Sri Chaitanya IIT Academy Past Paper PDF 10-08-2024

TG ~ @bohring_bot × @HeyitsyashXD Sri Chaitanya IIT Academy.,India.  A.P  T.S  KARNATAKA  TAMILNADU  MAHARASTRA  DELHI  RANCHI A right Choice for the Real Aspirant IC...

TG ~ @bohring_bot × @HeyitsyashXD Sri Chaitanya IIT Academy.,India.  A.P  T.S  KARNATAKA  TAMILNADU  MAHARASTRA  DELHI  RANCHI A right Choice for the Real Aspirant ICON Central Office - Madhapur - Hyderabad SEC: Sr.Super60_NUCLEUS&STERLING_BT JEE-MAIN Date: 10-08-2024 Time: 09.00Am to 12.00Pm RPTM-04/05 Max. Marks: 300 IMPORTANT INSTRUCTION: 1. Immediately fill in the Admission number on this page of the Test Booklet with Blue/Black Ball Point Pen only. 2. The candidates should not write their Admission Number anywhere (except in the specified space) on the Test Booklet/ Answer Sheet. 3. The test is of 3 hours duration. 4. The Test Booklet consists of 90 questions. The maximum marks are 300. 5. There are three parts in the question paper 1,2,3 consisting of Physics, Chemistry and Mathematics having 30 questions in each subject and subject having two sections. (I) Section –I contains 20 multiple choice questions with only one correct option. Marking scheme: +4 for correct answer, 0 if not attempt and -1 in all other cases. (II) Section-II contains 10 Numerical Value Type questions. Attempt any 5 questions only, if more than 5 questions attempted, First 5 attempted questions will be considered. ∎ The Answer should be within 0 to 9999. If the Answer is in Decimal then round off to the nearest Integer value (Example i,e. If answer is above 10 and less than 10.5 round off is 10 and If answer is from 10.5 and less than 11 round off is 11). To cancel any attempted question bubble on the question number box. For example: To cancel attempted question 21. Bubble on 21 as shown belo w. Question Answered for Marking Question Cancelled for Marking Marking scheme: +4 for correct answer, 0 if not attempt and -1 in all other cases. 6. Use Blue / Black Point Pen only for writing particulars / marking responses on the Answer Sheet. Use of pencil is strictly prohibited. 7. No candidate is allowed to carry any textual material, printed or written, bits of papers, mobile phone any electron device etc, except the Identity Card inside the examination hall. 8. Rough work is to be done on the space provided for this purpose in the Test Booklet only. 9. On completion of the test, the candidate must hand over the Answer Sheet to the invigilator on duty in the Hall. However, the candidate are allowed to take away this Test Booklet with them. 10. Do not fold of make any stray marks on the Answer Sheet Name of the Candidate (in Capital): ________________________________________________ Admission Number: Candidate’s Signature:________________ Invigilator’s Signature: ________________ TG ~ @bohring_bot × @HeyitsyashXD SRI CHAITANYA IIT ACADEMY, INDIA 10-08-24_ Sr.Super60_ NUCLEUS & STERLING_BT _Jee-Main_RPTM-04/05_Q.P 10-08-2024_Sr.Super60_ NUCLEUS & STERLING_BT _ Jee-Main_RPTM-04/05_Test Syllabus PHYSICS : Gravitation: Law of gravitation; Gravitational potential and field; Acceleration due to gravity; Kepler’s law, Motion of planets and satellites in circular orbits; Escape velocity, Geostationary orbits (Important for ADVANCED) COMPLETE RPTM-4 SYLLABUS (Except MAINS Deleted Topics) (20Q RPTM4+10Q CUMULATIVE) CHEMISTRY : Alcohols, Phenols , Ethers : Alcohols : Physical properties; Reactions: esterification, dehydration (formation of alkenes and ethers); Reactions with: sodium, phosphorus halides, ZnCl2/concentrated HCl, thionyl chloride; Conversion of alcohols into aldehydes, ketones and carboxylic acids Phenols: Physical properties; Preparation, Electrophilic substitution reactions of phenol (halogenation, nitration, sulphonation); Reimer-Tiemann reaction, Kolbe reaction; Esterification; Etherification; Aspirin synthesis; Oxidation and reduction reactions of phenol. Ethers : Preparation, Properties &Reactions COMPLETE RPTM-4 SYLLABUS (Except MAINS Deleted Topics) (20Q RPTM4+10Q CUMULATIVE) MATHEMATICS : Indefinite Integration COMPLETE RPTM-4 SYLLABUS (Except MAINS Deleted Topics) (20Q RPTM4+10Q CUMULATIVE) Sec: Sr.Super60_ NUCLEUS & STERLING_BT Page 2 TG ~ @bohring_bot × @HeyitsyashXD SRI CHAITANYA IIT ACADEMY, INDIA 10-08-24_ Sr.Super60_ NUCLEUS & STERLING_BT _Jee-Main_RPTM-04/05_Q.P PHYSICS Max Marks: 100 (SINGLE CORRECT ANSWER TYPE) This section contains 20 multiple choice questions. Each question has 4 options (1), (2), (3) and (4) for its answer, out of which ONLY ONE option can be correct. Marking scheme: +4 for correct answer, 0 if not attempted and –1 in all other cases. 1. Match List I with List II List I List II (A) 3 Translational degrees of freedom only (I) Monoatomic gases (B) 3 Translational, 2 rotational degrees of freedoms (II) Rigid polyatomic gases (C) 3 Translational, 2 rotational and 2 (III) Rigid diatomic gases vibrational degrees of freedom (D) 3 Translational, 3 rotational (IV) Non rigid diatomic gases degrees of freedom 1) A-I, B-III, C-IV, D-II 2) A-IV, B-III, C-II, D-I 3) A-IV, B-II, C-I, D-III 4) A-I, B-IV, C-III, D-II 2. If the earth rotates faster than its present speed, the apparent weight of an object will 1) Increase at the equator but remain unchanged at the poles 2) Decrease at the equator but remain unchanged at the poles 3) Remain unchanged at the equator but decrease at the poles 4) Remain unchanged at the equator but increase at the poles 3. The masses and radii of the earth and moon are M1, R1 and M 2 , R2 respectively. Their centres are distance d apart. The minimum velocity with which a particle of mass m should be projected from a point midway between their centres so that it escapes to infinity is G 2G 1) 2  M1  M 2  2) 2  M1  M 2  d d Gm Gm  M1  M 2  3) 2  M1  M 2  4) 2 d d  R1  R2  Sec: Sr.Super60_ NUCLEUS & STERLING_BT Page 3 TG ~ @bohring_bot × @HeyitsyashXD SRI CHAITANYA IIT ACADEMY, INDIA 10-08-24_ Sr.Super60_ NUCLEUS & STERLING_BT _Jee-Main_RPTM-04/05_Q.P 4. If mass of earth is M , radius is R , and gravitational constant is G , then work done to take 1kg mass slowly from earth surface to infinity will be (Neglect rotation of Earth) GM GM 2GM GM 1) 2) 3) 4) 2R R R 2R 5. A body of mass m kg starts falling from a point 2R above the Earth’s surface. Its kinetic energy when it has fallen to a point ' R ' above the Earth’s surface [R-Radius of Earth, M-Mass of Earth, G-Gravitational Constant] 1 GMm 1 GMm 2 GMm 1 GMm 1) 2) 3) 4) 2 R 6 R 3 R 3 R 6. In some region, the gravitational field is zero. The gravitational potential in this region 1) Must be variable 2) Must be constant 3) Cannot be zero 4) Must be zero 7. The figure shows the motion of a planet around the sun in an elliptical orbit with sun at the focus. The shaded areas A and B are also shown in the figure which can be assumed to be equal. If t1 and t 2 represent the time for the planet to move from a to b and d to c respectively, then 1) t1  t2 2) t1  t2 3) t1  t2 4) t1  t2 8. Imagine a light planet revolving around a very massive star in a circular orbit of radius R with a period of revolution T. If the gravitational force of attraction between planet and star 5   is proportional to R  2  then T 2 is proportional to 1) R3 2) R7 2 3) R5 2 4) R3 2 Sec: Sr.Super60_ NUCLEUS & STERLING_BT Page 4 TG ~ @bohring_bot × @HeyitsyashXD SRI CHAITANYA IIT ACADEMY, INDIA 10-08-24_ Sr.Super60_ NUCLEUS & STERLING_BT _Jee-Main_RPTM-04/05_Q.P 9. The magnitudes of the gravitational force on a point mass m at distances r1 and r2 from the centre of a uniform sphere of radius R and mass M are F1 and F2 respectively. Then the correct statement is/are F r Statement A: 1  1 if r1  R and r2  R F2 r2 F1 r12 Statement B:  if r  R and r2  R F2 r22 1 F1 r1 Statement C:  if r1  R and r2  R F2 r2 F1 r22 Statement D:  if r1  R and r2  R F2 r12 1)only A 2)only B 3)both A and B 4) both C and D 10. Assertion: In Young’s double slit experiment, the number of fringes observed in a given field of view is small with longer wavelength of light and is large with shorter wavelength of light. Reason: In the double slit experiment the fringe width is directly proportional to wavelength of light. 1) If both assertion and reason are true and the reason is the correct explanation of the assertion. 2) If both assertion and reason are true but reason is not the correct explanation of the assertion. 3) If assertion is true but reason is false. 4) If the assertion and reason both are false. 11. Two bodies of masses m1 and m2 are initially at rest at infinite distance apart. They are then allowed to move towards each other under mutual gravitational attraction. Their relative velocity of approach at a separation r between them is 12 12  (m  m2 )   2G 1)  2G 1  2)   m1  m2   r  r  12 12 G  G  3)   m1  m2   4)   m1  m2   r   2r  Sec: Sr.Super60_ NUCLEUS & STERLING_BT Page 5 TG ~ @bohring_bot × @HeyitsyashXD SRI CHAITANYA IIT ACADEMY, INDIA 10-08-24_ Sr.Super60_ NUCLEUS & STERLING_BT _Jee-Main_RPTM-04/05_Q.P 12. The diagram showing the variation of gravitational potential of earth of radius R with distance from the centre of earth is 1) 2) 3) 4) 13. A sphere of mass M and radius R2 has a concentric cavity of radius R1 as shown in figure. The force F exerted by the sphere on a particle of mass m located at a distance r from the centre of sphere varies as ( 0  r   ) (mass distribution is uniform) 1) 2) 3) 4) Sec: Sr.Super60_ NUCLEUS & STERLING_BT Page 6 TG ~ @bohring_bot × @HeyitsyashXD SRI CHAITANYA IIT ACADEMY, INDIA 10-08-24_ Sr.Super60_ NUCLEUS & STERLING_BT _Jee-Main_RPTM-04/05_Q.P 14. Assertion (A): Air pressure in a car tyre increases during driving. Reason (R): Absolute zero degree temperature is not zero energy temperature 1) Both A and R are true and R is the correct explanation of A 2) Both A and R true but R is NOT the correct explanation of A 3) A is true but R is false 4) A is false and R is also false 15. Statement A : There is no atmosphere on the surface of moon Statement B : RMS velocity of gas molecules is larger than escape velocity on the surface of moon 1) Both A and B are true. 2) A is true but B is false. 3) A is false but B is true. 4) Both A and B are false 16. The mass density of a planet of radius R varies with the distance r from its centre as  r2   (r )  o  1  2 . Then the gravitational field is maximum at :  R    3 1 5 1) r  R 2) r  R 3) r  R 4) r  R 4 3 9 17. Volume versus temperature graph of two moles of helium gas is as shown in figure. The ratio of heat absorbed and the work done by the gas in process 1-2 is 5 5 7 1) 3 2) 3) 4) 2 3 2 18. N moles of an ideal diatomic gas is in a closed cylinder at temperature T. Suppose on supplying heat to the gas, its temperature remain constant but n moles get dissociated into atoms. Heat supplied to the gas is (Neglect vibrational mode) 1 3 3 1) Zero 2) nRT 3) nRT 4)  N  n  RT 2 2 2 Sec: Sr.Super60_ NUCLEUS & STERLING_BT Page 7 TG ~ @bohring_bot × @HeyitsyashXD SRI CHAITANYA IIT ACADEMY, INDIA 10-08-24_ Sr.Super60_ NUCLEUS & STERLING_BT _Jee-Main_RPTM-04/05_Q.P 19. A double convex lens made of a material of refractive index 1 is placed inside two liquids or refractive indices 2 and 3 as shown. 2  1  3. A wide, parallel beam of light is incident on the lens from the left. The lens will give rise to 1) A single convergent beam 2) Two different convergent beams 3) Two different divergent beams 4) A convergent and a divergent beam 20. A straight rod of length L extends from x  a to x  L  a. The gravitational force it exerts on point mass ‘m’ at x = 0, if the mass per unit length of the rod is A  Bx 2 , is given by   1 1   1 1   1) Gm  A     BL  iˆ 2) Gm  A     BL  iˆ  aL a   a aL    1 1   1 1   3) Gm  A     BL  iˆ 4) Gm  A     BL  iˆ  aL a   a aL  (NUMERICAL VALUE TYPE) Section-II contains 10 Numerical Value Type questions. Attempt any 5 questions only. First 5 attempted questions will be considered if more than 5 questions attempted. The Answer should be within 0 to 9999. If the Answer is in Decimal then round off to the nearest Integer value (Example i,e. If answer is above 10 and less than 10.5 round off is 10 and If answer is from 10.5 and less than 11 round off is 11). Marking scheme: +4 for correct answer, 0 if not attempt and -1 in all other cases. 21. The value of ‘g’ at a particular point near the pole of Earth is 10m s 2. Suppose the earth suddenly shrinks uniformly to half its present radius without losing any mass. The value of ‘g’ at the same point(in m s 2 ) (assuming that the distance of the point from the centre of earth does not change) will now be 22. Assume that the acceleration due to gravity on the surface of the moon is 0.2 times the acceleration due to gravity on the surface of the earth. If Re is the maximum range of a projectile on the earth’s surface, the maximum range on the surface of the moon for the same velocity of projection is nRe. The value of n is (for small velocities) 23. A body is projected vertically upwards from the surface of a planet of radius R with a velocity equal to half the escape velocity for that planet. The maximum height attained by the body above surface is R n. The value of n is Sec: Sr.Super60_ NUCLEUS & STERLING_BT Page 8 TG ~ @bohring_bot × @HeyitsyashXD SRI CHAITANYA IIT ACADEMY, INDIA 10-08-24_ Sr.Super60_ NUCLEUS & STERLING_BT _Jee-Main_RPTM-04/05_Q.P 24. Four particles, each of mass M and equidistant from each other, move along a circle of radius R under the action of their mutual gravitational attraction. The speed of each 1 GM particle is (1  2 n ). The value of n is 2 R 25. A mass M is split into two parts, m and ( M  m ) , which are then separated by a certain distance. What ratio of M m maximizes the gravitational force between the two parts? 26. 2 kg of ice at – 20°C is mixed with 5 kg of water at 20°C in an insulating vessel having a negligible heat capacity. Calculate the final mass of water (in kg) remaining in the container. It is given that the specific heats of water and ice are 1 Kcal/kg / °C and 0.5K cal / kg / C while the latent heat of fusion of ice is 80 Kcal / kg 27. A slab of glass of thickness 6cm and refractive index 1.5 is placed in front of a concave mirror, the faces of the slab being perpendicular to the principal axis of the mirror. If the radius of curvature of the mirror is 40cm and the reflected image coincides with the object, then the distance of the object from the mirror (in cm) is 28. A point object O is placed on the principal axis of a convex lens of focal length 20cm at a distance of 40cm to the left of it. The diameter of the lens is 10cm. If the eye is placed 60cm to the right of the lens at a distance h below the principal axis, then the maximum value of h to see the image will be ‘x’ cm. The value of 10x is 29. In a two slit experiment with monochromatic light fringes are obtained on a screen placed at some distance from the sits. If the screen is moved by 5  102 m towards the slits, the change in fringe width is 3  105 m If separation between the slits is 103 m , the wavelength of light used is (in A ). 30. Two bodies of masses m and 4 m are placed at a distance r. The gravitational potential at a nGm point on the line joining them where the gravitational field is zero is . Find the value of r ‘n’. Sec: Sr.Super60_ NUCLEUS & STERLING_BT Page 9 TG ~ @bohring_bot × @HeyitsyashXD SRI CHAITANYA IIT ACADEMY, INDIA 10-08-24_ Sr.Super60_ NUCLEUS & STERLING_BT _Jee-Main_RPTM-04/05_Q.P CHEMISTRY Max Marks: 100 (SINGLE CORRECT ANSWER TYPE) This section contains 20 multiple choice questions. Each question has 4 options (1), (2), (3) and (4) for its answer, out of which ONLY ONE option can be correct. Marking scheme: +4 for correct answer, 0 if not attempted and –1 in all other cases. 31. Match the following Column I Column II Name reaction Mechanism followed (major pathway only) P Potassium tertiary butoxide ion with 2- chloro 1 Addition elimation mechanism propane Q Potassium tertiry butoxide ion with methyl iodide 2 SN 1 R Potassium hydroxide with para chloro nitro 3 E2 benzene S Solvolysis of tertiary butyl chloride 4 S N Ar 5 SN 2 1) P-3; Q-5 ;R-1; S-2,4 2) P-3; Q-3,4; R-2; S-2 3) P-3; Q-5; R-1; S-1,2 4) P-3; Q-5, R-1,4; S-2 32. The number of products (including stereo isomers) formed are 1) 6 2) 5 3) 4 4) 3 33. Assertion(A) : Nitration of benzene with nitric acid requires the use of concentrated sulphuric acid Reason(R) : The mixture of nitric acid and sulphuric acid produces the electrophile, NO2 1) Both A and R are correct and R is the correct explanation of A. 2) Both A and R are correct but R is not the correct explanation of A 3) Both A and R are not correct. 4) A is not correct but R is correct. Sec: Sr.Super60_ NUCLEUS & STERLING_BT Page 10 TG ~ @bohring_bot × @HeyitsyashXD SRI CHAITANYA IIT ACADEMY, INDIA 10-08-24_ Sr.Super60_ NUCLEUS & STERLING_BT _Jee-Main_RPTM-04/05_Q.P 34. Assertion : Phenol is a weaker acid than ethanol Reason: Sodium phenoxide can’t be prepared by the reaction of phenol with aqueous NaOH 1)Both A and R are correct and R is the correct explanation of A. 2)Both A and R are correct but R is not the correct explanation of A 3) Both A and R are not correct. 4) A is not correct but R is correct. 35. What is the correct order of reactivity of alcohols in the following reaction? ZnCl R  OH  HCl   2 R  Cl  H O 2 1) 10  20  30 2) 10  20  30 3) 30  20  10 4) 30  10  20 36. CH 3CH 2OH can be converted into CH 3CHO by ______. 1) catalytic hydrogenation 2) treatment with LiAlH 4 3) treatment with pyridinium chlorochromate 4) treatment with KMnO4 37. The reaction of propene with HOCl (Cl2  H 2O) proceeds through the intermediate: 1) CH 3  CH   CH 2  OH 2) CH 3  CH   CH 2  Cl 3) CH 3  CH (OH )  CH 2 4) CH 3  CHCl  CH 2 38. Among the following, the conformation that corresponds to the most stable conformation of meso-butane-2,3-diol is 1) 2) 3) 4) Sec: Sr.Super60_ NUCLEUS & STERLING_BT Page 11 TG ~ @bohring_bot × @HeyitsyashXD SRI CHAITANYA IIT ACADEMY, INDIA 10-08-24_ Sr.Super60_ NUCLEUS & STERLING_BT _Jee-Main_RPTM-04/05_Q.P 39. Which one of the following phenols does not give colour when condensed with phthalic anhydride in presence of conc. H 2 SO4 ? 1) 2) 3) 4) 40. The major product formed in the following reaction is: 1) 2) 3) 4) 41. In the following sequence of reactions, the final product D is: 1) 2) 3) 4) Sec: Sr.Super60_ NUCLEUS & STERLING_BT Page 12 TG ~ @bohring_bot × @HeyitsyashXD SRI CHAITANYA IIT ACADEMY, INDIA 10-08-24_ Sr.Super60_ NUCLEUS & STERLING_BT _Jee-Main_RPTM-04/05_Q.P 42. A : p-nitrophenol is more acidic than m-nitrophenol and o-nitrophenol. B : Ethanol will give immediate turbidity with Lucas reagent. C: Aminobenzene and aniline are same organic compound D: Aminobenzene and aniline are different organic compound In the light of the above statements, choose the correct answer from the options given below: 1) A,C only 2) A,B only 3) A,B,C only 4) A,B,C,D 43. In the following reaction ‘A’ is   H  Major Product 'A' OH 1) 2) 3) 4) 44. Cyclohexene is_____type of an organic compound. 1) Benzenoid aromatic 2) Benzenoid non-aromatic 3) Acylic 4) Homocyclic Sec: Sr.Super60_ NUCLEUS & STERLING_BT Page 13 TG ~ @bohring_bot × @HeyitsyashXD SRI CHAITANYA IIT ACADEMY, INDIA 10-08-24_ Sr.Super60_ NUCLEUS & STERLING_BT _Jee-Main_RPTM-04/05_Q.P 45. A: IUPAC name of HO  CH 2  (CH 2 )3  CH 2  COCH 3 is 7-Hydroxyheptan-2-one. B: 2-oxoheptan -7- ol is the correct IUPAC name for above compound. C: Tropone is a Benzenoid aromatic compound D: Napthalene is Non- Benzenoid aromatic compound In the light of the above statements, choose the correct answer from the options given below: 1) A only 2) A,C only 3) A,B,C only 4) A,B,C,D OCH 3   HBr 46. In the reaction the products are: Br OCH 3 and H2 Br and CH 3 Br 1) 2) Br and CH 3OH OH and CH 3 Br 3) 4) 47. Identify correct acidic strength order in the following compounds 1) II  I  III 2) II  III  I 3) I  III  II 4) III  I  II 48. The following two reactions (i ) B H H  H O HgSO CH3C  CCH3  2 6  ii) CH 3C  CCH 3  2 4 (ii ) H 2O2 OH 1) Same aldehyde 2) Different aldehydes 3) Same ketones 4) Different ketones Sec: Sr.Super60_ NUCLEUS & STERLING_BT Page 14 TG ~ @bohring_bot × @HeyitsyashXD SRI CHAITANYA IIT ACADEMY, INDIA 10-08-24_ Sr.Super60_ NUCLEUS & STERLING_BT _Jee-Main_RPTM-04/05_Q.P 49. When neopentyl alcohol is heated with acid, it slowly converted into an 85:15 mixture of alkenes A and B, respectively. What are those alkenes? 1) 2) 3) 4) 50. The correct order of SN 2 reactivity of 1) I  III  II  IV 2) I  II  III  IV 3) IV  III  II  I 4) III  II  I  IV (NUMERICAL VALUE TYPE) Section-II contains 10 Numerical Value Type questions. Attempt any 5 questions only. First 5 attempted questions will be considered if more than 5 questions attempted. The Answer should be within 0 to 9999. If the Answer is in Decimal then round off to the nearest Integer value (Example i,e. If answer is above 10 and less than 10.5 round off is 10 and If answer is from 10.5 and less than 11 round off is 11). Marking scheme: +4 for correct answer, 0 if not attempt and -1 in all other cases. 51. Total number of cyclic isomers (including stereo isomers) possible for C4 H8O which do not liberate H 2 gas with Na metal is____ 52 Degree of unsaturation of A  B  Sec: Sr.Super60_ NUCLEUS & STERLING_BT Page 15 TG ~ @bohring_bot × @HeyitsyashXD SRI CHAITANYA IIT ACADEMY, INDIA 10-08-24_ Sr.Super60_ NUCLEUS & STERLING_BT _Jee-Main_RPTM-04/05_Q.P 53. How many structures of final products are possible? (including stereo)  Conc. H 2 SO4   Br2 CCl4 C4 H 8 Br2 OH 54 The number of stereoisomers possible for 1,2-dimethyl cyclopopane is: 55. How many of the following compounds are aromatic A) B) C) D) E) F) 56. How many moles of AgNO3 will react with the given compound 57. How many moles of formaldehyde is formed by the complete hydrolysis of following compound with H 3O  Sec: Sr.Super60_ NUCLEUS & STERLING_BT Page 16 TG ~ @bohring_bot × @HeyitsyashXD SRI CHAITANYA IIT ACADEMY, INDIA 10-08-24_ Sr.Super60_ NUCLEUS & STERLING_BT _Jee-Main_RPTM-04/05_Q.P  58. 25 Optically pure ( S )  (  ) -2- bromooctane,  D  36 , reacts with aqueous NaOH in  25 acetone to give optically pure (R)-(-)-2-octanol,  D  10.3. With partially racemized   25 2 – bromooctane whose  D  30 , the  D 25 of the alcohol product is 6.0. Calculate the % of inversion? 59. How many of the following are more reactive than benzene for electrophilc substitution reaction? 1) Anisole 2) Acetanilide 3) Cabolic acid 4) TNT 5)Mesitylene 6) Benzaldehyde 7) Acetophenone 8) Meta-Xylene 9) Ethyl benzoate 60. Butanone  i ' X ' RMgX excess ii H O  Alcohol 3 Ethyl Ethanoate  i ' Y ' RMgX excess ii H O  Alcohol 3 Find X+Y Sec: Sr.Super60_ NUCLEUS & STERLING_BT Page 17 TG ~ @bohring_bot × @HeyitsyashXD SRI CHAITANYA IIT ACADEMY, INDIA 10-08-24_ Sr.Super60_ NUCLEUS & STERLING_BT _Jee-Main_RPTM-04/05_Q.P MATHEMATICS Max Marks: 100 (SINGLE CORRECT ANSWER TYPE) This section contains 20 multiple choice questions. Each question has 4 options (1), (2), (3) and (4) for its answer, out of which ONLY ONE option can be correct. Marking scheme: +4 for correct answer, 0 if not attempted and –1 in all other cases. 61. Consider the function g ( x) defined as  2     2025 1 2024 g ( x) x  1  ( x  1) ( x 2  1)( x 4  1).... x 2  1  1and x  1, The value of [g(2025)] equals Note: [k] denotes greatest integer which is less than or equal to the given real number k. 1) 1 2) 2 3) 3 4) 4 n  x  sin x 62. Give that  cos    n 1  2n  2n sin  x   n 2   n 1  x     lt  n tan  n , x  (0,  )     2  2 Let f ( x )  n  n 1 2  2   for x   2 Then which one of the following alternative is true?  1) f ( x ) has non –removable discontinuity of finite type at x  2  2) f ( x ) has missing point discontinuity at x  2  3) f ( x ) is continuous at x  2  4) f ( x ) has non-removable discontinuity of finite type at x  2 63. Let f : A B and B C be two functions and gof : A C is defined. Then which of the following statement is true? 1) if gof is onto then f must be onto 2) if f is into and g is onto then gof must be onto functions. 3) if gof is one-one then g is not necessarily one-one 4) If f is injective and g is surjective then gof must be bijective mapping. Sec: Sr.Super60_ NUCLEUS & STERLING_BT Page 18 TG ~ @bohring_bot × @HeyitsyashXD SRI CHAITANYA IIT ACADEMY, INDIA 10-08-24_ Sr.Super60_ NUCLEUS & STERLING_BT _Jee-Main_RPTM-04/05_Q.P 64. Let f be a function defined as  3   1  f :  0, e 2   ,   , f ( x )  (ln x ) 2  3ln x  2 then f 1( x ) equals   4     3  4 x  1   3  4 x  1  1) log   2) log    2   2  3 4 x 1 3 4 x 1 3) e 2 4) e 2  n  2 65. If cos 1   , then the minimum and maximum value, of integer n are respectively  2  3 1)  6 and  3 2)  6 and  4 3) 3 and 6 4) 4 and 6 66. Let f : R R be a function defined by: max{t 3  3t}; x  2    t  x   2  f ( x )   x  2 x  6; 2  x  3 [ x  3]  9; 3  x  5    2 x  1; x5    Where [k] is the greatest integer less than or equal to k. Let m be the number of points where   f is not differentiable and I ( x )   x  3 x dx , If I  0   0 , then the ordered pair of 3  m, I  1  is equal to:  27   23   5   23  1)  3,  2)  3,  3)  4,  4)  4,   4   4   4   4    1  x  x1 4  qx p q 67. If  cot  2 tan 1 dx   C ; ( where p and q are relatively prime and  1 4  p  1 x  x  C is constant of integration) , then which one of the following is INCORRECT? 1) p  5 2) p  2 q  13 3) p  q  1 4) q  5 Sec: Sr.Super60_ NUCLEUS & STERLING_BT Page 19 TG ~ @bohring_bot × @HeyitsyashXD SRI CHAITANYA IIT ACADEMY, INDIA 10-08-24_ Sr.Super60_ NUCLEUS & STERLING_BT _Jee-Main_RPTM-04/05_Q.P 32 12 1  4   4  68. If  dx  A 1    B 1    C then which one of thefollowing is x4 x2  4  x2   x2  correct? 1) A  B  0 2) A  B  0 3) A  3B  0 4) 3 A  B  0 cos 2 x 2 1 C  1  tan 2 x 69. If  sin x dx  A log cot x  cot x  1  log B 2  K then C  1  tan x 1) A  7, B  C  0 2) B  C  7, A  1 3) B  C  0, A  1 4) B  C  0, A  1 70. For real numbers  ,  ,  and  , if 1  x  1   x  1  tan 2 2     x   dx   x2  1    x 4  3 x 2  1 tan 1   x       x2  1    2   2  1  ( x  1) 1 x  1   log e  tan 1      tan     tan  C   x   x   x         Where C is an arbitrary constant, then the value of 2025       is equal to ____ 1) 1215 2) 1220 3) 5050 4) 0 x cot x  1 71. If  2 2 dx  f ( x)  C where lim f ( x)  0 and cos ecx  0 (where C is constant x 0 cos ec x  x of integration) Then which of the following is correct? f ( x) 1) f (1)  1 2) lim does not exist x 0 x tan x  3) f (1)  1 4) f ( x ) is periodic with period 2 2 Sec: Sr.Super60_ NUCLEUS & STERLING_BT Page 20 TG ~ @bohring_bot × @HeyitsyashXD SRI CHAITANYA IIT ACADEMY, INDIA 10-08-24_ Sr.Super60_ NUCLEUS & STERLING_BT _Jee-Main_RPTM-04/05_Q.P x 2 (1  ln x) 72.  ln 4 x  x4 dx equals (where x>0, C is constant of integration) 1  x  1 2 2 1  ln x  x  1 1  ln x  1) ln    ln(ln x  x )  C 2) ln    tan  C 2  ln x  4 4  ln x  x  2  x  1  ln x  x  1 1  ln x  1   ln x  x  1 1  ln x   3) ln    tan    C 4)  ln    tan    C 4  ln x  x  2  x  4   ln x  x  3  x  73. Let f ( x ) satisfies xf 2 ( x)  f ( x)  x  1 x  R and f (1)  0  f ( x)  f ( x )  f "( x )  If h( x)     f ( x) d ( x) then h (3)  h (2) equals  2  f '( x )  1) 1 2) 0 3) 1 4) 2  2tan x    74. The value of integral  e x   cot 2  x    dx is equal to: 1  tan x  4  (Where C is constant of integration) 4    1) e x tan   x   C 2) e x tan  x    C    4  3   3  3) e x tan   x  C 4) e x tan  x  C  4   4  ax 2  2bx  c 75. If I   dx (where B2  AC ) is a rational function then which one of the  Ax2  2Bx  C  2 following condition must be necessary? A C 2B 1) 2Bb  Ac  aC 2) Aa  Bb  Cc 3) Bb  aC  cA 4)   c a b 3cot 3x  cot x 76. If  tan x  3tan3x dx  pf ( x)  qg ( x)  c where ‘c’ is constant of integration, then: 1 3  tan x 1) p  1; q  ; f ( x)  x; g ( x)  ln 3 3  tan x 1 3  tan 2 x 2) p  1; q   ; f ( x)  x; g ( x)  ln 3 3  tan 2 x Sec: Sr.Super60_ NUCLEUS & STERLING_BT Page 21 TG ~ @bohring_bot × @HeyitsyashXD SRI CHAITANYA IIT ACADEMY, INDIA 10-08-24_ Sr.Super60_ NUCLEUS & STERLING_BT _Jee-Main_RPTM-04/05_Q.P 2 3  tan 2 x 3) p  1; q   ; f ( x)  x; g ( x)  ln 3 3  tan x 1 3  tan 3 x 4) p  1; q   ; f ( x)  x; g ( x )  ln 3 3  tan x cos5 x 77. The integral  sin x dx is equal to (where C is constant of integration) 1 1 1 1 1) log e sin x  cos 2 x  cos 4 x  C 2)  log e sin x  cos 2 x  cos 4 x  C 2 4 2 4 1 1 3) log e sin x  cos 2 x  cos 4 x  C 4) log e sin x  cos 2 x  cos 4 x  C 2 2 dx   x  78. Let   a loge tan  b    sec(c  x)   d , where a,b,c,d are constant and 2sin x  sec x   2    b, c   0,  , then which one of thefollowing is correct?  2 b 5 3  b 1 1) a 2   2) b  c  3) b  c  4) a 2    8 8 8  8 79. Consider f ( x)  ln x  kx 2 , x  0. Match the column I with the value of k in column II. Column I Column II a f ( x )  0 has two distinct solutions p k 0 b f ( x )  0 has four distinct solutions q 1 k 2e c f ( x )  0 has six distinct solutions r  1  k  ,   2e  d f ( x )  0 has no solutions s k   ,0  t  1  k   0,   2e  1) ( a-r,p) ( b-q) ( c-t) (d-s) 2) ( a-p) ( b-r,q) ( c-t) (d-s,r) 3) ( a-r,t) ( b-p) ( c-t) (d-s,q ) 4) ( a-q,p) ( b-q) ( c-t) (d-s,t) Sec: Sr.Super60_ NUCLEUS & STERLING_BT Page 22 TG ~ @bohring_bot × @HeyitsyashXD SRI CHAITANYA IIT ACADEMY, INDIA 10-08-24_ Sr.Super60_ NUCLEUS & STERLING_BT _Jee-Main_RPTM-04/05_Q.P 80. Statement-1:- Let f : R R be a real valued continuous function which assumes only rational values. If f  3   1729. Then  2012  f  2013   1729.  Statement-2:- If a continuous function f  x  assumes only rational values, then f  x  is a constant function. 1) Statement-1is True, Statement-2 is True; Statement-2 is a correct Explanation forStatement-1 2) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1 3) Statement-1 is True, Statement-2 is False 4) Statement-1 is False, Statement-2 is True (NUMERICAL VALUE TYPE) Section-II contains 10 Numerical Value Type questions. Attempt any 5 questions only. First 5 attempted questions will be considered if more than 5 questions attempted. The Answer should be within 0 to 9999. If the Answer is in Decimal then round off to the nearest Integer value (Example i,e. If answer is above 10 and less than 10.5 round off is 10 and If answer is from 10.5 and less than 11 round off is 11). Marking scheme: +4 for correct answer, 0 if not attempt and -1 in all other cases. x 1   81.  dx  f ( x ) and if f (0)  and f    P 2  Q  R  tan x  cot x  sec x  cos ecx 2 2 2 1 (P, Q, R are rational) , then the value of P  Q  R is 4  4 3 2  x  x  x  3x  x  2   f ( x)  82. If  e dx  e x    C then f (2) is equal to ( f ( x ) is a   x  1 2   x  1    polynomial) x 2  n( n  1) 83. If f ( x)   dx where f (0)  0, n  N then the value of  if for ( x sin x  n cos x ) 2     n  3, f     4    Sec: Sr.Super60_ NUCLEUS & STERLING_BT Page 23 TG ~ @bohring_bot × @HeyitsyashXD SRI CHAITANYA IIT ACADEMY, INDIA 10-08-24_ Sr.Super60_ NUCLEUS & STERLING_BT _Jee-Main_RPTM-04/05_Q.P 84. If the equation x3  3x  1  0 has three real roots x1, x2 , x3 , where x1  x2  x3 , Then the value of  2025 { x1}  { x2 }  { x3}  is equal to (Note: {} denotes the fractional part of x ) 1  7cos2 x g ( x) 85. Suppose  sin7 x cos2 x dx  sin7 x  C where C is an arbitrary constant of integration. Then   the value of g '(0)  g ''   is 4  1  sin     86. Let f ( )  sin  tan    , where  4    4.   cos 2  d Then the value of ( f ( )) is d (tan  ) 2 x x2 f ( x )e x 1 87. If  e   dx   C then the sum f ( x )  f 2 ( x )  f 3 ( x )....... at x  , is  x4 x4 2 [Note where ‘C’ is constant of integration.] x y x y 88. Let J   dy and I   dx where x and y are independent variables such that xy x2  y g    J  I and g (1)  1 then the value of  g ( e)  is [note : e  2.7 and [.] is G.I.F] x 89. If  (cot 2 x cot 3 x  tan 2 x tan 7 x) tan 5 x  a ln(tan 2 x )  b ln (sin 3 x)  c ln(sec5 x) d ln(cos7 x)  C and a, b, c, d  Q (set of rational numbers ) and C is the constant of m integration. If  a  b  c  d  can be expressed as in the lowest form, then (m + n)=. n f '( x ) x  f ( x)  f ( x)  x  90. If   f ( x)  x  f ( x) x  x 2 dx is equal to m tan 1  nx    c , where m, n  R and ' c '  constant of integration ( x  0) then m2  n2 is equal to Sec: Sr.Super60_ NUCLEUS & STERLING_BT Page 24 TG ~ @bohring_bot × @HeyitsyashXD SRI CHAITANYA IIT ACADEMY, INDIA 10-08-24_ Sr.Super60_ NUCLEUS & STERLING_BT _Jee-Main_RPTM-04/05_Q.P Sec: Sr.Super60_ NUCLEUS & STERLING_BT Page 25