( a = \lim_{{x \to 0}} \frac{\sqrt{1 + \sqrt{1 + x^4}} - \sqrt{2}}{x^4} \) and \( b = \lim_{{x \to 0}} \frac{\sin^2 x}{\sqrt{2} - \sqrt{1 + \cos x}} \), then the value of \( ab^3 \) is:

Let \(\vec{a} = \hat{i} + 2\hat{j} + \hat{k}\), \(\vec{b} = 3(\hat{i} - \hat{j} + \hat{k})\). Let \(\vec{c}\) be the vector such that \(\vec{a} \times \vec{c} = \vec{b}\) and \(\vec{a} \cdot \vec{c} = 3\). Then \(\vec{a} \cdot ((\vec{c} \times \vec{b}) - \vec{b} \cdot \vec{c})\) is equal to:

The length of the chord of the ellipse \(\frac{x^2}{25} + \frac{y^2}{16} = 1\), whose mid-point is \(\left(1, \frac{2}{5}\right)\), is equal to:

If \(S = \{z \in C : |z – i| = |z + i| = |z–1|\}\), then, n(S) is:

Let \( S = \{ 1, 2, 3, \ldots, 10 \} \). Suppose \( M \) is the set of all the subsets of \( S \), then the relation \( R = \{ (A, B): A \cap B \neq \phi; A, B \in M \} \) is:

Evaluate the integral  \(\int_{0}^{1} \frac{1}{\sqrt{3+x} + \sqrt{1+x}} \, dx\) Given that the integral can be expressed in the form \(a + b\sqrt{2} + c\sqrt{3}\), where \(a, b, c\) are rational numbers, find the value of \(2a + 3b - 4c\).