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FINAL JEE–MAIN EXAMINATION – JANUARY, 2024 (Held On Saturday 27th January, 2024) TIME : 9 : 00 AM to 12 : 00 NOON MATHEMATICS TEST PAPER WITH ANSWER...

FINAL JEE–MAIN EXAMINATION – JANUARY, 2024 (Held On Saturday 27th January, 2024) TIME : 9 : 00 AM to 12 : 00 NOON MATHEMATICS TEST PAPER WITH ANSWER SECTION-A 5. If A denotes the sum of all the coefficients in the 1. n −1 Cr = ( k 2 − 8 ) n Cr +1 if and only if : expansion of (1 – 3x + 10x2)n and B denotes the sum of all the coefficients in the expansion of (1) 2 2  k  3 (2) 2 3  k  3 2 (1 + x2)n, then : (3) 2 3  k  3 3 (4) 2 2  k  2 3 (1) A = B3 (2) 3A = B 3 (3) B = A (4) A = 3B 2. The distance, of the point (7, –2, 11) from the line x − 6 y − 4 z −8 6. The number of common terms in the progressions = = along the line 1 0 3 4, 9, 14, 19,...... , up to 25th term and 3, 6, 9, 12, x − 5 y −1 z − 5......., up to 37th term is : = = , is : 2 −3 6 (1) 9 (2) 5 (1) 12 (2) 14 (3) 7 (4) 8 (3) 18 (4) 21 7. If the shortest distance of the parabola y2 = 4x from 3. Let x = x(t) and y = y(t) be solutions of the the centre of the circle x2 + y2 – 4x – 16y + 64 = 0 dx is d, then d2 is equal to : differential equations + ax = 0 and dt (1) 16 (2) 24 dy (3) 20 (4) 36 + by = 0 respectively, a, b  R. Given that dt x(0) = 2; y(0) = 1 and 3y(1) = 2x(1), the value of t, 8. If the shortest distance between the lines for which x(t) = y(t), is : x − 4 y +1 z x −  y +1 z − 2 = = and = = is (1) log 2 2 (2) log 4 3 1 2 −3 2 4 −5 3 6 (3) log3 4 (4) log 4 2 , then the sum of all possible values of  is : 3 5 (1) 5 (2) 8 4. If (a, b) be the orthocentre of the triangle whose (3) 7 (4) 10 vertices are (1, 2), (2, 3) and (3, 1), and b b 1 I1 =  x sin ( 4x − x ) dx , I2 =  sin ( 4x − x ) dx 1 9. If  dx = a + b 2 + c 3 , where 2 2 a a 0 3 + x + 1+ x I1 a, b, c are rational numbers, then 2a + 3b – 4c is , then 36 is equal to : I2 equal to : (1) 72 (2) 88 (1) 4 (2) 10 (3) 80 (4) 66 (3) 7 (4) 8 1 10. Let S = {l, 2, 3,... , 10}. Suppose M is the set of all x 2 y2 the subsets of S, then the relation 15. The length of the chord of the ellipse + = 1, 25 16 R = {(A, B): A  B  ; A, B  M} is :  2 (1) symmetric and reflexive only whose mid point is 1,  , is equal to :  5 (2) reflexive only (3) symmetric and transitive only 1691 (1) (4) symmetric only 5 2009 (2) 11. If S = {z  C : |z – i| = |z + i| = |z–1|}, then, n(S) is: 5 (1) 1 (2) 0 (3) 3 (4) 2 1741 (3) 5 12. Four distinct points (2k, 3k), (1, 0), (0, 1) and 1541 (0, 0) lie on a circle for k equal to : (4) 5 2 3 5 1 (1) (2) (3) (4) 13 13 13 13 16. The portion of the line 4x + 5y = 20 in the first quadrant is trisected by the lines L1 and L2 passing 13. Consider the function. through the origin. The tangent of an angle  a ( 7x − 12 − x 2 ) between the lines L1 and L2 is :  , x3  b x − 7x + 12 2 8 25  (1) (2) sin ( x − 3) 5 41  f (x) =  2 x −[x ] , x 3 2 30  (3) (4) b , x =3 5 41     17. ( ) Let a = i + 2 j + k , b = 3 i − j + k. Let c be the Where [x] denotes the greatest integer less than or vector such that a  c = b and a  c = 3. Then equal to x. If S denotes the set of all ordered pairs (a, b) such that f(x) is continuous at x = 3, then the (( ) ) a  c  b − b − c is equal to : number of elements in S is : (1) 32 (2) 24 (1) 2 (2) Infinitely many (3) 20 (4) 36 (3) 4 (4) 1 14. Let a1, a2, ….. a10 be 10 observations such that 1+ 1+ x4 − 2 18. If a = lim and 10 x →0 x4 a k =1 k = 50 and a k  j k  a j = 1100. Then the sin 2 x b = lim , then the value of ab3 is : standard deviation of a1, a2,.., a10 is equal to : x →0 2 − 1 + cos x (1) 5 (2) 5 (1) 36 (2) 32 (3) 25 (4) 30 (3) 10 (4) 115 2  cos x − sin x 0 23. If the solution of the differential equation 19.  Consider the matrix f (x) = sin x cos x 0.    (2x + 3y – 2) dx + (4x + 6y – 7) dy = 0, y(0) = 3, is  0 0 1  x + y + 3 loge |2x + 3y – | = 6, then  + 2 + 3 Given below are two statements : is equal to _____. Statement I: f(–x) is the inverse of the matrix f(x). Statement II: f(x) f(y) = f(x + y). 24. Let the area of the region {(x, y) : x – 2y + 4  0, In the light of the above statements, choose the m correct answer from the options given below x + 2y2  0, x + 4y2  8, y  0} be , where m n (1) Statement I is false but Statement II is true and n are coprime numbers. Then m + n is equal to (2) Both Statement I and Statement II are false _____. (3) Statement I is true but Statement II is false (4) Both Statement I and Statement II are true 25. If 1 1 1 8 = 3+ ( 3 + p ) + 2 ( 3 + 2p ) + 3 ( 3 + 3p ) +.... , 4 4 4 20. The function f : N – {1} → N; defined by f(n) = then the value of p is _____. the highest prime factor of n, is : (1) both one-one and onto 26. A fair die is tossed repeatedly until a six is (2) one-one only obtained. Let X denote the number of tosses (3) onto only required and let a = P(X = 3), b = P(X  3) and c = (4) neither one-one nor onto b+c P(X  6 |X > 3). Then is equal to _____. a SECTION-B 21. The least positive integral value of , for which the 27. Let the set of all a  R such that the equation angle between the vectors i − 2 j + 2k and cos 2x + a sin x = 2a − 7 has a solution be [p, q] i + 2 j − 2k is acute, is _____. 1 and r = tan 9 − tan 27 − + tan 81 , then cot 63 pqr is equal to _____. 22. Let for a differentiable function f : (0, ) → R , x f (x) − f (y)  log e   + x − y ,  x, y  (0, ). Let f (x) = x + x f '(1) + xf ''(2) + f '''(3) , x  R. 3 2 y 28. 20  1  Then f '(10) is equal to _____. Then  n =1 f '  2  is equal to _____. n  3  2 0 1 30. If  satisfies the equation x2 + x + 1 = 0 and 29.   Let A = 1 1 0 , B = [B1, B2, B3], where B1,   (1 + )7 = A + B + C2, A, B, C  0, then 1 0 1  5(3A – 2B – C) is equal to _____. 1    B2, B3 are column matrices, and AB1 = 0 ,   0  2 3 AB2 =  3  , AB3 =  2     0  1  If  = |B| and  is the sum of all the diagonal elements of B, then 3 + 3 is equal to _____. 4  FINAL JEE–MAIN EXAMINATION – JANUARY, 2024 (Held On Saturday 27th January, 2024) TIME : 9 : 00 AM to 12 : 00 NOON PHYSICS TEST PAPER WITH ANSWER SECTION-A 34. A proton moving with a constant velocity passes 31. Position of an ant (S in metres) moving in Y-Z through a region of space without any change in its plane is given by S = 2t 2 ˆj + 5kˆ (where t is in velocity. If E and B represent the electric and second). The magnitude and direction of velocity magnetic fields respectively, then the region of of the ant at t = 1 s will be : space may have : (1) 16 m/s in y-direction (A) E = 0, B = 0 (B) E = 0, B  0 (2) 4 m/s in x-direction (3) 9 m/s in z-direction (C ) E  0, B = 0 (D) E  0, B  0 (4) 4 m/s in y-direction Choose the most appropriate answer from the options given below : 32. Given below are two statements : (1)(A), (B) and (C) only Statement (I) :Viscosity of gases is greater than (2) (A), (C) and (D) only that of liquids. Statement (II) : Surface tension of a liquid (3) (A), (B) and (D) only decreases due to the presence of insoluble (4) (B), (C) and (D) only impurities. In the light of the above statements, choose the 35. The acceleration due to gravity on the surface of most appropriate answer from the options given earth is g. If the diameter of earth reduces to half of below : its original value and mass remains constant, then (1) Statement I is correct but statement II is acceleration due to gravity on the surface of earth incorrect (2) Statement I is incorrect but Statement II is would be : correct (1) g/4 (2) 2g (3) Both Statement I and Statement II are incorrect (3) g/2 (4) 4g (4) Both Statement I and Statement II are correct 36. A train is moving with a speed of 12 m/s on rails 33. If the refractive index of the material of a prism is which are 1.5 m apart. To negotiate a curve radius A cot   , where A is the angle of prism then the 400 m, the height by which the outer rail should be 2 angle of minimum deviation will be raised with respect to the inner rail is  (Given, g = 10 m/s2) : (1)  − 2A (2) − 2A 2 (1) 6.0 cm (2) 5.4 cm  (3) 4.8 cm (4) 4.2 cm (3)  − A (4) −A 2 5 37. Which of the following circuits is reverse - biased ? 42. A rectangular loop of length 2.5 m and width 2 m is placed at 60° to a magnetic field of 4 T. The loop is removed from the field in 10 sec. The average emf induced in the loop during this time is (1) – 2V (2) + 2V (1) (2) (3) + 1V (4) – 1V 43. An electric charge 10–6C is placed at origin (0, 0) m of X –Y co-ordinate system. Two points (3) (4) P and Q are situated at ( 3, 3)m and ( 6,0)m respectively. The potential difference between the 38. Identify the physical quantity that cannot be points P and Q will be : measured using spherometer : (1) 3V (1) Radius of curvature of concave surface (2) 6V (2) Specific rotation of liquids (3) 0 V (3) Thickness of thin plates (4) 3 V (4) Radius of curvature of convex surface 44. A convex lens of focal length 40 cm forms an 39. Two bodies of mass 4 g and 25 g are moving with image of an extended source of light on a photo- equal kinetic energies. The ratio of magnitude of electric cell. A current I is produced. The lens is their linear momentum is : replaced by another convex lens having the same (1) 3 : 5 (2) 5 : 4 diameter but focal length 20 cm. The photoelectric (3) 2 : 5 (4) 4 : 5 current now is : I 40. 0.08 kg air is heated at constant volume through (1) (2) 4 I 2 5°C. The specific heat of air at constant volume is (3) 2 I (4) I 0.17 kcal/kg°C and J = 4.18 joule/cal. The change in its internal energy is approximately. 45. A body of mass 1000 kg is moving horizontally (1) 318 J (2) 298 J with a velocity 6 m/s. If 200 kg extra mass is (3) 284 J (4) 142 J added, the final velocity (in m/s) is: (1) 6 (2) 2 41. The radius of third stationary orbit of electron for (3) 3 (4) 5 Bohr's atom is R. The radius of fourth stationary orbit will be: 4 46. A plane electromagnetic wave propagating in (1) R x-direction is described by 3 16 Ey = (200 Vm–1) sin[1.5 × 107t – 0.05 x] ; (2) R 9 The intensity of the wave is : 3 (Use 0 = 8.85 × 10–12 C2N–1m–2) (3) R 4 (1) 35.4 Wm–2 9 (2) 53.1 Wm–2 (4) R 16 (3) 26.6 Wm–2 (4) 106.2 Wm–2 6  SECTION-B 47. Given below are two statements : 51. A particle starts from origin at t = 0 with a velocity Statement (I) : Planck's constant and angular 5iˆ m / s and moves in x-y plane under action of a momentum have same dimensions. force which produces a constant acceleration of Statement (II) : Linear momentum and moment of (3iˆ + 2ˆj)m / s2. If the x-coordinate of the particle force have same dimensions. at that instant is 84 m, then the speed of the particle In the light of the above statements, choose the at this time is  m / s. The value of  is ______. correct answer from the options given below : (1) Statement I is true but Statement II is false 52. A thin metallic wire having cross sectional area of (2) Both Statement I and Statement II are false 10–4 m2 is used to make a ring of radius 30 cm. A (3) Both Statement I and Statement II are true positive charge of 2 C is uniformly distributed (4) Statement I is false but Statement II is true over the ring, while another positive charge of 30 pC is kept at the centre of the ring. The tension 48. A wire of length 10 cm and radius 7  10 –4 m in the ring is ______ N ; provided that the ring connected across the right gap of a meter bridge. does not get deformed (neglect the influence of When a resistance of 4.5  is connected on the left gap 1 gravity). (given, = 9  10 9 SI units) by using a resistance box, the balance length is found 4  0 to be at 60 cm from the left end. If the resistivity of the wire is R × 10–7m, then value of R is : 53. Two coils have mutual inductance 0.002 H. (1) 63 (2) 70 The current changes in the first coil according to (3) 66 (4) 35 the relation i = i0 sin t, where i0 = 5A and  = 50 rad/s. The maximum value of emf in the  49. A wire of resistance R and length L is cut into second coil is V. The value of  is ____.  5 equal parts. If these parts are joined parallely, then resultant resistance will be : 8 1 54. Two immiscible liquids of refractive indices (1) R 5 25 3 1 and respectively are put in a beaker as shown in (2) R 2 5 the figure. The height of each column is 6 cm. A (3) 25 R coin is placed at the bottom of the beaker. For near (4) 5 R normal vision, the apparent depth of the coin is  cm. The value of 𝛼 is______. 50. The average kinetic energy of a monatomic 4 molecule is 0.414 eV at temperature : (Use KB = 1.38 × 10–23 J/mol-K) (1) 3000 K (2) 3200 K (3) 1600 K (4) 1500 K 7 55. In a nuclear fission process, a high mass nuclide 59. The charge accumulated on the capacitor (A  236) with binding energy 7.6 MeV/Nucleon connected in the following circuit is ____ C dissociated into middle mass nuclides (A  118), (Given C = 150 F) having binding energy of 8.6 MeV/Nucleon. The energy released in the process would be ____ MeV. 56. Four particles each of mass 1 kg are placed at four corners of a square of side 2 m. Moment of inertia of system about an axis perpendicular to its plane and passing through one of its vertex is _____ kgm2. 60. If average depth of an ocean is 4000 m and the bulk modulus of water is 2 × 109 Nm–2, then V fractional compression of water at the bottom 57. A particle executes simple harmonic motion with V an amplitude of 4 cm. At the mean position, of ocean is  × 10–2. The value of  is _____ velocity of the particle is 10 cm/s. The distance of the particle from the mean position when its speed (Given, g = 10 ms–2,  = 1000 kg m–3) becomes 5 cm/s is  cm, where  = ______. 58. Two long, straight wires carry equal currents in opposite directions as shown in figure. The separation between the wires is 5.0 cm. The magnitude of the magnetic field at a point P midway between the wires is ____ T (Given : 0= 4× 10–7 TmA–1) 8  FINAL JEE–MAIN EXAMINATION – JANUARY, 2024 (Held On Saturday 27th January, 2024) TIME : 9 : 00 AM to 12 : 00 NOON CHEMISTRY TEST PAPER WITH ANSWER SECTION-A 66. Which of the following has highly acidic hydrogen? 61. Two nucleotides are joined together by a linkage known as : (1) Phosphodiester linkage (1) (2) Glycosidic linkage (3) Disulphide linkage (4) Peptide linkage (2) 62. Highest enol content will be shown by : (3) (1) (2) (4) (3) (4) 67. A solution of two miscible liquids showing 63. Element not showing variable oxidation state is : (1) Bromine (2) Iodine negative deviation from Raoult's law will have : (3) Chlorine (4) Fluorine (1) increased vapour pressure, increased boiling point (2) increased vapour pressure, decreased boiling point 64. Which of the following is strongest Bronsted base? (3) decreased vapour pressure, decreased boiling point (4) decreased vapour pressure, increased boiling point (1) (2) 68. Consider the following complex ions P = [FeF6]3– Q = [V(H2O) 6]2+ (3) (4) R = [Fe(H2O)6] 2+ The correct order of the complex ions, 65. Which of the following electronic configuration according to their spin only magnetic moment would be associated with the highest magnetic values (in B.M.) is : moment ? (1) R < Q < P (2) R < P < Q (1) [Ar] 3d7 (2) [Ar] 3d8 (3) Q < R < P (4) Q < P < R (3) [Ar] 3d3 (4) [Ar] 3d6 9 69. Choose the polar molecule from the following : 72. The ascending order of acidity of –OH group in the (1) CCl4 (2) CO2 following compounds is : (A) Bu – OH (3) CH2 = CH2 (4) CHC13 (B) 70. Given below are two statements : Statement (I) : The 4f and 5f - series of elements (C) are placed separately in the Periodic table top reserve the principle of classification. (D) Statement (II) : S-block elements can be found in pure form in nature. In the light of the above (E) statements, choose the most appropriate answer from the options given below : Choose the correct answer from the options given below : (1) Statement I is false but Statement II is true (1) (A) < (D) < (C) < (B) < (E) (2) Both Statement I and Statement II are true (2) (C) < (A) < (D) < (B) < (E) (3) Statement 1 is true but Statement II is false (3) (C) < (D) < (B) < (A) < (E) (4) Both Statement 1 and Statement II are false (4) (A) < (C) < (D) < (B) < (E) 73. Given below are two statements : one is labelled as 71. Given below are two statements : Assertion (A) and the other is labelled as Reason (R) Statement (I) : p-nitrophenol is more acidic than Assertion (A) : Melting point of Boron (2453 K) m-nitrophenol and o-nitrophenol. is unusually high in group 13 elements. Reason (R) : Solid Boron has very strong Statement (II) : Ethanol will give immediate crystalline lattice. turbidity with Lucas reagent. In the light of the above statements, choose the In the light of the above statements, choose the most appropriate answer from the options given correct answer from the options given below : below ; (1) Both (A) and (R) are correct but (R) Is not (1) Statement I is true but Statement II is false the correct explanation of (A) (2) Both Statement I and Statement II are true (2) Both (A) and (R) are correct and (R) is the (3) Both Statement I and Statement II are false correct explanation of (A) (3) (A) is true but (R) is false (4) Statement I is false but Statement II is true (4) (A) is false but (R) is true 10  77. IUPAC name of following compound (P) is : 74. Cyclohexene is _________ type of an organic compound. (1) Benzenoid aromatic (2) Benzenoid non-aromatic (1) l-Ethyl-5, 5-dimethylcyclohexane (3) Acyclic (2) 3-Ethyl-1,1-dimethylcyclohexane (4) Alicyclic (3) l-Ethyl-3, 3-dimethylcyclohexane (4) l,l-Dimethyl-3-ethylcyclohexane 75. Yellow compound of lead chromate gets dissolved on treatment with hot NaOH solution. The product 78. NaCl reacts with conc. H2SO4 and K2Cr2O7 to give of lead formed is a : reddish fumes (B), which react with NaOH to give (1) Tetraanionic complex with coordination yellow solution (C). (B) and (C) respectively are ; number six (1) CrO2Cl2, Na2CrO4 (2) Na2CrO4, CrO2Cl2 (2) Neutral complex with coordination number (3) CrO2Cl2, KHSO4 (4) CrO2Cl2, Na2Cr2O7 four (3) Dianionic complex with coordination number 79. The correct statement regarding nucleophilic six substitution reaction in a chiral alkyl halide is ; (4) Dianionic complex with coordination number (1) Retention occurs in SNl reaction and inversion four occurs in SN2 reaction. (2) Racemisation occurs in SNl reaction and 76. Given below are two statements : retention occurs in SN2 reaction. Statement (I) : Aqueous solution of ammonium (3) Racemisation occurs in both SN1 and SN2 carbonate is basic. reactions. Statement (II) : Acidic/basic nature of salt (4) Racemisation occurs in SN1 reaction and solution of a salt of weak acid and weak base inversion occurs in SN2 reaction. depends on Ka and Kb value of acid and the base forming it. 80. The electronic configuration for Neodymium is : In the light of the above statements, choose the most [Atomic Number for Neodymium 60] appropriate answer from the options given below : (1) [Xe] 4f4 6s2 (2) [Xe] 5f47s2 (1) Both Statement I and Statement II are correct (3) [Xe] 4f6 6s2 (4) [Xe] 4f15d16s2 (2) Statement I is correct but Statement II is incorrect SECTION-B (3) Both Statement 1 and Statement II are incorrect 81. The mass of silver (Molar mass of Ag : 108 gmol–1 (4) Statement I is incorrect but Statement II is displaced by a quantity of electricity which correct displaces 5600 mL of O2 at S.T.P. will be _____ g. 11 82. Consider the following data for the given reaction 86. Among the given organic compounds, the total number of aromatic compounds is 2HI(g)→ H2(g) + I2(g) 1 2 3 (A) (B) –1 HI (mol L ) 0.005 0.01 0.02 –1 –4 –3 Rate (mol L s–1) 7.5 × 10 3.0 ×10 1.2×10–2 (C) (D) The order of the reaction is _________. 83. Mass of methane required to produce 22 g of CO2 87. Among the following, total number of meta after complete combustion is ______g. directing functional groups is (Integer based) (Given Molar mass in g mol–1 C = 12.0 – OCH3, –NO2, –CN, –CH3 –NHCOCH3, – COR, –OH, – COOH, –Cl H = 1.0 O = 16.0) 88. The number of electrons present in all the 1 completely filled sub shells having n=4 and s = + 2 84. If three moles of an ideal gas at 300 K expand is ______. isothermally from 30 dm3 to 45 dm3 against a (Where n = principal quantum number ands = spin constant opposing pressure of 80 kPa, then the quantum number) amount of heat transferred is_______ J. 89. Sum of bond order of CO and NO+ is _______. 85. 3-Methylhex-2-ene on reaction with HBr in 90. From the given list, the number of compounds with presence of peroxide forms an addition product + 4 oxidation state of Sulphur ________. (A). The number of possible stereoisomers for 'A' SO3, H2SO3, SOCl2, SF4, BaSO4, H2S2O7 is_________. 12