List of top Questions asked in JEE Main- 2021

Let 

, $x \in [0, \pi]$. Then the maximum value of $f(x)$ is equal to _________. 
 

Let F : [3, 5] $\rightarrow$ R be a twice differentiable function on (3, 5) such that
$F(x) = e^{-x} \int_3^x (3t^2 + 2t + 4F'(t))dt$.
If $F'(4) = \frac{\alpha e^\beta - 224}{(e^\beta-4)^2}$, then $\alpha+\beta$ is equal to _________.
Let $\vec{a}=\hat{i}+\hat{j}+\hat{k}$, $\vec{b}$ and $\vec{c}=\hat{j}-\hat{k}$ be three vectors such that $\vec{a} \times \vec{b} = \vec{c}$ and $\vec{a} \cdot \vec{b} = 1$. If the length of projection vector of the vector $\vec{b}$ on the vector $\vec{a} \times \vec{c}$ is $l$, then the value of $3l^2$ is equal to _________.
Let the domain of the function $f(x) = \log_4(\log_5(\log_3(18x-x^2-77)))$ be $(a, b)$. Then the value of the integral $\int_a^b \frac{\sin^3 x}{\sin^3 x + \sin^3(a+b-x)} dx$ is equal to _________.
If $\log_3 2, \log_3(2^x-5), \log_3(2^x-\frac{7}{2})$ are in an arithmetic progression, then the value of x is equal to _________.
For real numbers $\alpha$ and $\beta$, consider the following system of linear equations:
$x+y-z=2$
$x+2y+\alpha z = 1$
$2x-y+z = \beta$
If the system has infinite solutions, then $\alpha+\beta$ is equal to _________.

Let S = 1, 2, 3, 4, 5, 6, 7. Then the number of possible functions $f: S \rightarrow S$ such that $f(m \cdot n) = f(m) \cdot f(n)$ for every $m, n \in S$ and $m \cdot n \in S$ is equal to _________. 
 

If $y=y(x)$, $y \in [0, \pi/2)$ is the solution of the differential equation $\sec y \frac{dy}{dx} - \sin(x+y) - \sin(x-y) = 0$, with $y(0)=0$, then $5y'(\pi/2)$ is equal to _________.
Let $f:[0, 3] \rightarrow R$ be defined by $f(x) = \min\{x-[x], 1+[x]-x\}$ where $[x]$ is the greatest integer less than or equal to x. Let P denote the set containing all $x \in [0, 3]$ where f is discontinuous, and Q denote the set containing all $x \in [0, 3]$ where f is not differentiable. Then the sum of number of elements in P and Q is equal to _________.
A correct experiment is performed to measure the speed of sound in air using a resonance column tube of diameter 6 cm. The frequency of the tuning fork is 504 Hz. Speed of the sound is given as 336 m/s. The zero of the metre scale coincides with the top end of the resonance column tube. The reading of the water level in the column tube, when the first resonance occurs is :

In an octagon ABCDEFGH of equal side, what is the sum of $\vec{AB} + \vec{AC} + \vec{AD} + \vec{AE} + \vec{AF} + \vec{AG} + \vec{AH}$, if $\vec{AO} = 2i + 3j - 4k$?

Two satellites A and B of masses 200 kg and 400 kg are revolving round the earth at height of 600 km and 1600 km respectively. If $T_A$ and $T_B$ are the time periods of A and B respectively then the value of $T_B - T_A$ is : [$R_e = 6400$ km, $M_e = 6 \times 10^{24}$ kg]
 

A 5V battery is connected across the points X and Y. Assume D1 and D2 to be normal silicon diodes. Find the current supplied by the battery if the +ve terminal of the battery is connected to point X and -ve to point Y. 

The pitch of the screw gauge is 1 mm and there are 100 divisions on the circular scale. When nothing is put in between the jaws, the zero of the circular scale lies 8 divisions below the reference line. When a wire is placed between the jaws, the first linear scale division is clearly visible while 72nd division on circular scale coincides with the reference line. The radius of the wire is :
An $\alpha$ particle and a proton are accelerated from rest by a potential difference of 200 V. After this, their de Broglie wavelengths are $\lambda_\alpha$ and $\lambda_p$ respectively. The ratio $\lambda_p/\lambda_\alpha$ is :
Two coherent light sources having intensity in the ratio $2x$ produce an interference pattern. The ratio $(I_{max} - I_{min}) / (I_{max} + I_{min})$ will be :
An engine of a train, moving with uniform acceleration, passes the signal-post with velocity $u$ and the last compartment with velocity $v$. The velocity with which middle point of the train passes the signal post is :

Magnetic fields at two points on the axis of a circular coil at a distance of 0.05 m and 0.2 m from the centre are in the ratio 8 : 1. The radius of coil is _____________

The angular frequency of alternating current in a L-C-R circuit is 100 rad/s. The components connected are shown in the figure. Find the value of inductance of the coil and capacity of condenser. (Note: Based on typical values for this specific question where $V_L = V_C$ or resonance is implied). 

Two radioactive substances X and Y originally have $N_1$ and $N_2$ nuclei respectively. Half life of X is half of the half life of Y. After three half lives of Y, number of nuclei of both are equal. The ratio $N_1/N_2$ will be equal to :

A solid sphere of radius R gravitationally attracts a particle placed at 3R from its centre with a force $F_1$. Now a spherical cavity of radius (R/2) is made in the sphere (as shown in figure) and the force becomes $F_2$. The value of $F_1 : F_2$ is :

A monoatomic gas of mass 4.0 u is kept in an insulated container. Container is moving with velocity 30 m/s. If container is suddenly stopped then change in temperature of the gas (R= gas constant) is $\frac{x}{3R}$. Value of x is ________.
512 identical drops of mercury are charged to a potential of 2 V each. The drops are joined to form a single drop. The potential of this drop is ______ V.
A coil of inductance 2 H having negligible resistance is connected to a source of supply whose voltage is given by V = 3t volt. (where t is in second). If the voltage is applied when t=0, then the energy stored in the coil after 4 s is _________ J.
A small bob tied at one end of a thin string of length 1 m is describing a vertical circle so that the maximum and minimum tension in the string are in the ratio 5 : 1. The velocity of the bob at the highest position is ________ m/s. (Take g=10 m/s²)