List of top Questions asked in JEE Main- 2021

If \( S = \frac{7}{5} + \frac{9}{5^2} + \frac{13}{5^3} + \frac{19}{5^4} + ... \), then 160 S is equal to _________.
Let \( f(x) \) be a cubic polynomial with \( f(1) = -10 \), \( f(-1) = 6 \), and has a local minima at \( x = 1 \), and \( f'(x) \) has a local minima at \( x = -1 \). Then \( f(3) \) is equal to _________.
If \( \int \frac{\sin x}{\sin^3 x + \cos^3 x} dx = \alpha \log_e |1 + \tan x| + \beta \log_e |1 - \tan x + \tan^2 x| + \gamma \tan^{-1} \left( \frac{2 \tan x - 1}{\sqrt{3}} \right) + C \), when C is a constant of integration, then the value of \( 18(\alpha + \beta + \gamma^2) \) is _________.
If the line \( y = mx \) bisects the area enclosed by the lines \( x = 0, y = 0, x = \frac{3}{2} \) and the curve \( y = 1 + 4x - x^2 \), then 12 m is equal to _________.
Let B be the centre of the circle \( x^2 + y^2 - 2x + 4y + 1 = 0 \). Let the tangents at two points P and Q on the circle intersect at the point \( A(3, 1) \). Then \( 8 \cdot \frac{\text{area } \Delta APQ}{\text{area } \Delta BPQ} \) is equal to _________.
A tangent line L is drawn at the point \( (2, -4) \) on the parabola \( y^2 = 8x \). If the line L is also tangent to the circle \( x^2 + y^2 = a \), then 'a' is equal to _________.
Suppose the line $\frac{x - 2}{\alpha} = \frac{y - 2}{-5} = \frac{z + 2}{2}$ lies on the plane $x + 3y - 2z + \beta = 0$. Then $(\alpha + \beta)$ is equal to _________.
If e is the electronic charge, c is the speed of light in free space and h is Planck's constant, the quantity $\frac{1}{4\pi\epsilon_0}\frac{e^2}{hc}$ has dimensions of:
A stone is dropped from the top of a building. When it crosses a point 5 m below the top, another stone starts to fall from a point 25 m below the top. Both stones reach the bottom simultaneously. The height of the building is:

A sphere of radius 'a' and mass 'm' rolls along a horizontal plane with constant speed $v_0$. It encounters an inclined plane at angle $\theta$ and climbs upward. Assuming that it rolls without slipping, how far up the sphere will travel? 

 

The point A moves with a uniform speed along the circumference of a circle of radius 0.36 m and covers 30° in 0.1 s. The perpendicular projection 'P' from 'A' on the diameter MN represents the simple harmonic motion of 'P'. The restoration force per unit mass when P touches M will be: 

Thermodynamic process is shown below on a P-V diagram for one mole of an ideal gas. If $V_2=2V_1$ then the ratio of temperature $T_2/T_1$ is: The process is $PV^{1/2}$ = constant. 

 

Given below are two statements:
Statement I: In a diatomic molecule, the rotational energy at a given temperature obeys Maxwell's distribution.
Statement II: In a diatomic molecule, the rotational energy at a given temperature equals the translational kinetic energy for each molecule.

Two identical springs of spring constant '2k' are attached to a block of mass m and to fixed support (see figure). When the mass is displaced from equilibrium position on either side, it executes simple harmonic motion. The time period of oscillations of this system is: 

 

Y = A sin($\omega$t+$\phi_0$) is the time-displacement equation of a SHM. At t = 0 the displacement of the particle is Y = $\frac{A}{2}$ and it is moving along negative x-direction. Then the initial phase angle $\phi_0$ will be:
The number of integral values of m so that the abscissa of point of intersection of lines 3x + 4y = 9 and y = mx + 1 is also an integer, is :

A charge 'q' is placed at one corner of a cube as shown in figure. The flux of electrostatic field $\vec{E}$ through the shaded area is: 

 

An electron with kinetic energy $K_1$ enters between parallel plates of a capacitor at an angle '$\alpha$' with the plates. It leaves the plates at angle '$\beta$' with kinetic energy $K_2$. Then the ratio of kinetic energies $K_1 : K_2$ will be:
In a ferromagnetic material, below the curie temperature, a domain is defined as:
An LCR circuit contains resistance of 110 $\Omega$ and a supply of 220 V at 300 rad/s angular frequency. If only capacitance is removed from the circuit, current lags behind the voltage by 45°. If on the other hand, only inductor is removed the current leads by 45° with the applied voltage. The rms current flowing in the circuit will be:
The stopping potential for electrons emitted from a photosensitive surface illuminated by light of wavelength 491 nm is 0.710 V. When the incident wavelength is changed to a new value, the stopping potential is 1.43 V. The new wavelength is:
Consider the diffraction pattern obtained from the sunlight incident on a pinhole of diameter 0.1 $\mu$m. If the diameter of the pinhole is slightly increased, it will affect the diffraction pattern such that:
An electron of mass $m_e$ and a proton of mass $m_p = 1836 m_e$ are moving with the same speed. The ratio of their de Broglie wavelength $\frac{\lambda_{electron}}{\lambda_{proton}}$ will be:
The wavelength of the photon emitted by a hydrogen atom when an electron makes a transition from n=2 to n=1 state is:
If a message signal of frequency '$f_m$' is amplitude modulated with a carrier signal of frequency '$f_c$' and radiated through an antenna, the wavelength of the corresponding signal in air is: