JEE Main 2021 Question Paper with Solutions

A large block of wood of mass M = 5.99 kg is hanging from two long massless cords. A bullet of mass m = 10 g is fired into the block and gets embedded in it. The (block + bullet) then swing upwards, their centre of mass rising a vertical distance h = 9.8 cm before the (block + bullet) pendulum comes momentarily to rest at the end of its arc. The speed of the bullet just before collision is : (take g = 9.8 ms -2 )

A bimetallic strip consists of metals A and B. It is mounted rigidly as shown. The metal A has higher coefficient of expansion compared to that of metal B. When the bimetallic strip is placed in a cold bath, it will :

What will be the nature of flow of water from a circular tap, when its flow rate increased from 0.18 L/min to 0.48 L/min? The radius of the tap and viscosity of water are 0.5 cm and 10 − 3 Pa s, respectively. (Density of water : 10 3 kg/m 3 )

A mosquito is moving with a velocity v → = 0.5 t 2 i ^ + 3 t j ^ + 9 k ^ m/s and accelerating in uniform conditions. What will be the direction of mosquito after 2 s?

Statement I : A cyclist is moving on an unbanked road with a speed of 7 kmh − 1 and takes a sharp circular turn along a path of radius of 2m without reducing the speed. The static friction coefficient is 0.2. The cyclist will not slip and pass the curve. (g = 9.8 m/s 2 ) Statement II : If the road is banked at an angle of 45 ∘ , cyclist can cross the curve of 2m radius with the speed of 18.5 kmh − 1 without slipping. In the light of the above statements, choose the correct answer from the options given below.

Statement I : Sodium hydride can be used as an oxidising agent. Statement II : The lone pair of electrons on nitrogen in pyridine makes it basic. Choose the CORRECT answer from the options given below :

The correct statements about H 2 O 2 are : (A) used in the treatment of effluents. (B) used as both oxidising and reducing agents. (C) the two hydroxyl groups lie in the same plane. (D) miscible with water. Choose the correct answer from the options given below :

Identify the elements X and Y using the ionisation energy values given below : Ionization energy (kJ/mol) 1 ${st}$ 2 ${nd}$ X 495 4563 Y 731 1450

Match List - I with List - II : List - I Test/Reagents/Observation(s) List - II Species detected (a) Lassaigne's Test (i) Carbon (b) Cu(II) oxide (ii) Sulphur (c) Silver nitrate (iii) N, S, P, and halogen (d) The sodium fusion extract gives black precipitate with acetic acid and lead acetate (iv) Halogen Specifically The correct match is :

The characteristics of elements X, Y and Z with atomic numbers, respectively, 33, 53 and 83 are :

Consider a rectangle ABCD having 5, 7, 6, 9 points in the interior of the line segments AB, CD, BC, DA respectively. Let α be the number of triangles having these points from different sides as vertices and β be the number of quadrilaterals having these points from different sides as vertices. Then ( β − α ) is equal to :

The least value of |z| where z is complex number which satisfies the inequality $\exp \left( {{{(|z| + 3)(|z| - 1)} \over {||z| + 1|}}{{\log }_e}2} \right) \ge {\log _{\sqrt 2 }}|5\sqrt 7 + 9i|,i = \sqrt { - 1}$ , is equal to :

Let A = {2, 3, 4, 5, ....., 30} and ' $\simeq$ ' be an equivalence relation on A $\times$ A, defined by (a, b) $\simeq$ (c, d), if and only if ad = bc. Then the number of ordered pairs which satisfy this equivalence relation with ordered pair (4, 3) is equal to :

Consider three observations a, b, and c such that b = a + c. If the standard deviation of a + 2, b + 2, c + 2 is d, then which of the following is true?

Which of the following Boolean expression is a tautology?

The magnetic field in a region is given by $\overrightarrow B = {B_o}\left( {{x \over a}} \right)\widehat k$ . A square loop of side d is placed with its edges along the x and y axes. The loop is moved with a constant velocity $\overrightarrow v$ = v 0 $\widehat i$ . The emf induced in the loop is :

A charge Q is moving $\overrightarrow {dl}$ distance in the magnetic field $\overrightarrow {B}$ . Find the value of work done by $\overrightarrow {B}$ .

Find out the surface charge density at the intersection of point x = 3 m plane and x-axis, in the region of uniform line charge of 8 nC/m lying along the z-axis in free space.

Two identical antennas mounted on identical towers are separated from each other by a distance of 45 km. What should nearly be the minimum height of receiving antenna to receive the signals in line of sight? (Assume radius of earth is 6400 km)

The refractive index of a converging lens is 1.4. What will be the focal length of this lens if it is placed in a medium of same refractive index? Assume the radii of curvature of the faces of lens are R 1 and R 2 respectively.

If y = y(x) is the solution of the differential equation ${{dy} \over {dx}}$ + (tan x) y = sin x, $0 \le x \le {\pi \over 3}$ , with y(0) = 0, then $y\left( {{\pi \over 4}} \right)$ equal to :

Let f : S $\to$ S where S = (0, $\infty$ ) be a twice differentiable function such that f(x + 1) = xf(x). If g : S $\to$ R be defined as g(x) = log e f(x), then the value of |g''(5) $-$ g''(1)| is equal to :

Consider the integral $I = \int_0^{10} {{{[x]{e^{[x]}}} \over {{e^{x - 1}}}}dx}$ , where [x] denotes the greatest integer less than or equal to x. Then the value of I is equal to :

If the foot of the perpendicular from point (4, 3, 8) on the line ${L_1}:{{x - a} \over l} = {{y - 2} \over 3} = {{z - b} \over 4}$ , l $\ne$ 0 is (3, 5, 7), then the shortest distance between the line L 1 and line ${L_2}:{{x - 2} \over 3} = {{y - 4} \over 4} = {{z - 5} \over 5}$ is equal to :

Let A denote the event that a 6-digit integer formed by 0, 1, 2, 3, 4, 5, 6 without repetitions, be divisible by 3. Then probability of event A is equal to :

Identify the reagent(s) 'A' and condition(s) for the reaction

Which of the following reduction reaction CANNOT be carried out with coke?

In the above reaction, the reagent ''A'' is :

Arrange the following metal complex/compounds in the increasing order of spin only magnetic moment. Presume all the three, high spin system. (Atomic numbers Ce = 58, Gd = 64 and Eu = 63) (a) (NH 4 ) 2 [Ce(NO 3 ) 6 ] (b) Gd(NO 3 ) 3 and (c) Eu(NO 3 ) 3

The structure of X is :