List of top Questions asked in JEE Main

Let \( y = y(x) \) be the solution of the differential equation \[ \frac{dy}{dx} + 2y \sec^2 x = 2 \sec^2 x + 3 \tan x \cdot \sec^2 x \] such that \( y(0) = \frac{5}{4} \). Then \[ 12 \left( y\left( \frac{\pi}{4} \right) - e^{-2} \right) \] is equal to _____.

The maximum speed of a boat in still water is 27 km/h. Now this boat is moving downstream in a river flowing at 9 km/h. A man in the boat throws a ball vertically upwards with speed of 10 m/s. Range of the ball as observed by an observer at rest on the river bank is cm. (Take g = 10 m/s²).
What is the sum of the infinite series \( S = \sum_{n=0}^{\infty} \frac{1}{3^n} \)?
If a square is divided into \( 4 \times 4 \) squares. If two squares are chosen randomly, then the probability that the squares don't share a common side is:
Three defective oranges are accidentally mixed with seven good ones and on looking at them, it is not possible to differentiate between them. Two oranges are drawn at random from the lot. If \( x \) denotes the number of defective oranges, then the variance of \( x \) is:

The correct order of stability of following carbocations is: 


 

Match List - I with List - II: List - I:
  • (A) Electric field inside (distance \( r > 0 \) from center) of a uniformly charged spherical shell with surface charge density \( \sigma \), and radius \( R \).
  • (B) Electric field at distance \( r > 0 \) from a uniformly charged infinite plane sheet with surface charge density \( \sigma \).
  • (C) Electric field outside (distance \( r > 0 \) from center) of a uniformly charged spherical shell with surface charge density \( \sigma \), and radius \( R \).
  • (D) Electric field between two oppositely charged infinite plane parallel sheets with uniform surface charge density \( \sigma \).
List - II:
  • (I) \( \frac{\sigma}{\epsilon_0} \)
  • (II) \( \frac{\sigma}{2\epsilon_0} \)
  • (III) 0
  • (IV) \( \frac{\sigma}{\epsilon_0 r^2} \)
Choose the correct answer from the options given below:
A transparent film of refractive index 2.0 is coated on a glass slab of refractive index 1.45. What is the minimum thickness of transparent film to be coated for the maximum transmission of green light of wavelength 550 nm?
A parallel plate capacitor of area \( A = 16 \, \text{cm}^2 \) and separation between the plates \( 10 \, \text{cm} \), is charged by a DC current. Consider a hypothetical plane surface of area \( A_0 = 3.2 \, \text{cm}^2 \) inside the capacitor and parallel to the plates. At an instant, the current through the circuit is 6A. At the same instant the displacement current through \( A_0 \) is \(\_\_\_\_\_ \)mA.
Identify the number of structure/s from the following which can be correlated to D-glyceraldehyde. 
D-glyceraldehyde

Consider the following statements: 
A. The junction area of a solar cell is made very narrow compared to a photodiode.
B. Solar cells are not connected with any external bias. 
C. LED is made of lightly doped p-n junction.
D. Increase of forward current results in a continuous increase in LED light intensity. 
E. LEDs have to be connected in forward bias for emission of light.

One mole of the octahedral complex compound \( \text{Co(NH}_3)_5\text{Cl}_3 \) gives 3 moles of ions on dissolution in water. One mole of the same complex reacts with excess of \( \text{AgNO}_3 \) solution to yield two moles of \( \text{AgCl}(s) \). The structure of the complex is:
The square of the distance of the point \(\left( \frac{15}{7}, \frac{32}{7}, 7 \right)\) from the line \(\frac{x+1}{3} = \frac{y+3}{5} = \frac{z+5}{7}\) in the direction of the vector \(\mathbf{i} + 4\mathbf{j} + 7\mathbf{k}\) is:
Let \( A = \{1, 2, 3, \dots, 10\} \) and \( B = \left\{ \frac{m}{n} : m, n \in A, m <n \text{ and } \gcd(m, n) = 1 \right\} \). Then \( n(B) \) is equal to :
Let \( \vec{c} \) be the projection vector of \( \mathbf{b} = \lambda \hat{i} + 4 \hat{k}, \lambda > 0 \), on the vector \( \vec{a} = \hat{i} + 2 \hat{j} + 2 \hat{k} \). If \( |\vec{a} + \vec{c}| = 7 \), then the area of the parallelogram formed by the vectors \( \vec{b} \) and \( \vec{c} \) is:
Given below are two statements: Statement I : In a vernier callipers, one vernier scale division is always smaller than one main scale division. Statement II : The vernier constant is given by one main scale division multiplied by the number of vernier scale divisions. In light of the above statements, choose the correct answer from the options given below.
Two spherical bodies of same materials having radii 0.2 m and 0.8 m are placed in same atmosphere. The temperature of the smaller body is 800 K and temperature of bigger body is 400 K. If the energy radiate from the smaller body is E, the energy radiated from the bigger body is (assume, effect of the surrounding to be negligible):

Match List-I with List-II:

Consider a circular disc of radius 20 cm with center located at the origin. A circular hole of radius 5 cm is cut from this disc in such a way that the edge of the hole touches the edge of the disc. The distance of the center of mass of the residual or remaining disc from the origin will be:
Let $ A = [a_{ij}] $ be a $3 \times 3 $ matrix such that: $$ A = \begin{bmatrix} 0 & 0 & 4 \\ 1 & 0 & 0 \\ 0 & 1 & 3 \\ \end{bmatrix}, A^{-1} = \begin{bmatrix} 0 & 1 & 0 \\ 1 & 3 & 0 \\ 2 & 1 & 0 \end{bmatrix}. $$ Then $ a_{23} $ equals:
Let \( [x] \) denote the greatest integer function, and let \( m \) and \( n \) respectively be the numbers of the points, where the function \( f(x) = [x] + |x - 2| \), \( -2<x<3 \), is not continuous and not differentiable. Then \( m + n \) is equal to:
Let \( f: (0, \infty) \to \mathbb{R} \) be a function which is differentiable at all points of its domain and satisfies the condition \( x^2 f'(x) = 2f(x) + 3 \), with \( f(1) = 4 \). Then \( 2f(2) \) is equal to:
The elemental composition of a compound is 54.2%C, 9.2%H, and 36.6%O. If the molar mass of the compound is 132 g/mol, the molecular formula of the compound is:

For hydrogen atom, the orbital/s with lowest energy is/are:

If \( \lim_{t \to \infty} \left( \int_0^{1} \left( 3x + 5 \right)^t dx \right) = \frac{\alpha}{5e} \left( \frac{8}{5} \right)^{\frac{3}{2}}, \) then \( \alpha \) is equal to ____ :