JEE Advanced 2021 Question Paper with Solutions

A projectile is thrown from a point O on the ground at an angle 45 $^\circ$ from the vertical and with a speed 5 $\sqrt 2$ m/s. The projectile at the highest point of its trajectory splits into two equal parts. One part falls vertically down to the ground, 0.5 s after the splitting. The other part, t seconds after splitting, falls to the ground at a distance x meters from the point O. The acceleration due to gravity g = 10 m/s 2 . The value of x is _______________.

A thin rod of mass M and length a is free to rotate in horizontal plane about a fixed vertical axis passing through point O. A thin circular disc of mass M and of radius a/4 is pivoted on this rod with its center at a distance a/4 from the free end so that it can rotate freely about its vertical axis, as shown in the figure. Assume that both the rod and the disc have uniform density and they remain horizontal during the motion. An outside stationary observer finds the rod rotating with an angular velocity Ω and the disc rotating about its vertical axis with angular velocity 4 Ω . The total angular momentum of the system about the point O is ( M a 2 Ω 48 ) n . The value of n is ___________.

The smallest division on the main scale of a Vernier calipers is 0.1 cm. Ten divisions of the Vernier scale correspond to nine divisions of the main scale. The figure below on the left shows the reading of this calipers with no gap between its two jaws. The figure on the right shows the reading with a solid sphere held between the jaws. The correct diameter of the sphere is

A particle of mass M = 0.2 kg is initially at rest in the xy-plane at a point (x = − l, y = − h), where l = 10 m and h = 1 m. The particle is accelerated at time t = 0 with a constant acceleration a = 10 m/s 2 along the positive x-direction. Its angular momentum and torque with respect to the origin, in SI units, are represented by L → and τ → respectively. i ^ , j ^ and k ^ are unit vectors along the positive x, y and z-directions, respectively. If k ^ = i ^ × j ^ then which of the following statement(s) is(are) correct?

An ideal gas undergoes a four step cycle as shown in the P-V diagram below. During this cycle, heat is absorbed by the gas in

Among the following, the conformation that corresponds to the most stable conformation of meso-butane-2,3-diol is

The value of y is ________.

The value of standard enthalpy, $\Delta$ H o (in kJ mol $-$ 1 ) for the given reaction is _______.

The major product formed in the following reaction of

The value of $\Delta$ S $\theta$ (in J K $-$ 1 mol $-$ 1 ) for the given reaction, at 1000 K is _________.

For x $\in$ R, the number of real roots of the equation $3{x^2} - 4\left| {{x^2} - 1} \right| + x - 1 = 0$ is ________.

For any complex number w = c + id, let arg ⁡ ( w ) ∈ ( − π , π ] , where i = − 1 . Let α and β be real numbers such that for all complex numbers z = x + iy satisfying arg ⁡ ( z + α z + β ) = π 4 , the ordered pair (x, y) lies on the circle x 2 + y 2 + 5 x − 3 y + 4 = 0 , Then which of the following statements is (are) TRUE?

Consider a triangle Δ whose two sides lie on the x-axis and the line x + y + 1 = 0. If the orthocenter of Δ is (1, 1), then the equation of the circle passing through the vertices of the triangle Δ is

Consider the lines L 1 and L 2 defined by ${L_1}:x\sqrt 2 + y - 1 = 0$ and ${L_2}:x\sqrt 2 - y + 1 = 0$ For a fixed constant $\lambda$ , let C be the locus of a point P such that the product of the distance of P from L 1 and the distance of P from L 2 is $\lambda$ 2 . The line y = 2x + 1 meets C at two points R and S, where the distance between R and S is $\sqrt {270}$ . Let the perpendicular bisector of RS meet C at two distinct points R' and S'. Let D be the square of the distance between R' and S'. The value of D is __________.

Let E denote the parabola y 2 = 8x. Let P = ( − 2, 4), and let Q and Q' be two distinct points on E such that the lines PQ and PQ' are tangents to E. Let F be the focus of E. Then which of the following statements is(are) TRUE?

A heavy nucleus Q of half-life 20 minutes undergoes alpha-decay with probability of 60% and beta-decay with probability of 40%. Initially, the number of Q nuclei is 1000. The number of alpha-decays of Q in the first one hour is

Two point charges $-$ Q and +Q/ $\sqrt 3$ are placed in the xy-plane at the origin (0, 0) and a point (2, 0), respectively, as shown in the figure. This results in an equipotential circle of radius R and potential V = 0 in the xy-plane with its center at (b, 0). All lengths are measured in meters. The value of R is __________ meter.

In the circuit shown below, the switch S is connected to position P for a long time so that the charge on the capacitor becomes q 1 μ C. Then S is switched to position Q. After a long time, the charge on the capacitor is q 2 μ C. The magnitude of q 2 is ________________.

Two point charges $-$ Q and +Q/ $\sqrt 3$ are placed in the xy-plane at the origin (0, 0) and a point (2, 0), respectively, as shown in the figure. This results in an equipotential circle of radius R and potential V = 0 in the xy-plane with its center at (b, 0). All lengths are measured in meters. The value of b is __________ meter.

An extended object is placed at point O, 10 cm in front of a convex lens L 1 and a concave lens L 2 is placed 10 cm behind it, as shown in the figure. The radii of curvature of all the curved surfaces in both the lenses ae 20 cm. The refractive index of both the lenses is 1.5. The total magnification of this lens system is

Three numbers are chosen at random, one after another with replacement, from the set S = {1, 2, 3, ......, 100}. Let p 1 be the probability that the maximum of chosen numbers is at least 81 and p 2 be the probability that the minimum of chosen numbers is at most 40. The value of ${{125} \over 4}{p_2}$ is ___________.

Consider three sets E 1 = {1, 2, 3}, F 1 = {1, 3, 4} and G 1 = {2, 3, 4, 5}. Two elements are chosen at random, without replacement, from the set E 1 , and let S 1 denote the set of these chosen elements. Let E 2 = E 1 $-$ S 1 and F 2 = F 1 $\cup$ S 1 . Now two elements are chosen at random, without replacement, from the set F 2 and let S 2 denote the set of these chosen elements. Let G 2 = G 1 $\cup$ S 2 . Finally, two elements are chosen at random, without replacement, from the set G 2 and let S 3 denote the set of these chosen elements. Let E 3 = E 2 $\cup$ S 3 . Given that E 1 = E 3 , let p be the conditional probability of the event S 1 = {1, 2}. Then the value of p is

Let f : R $\to$ R be defined by $f(x) = {{{x^2} - 3x - 6} \over {{x^2} + 2x + 4}}$ Then which of the following statements is (are) TRUE?

Three numbers are chosen at random, one after another with replacement, from the set S = {1, 2, 3, ......, 100}. Let p 1 be the probability that the maximum of chosen numbers is at least 81 and p 2 be the probability that the minimum of chosen numbers is at most 40. The value of 625 4 p 1 is ___________.

The area of the region { ( x , y ) : 0 ≤ x ≤ 9 4 , 0 ≤ y ≤ 1 , x ≥ 3 y , x + y ≥ 2 } is

The reaction of Q with PhSNa yields an organic compound (major product) that gives positive Carius test on treatment with Na 2 O 2 followed by addition of BaCl 2 . The correct option(s) for Q is (are)

For the given close packed structure of a salt made of cation X and anion Y shown below (ions of only one face are shown for clarity), the packing fraction is approximately (packing fraction = ${{packing\,efficiency} \over {100}}$ )

The correct statement(s) related to the metal extraction processes is (are) :

The value of | y | is ________.

Given : The compound(s), which on reaction with HNO 3 will give the product having degree of rotation, [ $\alpha$ ] D = $-$ 52.7 $^\circ$ is (are)