JEE Advanced 2023 Paper 1

Le t (0,1) (1,2) (3,4)S and {0,1, 2, 3}T . Then which of the following statements is(are) true?

1. There are infinitely many functions from S to T
2. There are infinitely many strictly increasing functions fro m S to T
3. T h e number of continuous functions from S to Tis at most 120
4. Every continuous function from S to T is differentiable Mathematics Q. 2 Let 1T and 2T be two distinct common tangents to the ellipse 22 :163xyE and the parabola 2:1 2Pyx . Suppose that the tangent 1T touches P and E at the points 1A and 2A, respectively and the tangent 2T touches P and E at the points 4A and 3A, respectively. Then which of the following s tatements is(are) true ? (A) The area of the quadrilateral 1234AAAA is 35 square units
5. The area of the quadrilateral 1234AAAA is 36 square units
6. The tangents 1T and 2T meet the x-axis at the point (3 , 0 )
7. The tangents 1T and 2T meet the x-axis at the point (6 , 0 ) Q. 3 Let :[0,1 ] [0,1 ]f be the function defined by 3 251 7()39 3 6xfx x x . Consider the squ are reg ion [0,1] [0,1]S . Let {( , ) : ( )}Gx y S y f x be called the green region and {( , ) : ( )}Rx ySyf x   be called the red region. Let {( , ) : [0,1]}hLx h S x be the horizontal line drawn at a height [0,1]h . Then which of the following statements is(are) true? (A) There exists an 12,43h such that the area of the green region above the line hLequals the area of the green region below the line hL
8. There exists an 12,43h such that the area of the red region above the line hLequals the area of the red region below the line hL
9. There exists an 12,43h such that the area of the g reen region above the line hLequals the area of the red region below the line hL
10. There exists an 12,43h such that the area of the red region above the line hLequals the area of the green region below the line hL SE CTION 2 (Maximum Marks: 12) This section contains FOUR (04) questions. Each question has FOUR options (A),
11. ,
12. and
13. . Q. 4 Let :( 0 , 1 )f be the function defined as ()fxn if 11,1xnn  where n. Let :( 0 , 1 )g be a function such that 21() 2x xtdt g x xt for all (0,1)x . Then 0lim ( ) ( ) xfxgx  (A) does NO T exist
14. is equal to 1
15. is equal to 2
16. is equal to 3

Let Q be the cube with the set of vertices  3 123 123(, , ) : , , { 0 , 1 }xxx xxx . LetFbe the set of all twelve lines containing the diagonals of the six faces o f the cube Q. Let S be the set of all four lines containing the ma in diagonals of the cube Q; for instance, the lin e passing through the vertices (0,0,0) and (1,1,1) is in S. For lines 1 and 2, let 12(, )d denote the shortest distance between them. Then the maximum value of 12(, )d , as 1 varies over F and 2 varies over S, is

1. 1 6
2. 1 8
3. 1 3
4. 1 12 Q. 6 Let 22 2( , ) : 1 and 582 0xyXx y yx     . Three distinct points P, Q and R are randomly chosen from X. Then the probability that P, Q and R form a triangle whose area is a positive integer, is (A) 71 220
5. 73 220
6. 79 220
7. 83 220 Q. 7 Let Pbe a point on the parabola 24 ya x , where 0a. The normal to the parabola at P me ets the x-axis at a point Q. The area of the triangle PFQ, where F is the focus of the parabola, is 120. If the slope m of the normal and a are both positive integers, then the pair (, )am is (A) (2,3)
8. (1, 3)
9. (2,4)
10. (3, 4) Q. 8 Let 1tan ( ) , ,22x forx. Then the number of real solutions of the equation 11 cos(2 ) 2 tan (tan ) x x in the set 33,, ,22 2 2 2 2                   is equal to

Let 2nbe a natural number and :[0,1 ]f be the function defined by 1(1 2 ) if 02 132( 2 1 ) i f 24()314( 1 ) i f 4 11 i f 11nn x xn nn x xnnfx nn x xnn nnx xnn       If n is such that the area of the region bounded by the curves 0x, 1x, 0y and () yf x is 4 , then the maximum value of the function f is

Let  75 57r  denote the (2 )r digit number where the first a nd the last digits are 7 and the remaining r digits are 5. Consider the sum 98 77 757 7557 75 57.S    If 99 75 57,mSn where m and n are natural numbers less than 3000, then the value of mn is Q. 15 Consider the given data with frequency distribution 3 8 11 10 5 4 52 3 2 44i ix f Match each e ntry in List-I to the correct entries in List-II. List-I List-II (P) The mean of the above data is (1) 2.5 (Q ) The median of the above data is (2) 5 (R) The mean deviation about the mean of the above data is (3) 6 ( S) The mean deviation about the median of the above data is (4) 2.7 (5) 2.4 Th e correct option is:

1. ( ) (3) ( ) (2) ( ) (4) ( ) (5)PQRS
2. ( ) (3) ( ) (2) ( ) (1) ( ) (5)PQRS
3. ( ) (2) ( ) (3) ( ) (4) ( ) (1)PQRS
4. ( ) (3) ( ) (3) ( ) (5) ( ) (5)PQRS Q. 16 Let 1 and 2 be the lines 1ˆ ˆˆ() ri j k  and 2ˆˆˆˆ() ()rj k i k   , respectively. Let X be the s et of all the planes H that contain the line 1. For a plane H, let ()dH denote the smallest possible distance between the points of 2 and H. Let 0H be a plane in X for which 0()dH is the maximum value of ()dH as H varies over all planes in X. Match each entry in List-I to the correct entries in List-II. List-I List-II (P) The value of 0()dH is (1) 3 (Q) The distance of the point (0,1,2) from 0H is (2)1 3 (R) The distance of origin from 0H is (3) 0 (S) The di stance of origin from the point of intersection of planes yz, 1x and 0H is (4) 2 (5) 1 2 Th e correct option is: (A) ( ) (2) ( ) (4) ( ) (5) ( ) (1)PQRS
5. ( ) (5) ( ) (4) ( ) (3) ( ) (1)PQRS
6. ( ) (2) ( ) (1) ( ) (3) ( ) (2)PQ RS
7. ( ) (5) ( ) (1) ( ) (4) ( ) (2)PQRS

Let z be a complex number satisfying 32|| 2 4 8 0zz z  , where z denotes the complex conjugate of z. Let the imaginary part of z be nonzero . Match each entry in List-I to the correct entries in List-II. List-I List-II (P) 2||zis equal to (1) 12 (Q) 2||zz is equal to (2) 4 (R) 22|| | |zz z is equal to (3) 8 (S) 2|1 |z is equal to (4) 10 ( 5)7 The correct option is:

1. ( ) (1) ( ) (3) ( ) (5) ( ) (4)PQ R S
2. ( ) (2) ( ) (1) ( ) (3) ( ) (5)PQ RS
3. ( ) (2) ( ) (4) ( ) (5) ( ) (1)PQRS
4. ( ) (2) ( ) (3) ( ) (5) ( ) (4)PQRS END OF THE QUESTION PAPER

A slide with a frictionless curved surface, which becomes horizontal at its lower end, is fixed on the terrace of a building of height 3ℎ from the ground, as shown in the figure. A spherical ball of mass 𝑚 is released on the slide from rest at a height ℎ from the top of the terrace. The ball leaves the slide with a velocity 𝑢⃗ 0=𝑢0𝑥̂ and falls on the ground at a distance 𝑑 from the building making an angle 𝜃 with the horizontal. It bounces off with a velocity v⃗ and reaches a maximum height ℎ1. The acceleration due to gravity is 𝑔 and the coefficient of restitution of the ground is 1√3⁄. Which of the following statement(s) is(are) correct?

1. u⃗ 0=√2𝑔ℎ𝑥̂
2. v⃗ =√2𝑔ℎ(𝑥̂−𝑧̂)
3. 𝜃=60°
4. 𝑑/ℎ1=2√3

A plane polarized blue light ray is incident on a prism such that there is no reflection from the surface of the prism . The angle of deviation of the emergent ray is 𝛿=60° (see Figure-1). The angle of minimum deviation for red light from the same prism is 𝛿min= 30° (see Figure -2). The refractive index of the prism material for blue light is √3. Which of the following statement(s) is(are) correct?

1. The blue light is polarized in the plane of incidence.
2. The angle of the prism is 45°.
3. The refractive index of the material of the prism for red light is √2.
4. The angle of refraction for blue light in air at the exit plane of the prism is 60°.

In a circuit shown in the figure, t he capacitor 𝐶 is initially uncharged and the key 𝐾 is open. In this condition, a current of 1 A flows through the 1 Ω resistor. The key is closed at time 𝑡 = 𝑡0. Which of the following statement(s) is(are) correct ? [Given: 𝑒−1=0.36]

1. The value of the resistance 𝑅 is 3 Ω.
2. For 𝑡< 𝑡0, the value of current 𝐼1 is 2 A.
3. At 𝑡 =𝑡0+7.2 𝜇s, the current in the capacitor is 0.6 A.
4. For 𝑡→∞, the charge on the capacitor is 12 𝜇C.

A bar of mass 𝑀=1.00 kg and length 𝐿=0.20 m is lying on a horizontal frictionless surface. One end of the bar is pivoted at a point about which it is free to rotate. A small mass 𝑚=0.10 kg is moving on the same horizontal surface with 5.00 m s⁻¹ speed on a path perpendicular to the bar. It hits the bar at a distance 𝐿/2 from the pivoted end and returns back on the same path with speed v. After this elastic collision, the b ar rotates with an angular velocity 𝜔. Which of the following statement is correct?

1. 𝜔=6.98 rad s⁻¹ and v=4.30 m s⁻¹
2. 𝜔=3.75 rad s⁻¹ and v=4.30 m s⁻¹
3. 𝜔=3.75 rad s⁻¹ and v=10.0 m s⁻¹
4. 𝜔=6.80 rad s⁻¹ and v=4.10 m s⁻¹

A container has a base of 50 cm×5 cm and height 50 cm, as shown in the figure. It has two parallel electrically conducting walls each of area 50 cm×50 cm. The remaining walls of the container are thin and non-conducting. The container is being filled with a liquid of dielectric constant 3 at a uniform rate of 250 cm³ s⁻¹. What is t he value of the capacitance of the container after 10 seconds ? [Given: Permittivity of free space 𝜖0=9×10−12 C2N−1m−2, the effects of the non -conducting walls on the capacitance are negligible ]

1. 27 pF
2. 63 pF
3. 81 pF
4. 135 pF

wOne mole of an ideal gas expands adiabatically from an initial state (𝑇A,𝑉0) to final state (𝑇f,5𝑉0). Another mole of the same gas expands isothermally from a different initial state (𝑇B,𝑉0) to the same final state (𝑇f,5𝑉0). The ratio of the specific heats at constant pressure and constant volume of this ideal gas i s 𝛾. What is the ratio 𝑇A/𝑇B?

1. 5𝛾−1
2. 51−𝛾
3. 5𝛾
4. 51+𝛾

Two satellites P and Q are moving in different circular orbits around the Earth (radius 𝑅). The height s of P and Q from the Earth surface are ℎP and ℎQ, respectively , where ℎP=𝑅/3. The accelerations of P and Q due to Earth’s gravity are 𝑔P and 𝑔Q, respectively. If 𝑔P/𝑔Q=36/25, what is the value of ℎQ?

1. 3𝑅/5
2. 𝑅/6
3. 6𝑅/5
4. 5𝑅/6

A Hydrogen -like atom has atomic number 𝑍. Photons emitted in the electronic transitions from level 𝑛=4 to level 𝑛=3 in these atom s are used to perform photoelectric effect experiment on a target m etal. The maximum kinetic energy of the photoelectrons generated is 1 .95 eV. If the photoelectric threshold wavelength for the target metal is 310 nm, the value of 𝑍 is _______ . [Given: ℎ𝑐=1240 eV-nm and 𝑅ℎ𝑐=13.6 eV, where 𝑅 is the Rydberg constant, ℎ is the Planck’s constant and 𝑐 is the speed of light in vacuum ]

Numerical answer type

An optical arrangement consists of two c oncave mirrors M1 and M2, and a convex lens L with a common principal axis, as shown in the figure. The focal length of L is 10 cm. The radii of curvature of M1 and M2 are 20 cm and 24 cm, respectively. The distance between L and M2 is 20 cm. A point object S is placed at the mid-point between L and M2 on the axis. When the distance between L and M1 is 𝑛/7 cm, one of the images coincides with S. The value of 𝑛 is _______.

Numerical answer type

In an experiment for determination of the focal length of a thin convex lens, the distance of the object from the lens is 10±0.1 cm and the distance of its real image from the lens is 20±0.2 cm. The error in the determination of focal length of the lens is 𝑛 %. The value of 𝑛 is _______.

Numerical answer type

A closed c ontainer contains a homogeneous mixture of two moles of an ideal monatomic gas (𝛾=5/3) and one mole of an ideal diatomic gas (𝛾=7/5). Here, 𝛾 is the ratio of the specific heats at constant pressure and constant volume of an ideal gas. The gas mixture does a work of 66 Joule when heated at constant pressure. The change in its internal energy is ________ Joule .

Numerical answer type

A person of height 1.6 m is walking away from a lamp post of height 4 m along a straight path on the flat ground . The lamp post and the person are always perpendicular to the ground . If the speed of the person is 60 cm s⁻¹, the speed of the tip of the person’s shadow on the ground with respect to the person is _______ cm s⁻¹.

Numerical answer type

Two point -like objects of mass es 20 gm and 30 gm are fixed at the two ends of a rigid massless rod of length 10 cm. This system is suspended vertically from a rigid ceiling using a thin wire attached to its center of mass , as shown in the figure. The resulting torsional pendulum undergoes small oscillations . The torsional constant o f the wire is 1.2×10−8 N m rad−1. The angular frequency of the oscillations in 𝑛×10⁻³ rad s⁻¹. The value of 𝑛 is _____.

Numerical answer type

List-I shows different radioactive decay processes and List-II provides possible emitted particles. Match each entry in List -I with an appropriate entry from List -II, and choose the correct option. List-I List-II (P) 𝑈→𝑃𝑎91234 92238(1) one 𝛼 particle and one 𝛽+ particle (Q) 𝑃𝑏→𝑃𝑏82210 82214(2) three 𝛽− particle s and one 𝛼 particle (R) 𝑇𝑙→𝑃𝑏82206 81210(3) two 𝛽− particles and one 𝛼 particle (S) 𝑃𝑎→𝑅𝑎88224 91228(4) one 𝛼 particle and one 𝛽− particle (5) one 𝛼 particle and two 𝛽+ particles

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

Match the temperature of a black body given in List -I with an appropriate statement in List -II, and choose the correct option. [Given: Wien’s constant as 2.9×10⁻³ m-K and ℎ𝑐 𝑒=1.24×10−6 V-m] List-I List-II (P) 2000 K (1) The radiation at peak wavelength can lead to emission of photoelectrons from a metal of work function 4 eV. (Q) 3000 K (2) The radiation at peak wavelength is visible to human eye. (R) 5000 K (3) The radiation at peak emission wavelength will result in the widest central maximum of a single slit diffraction. (S) 10000 K (4) The power emitted per unit area is 1/16 of that emitted by a blackbody at temperature 6000 K. (5) The radiation at peak emiss ion wavelength can be used to image human bones .

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

A series LCR circuit is connected to a 45sin(𝜔𝑡) Volt source. The resonant angular frequency of the circuit is 105 rad s⁻¹ and current amplitude at resonance is 𝐼0. When the angular frequency of the source is 𝜔=8×104 rad s⁻¹, the current amplitude in the circuit is 0.05 𝐼0. If 𝐿=50 mH, match each entry in List-I with an appropriate value from List -II and choose the correct option. List-I List-II (P) 𝐼0 in mA (1) 44.4 (Q) The quality factor of the circuit (2) 18 (R) The bandwidth of the circuit in rad s⁻¹(3) 400 (S) The peak power dissipated at resonance in W att (4) 2250 (5) 500

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

A thin conducting rod MN of mass 20 gm, length 25 cm and resistance 10 Ω is held on frictionless, long, perfectly conducting vertical rails as shown in the figure . There is a uniform magnetic field 𝐵0=4 T directed perpendicular to the plane of the rod -rail arrangement. The rod is released from rest at time 𝑡=0 and it moves down along the rails. Assume air drag is negligible. M atch each quantity in List -I with an appropriate value from List-II, and choose the correct option. [Given: The acceleration due to gravity 𝑔=10 m s−2 and 𝑒−1=0.4] List-I List-II (P) At 𝑡=0.2 s, the magnitude of the induced emf in Volt (1) 0.07 (Q) At 𝑡=0.2 s, the magnitude of the magnetic force in Newton (2) 0.14 (R) At 𝑡=0.2 s, the power dissipated as heat in Watt (3) 1.20 (S) The magnitude of terminal velocity of the rod in m s⁻¹(4) 0.12 (5) 2.00

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

The c orrect statement (s) related to processes invol ved in the extraction of metals is(are)

1. Roasting of Malachite produces C uprite.
2. Calcination of Calamine produces Z incite.
3. Copper pyrites is heated with silica in a reverberatory furnace to remove iron .
4. Impure silver is treated with aqueous KCN in the presence of oxygen followed by reduction with zinc metal . Chemistry

) In the following reactions, P, Q, R, and S are the major products . The correct statement(s) about P, Q, R, and S is(are)

1. Both P and Q have asymmetric carbon(s) .
2. Both Q and R have asymmetric carbon(s) .
3. Both P and R have asymmetric carbon(s) .
4. P has asymmetric carbon(s), S does not have any asymmetric carbon .

Consider the following reaction scheme and choose the correct option (s) for the major products Q, R and S.

In the s cheme given below , X and Y, respectively, are

1. CrO 42 and Br₂
2. MnO 42 and Cl 2
3. MnO 4 and Cl 2
4. MnSO4 and HOCl

Plotting 1/ Λm against cΛm for aqueous solutions of a monobasic weak acid (HX) resulted in a straight line with y -axis intercept of P and slope of S. The ratio PS⁄ is [Λm = molar conductivity m = limiting molar conductivity c = molar concentration Ka = dissociation constant of HX]

1. Ka m
2. Ka m/2
3. 2 Ka m
4. 1 / (Ka m)

On decreasing the 𝑝H from 7 to 2 , the s olubility of a sparingly soluble salt (MX) of a weak acid (HX) increased from 10−4 mol L⁻¹ to 10⁻³ mol L⁻¹. The 𝑝Ka of HX is

1. 3
2. 4
3. 5
4. 2

In the given reaction scheme, P is a phenyl alkyl ether , Q is an aromatic compound ; R and S are the major product s. The correct statement about S is

1. It primarily inhibit s noradrenaline degrad ing enzyme s.
2. It inhibit s the synthesis of prostaglandin .
3. It is a narcotic drug .
4. It is ortho-acetyl benzoic acid .

The stoichiometric reaction of 516 g of dimethyldichlorosilane with water results in a tetrameric cyclic product X in 75% yield. The weight (in g) of X obtained is___. [Use, molar mass (g mol⁻¹): H = 1, C = 12, O = 16, Si = 28, Cl = 35.5]

A gas has a compressibility factor of 0.5 and a molar volume of 0.4 dm³ mol⁻¹ at a temperature of 800 K and pressure x atm. If it shows ideal gas behaviour at the same temperature and pressure, the molar volume will be y dm³ mol⁻¹. The value of 𝐱/𝐲 is ___. [Use: Gas constant, R = 8 ×10−2 L atm K⁻¹ mol⁻¹]

Numerical answer type

The plot of log𝑘𝑓 versus 1𝑇⁄ for a reversible reaction A (g)⇌P (g) is shown. Pre-exponential factors for the forward and backward reactions are 1015 s⁻¹ and 1011 s⁻¹, respectively. If the value of log𝐾 for the reaction at 500 K is 6, the value of |log𝑘𝑏| at 250 K is ___. [K = equilibrium constant of the reaction 𝑘𝑓 = rate constant of forward reaction 𝑘𝑏 = rate constant of backward reaction]

Numerical answer type

One mole of an ideal monoatomic gas undergoes two reversible processes (A  B and B  C) as shown in the given figure: A  B is an adiabatic process. If the total heat absorbed in the entire process (A  B and B  C) is R𝑇2ln10, the value of 2log𝑉3 is ___. [Use, molar heat capacity of the gas at constant pressure, 𝐶p,m=5 2R]

Numerical answer type

In a one-litre flask, 6 moles of A undergoes the reaction A (g)⇌P (g). The progress of product formation at two temperatures (in Kelvin) , T1 and T2, is shown in the figure: If T1=2T2 and (∆G2Θ−∆G1Θ) = RT2lnx, then the value of x is ___. [∆G1Θ and ∆G2Θ are standard Gibb’s free energy change for the reaction at temperature s T1 and T 2, respectively. ]

Numerical answer type

The total number of sp2 hybridi sed carbon atoms in the major product P (a non -heterocyclic compound) of the following reaction is ___.

Numerical answer type

Match the reactions (in the given stoichiometry of the reactants ) in List-I with one of their products given in List -II and choose the correct option. List-I List-II (P) P2O3 + 3H₂O  (1) P(O)(OCH 3)Cl 2 (Q) P₄ + 3NaOH + 3H 2O  (2) H3PO 3 (R) PCl 5 + CH 3COOH  (3) PH 3 (S) H3PO 2 + 2H₂O + 4AgNO 3  (4) POCl 3 (5) H3PO 4

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

Match the electronic configurations in List -I with appropriate metal complex ions in List -II and choose the correct option. [Atomic Number: Fe = 26, Mn = 25, Co = 27] List-I List-II (P) t2g 6eg0 (1) [Fe(H 2O)6]2+ (Q) t2g3 eg2 (2) [Mn(H 2O)6]2+ (R) e2 t23 (3) [Co(NH 3)6]3+ (S) t2g 4eg2 (4) [FeCl 4] (5) [CoCl 4]2

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

Match the reactions in List-I with the features of their products in List-II and choose the correct option . List-I List-II (P) (1) Inversion of configuration (2) Retention of configuration (3) Mixture of enantiomers (4) Mixture of structural isomers (5) Mixture of diastereom ers (Q) (R) (S)

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

The major products obtained from the reactions in List-II are the reactants for the named reactions mentioned in List-I. Match List-I with List -II and choose the correct option . List-I List-II (P) Etard reaction (1) (Q) Gatterman n reaction (2) (R) Gatterman n-Koch reaction (3) (S) Rosenmund reduction (4) (5)

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