FIITJEE JEE (Main) 2024 Past Paper PDF

FIITJEE ALL INDIA TEST SERIES FULL TEST – I JEE (Main)-2024 TEST DATE: 28-12-2023 Time Allotted: 3 Hours...

FIITJEE ALL INDIA TEST SERIES FULL TEST – I JEE (Main)-2024 TEST DATE: 28-12-2023 Time Allotted: 3 Hours Maximum Marks: 300 General Instructions:  The test consists of total 90 questions.  Each subject (PCM) has 30 questions.  This question paper contains Three Parts.  Part-A is Physics, Part-B is Chemistry and Part-C is Mathematics.  Each part has only two sections: Section-A and Section-B.  Section – A : Attempt all questions.  Section – B : Do any five questions out of 10 questions. Section-A (01 – 20, 31 – 50, 61 – 80) contains 60 multiple choice questions which have only one correct answer. Each question carries +4 marks for correct answer and –1 mark for wrong answer. Section-B (21 – 30, 51 – 60, 81 – 90) contains 30 Numerical based questions. The answer to each question is rounded off to the nearest integer value. Each question carries +4 marks for correct answer and –1 mark for wrong answer. FIITJEE Ltd., FIITJEE House, 29-A, Kalu Sarai, Sarvapriya Vihar, New Delhi -110016, Ph 46106000, 26569493, Fax 26513942 website: www.fiitjee.com AITS-FT-I-PCM-JEE(Main)/2024 2 Physics PART – A SECTION – A (One Options Correct Type) This section contains 20 multiple choice questions. Each question has four choices (A), (B), (C) and (D), out of which ONLY ONE option is correct. 1. Two bodies A and B each of mass M are fixed F together by a massless spring. A force F acts on the A B mass B as shown in figure. At the instant shown the mass A has acceleration a. What is the acceleration of mass B? (A) (F/M) – a (B) a (C) –a (D) (F/M) 2. Potential energy of a body of mass ‘m’ in a conservative force field can be expressed U = x – y as where x and y are position coordinates of the body. Acceleration of the body can be expressed as: 1/ 2  2  2   2  2  (A) (B)  2  m  m     (C) (D) m m 3. A small ball falling vertically downward with constant velocity 4 m/s strikes elastically a massive inclined cart moving with 4 m/s velocity 4 m/s horizontally as shown. The speed of the rebound of the ball is (A) 4 2 m/s (B) 4 3 m/s (C) 4 m/s (D) 4 5 m/s 4 m/s 45 4. A cylindrical drum is pushed along by a board of  length l. The drum rolls forward on the ground a   distance of. There is no slipping at any instant. 2 2 During the process of pushing the board, the distance moved by the man on the ground is:  3 (A) (B) 2 4 (C)  (D) none of these 5. If resistance of each wire in the network shown is r, the equivalent resistance between A & C is equal to E D r C (A) r (B) 2 2r 3r A B (C) (D) 3 2 FIITJEE Ltd., FIITJEE House, 29-A, Kalu Sarai, Sarvapriya Vihar, New Delhi -110016, Ph 46106000, 26569493, Fax 26513942 website: www.fiitjee.com  3 AITS-FT-I-PCM-JEE(Main)/2024 6. A conducting rod AB of length l = 1m is moving at a velocity v =       4m/s making an angle 300 with its length. C is the middle point of the rod. A uniform magnetic field B = 2T exists in a direction    300   A C B perpendicular to the plane of motion, then       (A) VA –VB = 8 V (B) VA –VB = 4 V (C) VB –VC = 2 V (D) VB –VC = 4 V 7. Two particles A and B are projected simultaneously in the directions vB= 10 3 m/s shown in figure with velocities v A  25 m/s and v B  10 3 m/s. If vA=25m/s they collide in air after 2s, the angle  is (A) 30° (B) 45°  60° (C) 53° (D) 37° x 8. Three capillaries of length L, L/2 and L/3 are connected in series. Their radii are r, r/2 and r/3 respectively. Then if stream-line flow is to be maintained and the pressure across the first capillary is P, then (A) the pressure difference across the ends of second capillary is 8P (B) the pressure difference across the third capillary is 81P (C) the pressure difference across the ends of the second capillary is 16P (D) the pressure difference across the third capillary is 9P 9. An electron in hypothetical hydrogen atom is in its 3rd excited state and makes transition from 3rd to 2nd excited, then to 1st excited state and then to ground state. If the amount of time spent by the  1  electron in any state of quantum number n, is proportional to   , then the ratio of no. of  n 1  revolutions completed by the electron in 1st excited state to that in the 2nd excited state will be 27 (A) 2 (B) 8 27 27 (C) (D) 4 6 10. A ray of light passes through four transparent media with D refractive indices 1, 2, 3 and 4 as shown in the figure. The 1 2 surface of all media are parallel. If the emergent ray CD is parallel 3 to the incident ray AB, then B 4 C (A) 1 = 2 (B) 2 = 3 (C) 3 = 4 (D) 4 = 1 A 1 1 11. The mass of the moon is of the earth but the gravitational pull is of the earth. It is due to 81 6 the fact that 81 9 (A) The radius of the moon is of the earth (B) The radius of the earth is of the moon 6 6 (C) Moon is the satellite of the earth (D) None of these 12. The potential energy of a particle of mass 1 kg in motion along the x-axis is given by U  4(1  cos 2 x) J. Here x is in metres. The period of small oscillations (in sec) is (A) 2 (B)   (C) (D) 2x 2 FIITJEE Ltd., FIITJEE House, 29-A, Kalu Sarai, Sarvapriya Vihar, New Delhi -110016, Ph 46106000, 26569493, Fax 26513942 website: www.fiitjee.com AITS-FT-I-PCM-JEE(Main)/2024 4 1 1 13. For of the distance between two stations a train is uniformly accelerated and of the m n distance it is uniformly retarded. It starts from rest at one station and comes to rest at the other. The ratio of the greatest velocity of the average velocity will be:  1 1  1 1 (A)  1    (B)  1     m n   m n 1 1   1 1 (C)    1 (D)    m n  m n 14. A particle starts from rest. Its acceleration (a) versus time (t) a(m / s2 ) is as shown in the figure. The maximum speed of the particle 10 will be (A) 110 m/s (B) 55 m/s (C) 550 m/s (D) 660 m/s t(s) 15. Two particles of mass ‘m’ each are tied at the ends of a light F string of length ‘2a’. The whole system is kept on frictionless horizontal surface with the string held tight so that each mass is at a distance ‘a’ from the centre P (as shown in figure). a a Now, the mid-point of the string is pulled vertically upwards with a small but constant force F. As a result, the particles move towards each other on the surface. The magnitude of acceleration, when the separation between them becomes 2x is: F a F x (A) (B) 2m a 2  x 2 2m a 2  x 2 F x F a2  x 2 (C) (D) 2m a 2m x 16. A charged ball hangs from a silk thread of length l. It makes an angle  with a + large charged conducting sheet P as shown in the figure. The surface charge + S density  of the sheet is proportional to + +  (A) cos  (B) cot  + (C) sin  (D) tan  + 17. What is the ratio of the circumference of the first Bohr orbit for the electron in the hydrogen atom to the de-Broglie wavelength of electrons having the same velocity as the electron in the first Bohr orbit of the hydrogen atom? (A) 1 : 1 (B) 1 : 2 (C) 1 : 4 (D) 2 : 1 18. A glass prism of refractive index 1.5 is immersed in water ( = 4/3). A light beam incident normally on the face AB is totally reflected to B A reach the face BC, if  8 2 (A) sin   (B) sin   9 3 4  8 (C) sin   (D)  sin   C 5 3 9 FIITJEE Ltd., FIITJEE House, 29-A, Kalu Sarai, Sarvapriya Vihar, New Delhi -110016, Ph 46106000, 26569493, Fax 26513942 website: www.fiitjee.com  5 AITS-FT-I-PCM-JEE(Main)/2024 19. There is a uniform magnetic field perpendicular to        the plane of paper. A positive charged particle enters in the region, perpendicularly and collides   Q O N    M inelastically at point Q to the rigid wall MN. If Co- efficient of restitution e = 1/2 then particle   a/2  a/2     (A) Retraces its path a (B) Strikes at point O        (C) Strikes at point M        (D) strikes at point N 20. A tuning fork of frequency 340 Hz is vibrated just above a cylindrical tube of length 120 cm. Water is slowly poured in the tube. If the speed of sound is 340 m/s then the minimum height of water required for resonance is (A) 25 cm (B) 45 cm (C) 75 cm (D) 95 cm SECTION – B (Numerical Answer Type) This section contains 10 Numerical based questions. The answer to each question is rounded off to the nearest integer value. 21. Three parallel conducting plates (A, B, C) are given initial charges A C B Q, 2Q, 4Q respectively. If the switch S is closed, the charge flowing to earth is xQ. Value of x is S 22. In the figure shown the four rods have  = 0.5 /m resistance per unit  length. The arrangement is kept in a magnetic field of constant magnitude B B = 2T and directed perpendicular to the plane of the figure and directed v inwards. Initially the rods form a square of side length l = 15 m as shown. l vvv Now each wire starts moving with constant velocity v = 5 m/s towards opposite wire. find the force required in newton on each wire to keep its velocity constant at t = 1. l 23. In the circuit shown, the value of L is 5 henry and the R L C power factor of the circuit is 0.8. It is also given that the 2 voltage drop across capacitor is times the voltage drop 5 across the inductor. Find impendence (in ohm) of the ~ circuit. 220 sin 314 t 24. Two bodies of same mass tied with an inelastic string of length  lie together. One of them is projected vertically upwards with velocity 6g. The maximum height up to which the centre of mass of system of the two masses rises is k. Find the value of k. FIITJEE Ltd., FIITJEE House, 29-A, Kalu Sarai, Sarvapriya Vihar, New Delhi -110016, Ph 46106000, 26569493, Fax 26513942 website: www.fiitjee.com AITS-FT-I-PCM-JEE(Main)/2024 6 25. A metal rod AB of length 10x has its one end A in ice at 0°C and the other end B in water at 100°C. If a point P on the rod is maintained at 400°C, then it is found that equal amounts of water and ice evaporate and melt per unit time. The latent heat of evaporation of water is 540 cal/g and latent heat of melting of ice is 80 cal/g. If the point P is at a distance of x from the ice end A. The value of  is (Neglect any heat loss to the surroundings.) 26. A dielectric disc has uniform charge density  only in shaded portion. Now the electric field 4R1 16R1 intensity at point P is equal to ………. (If   200 SI unit, d  , R2  ) 3 9 27. A particle starts from rest at the origin and moves along the x-axis under the action of a force F = (12 – 2t) Newton. Find the time after which it will be at the origin again. 28. An -particle is accelerated by a potential difference of 104V. Find the change in its direction (in degree) of motion as angle measured in degree if it enters normally in a region of thickness 0.1 m having transverse magnetic induction of 0.1 T -27 [given mass of -particle = 6.4  10 kg]. 29. A glass rod of diameter d1 = 1.5 mm is inserted symmetrically into a glass capillary of inside diameter d2 = 2 mm. Then the whole arrangement is vertically oriented and brought in contact with the surface of water. To what height (in mm) will the water rise in the capillary? (Density of water = 1 gm/cc, surface tension = 0.07 N/m, angle of contact = 00, g = 10 m/s2) 30. In the given AC circuit, when switch S is at position 1, the  source emf leads current by. Now, if the switch is at 6  position 2, then source emf leads current by. Find the x value of x. FIITJEE Ltd., FIITJEE House, 29-A, Kalu Sarai, Sarvapriya Vihar, New Delhi -110016, Ph 46106000, 26569493, Fax 26513942 website: www.fiitjee.com  7 AITS-FT-I-PCM-JEE(Main)/2024 Chemistry PART – B SECTION – A (One Options Correct Type) This section contains 20 multiple choice questions. Each question has four choices (A), (B), (C) and (D), out of which ONLY ONE option is correct. 31. The structure of Q in the following reaction scheme is CHO HO H H OH PhNHNH  excess  PhNHNH  excess  H ,Ni 2  2 N  2 Q  P  H OH CH2 OH M(enantiopure) (A) CHO (B) CH2 OH HO H H OH H OH H OH H OH H OH CH2 OH CH2 OH (C) CH2 OH (D) CH2 OH H OH HO H HO H H OH H OH HO H CH2 OH CH2 OH 32. The magnetic moment (in units of BM) of copper in [Cu(H2O)4]2+ and [Cu(NH3)4]2+ respectively is (A) 1.73 and 0 (B) 1.73 and 1.73 (C) 2.83 and 2.83 (D) 0 and 2.83 33. The transition metal (M) complex that can have all isomers (geometric, linkage, and ionization) is (A) [M(NH3)4Br2]SCN (B) [M(NH3)4Cl2]Br (C) [M(NH3)4(H2O)2]Cl3 (D) [M(NH3)4 (H2O)2] (SCN) 34. The products X and Y in the following reaction sequence, respectively, are NH4 Cl in C6H5 Cl NaBH4 BCl3 150 ºC X  Y (A) B3N3Cl6 and B3N3H6 (B) B3N3H3Cl3 and B3N3H6 (C) B3N3H3Cl3 and B3N3H12 (D) B3N3H9Cl3 and B3N3H12 FIITJEE Ltd., FIITJEE House, 29-A, Kalu Sarai, Sarvapriya Vihar, New Delhi -110016, Ph 46106000, 26569493, Fax 26513942 website: www.fiitjee.com AITS-FT-I-PCM-JEE(Main)/2024 8 35. Following is the reaction flow chart for manganese oxidocomplexes under different alkaline pH conditions. Compounds (S) and (T) respectively are (A) S = MnO (OH)2; T = Mn(OH)2 (B) S = MnO2; T = MnO(OH) 2– 2– (C) S = MnO4 ; T = MnO(OH) (D) T = MnO4 ; S = MnO2 36. Number of different types of dipeptides produced using a mixture of glycine and L-valine, and number of optically active dipeptides formed in this mixture will be (A) Four dipeptides, all optically active (B) Two dipeptides, all optically active (C) Four dipeptides, three optically active (D) Two dipeptides, none optically active 37. Latimer diagram are the compact representations of electrochemical equilibria in substances of multiple oxidation states. The value of the potential, x, in the Latimer diagram of gold (at pH = 1.0) is x 1.83 Au Au3  Au 1.517 (A) 2.72 V (B) 3.18 V (C) –3.18 (D) 1.36 V 38. An organic compound on reaction with 2,4-dinitrophenylhydrazine (2,4-DNP) gives a yellow precipitate. It also gives silver mirror on reaction with ammoniacal AgNO3. It gives an alcohol and sodium salt of a carboxylic acid on reaction with concentrated NaOH. It yields benzene-1,2- dicarboxylic acid on heating with alkaline KMnO4. The structure of the compound among the following is (A) (B) (C) (D) – + 39. The pH of an aqueous buffer prepared using CH3COOH and CH3COO Na IS 4.80. CH3 COO    CH3 COOH The quantity  is________ CH3COOH (round off to three decimal places [Given: pKa of CH3COOH in water is 4.75] 0.05 Given 10 = 1.1220 (A) 0.122 (B) 0.125 (C) 0.120 (D) 0.123 FIITJEE Ltd., FIITJEE House, 29-A, Kalu Sarai, Sarvapriya Vihar, New Delhi -110016, Ph 46106000, 26569493, Fax 26513942 website: www.fiitjee.com  9 AITS-FT-I-PCM-JEE(Main)/2024 40. The molar heat capacity of a substance is represented in the temperature range 298 K to 400 K –1 –1 by the empirical reaction Cp,m = 14 + bT J K mol , where b is a constant. The molar enthalpy –1 change when the substance is heated from 300 K to 350 K is 2 kJ mol. The value of b nearest –2 to_______J K mol (A) 0.08 (B) 0.12 (C) 0.05 (D) 0.10 41. The observed magnetic moment of octahedral Mn3+, Fe3+ and Co3+ complexes are 4.95, 6.06 and 3+ 3+ 3+ 0.00 BM, respectively. The correct option for the electronic configuration of Mn , Fe and Co metal ion in these complexes, respectively, is 4 (A) t 2g eg0 ,t 32g eg2 and t 2g 4 eg2 (B) t 32g e1g ,t 2g 5 e0g and t 62g eg0 (C) t 32g e1g ,t 2g 3 eg2 and t 62g eg0 (D) t 22g e1g ,t 2g 3 eg2 and t 2g 4 eg2 42. The complementary strand for the following single strand of DNA is (A) (B) (C) (D) 43. The correct order of pKa for the following compounds is (A) II > I > III > IV (B) II > I > IV > III (C) III > IV > I > II (D) IV > II > I > III 44. The rate constant values for the decay of radioisotopes X and Y, used in ratio-medicine are 0.05 h–1 and 0.025 h–1 respectively. In a hospital, at a time ‘t0’ the activity of a sample of X was found to be twice that of Y. The activities of the two radioisotopes will be approximately equal when the time elapsed is: (A) twice the half-life of Y (B) twice and half-life of X (C) equal to the half-life of X (D) equal to 1/2 the half-life of Y 45. Sea water containing 1 M NaCl has to be desalinated at 300 K using a membrane permeable only to water. The minimum pressure (in bars) required on the sea-water side of the membrane is________ (R = 8.3 J mol-1 K-1, 1 bar = 105 N/m2) (A) 24.94 (B) 49.88 (C) 2490 (D) 4980 46. The property of radiation that is not different at various regions of the electromagnetic spectrum is (A) energy (B) frequency (C) velocity (D) wavelength FIITJEE Ltd., FIITJEE House, 29-A, Kalu Sarai, Sarvapriya Vihar, New Delhi -110016, Ph 46106000, 26569493, Fax 26513942 website: www.fiitjee.com AITS-FT-I-PCM-JEE(Main)/2024 10 47. Among the following species, the one that has pentagonal shape is (A) XeOF4 (B) IF5 – ]– (C) [SF5] (D) [XeF5 48. Consider the electrochemical cell M  s  | MI2  s  | MI2  aq | M  s  where ‘M’ is a metal. At 298 K, the standard reduction potential are E0M2  aq /M s  0.12 V, E0MI2  s /M s  0.36 V and the temperature coefficient is  E0cell  4 1    1.5  10 V K. At this temperature the standard enthalpy change for the overall cell  T P reaction r H0 , is_________kJ mol–1 (A) –37.60 (B) 37.60 (C) 3.76 (D) –3.76 49. Consider the following two parallel irreversible first order reactions at temperature T where 1 and 2 are the rate constants and their values are 5 × 10–2 and 15 × 10–2 min–1, respectively, at temperature T. If the initial concentration of the reactant ‘P’ is 4 mol L–1, then the concentration of product ‘R’ after 10 min of reaction is mol L–1. –2 Given e = 0.135 (A) 1.59 (B) 2.59 (C) 3.59 (D) 4.59 50. Gas phase bond length and dipole moment of a compound (MX) is 3 Å and 10.8 D, respectively. The ionic character in gas phase MX is %. (1D = 3.336 × 10 –30 C m) (A) 64.50 (B) 74.50 (C) 84.50 (D) 90.50 SECTION – B (Numerical Answer Type) This section contains 10 Numerical based questions. The answer to each question is rounded off to the nearest integer value. 51. The total number of  and  particles emitted in the following radioactive decay is x. Find the x5 value of. 11 238 210 92 U  82 Pb k2 k1 52. For the elementary reaction C  A  B, k1  2k 2. At time t = 0, [A] = A0 and [B] = [C] = 0. At a later time t, the value of [B]/[C] is_____ (round off to the nearest integer) FIITJEE Ltd., FIITJEE House, 29-A, Kalu Sarai, Sarvapriya Vihar, New Delhi -110016, Ph 46106000, 26569493, Fax 26513942 website: www.fiitjee.com  11 AITS-FT-I-PCM-JEE(Main)/2024 53. For the given general reactions P  g  Q  g  K C  10 Q g  B g KC  2 B g  D g K C  0.01 Calculate the value of KC for the reaction D  g   P  g 54. The number of unpaired electron in [Co(H2O)6]2+ is_________ 55. The volume of water (in mL) required to be added to a 100 mL solution (aq. 0.1 M) of a weak acid (HA) at 25ºC to double its degree of dissociation is x, find the value of x/100 –5 [Given: Ka of HA at 25 ºC = 1.8  10 ] 56. The ratio of the 2p 1s transition energy in He+ to that in the H atom is closest to 57. Total number of paramagnetic species NO, O2, B2 and C2 in their ground state is 58. Half-life (t1/2) of a chemical reaction varies with the initial concentration of reactant (Ao) as given below Ao (mol L–1) 5  10–2 4  10–2 3  10–2 t1/2 360 450 600 The order of the reaction is__________ 59. For the reaction H2PO2  aq  OH  aq  HPO32   aq   H2  g 2 the rate expression is k H2PO2  OH . If the concentration of H2PO2 is doubled, the new rate is_____ old rate 60. On heating a sample of 25 mg hydrated compound (molecular weight = 250 g/mol) in thermogravimetric analysis, 16 mg of dehydrated compound remains. The number of water molecules lost per molecule of hydrated compound is. (Molecular weight of water = 18 g/mol) FIITJEE Ltd., FIITJEE House, 29-A, Kalu Sarai, Sarvapriya Vihar, New Delhi -110016, Ph 46106000, 26569493, Fax 26513942 website: www.fiitjee.com AITS-FT-I-PCM-JEE(Main)/2024 12 Mathematics PART – C SECTION – A (One Options Correct Type) This section contains 20 multiple choice questions. Each question has four choices (A), (B), (C) and (D), out of which ONLY ONE option is correct.   2 3 12    61. The value of sin1 cot  sin 1  cos 1  sec 1 2   is equal to    4 4    (A) 1 (B) 0 (C) –1 (D) 2 tan 3 sin 3 62. If  4, then equals tan  sin  (A) 3/5 (B) 4/5 (C) 3/4 (D) none of these 63. Let A  6,10,14,....,1002 and B is set of divisors of the integer 360, then S(A  B) is [S(A) denotes sum of elements in set A] (A) 148 (B) 154 (C) 156 (D) 162 64. Number of integral values of  for which f(x)  ln(2 cos x  5) is defined for all x  R is – (A) 5 (B) 4 (C) 6 (D) 11 65. The solution of the differential equation {y(1  x 1 )  sin y}dx  (x  ln x  x cos y)dy  0 is – (A) x  y ln x  y sin x  c (B) xy  yln x  x sin y  c (C) y  x ln x  x sin y  c (D) xy  xln x  y sin y  c (where c is arbitrary constant)  1 2  11 4  66. If in triangle ABC, A  (1, 10), circumcentre    ,  and orthocenter   ,  then the co-  3 3  3 3 ordinates of mid-point of side opposite to A is  11  (A)  1,   (B) (1, 5)  3 (C) (1, –3) (D) (1, 6) 67. Consider a circle with centre lying on the focus of the parabola y2 = 2px such that it touches the directix of the parabola. Then a point of intersection of the circle and the parabola is p  p p (A)  , p  (B)  ,   2  2 2  p   p  (C)   , p  (D)   ,  p   2   2  FIITJEE Ltd., FIITJEE House, 29-A, Kalu Sarai, Sarvapriya Vihar, New Delhi -110016, Ph 46106000, 26569493, Fax 26513942 website: www.fiitjee.com  13 AITS-FT-I-PCM-JEE(Main)/2024 68. If the range of values of a such that a circle passes through the points of intersection of y = x 2 and x 2  y2  a2 is (, ) then    equals (A) 0 (B) 1 (C) 2 (D) 3   69. Let   a î  b ĵ  ck̂ and   b î  c ĵ  ak̂ , where a, b, c  R. If ‘’ be the angle between    and  then, (A)   (0, /2) (B)   [0, 2/3] (C)   (2/3,] (D) none of these a x  ax  2 a x  a x  2 1 70. If a, b, c > 0 and x, y, z  R, then the determinant b y  b y  2 b y  b y  2 1 is equal to c z c  z 2 c z c z 2  1 (A) ax + by + cz (B) a-x b–y c-z (C) a2x b2y c2z (D) 0 71. A and B are two events such that P(A) = 0.2 and P(AB) = 0.7. If A and B are independent events then P(B) equals (A) 2/7 (B) 7/9 (C) 3/8 (D) none of these 1024 72. The value of  log r  is equal to, ([.] denotes the greatest integer function) r 1 2 (A) 8192 (B) 8204 (C) 8194 (D) none of these 73. Number of natural numbers < 2.104 which can be formed with the digits 1, 2, 3 only is equal to 35  2.34  3 36  2.34  3 (A) (B) 2 2 37  1 (C) (D) none of these 2 2 3 4 -1 74. The value of i log(x – i) + i  +i log(x +i) + i ( 2 tan x), x> 0 ( where i =  1 ) is (A) 0 (B) 1 (C) 2 (D) 3 75. If 3x2 – 2(a – d) x + (a2 + 2(b2 +c2) + d2) = 2(ab + bc + cd), then (A) a, b, c, d are in G.P. (B) a, b, c, d are in H.P. (C) a, b, c, d are in A.P. (D) None of these 76. If a sinx + bcos(C + x) + bcos (C –x) = , then the minimum value of |cosC| is  2  a2  2  a2 (A) (B) b2 4b 2  2  a2 (C) (D) none of these 16b 2 FIITJEE Ltd., FIITJEE House, 29-A, Kalu Sarai, Sarvapriya Vihar, New Delhi -110016, Ph 46106000, 26569493, Fax 26513942 website: www.fiitjee.com AITS-FT-I-PCM-JEE(Main)/2024 14 77. The solution of the differential equation ydx + xy2 dx = xdy, is x x x2 (A)  x 2   (B)   y y 2 x x2 (C) 2   (D) None of these 2y 4 2 78. The value of   sin x  cos x  dx 0 is equal to where [.] represents GIF  (A) (B)  2 3 (C) (D) 2 2 3 2 79. If 3(a + 2c) = 4(b + 3d) then the equation ax + bx + cx + d = 0 will have (A) no real solution (B) atleast one real root in (–1, 0) (C) atleast one real root in (0, 1) (D) none of these 80. The mean and variance of seven observations are 8 and 16, respectively. If 5 of the observations are 2, 4, 10, 12, 14, then the product of the remaining two observations is : (A) 40 (B) 45 (C) 49 (D) 48 SECTION – B (Numerical Answer Type) This section contains 10 Numerical based questions. The answer to each question is rounded off to the nearest integer value. x 2 1 dt dt 81. If f  x     f  t  2 and   f  t  2   6  3 then 0 0 f(9) is equal to 82. Consider an ellipse, whose centre is at the origin and its major axis is along the x-axis. If its 3 eccentricity is and the distance between it foci is 6, then the area (in sq. units) of the 5 quadrilateral inscribed in the ellipse, with the vertices as the vertices of the ellipse, is :  8 2 2 1 83. If M, N are 3  2, 2  3 matrices such that MN   2 5 4  , then .det  NM  is equal to ___  10  2 4 5  (NM is invertiable and [.] denotes the greatest integer function) 84. Number of positive integers n less than 15, for which n! + (n+1)! + (n+2)! is an integral multiple of 49, is 11  1 85. The sum of coefficients of even powers of x in the expansion of  x   is  x FIITJEE Ltd., FIITJEE House, 29-A, Kalu Sarai, Sarvapriya Vihar, New Delhi -110016, Ph 46106000, 26569493, Fax 26513942 website: www.fiitjee.com  15 AITS-FT-I-PCM-JEE(Main)/2024 86. The number of points of discontinuity of f (x) in [0, 2] where [cos x], x 1 f (x) =  , ([.] denotes the greatest integer function), are | 2x  3 | [x  2], x 1 87. The curve x + y – ln (x + y) = 2x + 5 has a vertical tangent at the point (, ). Then  +  is equal to  x2 6 8   2x 3 5     88. Let A =  3 y2 9  , B =  2 2y 6 . If trace A = trace B then x + y + z is equal to 4 5 z 2   1 4 2z  3   (trace equal to sum of principal diagonal elements in a matrix) 89. The area enclosed by the curve |y| = sin 2x, when x [0, 2] is 1 90. If the length of shortest distance the two lines  x  1  1  y  3   z  2 and 3x – y – 2z + 4 = 0 2 4 = 2x + y + z + 1 is s 14. Then the value of s is _______ FIITJEE Ltd., FIITJEE House, 29-A, Kalu Sarai, Sarvapriya Vihar, New Delhi -110016, Ph 46106000, 26569493, Fax 26513942 website: www.fiitjee.com