JEE Advanced 2024 Paper 1

A dimensionless quantity is constructed in terms of electronic charge e, permittivity of free space 0, Planck's constant h, and speed of light c. If the dimensionless quantity is written as e 0 h c and n is a non-zero integer, then ( , , , ) is given by

1. (2n,-n,-n,-n)
2. (n,-n,-2n,-n)
3. (n,-n,-n,-2n)
4. (2n,-n,-2n,-2n)

An infinitely long wire, located on the z-axis, carries a current I along the +z-direction and produces the magnetic field B. The magnitude of the line integral B dl along a straight line from the point (- 3a, a, 0) to (a, a, 0) is given by [ 0 is the magnetic permeability of free space.]

1. 7 0 I/24
2. 7 0 I/12
3. 0 I/8
4. 0 I/6

Two beads, each with charge q and mass m, are on a horizontal, frictionless, non-conducting, circular hoop of radius R. One of the beads is glued to the hoop at some point, while the other one performs small oscillations about its equilibrium position along the hoop. The square of the angular frequency of the small oscillations is given by [ 0 is the permittivity of free space.]

1. q2/(4 0 R3m)
2. q2/(32 0 R3m)
3. q2/(8 0 R3m)
4. q2/(16 0 R3m)

A block of mass 5 kg moves along the x-direction subject to the force F=(-20x+10) N, with the value of x in metre. At time t=0 s, it is at rest at position x=1 m. The position and momentum of the block at t=( /4) s are

1. -0.5 m, 5 kg m/s
2. 0.5 m, 0 kg m/s
3. 0.5 m, -5 kg m/s
4. -1 m, 5 kg m/s

A particle of mass m is moving in a circular orbit under the influence of the central force F(r)=-kr, corresponding to the potential energy V(r)=kr2/2, where k is a positive force constant and r is the radial distance from the origin. According to the Bohr's quantization rule, the angular momentum of the particle is given by L=n , where =h/(2 ), h is the Planck's constant, and n a positive integer. If v and E are the speed and total energy of the particle, respectively, then which of the following expression(s) is(are) correct?

1. r2=n 1mk
2. v2=n km3
3. Lmr2= km
4. E= n 2 km

The specific heat capacity of a substance is temperature dependent and is given by the formula C=kT, where k is a constant of suitable dimensions in SI units, and T is the absolute temperature. If the heat required to raise the temperature of 1 kg of the substance from -73 to 27 is nk, the value of n is . [Given: 0 K=-273 .]

A point source S emits unpolarized light uniformly in all directions. At two points A and B, the ratio r=IA/IB of the intensities of light is 2. If a set of two polaroids having 45 angle between their pass-axes is placed just before point B, then the new value of r will be .

Two large, identical water tanks, 1 and 2, kept on the top of a building of height H, are filled with water up to height h in each tank. Both the tanks contain an identical hole of small radius on their sides, close to their bottom. A pipe of the same internal radius as that of the hole is connected to tank 2, and the pipe ends at the ground level. When the water flows from the tanks 1 and 2 through the holes, the times taken to empty the tanks are t1 and t2, respectively. If H= ( 169 )h, then the ratio t1/t2 is .

A light ray is incident on the surface of a sphere of refractive index n at an angle of incidence 0. The ray partially refracts into the sphere with angle of refraction 0 and then partly reflects from the back surface. The reflected ray then emerges out of the sphere after a partial refraction. The total angle of deviation of the emergent ray with respect to the incident ray is . Match the quantities mentioned in List-I with their values in List-II and choose the correct option.

1. P → 5; Q → 2; R → 1; S → 4
2. P → 5; Q → 1; R → 2; S → 4
3. P → 3; Q → 2; R → 1; S → 4
4. P → 3; Q → 1; R → 2; S → 5

Two uniform strings of mass per unit length and 4 , and length L and 2L, respectively, are joined at point O, and tied at two fixed ends P and Q, as shown in the figure. The strings are under a uniform tension T. If we define the frequency 0= 12L T , which of the following statement(s) is(are) correct? [figure]

1. With a node at O, the minimum frequency of vibration of the composite string is 0.
2. With an antinode at O, the minimum frequency of vibration of the composite string is 2 0.
3. When the composite string vibrates at the minimum frequency with a node at O, it has 6 nodes, including the end nodes.
4. No vibrational mode with an antinode at O is possible for the composite string.

A glass beaker has a solid, plano-convex base of refractive index 1.60, as shown in the figure. The radius of curvature of the convex surface (SPU) is 9 cm, while the planar surface (STU) acts as a mirror. This beaker is filled with a liquid of refractive index n up to the level QPR. If the image of a point object O at a height of h (OT in the figure) is formed onto itself, then, which of the following option(s) is(are) correct? [figure]

1. For n=1.42, h=50 cm.
2. For n=1.35, h=36 cm.
3. For n=1.45, h=65 cm.
4. For n=1.48, h=85 cm.

A disc of mass M and radius R is free to rotate about its vertical axis as shown in the figure. A battery operated motor of negligible mass is fixed to this disc at a point on its circumference. Another disc of the same mass M and radius R/2 is fixed to the motor's thin shaft. Initially, both the discs are at rest. The motor is switched on so that the smaller disc rotates at a uniform angular speed . If the angular speed at which the large disc rotates is /n, then the value of n is . [figure]

A source (S) of sound has frequency 240 Hz. When the observer (O) and the source move towards each other at a speed v with respect to the ground (as shown in Case 1 in the figure), the observer measures the frequency of the sound to be 288 Hz. However, when the observer and the source move away from each other at the same speed v with respect to the ground (as shown in Case 2 in the figure), the observer measures the frequency of sound to be n Hz. The value of n is . [figure]

A thin uniform rod of length L and certain mass is kept on a frictionless horizontal table with a massless string of length L fixed to one end (top view is shown in the figure). The other end of the string is pivoted to a point O. If a horizontal impulse P is imparted to the rod at a distance x=L/n from the mid-point of the rod (see figure), then the rod and string revolve together around the point O, with the rod remaining aligned with the string. In such a case, the value of n is . [figure]

One mole of a monatomic ideal gas undergoes the cyclic process J K L M J, as shown in the P-T diagram. [figure] Match the quantities mentioned in List-I with their values in List-II and choose the correct option. [ R is the gas constant.]

1. P → 1; Q → 3; R → 5; S → 4
2. P → 4; Q → 3; R → 5; S → 2
3. P → 4; Q → 1; R → 2; S → 2
4. P → 2; Q → 5; R → 3; S → 4

Four identical thin, square metal sheets, S1, S2, S3 and S4, each of side a are kept parallel to each other with equal distance d ( a) between them, as shown in the figure. Let C0= 0 a2/d, where 0 is the permittivity of free space. [figure] Match the quantities mentioned in List-I with their values in List-II and choose the correct option.

1. P → 3; Q → 2; R → 4; S → 5
2. P → 2; Q → 3; R → 2; S → 1
3. P → 3; Q → 2; R → 4; S → 1
4. P → 3; Q → 2; R → 2; S → 5

The circuit shown in the figure contains an inductor L, a capacitor C0, a resistor R0 and an ideal battery. The circuit also contains two keys K1 and K2. Initially, both the keys are open and there is no charge on the capacitor. At an instant, key K1 is closed and immediately after this the current in R0 is found to be I1. After a long time, the current attains a steady state value I2. Thereafter, K2 is closed and simultaneously K1 is opened and the voltage across C0 oscillates with amplitude V0 and angular frequency 0. [figure] Match the quantities mentioned in List-I with their values in List-II and choose the correct option.

1. P → 1; Q → 3; R → 2; S → 5
2. P → 1; Q → 2; R → 3; S → 5
3. P → 1; Q → 3; R → 2; S → 4
4. P → 2; Q → 5; R → 3; S → 4

A closed vessel contains 10 g of an ideal gas X at 300 K, which exerts 2 atm pressure. At the same temperature, 80 g of another ideal gas Y is added to it and the pressure becomes 6 atm. The ratio of root mean square velocities of X and Y at 300 K is

1. 2 2 : 3
2. 2 2 : 1
3. 1 : 2
4. 2 : 1

At room temperature, disproportionation of an aqueous solution of in situ generated nitrous acid (HNO2) gives the species

1. H3O+, NO3- and NO
2. H3O+, NO3- and NO2
3. H3O+, NO- and NO2
4. H3O+, NO3- and N2O

Among the following options, select the option in which each complex in Set-I shows geometrical isomerism and the two complexes in Set-II are ionization isomers of each other. [en = H2NCH2CH2NH2]

1. Set-I: [Ni(CO)4] and [PdCl2(PPh3)2] Set-II: [Co(NH3)5Cl]SO4 and [Co(NH3)5(SO4)]Cl
2. Set-I: [Co(en)(NH3)2Cl2] and [PdCl2(PPh3)2] Set-II: [Co(NH3)6][Cr(CN)6] and [Cr(NH3)6][Co(CN)6]
3. Set-I: [Co(NH3)3(NO2)3] and [Co(en)2Cl2] Set-II: [Co(NH3)5Cl]SO4 and [Co(NH3)5(SO4)]Cl
4. Set-I: [Cr(NH3)5Cl]Cl2 and [Co(en)(NH3)2Cl2] Set-II: [Cr(H2O)6]Cl3 and [Cr(H2O)5Cl]Cl2 H2O

Among the following, the correct statement(s) for electrons in an atom is(are)

1. Uncertainty principle rules out the existence of definite paths for electrons.
2. The energy of an electron in 2s orbital of an atom is lower than the energy of an electron that is infinitely far away from the nucleus.
3. According to Bohr’s model, the most negative energy value for an electron is given by n = 1, which corresponds to the most stable orbit.
4. According to Bohr’s model, the magnitude of velocity of electrons increases with increase in values of n.

The option(s) in which at least three molecules follow Octet Rule is(are)

1. CO2, C2H4, NO and HCl
2. NO2, O3, HCl and H2SO4
3. BCl3, NO, NO2 and H2SO4
4. CO2, BCl3, O3 and C2H4

Consider the following volume−temperature (V−T) diagram for the expansion of 5 moles of an ideal monoatomic gas. [figure] Considering only P-V work is involved, the total change in enthalpy (in Joule) for the transformation of state in the sequence X→Y→Z is . [Use the given data: Molar heat capacity of the gas for the given temperature range, CV,m = 12 J K-1 mol-1 and gas constant, R = 8.3 J K-1 mol-1]

Consider the following reaction, 2H2(g) + 2NO(g) N2(g) + 2H2O(g) which follows the mechanism given below: 2NO(g) k-1 k1 N2O2(g) (fast equlibrium) N2O2(g) + H2(g) k2 N2O(g) + H2O(g) (slow reaction) N2O(g) + H2(g) k3 N2(g) + H2O(g) (fast reaction) The order of the reaction is .

Among V(CO)6, Cr(CO)5, Cu(CO)3, Mn(CO)5, Fe(CO)5, [Co(CO)3]3-, [Cr(CO)4]4-, and Ir(CO)3, the total number of species isoelectronic with Ni(CO)4 is . [Given, atomic number: V = 23, Cr = 24, Mn = 25, Fe = 26, Co = 27, Ni = 28, Cu = 29, Ir = 77]

Among the following complexes, the total number of diamagnetic species is . [Mn(NH3)6]3+, [MnCl6]3-, [FeF6]3-, [CoF6]3-, [Fe(NH3)6]3+, and [Co(en)3]3+ [Given, atomic number: Mn = 25, Fe = 26, Co = 27; en = H2NCH2CH2NH2]

In a conductometric titration, small volume of titrant of higher concentration is added stepwise to a larger volume of titrate of much lower concentration, and the conductance is measured after each addition. The limiting ionic conductivity ( 0) values (in mS m2 mol-1) for different ions in aqueous solutions are given below: array|c|c|c|c|c|c|c|c|c|c| Ions & Ag+ & K+ & Na+ & H+ & NO3- & Cl- & SO42- & OH- & CH3COO- \ 0 & 6.2 & 7.4 & 5.0 & 35.0 & 7.2 & 7.6 & 16.0 & 19.9 & 4.1 \ array For different combinations of titrates and titrants given in List-I, the graphs of ‘conductance’ versus ‘volume of titrant’ are given in List-II. Match each entry in List-I with the appropriate entry in List-II and choose the correct option.

1. P-4, Q-3, R-2, S-5
2. P-2, Q-4, R-3, S-1
3. P-3, Q-4, R-2, S-5
4. P-4, Q-3, R-2, S-1

Based on VSEPR model, match the xenon compounds given in List-I with the corresponding geometries and the number of lone pairs on xenon given in List-II and choose the correct option.

1. P-5, Q-2, R-3, S-1
2. P-5, Q-3, R-2, S-4
3. P-4, Q-3, R-2, S-1
4. P-4, Q-2, R-5, S-3

Aspartame, an artificial sweetener, is a dipeptide aspartyl phenylalanine methyl ester. The structure of aspartame is [figure]

1. [figure]
2. [figure]
3. [figure]
4. [figure]

Reaction of iso-propylbenzene with O2 followed by the treatment with H3O+ forms phenol and a by-product P. Reaction of P with 3 equivalents of Cl2 gives compound Q. Treatment of Q with Ca(OH)2 produces compound R and calcium salt S. The correct statement(s) regarding P, Q, R and S is(are)

1. Reaction of P with R in the presence of KOH followed by acidification gives [figure]
2. Reaction of R with O2 in the presence of light gives phosgene gas
3. Q reacts with aqueous NaOH to produce Cl3CCH2OH and Cl3CCOONa
4. S on heating gives P

Complete reaction of acetaldehyde with excess formaldehyde, upon heating with conc. NaOH solution, gives P and Q. Compound P does not give Tollens’ test, whereas Q on acidification gives positive Tollens’ test. Treatment of P with excess cyclohexanone in the presence of catalytic amount of p-toluenesulfonic acid (PTSA) gives product R. Sum of the number of methylene groups (-CH2-) and oxygen atoms in R is .

List-I contains various reaction sequences and List-II contains the possible products. Match each entry in List-I with the appropriate entry in List-II and choose the correct option.

1. P-3, Q-5, R-4, S-1
2. P-3, Q-2, R-4, S-1
3. P-3, Q-5, R-1, S-4
4. P-5, Q-2, R-4, S-1

List-I contains various reaction sequences and List-II contains different phenolic compounds. Match each entry in List-I with the appropriate entry in List-II and choose the correct option.

1. P-2, Q-3, R-4, S-5
2. P-2, Q-3, R-5, S-1
3. P-3, Q-5, R-4, S-1
4. P-3, Q-2, R-5, S-4

In the following reaction sequence, the major product P is formed. [figure] Glycerol reacts completely with excess P in the presence of an acid catalyst to form Q. Reaction of Q with excess NaOH followed by the treatment with CaCl2 yields Ca-soap R, quantitatively. Starting with one mole of Q, the amount of R produced in gram is . [Given, atomic weight: H = 1, C = 12, N = 14, O = 16, Na = 23, Cl = 35, Ca = 40]

Let f(x) be a continuously differentiable function on the interval (0, ) such that f(1)=2 and t x t10f(x)-x10f(t)t9-x9=1 for each x>0. Then, for all x>0, f(x) is equal to

1. 3111x- 911x10
2. 911x+ 1311x10
3. -911x+ 3111x10
4. 1311x+ 911x10

A student appears for a quiz consisting of only true-false type questions and answers all the questions. The student knows the answers of some questions and guesses the answers for the remaining questions. Whenever the student knows the answer of a question, he gives the correct answer. Assume that the probability of the student giving the correct answer for a question, given that he has guessed it, is 12. Also assume that the probability of the answer for a question being guessed, given that the student’s answer is correct, is 16. Then the probability that the student knows the answer of a randomly chosen question is

1. 112
2. 17
3. 57
4. 512

Let 2<x< be such that x= -5 11. Then ( 11x2 ) ( 6x- 6x )+ ( 11x2 ) ( 6x+ 6x ) is equal to

1. 11-12 3
2. 11+12 3
3. 11+13 2
4. 11-13 2

Consider the ellipse x29+ y24=1. Let S(p,q) be a point in the first quadrant such that p29+ q24>1. Two tangents are drawn from S to the ellipse, of which one meets the ellipse at one end point of the minor axis and the other meets the ellipse at a point T in the fourth quadrant. Let R be the vertex of the ellipse with positive x-coordinate and O be the center of the ellipse. If the area of the triangle ORT is 32, then which of the following options is correct?

1. q=2, p=3 3
2. q=2, p=4 3
3. q=1, p=5 3
4. q=1, p=6 3

Let S= \a+b 2 : a,b Z \, T1= \ (-1+ 2 )n : n N \, and T2= \ (1+ 2 )n : n N \. Then which of the following statements is (are) TRUE?

1. Z T1 T2 S
2. T1 (0, 12024 )= , where denotes the empty set.
3. T2 (2024, )
4. For any given a,b Z, ( (a+b 2 ) )+i ( (a+b 2 ) ) Z if and only if b=0, where i= -1.

Let R2 denote R R. Let S= \(a,b,c) : a,b,c R and ax2+2bxy+cy2>0 for all (x,y) R2-\(0,0)\ \. Then which of the following statements is (are) TRUE?

1. (2, 72,6 ) S
2. If (3,b, 112 ) S, then |2b|<1.
3. For any given (a,b,c) S, the system of linear equations ax+by=1 bx+cy=-1 has a unique solution.
4. For any given (a,b,c) S, the system of linear equations (a+1)x+by=0 bx+(c+1)y=0 has a unique solution.

Let R3 denote the three-dimensional space. Take two points P=(1,2,3) and Q=(4,2,7). Let dist(X,Y) denote the distance between two points X and Y in R3. Let S= \X R3 : (dist(X,P) )2- (dist(X,Q) )2=50 \ and T= \Y R3 : (dist(Y,Q) )2- (dist(Y,P) )2=50 \. Then which of the following statements is (are) TRUE?

1. There is a triangle whose area is 1 and all of whose vertices are from S.
2. There are two distinct points L and M in T such that each point on the line segment LM is also in T.
3. There are infinitely many rectangles of perimeter 48, two of whose vertices are from S and the other two vertices are from T.
4. There is a square of perimeter 48, two of whose vertices are from S and the other two vertices are from T.

Let a=3 2 and b= 151/6 6. If x,y R are such that 3x+2y= a (18 ) 54 and 2x-y= b ( 1080 ), then 4x+5y is equal to .

Let f(x)=x4+ax3+bx2+c be a polynomial with real coefficients such that f(1)=-9. Suppose that i 3 is a root of the equation 4x3+3ax2+2bx=0, where i= -1. If 1, 2, 3, and 4 are all the roots of the equation f(x)=0, then | 1|2+| 2|2+| 3|2+| 4|2 is equal to .

Let S= \A= pmatrix 0 & 1 & c \ 1 & a & d \ 1 & b & e pmatrix : a,b,c,d,e \0,1\ and |A| \-1,1\ \, where |A| denotes the determinant of A. Then the number of elements in S is .

A group of 9 students, s1,s2, ,s9, is to be divided to form three teams X,Y, and Z of sizes 2,3, and 4, respectively. Suppose that s1 cannot be selected for the team X, and s2 cannot be selected for the team Y. Then the number of ways to form such teams, is .

Let OP= -1 i+ j+ k, OQ= i+ -1 j+ k and OR= i+ j+ 12 k be three vectors, where , R-\0\ and O denotes the origin. If ( OP OQ ) OR=0 and the point ( , ,2) lies on the plane 3x+3y-z+l=0, then the value of l is .

Let X be a random variable, and let P(X=x) denote the probability that X takes the value x. Suppose that the points (x,P(X=x) ), x=0,1,2,3,4, lie on a fixed straight line in the xy-plane, and P(X=x)=0 for all x R-\0,1,2,3,4\. If the mean of X is 52, and the variance of X is , then the value of 24 is .

Let and be the distinct roots of the equation x2+x-1=0. Consider the set T=\1, , \. For a 3 3 matrix M= (aij )3 3, define Ri=ai1+ai2+ai3 and Cj=a1j+a2j+a3j for i=1,2,3 and j=1,2,3. Match each entry in List-I to the correct entry in List-II.

1. (P) → (4) (Q) → (2) (R) → (5) (S) → (1)
2. (P) → (2) (Q) → (4) (R) → (1) (S) → (5)
3. (P) → (2) (Q) → (4) (R) → (3) (S) → (5)
4. (P) → (1) (Q) → (5) (R) → (3) (S) → (4)

Let the straight line y=2x touch a circle with center (0, ), >0, and radius r at a point A1. Let B1 be the point on the circle such that the line segment A1B1 is a diameter of the circle. Let +r=5+ 5. Match each entry in List-I to the correct entry in List-II.

1. (P) → (4) (Q) → (2) (R) → (1) (S) → (3)
2. (P) → (2) (Q) → (4) (R) → (1) (S) → (3)
3. (P) → (4) (Q) → (2) (R) → (5) (S) → (3)
4. (P) → (2) (Q) → (4) (R) → (3) (S) → (5)

Let R be such that the lines L1: x+111= y+212= z+293 and L2: x+163= y+112= z+4 intersect. Let R1 be the point of intersection of L1 and L2. Let O=(0,0,0), and n denote a unit normal vector to the plane containing both the lines L1 and L2. Match each entry in List-I to the correct entry in List-II.

1. (P) → (3) (Q) → (4) (R) → (1) (S) → (2)
2. (P) → (5) (Q) → (4) (R) → (1) (S) → (2)
3. (P) → (3) (Q) → (4) (R) → (1) (S) → (5)
4. (P) → (3) (Q) → (1) (R) → (4) (S) → (5)

Let f: R R and g: R R be functions defined by f(x)= cases x|x| ( 1x ), & x 0, \ 0, & x=0, cases and g(x)= cases 1-2x, & 0 x 12, \ 0, & otherwise. cases Let a,b,c,d R. Define the function h: R R by h(x)=a f(x)+b (g(x)+g ( 12-x ) )+c (x-g(x) )+d g(x), x R. Match each entry in List-I to the correct entry in List-II.

1. (P) → (4) (Q) → (3) (R) → (1) (S) → (2)
2. (P) → (5) (Q) → (2) (R) → (4) (S) → (3)
3. (P) → (5) (Q) → (3) (R) → (2) (S) → (4)
4. (P) → (4) (Q) → (2) (R) → (1) (S) → (3)