( )(1 ST YEAR_REGULAR) IIT-2015 (P1)_SET A DATE: Time: 3 hours Maximum Marks: 264

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1 FIITJEE RET 13 (17 19)(1 ST YEAR_REGULAR) IIT-15 (P1)_SET A DATE: Time: 3 hours Maximum Marks: 64 INSTRUCTIONS: A. General 1. This booklet is your Question Paper containing 6 questions.. Blank papers, clipboards, log tables, slide rules, calculators, cellular phones, pagers and electronic gadgets in any form are not allowed to be carried inside the examination hall. 3. Fill in the boxes provided for Name and Enrolment No. 4. The answer sheet, a machine-readable Objective Response (ORS), is provided separately. 5. DO NOT TAMPER WITH / MULTILATE THE ORS OR THE BOOKLET. B. Filling in the OMR: 6. The instructions for the OMR sheet are given on the OMR itself. C. Question paper format: 7. The question paper consists of 3 parts (Physics, Chemistry and Mathematics). Each part consists of two sections. 8. Section I contains 8 questions. The answer to each question is a single digit integer, ranging from to 9 (both inclusive). 9. Section II contains 1 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONE or MORE are correct. 1. Section III contains Match the following type questions and you will have to match entries in Column I with the entries in Column II D. Marking Scheme 11. For each question in Section I, you will be awarded 4 marks if you darken ALL the bubble(s) corresponding to the correct answer(s) ONLY. In all other cases zero () marks will be awarded. No negative marks will be awarded for incorrect answers in this section. 1. For each question in Section II, you will be awarded 4 marks if you darken ALL the bubble(s) corresponding to the correct answer(s) ONLY. In all other cases zero () marks will be awarded. marks will be awarded for incorrect answers in this section. 13. For each question in Section III, you will be awarded marks for each entry in Column I; if you darken ALL the bubble(s) corresponding to the correct answer(s) ONLY. In all other cases zero () marks will be awarded. 1 marks will be awarded for incorrect answers in this section. Don t write / mark your answers in this question booklet. If you mark the answers in question booklet, you will not be allowed to continue the exam. NAME: ENROLLMENT NO.:

2 PAPER I PART I: PHYSICS SECTION 1 (Maximum Marks: 3) This section contains EIGHT questions. The answer to each question is a SINGLE DIGIT INTEGER ranging from to 9, both inclusive. For each question, darken the bubble corresponding to the correct integer in the ORS. Marking scheme: +4 If the bubble corresponding to the answer is darkened. In all other cases. 1. A solid sphere of radius R has moment of inertia I about its geometrical axis. It is melted into a disc of radius r and thickness t. If it s moment of inertia about the tangential axis (which is perpendicular to plane of the disc) is also equal to k I, then the value of r = R. Find K. 15. A circular plate of uniform thickness has a radius of 56 cm. A circular portion of radius 4 cm is removed from one edge of the plate as shown. At what distance (in cm) to the left from the centre of the disc is the centre of mass of the remaining portion? 56 cm O 4 cm 3. Two spheres each of mass M and radius R/ are connected with a massless rod of length R as shown in the figure. The moment of inertia of the system about an axis passing through the centre of one of the spheres and perpendicular 3 to the rod is C MR 5 where C is equal to P R M R Q R Y Y M

3 4. A wire of length land mass m is bent in the form of a rectangle ABCD) with AB BC =. The moment of inertia of this wire frame about the side BC is k m 16, where k is equal to 5. A thin uniform metallic triangular sheet of mass. M has sides AB = BC =L. Its moment of inertia about axis AC lying in the A ML plane of the sheet is. Find the value of α Axis of rotation B C 6. A rigid body can be hinged about any point on x -axis. When it is hinged at x, moment of inertia about axis through hinge is given by I 3x 18x 15. x -coordinate of centre of mass is 7. Four solid spheres, each of diameter 5 cm and mass.5 kg are placed with their centres at the corners of a square of side 4 cm. The moment of inertia of the system about the diagonal of the square is N x 1 4 kg m, then N is 8. A thin wire of length L and uniform linear mass density is bent into a circular loop with centre at O as shown in figure. The moment of inertia of the loop about the 3 L axis XX' is. Find the value of α? 8 x 9 << O x'

4 SECTION (Maximum Marks: 4) This section contains TEN questions. Each question has FOUR options (A), (B), (C) and (D). ONE OR MORE THAN ONE of these four option(s) is (are) correct. For each question, darken the bubble(s) corresponding to all the correct option(s) in the ORS. Marking scheme: +4 If only the bubble(s) corresponding to all the correct option(s) is (are) darkened. If none of the bubbles is darkened. In all other cases 9. Semi circular ring of mass 3 m and semi circular disc of mass m of same radius R and placed as shown in figure. If centre of them is chosen as origin, centre of mass of system is R (A), 3, R (B) R, 3 R (C), (D) y R m /3 m x 1. A paraboloid shaped solid object is formed by rotating on parabola y x about y-axis as shown in figure. If the height of the body is h then the position of centre of mass from origin. (Assume density to be uniform throughout). h h (A) (B) 3 4 (C) h 3 (D) h

5 11. From the circular disc of radius 4R, two small discs of same radius R, are cut off as shown in the figure. The centre of mass of the new structure will be : R (A) ˆ ˆj 5 i (B) 3R ˆ ˆj 15 -i (C) R ˆ ˆj 5 i (D) 3R ˆ ˆj 14 i R 4R y R x 1. The moment of inertia of a thin square plate ABCD of uniform thickness about an axis passing through the centre O and perpendicular to plate is (A) I 1 + I (B) I + I 3 (C) I 1 + I 3 (D) I 3 + I 4 A D O 4 B C If a circular concentric hole is made in a disc then about an axis passing through the centre of the disc and perpendicular to its plane (A) moment of inertia decreases (B) moment of inertia increases (C) radius of gyration increases (D) radius of gyration decreases 14. Four particles of mass m each are placed at four corners of a square ABCD of side a. Point O is the centre of the square. Moment of inertia of all four particles about an axis passing through (A) A and B is ma (B) A and C is ma (C) O and perpendicular to plane of square is ma (D) O and parallel to CD is ma

6 15. A wire of length is bent into the shape of an n sided regular polygon mass be m, then its m 1 (A) the M.I. about its centre(axis perpendicular to the plane) is cot 3 4n 3 n m 1 (B) the M.I. about its centre (axis perpendicular to the plane) is cot 4n 3 n m (C) the M.I. in the limiting case is an if n then M.I. is 4 m (D) the M.I. in the limiting case is an if n then M.I. is If l represents the M.I of a solid sphere about an axis parallel to the diameter of the solid sphere at a distance x from it, then which of the following graphs represents the variation of I with x? (A) (B) (C) (D) 17. A square plate of mass M and edge l is shown in fig. The moment of inertia of the plate about the axis in the plane of plate and passing through one of its vertex o making an angle 15 from horizontal is : (A) (C) ML 1 7ML 1 (B) 9ML 4 (D) None of these 18. Moment of inertia of a uniform annular disc of internal radius r and external radius R and mass M about an axis through its centre and perpendicular to its plane is (A) ½ M(R r ) (B) ½ M(R + r 4 4 M(R r ) ) (C) (R r ) (D) M(R r ) (R r )

7 SECTION 3 (Maximum Marks: 16) This section contains TWO questions. Each question contains two columns, Column I and Column II Column I has four entries (A), (B), (C) and (D) Column II has five entries (P), (Q), (R), (S) and (T) Match the entries in Column I with the entries in Column II One or more entries in Column I may match with one or more entries in Column II. The ORS contains a 4 5 matrix whose layout will be similar to the one shown below: (A) (P) (Q) (R) (S) (T) (B) (P) (Q) (R) (S) (T) (C) (P) (Q) (R) (S) (T) (D) (P) (Q) (R) (S) (T) For each entry in Column I, darken the bubbles of all the matching entries. For example, if entry (A) in Column I matches with entries (Q), (R) and (T), then darken these three bubbles in the ORS. Similarly, for entries (V), (C) and (D). Marking entry in Column I. + If only the bubble(s) corresponding to all the correct match (s) is (are) darkened. If none of the bubbles is darkened. 1 In all other cases. 19. The moment of inertia of some rigid bodies of equal mass M are given along its diameter and other also having equal mass M about given axis are given below. Match the following columns Column I A uniform circular hoop of inner radius R and outer radius R Column II (A) (p) 5 4 MR (B) ( XX ' is along the diameter of the hook) A hollow sphere of inner radius R and external radius R ( XX ' is along the diameter of the hollow sphere) A square lamina of side R. (q) MR 1 (C) (r) 6 MR 35 (D) ( XX ' is along the diagonal of the lamina) A uniform right conical body of radius R s) ( XX ' is along the axis of the conical body) (s) (t) 3 1 MR 5 4 MR

8 . Different objects are given in column I and corresponding M.I are given in column II Column I Column II (A) A semi circular disc of mass m and radius R axis is passing through O and perpendicular to (p) MR plane O A hollow cylinder is cut as shown (B) in figure mass of the object is m (q) MR axis and radius is R (C) Three rods of each mass m and length R are joined to form an equilateral triangle. M.I. about an axis passing through corner and perpendicular to plane O (r) MR (D) For a semi circular ring of mass M and radius R, M.I about an axis perpendicular to plane and passing through O O (s) 3 MR (t) None of these

9 PART II: CHEMISTRY SECTION 1 (Maximum Marks: 3) This section contains EIGHT questions. The answer to each question is a SINGLE DIGIT INTEGER ranging from to 9, both inclusive. For each question, darken the bubble corresponding to the correct integer in the ORS. Marking scheme: +4 If the bubble corresponding to the answer is darkened. In all other cases. 1. The given P-V isotherm represents the variation of P vs V at certain temperatures. Each of these isotherms have their respective temperatures marked against them. The isotherm, corresponding to which temperature represents the critical isotherm of the given real gas, which would be utilized in identification of critical temperature. P V 5. 4% of a mixture of. mole of N and.6 mole of H react to given NH 3 according to the equation. N + 3H NH 3 at constant temperature and pressure. Then the ratio of final volume to the initial volume of gases is x. Then 5x is equal to.

10 3. How many of the following statements are correct in context to the given question? The given figure bulb contains an excess of volatile solid A. At 5 C equilibrium formed as shown below. The following two equilibriums are simultaneously existing in the bulb. ' Asolid Agas HS gas Asolid A gas At equilibrium; total pressure inside bulb is 1 mm whereas partial pressure of HS inside bulb is 4 mm. If some A(g) is removed from bulb at equilibrium I. decomposition equilibria will shift forward II. sublimation equilibria will shift forward III. total pressure when equilibrium reestablished will be equal to 1 mm. IV. no change will happen in the state of equilibrium 4. Solid phosphorus melts and vapourizes at high temperature. Gaseous phosphorus effuses at a rate that is.567 times that of Ne in the same apparatus under the same conditions. How many atoms are in a molecule of gaseous phosphorus. 5. The number of distinct conformers generated by the rotation of C-C bond in H 3 C-CH 3 if is x & the number of distinct conformers generated due to rotation of C-C bond in DH C-CH D if is y, then what is the value of (x + y)? 6. For the given molecular formula C 7 H 8 O, what is the total number of benzenoid isomers possible

11 7. How many of the following compounds will exhibit tautomerism? O CH 3 O OH OH O O N N O NH N N H N H N N HCNO OH O 8. In how many of the following compounds, the gauche form generated due to rotation about C-C3 or C1- C is more stable than the anti-form generated during the rotation about the same C-C bond? Ethane, Propane, 1,-ethanediol, -fluoroethanol, 1-fluoroethanol, 1,-difluoroethane, -amino propan-1- ol,,3-butanediol, -fluorobutan-1-ol, 3-nitrobutan--ol SECTION (Maximum Marks: 4) This section contains TEN questions. Each question has FOUR options (A), (B), (C) and (D). ONE OR MORE THAN ONE of these four option(s) is (are) correct. For each question, darken the bubble(s) corresponding to all the correct option(s) in the ORS. Marking scheme: +4 If only the bubble(s) corresponding to all the correct option(s) is (are) darkened. If none of the bubbles is darkened. In all other cases 9. Which of the following statements is/are correct for an ideal gas? (A) all gas molecules move with the same speed (B) kinetic energy of molecules remains unchanged due to collisions with other molecules (C) kinetic energy of gas molecules is dependent only on temperature (D) speed of molecules depends both on molecular mass and temperature

12 3. Which statements is/are correct for the compounds X & Y: O O HC X CH 3 C H 3 (A) X & Y are chain isomers (B) X & Y are functional isomers (C) X & Y have the same index of hydrogen deficiency (D) X & Y can undergo tautomerism Y H 31. For which of the following reactions, the degree of dissociation cannot be calculated from the vapour density data (A) HI(g) H (g) + I (g) (B) NH 3 (g) N (g)+3h (g) (C) NO(g) N (g) + O (g) (D)PCl 5 (g) PCl 3 (g) + Cl (g) 3. Two bulbs A and B of equal capacity are filled with He and SO respectively at the same temperature. Which of the following is incorrect? (A) At 1 K, the V rms of SO becomes half of the V rms of He molecules at 7 o C. (B) The velocity of SO decreases if volume of B becomes four times. (C) There is no change in velocity if half of the molecules of SO are removed from B. (D) The velocity of SO becomes double if half of SO molecules are removed from B. 33. Molecules with the molecular formula C 4 H 1 O exhibit which of the following type of isomerism (A) Chain (B) Positional (C) Functional (D) Metamerism 34. Which of the following can exhibit geometrical isomerism (A) Azobenzene (B) Cyclohexene (C) H 3 C N OH H (D) Decalin 35. Keto-Enol tautomerism is exhibited by (A) Acetyl chloride (C) Trifluoro acetic acid (B) N-methyl acetamide (D) p-nitroso phenol

13 36. Which of the following statements is/are correct? (A) trans alkenes are generally more stable than cis alkenes (B) the boiling point of cis alkene is generally more than trans alkene (C) the melting point of cis alkene is generally less than trans alkene (D) the dipole moment of cis alkene is generally more than trans alkene 37. Which of the following statements is/are incorrect? H 3 C Cl H Br This compound can be named as: (A) only cis-1-bromo-1-chloro propene (C) (Z)-1-bromo-1-chloro propene (B) only trans-1-bromo-1-chloro propene (D) (E)-1-bromo-1-chloro propene 38. The NH 3 gas is liquefied more easily than N. Hence (A) Van der Wall s constant a and b of NH 3 > that of N (B) a and b of NH 3 < that of N. (C) a(nh 3 ) > a(n ) but b(nh 3 ) < b(n ) (D) the intermolecular force of attraction between the NH 3 molecules > that between the N molecules SECTION 3 (Maximum Marks: 16) This section contains TWO questions. Each question contains two columns, Column I and Column II Column I has four entries (A), (B), (C) and (D) Column II has five entries (P), (Q), (R), (S) and (T) Match the entries in Column I with the entries in Column II One or more entries in Column I may match with one or more entries in Column II. The ORS contains a 4 5 matrix whose layout will be similar to the one shown below: (A) (P) (Q) (R) (S) (T) (B) (P) (Q) (R) (S) (T) (C) (P) (Q) (R) (S) (T) (D) (P) (Q) (R) (S) (T) For each entry in Column I, darken the bubbles of all the matching entries. For example, if entry (A) in Column I matches with entries (Q), (R) and (T), then darken these three bubbles in the ORS. Similarly, for entries (V), (C) and (D). Marking entry in Column I. + If only the bubble(s) corresponding to all the correct match (s) is (are) darkened. If none of the bubbles is darkened. 1 In all other cases.

14 39. Match the contents of Column I with contents of Column II for 1 mole of gas. Column I Column II (A) When force of attraction among gas PV = RT (p) molecules is negligible (B) At STP (q) (P + a/v )(V b) = RT (C) At low pressure and high temperature (r) PV = RT a/v (D) When volume of gas molecules is PV = RT + Pb (s) negligible 4. Match the following: Column I (A change has been made to the given system at equilibrium) (A) P is increased for (B) (C) (D) Column II (Effect of change on the system) Ice S H O L (p) Entropy of the system will increase Volume of the container is increased for NH g HCl g NH Cl S 3 4 Inert gas is added at constant volume for Fe S 4 H O S Fe O S 4H g 3 4 (q) Temperature is increased for CaCO S CaO S CO H (s) 3 (r) Free energy of the system will decrease Equilibrium position will shift to right Magnitude of K b (rate constant for backward direction) will rise. PART III: MATHEMATICS SECTION 1 (Maximum Marks: 3) This section contains EIGHT questions. The answer to each question is a SINGLE DIGIT INTEGER ranging from to 9, both inclusive. For each question, darken the bubble corresponding to the correct integer in the ORS. Marking scheme: +4 If the bubble corresponding to the answer is darkened. In all other cases. 41. If n(u) = 6, n(a) = 35, n(b) = 4, n(a B) = 1, then n(a B) = 4. In a group of 6 people, 8 like pepsi but not coke and 16 like pepsi. Number of people who like coke but not pepsi is k, then k 1 is

15 43. If U = {1,, 3, 4, 5, 6, 7, 8, 9, 1}, A = {1,, 3, 4}, B = {, 4, 6, 8, 1}, then n((a B)) = 44. The number of non-empty subsets of set {1,, 3, 4} is k, then k 7 is 45. If x1 y x y 5 15, then the value of x 1 + x + x 3 y 1 y y 3 is 1 1 x3 y Let A = and (1)B = 4 5. If B is the inverse of A, then the value of is If A, B and C are n n matrix and det (A) =, det (B) = 3, and det (C) = 5, then the value of [det (A BC)] is k, then k 3 is 48. If A = 3A I, and A 8 = pa + qi, then p + q is SECTION (Maximum Marks: 4) This section contains TEN questions. Each question has FOUR options (A), (B), (C) and (D). ONE OR MORE THAN ONE of these four option(s) is (are) correct. For each question, darken the bubble(s) corresponding to all the correct option(s) in the ORS. Marking scheme: +4 If only the bubble(s) corresponding to all the correct option(s) is (are) darkened. If none of the bubbles is darkened. In all other cases sin cos sin sin cos 49. If = coscos cos sin sin, then sinsin sin cos (A) is independent of (B) is independent of (C) is a constant (D) =

16 5. If (, ) = cos sin 1 sin cos 1, then cos sin 1 (A) f(3, ) = f(4, ) (B) f(, 4) = f(, 6) (C) f(1, ) = f(, ) (D) none of these 51. The determinant = 5. If f(x) = 53. If (x) = a x ab ac ab b x bc ac bc c x is divisible (A) x (B) x (C) x 3 (D) none of these a 1 ax a 1, then f(x) f(x) is divisible by ax ax a (A) x (B) a (C) a + 3x (D) x x 4x 3 x 4 13 x 5x 9 4x 5 6 8x 6x 1 16x 6 14 = ax 3 + bx + cx + d, then (A) a = 3 (B) b = (C) c = (D) none of these 54. If determinant cos sin cos sin cos sin cos sin cos (A) positive (B) independent of (C) independent of (D) none of these is

17 1 1 a If A = 1, B = b 1, and (A + B) = A + B + AB, then (A) a = 1 (B) a = 1 (C) b = (D) b = If A = 1 3 is an orthogonal matrix, then a b (A) a = (B) a =, b = 1 (C) b = 1 (D) b = tan 1 tan a b 57. If tan 1 tan 1 b a, then (A) a = cos (B) a = 1 (C) b = sin (D) b = If A 1 = 3 1, then 1/ (A) A = 1 (B) adj A = 3 1 1/ 3 1 1/ / 3 7 (C) A = 1/ 3 1 (D) A = SECTION 3 (Maximum Marks: 16) This section contains TWO questions. Each question contains two columns, Column I and Column II Column I has four entries (A), (B), (C) and (D) Column II has five entries (P), (Q), (R), (S) and (T) Match the entries in Column I with the entries in Column II One or more entries in Column I may match with one or more entries in Column II. The ORS contains a 4 5 matrix whose layout will be similar to the one shown below: (A) (P) (Q) (R) (S) (T) (B) (P) (Q) (R) (S) (T) (C) (P) (Q) (R) (S) (T) (D) (P) (Q) (R) (S) (T) For each entry in Column I, darken the bubbles of all the matching entries. For example, if entry (A) in Column I matches with entries (Q), (R) and (T), then darken these three bubbles in the ORS. Similarly, for entries (V), (C) and (D). Marking entry in Column I. + If only the bubble(s) corresponding to all the correct match (s) is (are) darkened. If none of the bubbles is darkened. 1 In all other cases.

18 59. Column I (A) 1 1 a b c c c b c 1 1 a a a b b c a b c b a b a c ac ac (B) sinacosb sina sinb cosa (C) (D) cosacosb cosa sinb sina is sina sinb sinacosb sinacosb sina sinb cosa cosa cosa sina sin acosb sin a sinb cos a sinb cosb sinacos b sina sin b If a, b and c are the sides of a triangle and A, B and C are the angles opposite to a, b and c, a bsina c sina respectively, then = is bsina 1 cos A c sina cos A 1 Column II p. Independent of a q. Independent of b r. Independent of c s. Dependent on a, b 6. Column I (A, B, C are matrices) (A) If A =, then A 1 = (where A is of order 3) (B) If A = 1, then adj (adj (A)) = 8 p. 1 q. 4 Column II (where A is of order 3) (C) If (A + B) = A + B, and A =, then B = (where A and B are of odd order) (D) If A =, B = 3 and C = 4, then ABC is equal to r. 4 s. t. Does not exist

19 PHYSICS CHEMISTRY FIITJEE RET 13 (17 19)(1 ST YEAR_REGULAR) IIT-15 (P1)_SET A DATE: ANSWERS D 1. C 11. D 1. A,B,C,D 13. A,C 14. A,B,C,D 15. BONUS 16. C 17. B 18. B 19. At, Br, Cq, Ds. Ar, Bp, Cs, Dt B,C 3. B, C, D 31. A, C 3. B, D 33. A, B, C, D 34. A, C, D 35. A, B, D 36. A, B, C, D 37. A, B, C 38. C,D 39. A s ; B p,q ; C p ; D r 4. A p, q, r; B p, s ; C p ; D p, q, r, s MATHEMATICS B 5. AC 51. AB 5. ABC 53. BC 54. AB 55. BONUS 56. AC 57. AC 58. BONUS 59. A p, q, r; B q; C s; D p, q, r 6. A q; B p; C s; D r

20 FIITJEE RET 13 (17 19)(1 ST YEAR_REGULAR) IIT-15 (P1)_SET B DATE: Time: 3 hours Maximum Marks: 64 INSTRUCTIONS: A. General 1. This booklet is your Question Paper containing 6 questions. 6. Blank papers, clipboards, log tables, slide rules, calculators, cellular phones, pagers and electronic gadgets in any form are not allowed to be carried inside the examination hall. 7. Fill in the boxes provided for Name and Enrolment No. 8. The answer sheet, a machine-readable Objective Response (ORS), is provided separately. 9. DO NOT TAMPER WITH / MULTILATE THE ORS OR THE BOOKLET. B. Filling in the OMR: 6. The instructions for the OMR sheet are given on the OMR itself. C. Question paper format: 14. The question paper consists of 3 parts (Physics, Chemistry and Mathematics). Each part consists of two sections. 15. Section I contains 8 questions. The answer to each question is a single digit integer, ranging from to 9 (both inclusive). 16. Section II contains 1 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONE or MORE are correct. 17. Section III contains Match the following type questions and you will have to match entries in Column I with the entries in Column II D. Marking Scheme 18. For each question in Section I, you will be awarded 4 marks if you darken ALL the bubble(s) corresponding to the correct answer(s) ONLY. In all other cases zero () marks will be awarded. No negative marks will be awarded for incorrect answers in this section. 19. For each question in Section II, you will be awarded 4 marks if you darken ALL the bubble(s) corresponding to the correct answer(s) ONLY. In all other cases zero () marks will be awarded. marks will be awarded for incorrect answers in this section.. For each question in Section III, you will be awarded marks for each entry in Column I; if you darken ALL the bubble(s) corresponding to the correct answer(s) ONLY. In all other cases zero () marks will be awarded. 1 marks will be awarded for incorrect answers in this section. Don t write / mark your answers in this question booklet. If you mark the answers in question booklet, you will not be allowed to continue the exam. NAME: ENROLLMENT NO.:

21 PAPER I PART I: PHYSICS SECTION 1 (Maximum Marks: 3) This section contains EIGHT questions. The answer to each question is a SINGLE DIGIT INTEGER ranging from to 9, both inclusive. For each question, darken the bubble corresponding to the correct integer in the ORS. Marking scheme: +4 If the bubble corresponding to the answer is darkened. In all other cases. 1. A thin uniform metallic triangular sheet of mass. M has sides AB = BC =L. Its moment of inertia about axis AC lying in the A ML plane of the sheet is. Find the value of α Axis of rotation B C. A rigid body can be hinged about any point on x -axis. When it is hinged at x, moment of inertia about axis through hinge is given by I 3x 18x 15. x -coordinate of centre of mass is 3. Four solid spheres, each of diameter 5 cm and mass.5 kg are placed with their centres at the corners of a square of side 4 cm. The moment of inertia of the system about the diagonal of the square is N x 1 4 kg m, then N is 4. A thin wire of length L and uniform linear mass density is bent into a circular loop with centre at O as shown in figure. The moment of inertia of the loop about the 3 L axis XX' is. Find the value of α? 8 x 9 << O x'

22 5. A solid sphere of radius R has moment of inertia I about its geometrical axis. It is melted into a disc of radius r and thickness t. If it s moment of inertia about the tangential axis (which is perpendicular to plane of the disc) is also equal to k I, then the value of r = R. Find K A circular plate of uniform thickness has a radius of 56 cm. A circular portion of radius 4 cm is removed from one edge of the plate as shown. At what distance (in cm) to the left from the centre of the disc is the centre of mass of the remaining portion? 56 cm O 4 cm 7. Two spheres each of mass M and radius R/ are connected with a massless rod of length R as shown in the figure. The moment of inertia of the system about an axis passing through the centre of one of the spheres and perpendicular 3 to the rod is C MR 5 where C is equal to P R M R Q R Y Y M 8. A wire of length land mass m is bent in the form of a rectangle ABCD) with AB BC =. The moment of inertia of this wire frame about the side BC is k m 16, where k is equal to

23 SECTION (Maximum Marks: 4) This section contains TEN questions. Each question has FOUR options (A), (B), (C) and (D). ONE OR MORE THAN ONE of these four option(s) is (are) correct. For each question, darken the bubble(s) corresponding to all the correct option(s) in the ORS. Marking scheme: +4 If only the bubble(s) corresponding to all the correct option(s) is (are) darkened. If none of the bubbles is darkened. In all other cases 9. Four particles of mass m each are placed at four corners of a square ABCD of side a. Point O is the centre of the square. Moment of inertia of all four particles about an axis passing through (A) A and B is ma (B) A and C is ma (C) O and perpendicular to plane of square is ma (D) O and parallel to CD is ma 1. A wire of length is bent into the shape of an n sided regular polygon mass be m, then its m 1 (A) the M.I. about its centre(axis perpendicular to the plane) is cot 3 4n 3 n m 1 (B) the M.I. about its centre (axis perpendicular to the plane) is cot 4n 3 n m (C) the M.I. in the limiting case is an if n then M.I. is 4 m (D) the M.I. in the limiting case is an if n then M.I. is If l represents the M.I of a solid sphere about an axis parallel to the diameter of the solid sphere at a distance x from it, then which of the following graphs represents the variation of I with x? (A) (B) (C) (D)

24 1. A square plate of mass M and edge l is shown in fig. The moment of inertia of the plate about the axis in the plane of plate and passing through one of its vertex o making an angle 15 from horizontal is : (A) (C) ML 1 7ML 1 (B) 9ML 4 (D) None of these 13. Moment of inertia of a uniform annular disc of internal radius r and external radius R and mass M about an axis through its centre and perpendicular to its plane is (A) ½ M(R r ) (B) ½ M(R + r 4 4 M(R r ) ) (C) (R r ) (D) M(R r ) (R r ) 14. Semi circular ring of mass 3 m and semi circular disc of mass m of same radius R and placed as shown in figure. If centre of them is chosen as origin, centre of mass of system is R (A) 3, R (C), (B) (D), R, 3 R y R m /3 m x 15. A paraboloid shaped solid object is formed by rotating on parabola y x about y-axis as shown in figure. If the height of the body is h then the position of centre of mass from origin. (Assume density to be uniform throughout). h h (A) (B) 3 4 (C) h 3 (D) h

25 16. From the circular disc of radius 4R, two small discs of same radius R, are cut off as shown in the figure. The centre of mass of the new structure will be : R (A) ˆ ˆj 5 i (B) 3R ˆ ˆj 15 -i (C) R ˆ ˆj 5 i (D) 3R ˆ ˆj 14 i R 4R y R x 17. The moment of inertia of a thin square plate ABCD of uniform thickness about an axis passing through the centre O and perpendicular to plate is (A) I 1 + I (B) I + I 3 (C) I 1 + I 3 (D) I 3 + I 4 A D O 4 B C If a circular concentric hole is made in a disc then about an axis passing through the centre of the disc and perpendicular to its plane (A) moment of inertia decreases (B) moment of inertia increases (C) radius of gyration increases (D) radius of gyration decreases

26 SECTION 3 (Maximum Marks: 16) This section contains TWO questions. Each question contains two columns, Column I and Column II Column I has four entries (A), (B), (C) and (D) Column II has five entries (P), (Q), (R), (S) and (T) Match the entries in Column I with the entries in Column II One or more entries in Column I may match with one or more entries in Column II. The ORS contains a 4 5 matrix whose layout will be similar to the one shown below: (A) (P) (Q) (R) (S) (T) (B) (P) (Q) (R) (S) (T) (C) (P) (Q) (R) (S) (T) (D) (P) (Q) (R) (S) (T) For each entry in Column I, darken the bubbles of all the matching entries. For example, if entry (A) in Column I matches with entries (Q), (R) and (T), then darken these three bubbles in the ORS. Similarly, for entries (V), (C) and (D). Marking entry in Column I. + If only the bubble(s) corresponding to all the correct match (s) is (are) darkened. If none of the bubbles is darkened. 1 In all other cases. 19. Different objects are given in column I and corresponding M.I are given in column II Column I Column II (A) A semi circular disc of mass m and radius R axis is passing through O and perpendicular to (p) MR plane O A hollow cylinder is cut as shown (B) in figure mass of the object is m (q) MR axis and radius is R (C) Three rods of each mass m and length R are joined to form an equilateral triangle. M.I. about an axis passing through corner and perpendicular to plane O (r) MR (D) For a semi circular ring of mass M and radius R, M.I about an axis perpendicular to plane and passing through O O (s) 3 MR (t) None of these

27 . The moment of inertia of some rigid bodies of equal mass M are given along its diameter and other also having equal mass M about given axis are given below. Match the following columns Column I A uniform circular hoop of inner radius R and outer radius R Column II (A) (p) 5 4 MR ( XX ' is along the diameter of the hook) A hollow sphere of inner radius R and external radius R (B) ( XX ' is along the diameter of the hollow sphere) (q) MR 1 A square lamina of side R. (C) (r) 6 MR 35 ( XX ' is along the diagonal of the lamina) A uniform right conical body of radius R s) (D) (s) 3 1 MR ( XX ' is along the axis of the conical body) (t) 5 4 MR

28 PART II: CHEMISTRY SECTION 1 (Maximum Marks: 3) This section contains EIGHT questions. The answer to each question is a SINGLE DIGIT INTEGER ranging from to 9, both inclusive. For each question, darken the bubble corresponding to the correct integer in the ORS. Marking scheme: +4 If the bubble corresponding to the answer is darkened. In all other cases. 1. The number of distinct conformers generated by the rotation of C-C bond in H 3 C-CH 3 if is x & the number of distinct conformers generated due to rotation of C-C bond in DH C-CH D if is y, then what is the value of (x + y)?. For the given molecular formula C 7 H 8 O, what is the total number of benzenoid isomers possible 3. How many of the following compounds will exhibit tautomerism? O CH 3 O OH OH O O N N O NH N N H N H N N HCNO OH O 4. In how many of the following compounds, the gauche form generated due to rotation about C-C3 or C1- C is more stable than the anti-form generated during the rotation about the same C-C bond? Ethane, Propane, 1,-ethanediol, -fluoroethanol, 1-fluoroethanol, 1,-difluoroethane, -amino propan-1- ol,,3-butanediol, -fluorobutan-1-ol, 3-nitrobutan--ol

29 5. The given P-V isotherm represents the variation of P vs V at certain temperatures. Each of these isotherms have their respective temperatures marked against them. The isotherm, corresponding to which temperature represents the critical isotherm of the given real gas, which would be utilized in identification of critical temperature. P V % of a mixture of. mole of N and.6 mole of H react to given NH 3 according to the equation. N + 3H NH 3 at constant temperature and pressure. Then the ratio of final volume to the initial volume of gases is x. Then 5x is equal to. 7. How many of the following statements are correct in context to the given question? The given figure bulb contains an excess of volatile solid A. At 5 C equilibrium formed as shown below. The following two equilibriums are simultaneously existing in the bulb. ' Asolid Agas HS gas Asolid A gas At equilibrium; total pressure inside bulb is 1 mm whereas partial pressure of HS inside bulb is 4 mm. If some A(g) is removed from bulb at equilibrium I. decomposition equilibria will shift forward II. sublimation equilibria will shift forward III. total pressure when equilibrium reestablished will be equal to 1 mm. IV. no change will happen in the state of equilibrium

30 8. Solid phosphorus melts and vapourizes at high temperature. Gaseous phosphorus effuses at a rate that is.567 times that of Ne in the same apparatus under the same conditions. How many atoms are in a molecule of gaseous phosphorus. SECTION (Maximum Marks: 4) This section contains TEN questions. Each question has FOUR options (A), (B), (C) and (D). ONE OR MORE THAN ONE of these four option(s) is (are) correct. For each question, darken the bubble(s) corresponding to all the correct option(s) in the ORS. Marking scheme: +4 If only the bubble(s) corresponding to all the correct option(s) is (are) darkened. If none of the bubbles is darkened. In all other cases 9. Which of the following can exhibit geometrical isomerism (A) Azobenzene (B) Cyclohexene N OH (C) H 3 C H (D) Decalin 3. Keto-Enol tautomerism is exhibited by (A) Acetyl chloride (C) Trifluoro acetic acid (B) N-methyl acetamide (D) p-nitroso phenol 31. Which of the following statements is/are correct? (A) trans alkenes are generally more stable than cis alkenes (B) the boiling point of cis alkene is generally more than trans alkene (C) the melting point of cis alkene is generally less than trans alkene (D) the dipole moment of cis alkene is generally more than trans alkene 3. Which of the following statements is/are incorrect? H 3 C Cl H Br This compound can be named as: (A) only cis-1-bromo-1-chloro propene (C) (Z)-1-bromo-1-chloro propene (B) only trans-1-bromo-1-chloro propene (D) (E)-1-bromo-1-chloro propene

31 33. The NH 3 gas is liquefied more easily than N. Hence (A) Van der Wall s constant a and b of NH 3 > that of N (B) a and b of NH 3 < that of N. (C) a(nh 3 ) > a(n ) but b(nh 3 ) < b(n ) (D) the intermolecular force of attraction between the NH 3 molecules > that between the N molecules 34. Which of the following statements is/are correct for an ideal gas? (A) all gas molecules move with the same speed (B) kinetic energy of molecules remains unchanged due to collisions with other molecules (C) kinetic energy of gas molecules is dependent only on temperature (D) speed of molecules depends both on molecular mass and temperature 35. Which statements is/are correct for the compounds X & Y: O O HC X CH 3 C H 3 (A) X & Y are chain isomers (B) X & Y are functional isomers (C) X & Y have the same index of hydrogen deficiency (D) X & Y can undergo tautomerism Y H 36. For which of the following reactions, the degree of dissociation cannot be calculated from the vapour density data (A) HI(g) H (g) + I (g) (B) NH 3 (g) N (g)+3h (g) (C) NO(g) N (g) + O (g) (D)PCl 5 (g) PCl 3 (g) + Cl (g) 37. Two bulbs A and B of equal capacity are filled with He and SO respectively at the same temperature. Which of the following is incorrect? (A) At 1 K, the V rms of SO becomes half of the V rms of He molecules at 7 o C. (B) The velocity of SO decreases if volume of B becomes four times. (C) There is no change in velocity if half of the molecules of SO are removed from B. (D) The velocity of SO becomes double if half of SO molecules are removed from B. 38. Molecules with the molecular formula C 4 H 1 O exhibit which of the following type of isomerism (A) Chain (B) Positional (C) Functional (D) Metamerism

32 SECTION 3 (Maximum Marks: 16) This section contains TWO questions. Each question contains two columns, Column I and Column II Column I has four entries (A), (B), (C) and (D) Column II has five entries (P), (Q), (R), (S) and (T) Match the entries in Column I with the entries in Column II One or more entries in Column I may match with one or more entries in Column II. The ORS contains a 4 5 matrix whose layout will be similar to the one shown below: (A) (P) (Q) (R) (S) (T) (B) (P) (Q) (R) (S) (T) (C) (P) (Q) (R) (S) (T) (D) (P) (Q) (R) (S) (T) For each entry in Column I, darken the bubbles of all the matching entries. For example, if entry (A) in Column I matches with entries (Q), (R) and (T), then darken these three bubbles in the ORS. Similarly, for entries (V), (C) and (D). Marking entry in Column I. + If only the bubble(s) corresponding to all the correct match (s) is (are) darkened. If none of the bubbles is darkened. 1 In all other cases. 39. Match the following: Column I (A change has been made to the given system at equilibrium) (A) P is increased for (B) (C) (D) Column II (Effect of change on the system) Ice S H O L (p) Entropy of the system will increase Volume of the container is increased for NH g HCl g NH Cl S 3 4 Inert gas is added at constant volume for Fe S 4 H O S Fe O S 4H g 3 4 (q) Temperature is increased for CaCO S CaO S CO H (s) 3 (r) Free energy of the system will decrease Equilibrium position will shift to right Magnitude of K b (rate constant for backward direction) will rise. 4. Match the contents of Column I with contents of Column II for 1 mole of gas. Column I Column II (A) When force of attraction among gas PV = RT (p) molecules is negligible (B) At STP (q) (P + a/v )(V b) = RT (C) At low pressure and high temperature (r) PV = RT a/v (D) When volume of gas molecules is PV = RT + Pb (s) negligible

33 PART III: MATHEMATICS SECTION 1 (Maximum Marks: 3) This section contains EIGHT questions. The answer to each question is a SINGLE DIGIT INTEGER ranging from to 9, both inclusive. For each question, darken the bubble corresponding to the correct integer in the ORS. Marking scheme: +4 If the bubble corresponding to the answer is darkened. In all other cases. 41. If x1 y x y 5 15, then the value of x 1 + x + x 3 y 1 y y 3 is 1 1 x3 y Let A = and (1)B = 4 5. If B is the inverse of A, then the value of is If A, B and C are n n matrix and det (A) =, det (B) = 3, and det (C) = 5, then the value of [det (A BC)] is k, then k 3 is 44. If A = 3A I, and A 8 = pa + qi, then p + q is 45. If n(u) = 6, n(a) = 35, n(b) = 4, n(a B) = 1, then n(a B) = 46. In a group of 6 people, 8 like pepsi but not coke and 16 like pepsi. Number of people who like coke but not pepsi is k, then k 1 is 47. If U = {1,, 3, 4, 5, 6, 7, 8, 9, 1}, A = {1,, 3, 4}, B = {, 4, 6, 8, 1}, then n((a B)) = 48. The number of non-empty subsets of set {1,, 3, 4} is k, then k 7 is

34 SECTION (Maximum Marks: 4) This section contains TEN questions. Each question has FOUR options (A), (B), (C) and (D). ONE OR MORE THAN ONE of these four option(s) is (are) correct. For each question, darken the bubble(s) corresponding to all the correct option(s) in the ORS. Marking scheme: +4 If only the bubble(s) corresponding to all the correct option(s) is (are) darkened. If none of the bubbles is darkened. In all other cases 49. If determinant cos sin cos sin cos sin cos sin cos (A) positive (B) independent of (C) independent of (D) none of these If A = 1, B = a 1 b 1, and (A + B) = A + B + AB, then (A) a = 1 (B) a = 1 (C) b = (D) b = If A = 1 3 is an orthogonal matrix, then a b (A) a = (B) a =, b = 1 (C) b = 1 (D) b = tan 1 tan a b 5. If tan 1 tan 1 b a, then (A) a = cos (B) a = 1 (C) b = sin (D) b = 1 is

35 If A 1 = 3 1, then 1/ (A) A = 1 (B) adj A = 3 1 1/ 3 1 1/ / 3 7 (C) A = 1/ 3 1 (D) A = If = sin cos sin sin cos coscos cos sin sin, then sinsin sin cos (A) is independent of (B) is independent of (C) is a constant (D) = 55. If (, ) = cos sin 1 sin cos 1, then cos sin 1 (A) f(3, ) = f(4, ) (B) f(, 4) = f(, 6) (C) f(1, ) = f(, ) (D) none of these 56. The determinant = 57. If f(x) = a x ab ac ab b x bc ac bc c x is divisible (A) x (B) x (C) x 3 (D) none of these a 1 ax a 1, then f(x) f(x) is divisible by ax ax a (A) x (B) a (C) a + 3x (D) x

36 58. If (x) = x 4x 3 x 4 13 x 5x 9 4x 5 6 8x 6x 1 16x 6 14 = ax 3 + bx + cx + d, then (A) a = 3 (B) b = (C) c = (D) none of these SECTION 3 (Maximum Marks: 16) This section contains TWO questions. Each question contains two columns, Column I and Column II Column I has four entries (A), (B), (C) and (D) Column II has five entries (P), (Q), (R), (S) and (T) Match the entries in Column I with the entries in Column II One or more entries in Column I may match with one or more entries in Column II. The ORS contains a 4 5 matrix whose layout will be similar to the one shown below: (A) (P) (Q) (R) (S) (T) (B) (P) (Q) (R) (S) (T) (C) (P) (Q) (R) (S) (T) (D) (P) (Q) (R) (S) (T) For each entry in Column I, darken the bubbles of all the matching entries. For example, if entry (A) in Column I matches with entries (Q), (R) and (T), then darken these three bubbles in the ORS. Similarly, for entries (V), (C) and (D). Marking entry in Column I. + If only the bubble(s) corresponding to all the correct match (s) is (are) darkened. If none of the bubbles is darkened. 1 In all other cases. 59. Column I (A, B, C are matrices) (A) If A =, then A 1 = (where A is of order 3) (B) If A = 1, then adj (adj (A)) = 8 p. 1 q. 4 Column II (where A is of order 3) (C) If (A + B) = A + B, and A =, then B = (where A and B are of odd order) (D) If A =, B = 3 and C = 4, then ABC is equal to r. 4 s. t. Does not exist

37 6. Column I (A) 1 1 a b c c c b c 1 1 a a a b b c a b c b a b a c ac ac (B) sinacosb sina sinb cosa (C) (D) cosacosb cosa sinb sina is sina sinb sinacosb sinacosb sina sinb cosa cosa cosa sina sin acosb sin a sinb cos a sinb cosb sinacos b sina sin b If a, b and c are the sides of a triangle and A, B and C are the angles opposite to a, b and c, a bsina c sina respectively, then = is bsina 1 cos A c sina cos A 1 Column II p. Independent of a q. Independent of b r. Independent of c s. Dependent on a, b

38 PHYSICS CHEMISTRY FIITJEE RET 13 (17 19)(1 ST YEAR_REGULAR) IIT-15 (P1)_SET B DATE: ANSWERS A,B,C,D 1. B,C 11. C 1. B 13. B 14. D 15. C 16. D 17. A,B,C,D 18. A,C 19.. At, Br, Cq, Ds A, C, D 3. A, B, D 31. A, B, C, D 3. A, B, C 33. C,D 34. B,C 35. B, C, D 36. A, C 37. B, C, D 38. A, B, C, D 39. A p, q, r; B p, s ; C p ; D p, q, r, s 4. A s ; B p,q ; C p ; D r MATHEMATICS AB 5. AD 51. AC 5. AC 53. ABC 54. B 55. AC 56. AB 57. ABC 58. BC 59. A q; B p; C s; D r 6. A p, q, r; B q; C s; D p, q, r

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