HALF COURSE TEST VII. Time Allotted: 3 Hours Maximum Marks: FIITJEE Students from All Programs have qualified in JEE (Advanced), 2015.

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1 FIITJEE Students From All Programs have bagged 33 in Top 100, 77 in Top 00 and 05 in Top 500 All India Ranks. FIITJEE Performance in JEE (Advanced), 015: 4551 FIITJEE Students from All Programs have qualified in JEE (Advanced), 015. FIITJEE JEE (Advanced), 016 HALF COURSE TEST VII ALL INDIA INTEGRATED TEST SERIES Paper 1 Time Allotted: 3 Hours Maximum Marks: 46 Pleas e r ead th e instructions carefully. You are allotted 5 minutes specifically for this purpose. You are not allowed to leav e th e Exam ination Hall before the end of the test. A. General Instructions INSTRUCTIONS 1. Attempt ALL the questions. Answers have to be marked on the OMR sheets.. This question paper contains Three Parts. 3. Part-I is Physics, Part-II is Chemistry and Part-III is Mathematics. 4. Each part is further divided into three sections: Section-A, Section-B & Section-C 5. Rough spaces are provided for rough work inside the question paper. No additional sheets will be provided for rough work. 6. Blank Papers, clip boards, log tables, slide rule, calculator, cellular phones, pagers and electronic devices, in any form, are not allowed. B. Filling of OMR Sheet 1. Ensure matching of OMR sheet with the Question paper before you start marking your answers on OMR sheet.. On the OMR sheet, darken the appropriate bubble with black pen for each character of your Enrolment No. and write your Name, Test Centre and other details at the designated places. 3. OMR sheet contains alphabets, numerals & special characters for marking answers. C. Marking Scheme For All Three Parts. 1. Section A (01 07) contains 7 multiple choice questions which have only one correct answer. Each question carries +3 marks for correct answer and 1 mark for wrong answer. Section A (08 11) contains 4 multiple choice questions which have one or more than one correct answer. Each question carries +3 marks for correct answer. There is no negative marking. Section-A (1 16) contains paragraphs. Based upon paragraph, and 3 multiple choice questions have to be answered. Each question has only one correct answer and carries +3 marks for correct answer and 1 mark for wrong answer.. Section B (01 0) contains Matrix Match Type questions containing statements given in columns. Statements in the first column have to be matched with statements in the second column. Each question carries +8 marks for all correct answer. For each correct row + marks will be awarded. There may be one or more than one correct matching. No marks will be given for any wrong matching in any question. There is no negative marking. 3. Section-C (01 06) contains 6 Numerical based questions with answers as numerical value from 0 to 9 and each question carries +3 marks for correct answer. There is no negative marking. Name of the Candidate Enrolment No.

2 AIITS-HCT-VII (Paper-1)-PCM-JEE(Advanced)/16 Useful Data PHYSICS Acceleration due to gravity g = 10 m/s Planck constant Charge of electron Mass of electron h = J-s e = C m e = kg Permittivity of free space 0 = C /N-m Density of water water = 10 3 kg/m 3 Atmospheric pressure P a = 10 5 N/m Gas constant R = J K 1 mol 1 CHEMISTRY Gas Constant R = J K 1 mol 1 = Lit atm K 1 mol 1 = Cal K 1 mol 1 Avogadro's Number N a = Planck s constant h = Js = ergs 1 Faraday = coulomb 1 calorie = 4. joule 1 amu = kg 1 ev = J Atomic No: H=1, He =, Li=3, Be=4, B=5, C=6, N=7, O=8, N=9, Na=11, Mg=1, Si=14, Al=13, P=15, S=16, Cl=17, Ar=18, K =19, Ca=0, Cr=4, Mn=5, Fe=6, Co=7, Ni=8, Cu = 9, Zn=30, As=33, Br=35, Ag=47, Sn=50, I=53, Xe=54, Ba=56, Pb=8, U=9. Atomic masses: H=1, He=4, Li=7, Be=9, B=11, C=1, N=14, O=16, F=19, Na=3, Mg=4, Al = 7, Si=8, P=31, S=3, Cl=35.5, K=39, Ca=40, Cr=5, Mn=55, Fe=56, Co=59, Ni=58.7, Cu=63.5, Zn=65.4, As=75, Br=80, Ag=108, Sn=118.7, I=17, Xe=131, Ba=137, Pb=07, U=38.

3 3 AIITS-HCT-VII (Paper-1)-PCM-JEE(Advanced)/16 Physics PART I SECTION A Straight Objective Type This section contains 7 multiple choice questions numbered 1 to 7. Each question has 4 choices (A), (B), (C) and (D), out of which ONLY ONE is correct 1. At the middle of the mercury barometer tube there is a little column of air with the length l 0 and there is vacuum at the top as shown. Under the normal atmospheric pressure and the temperature of 300 kelvin, l 0 = 10 cm. Neglect expansion of the tube. The length of the air column if the temperature rises to 330 Kelvin will be: (A) 100 cm (B) 11 cm 11 (C) 10 cm (D) 1 cm air column l 0 vacuum. Two cars A and B are moving towards each other with same speed 5 m/s. Wind is blowing with speed 5 m/s in the direction of motion of car A. The car A blows horn of frequency 300 Hz and the sound is reflected from car B. The wavelength of the reflected sound received by the driver of car A is (velocity of sound in air = 330 m/s) (A) 5 m 36 (C) 35 m 36 (B) 31 m 36 (D) 37 m A particle is moving along the x axis is acted upon by a single force F = F 0 e kx, where F 0 and k are constants. The particle is released from rest at x = 0. It will attain a maximum kinetic energy of : F0 F0 (A) (B) k k ke kf0 (C) (D) ke k F k 0 e

4 AIITS-HCT-VII (Paper-1)-PCM-JEE(Advanced)/ First 10 cm piece of a uniform meter strick is cut and kept on the other end as shown in figure. Find the shift in the x-coordinate of center of mass (A) 5 cm (B) 10 cm (C) 9 cm (D) 45 cm 5. A satellite is in a circular orbit very close to the surface of a planet with speed v 0. At some point it is given an impulse along its direction of motion, causing its velocity to increase times (that is, its speed becomes v 0 ). It now goes into an elliptical orbit. The maximum possible value of for satellite to move in the elliptical orbit is: (A) (B) 1 (C) + 1 (D) 6. An elevator is moving upward with an acceleration a. It has velocity v at t = 0. A man inside the elevator lifts a body of mass m through a height h in interval t = 0 to t = t 0. The average power developed by the man is in elevator ref. frame : m(g a)h (A) (B) m(g + a) (v + 1 t at 0) 0 mgh (C) (D) mg (v + 1 t0 at 0) 7. A thermally insulated vessel contains some water at 0 C. The vessel is connected to a vacuum pump to pump out water vapour. This results in some water getting frozen. It is given Latent heat of vaporization of water at 0 C = J/kg and latent heat of freezing of water = J/kg. The maximum percentage amount of water that will be solidified in this manner will be : (A) 86.% (B) 33.6% (C) 1% (D) 4.36% 1

5 5 AIITS-HCT-VII (Paper-1)-PCM-JEE(Advanced)/16 Multiple Correct Choice Type This section contains 4 multiple choice questions. Each question has 4 choices (A), (B), (C) and (D) for its answer, out of which ONE OR MORE is/are correct 8. As a wave propagates : (A) The wave intensity remains constant for a plane wave (B) The wave intensity decreases as the inverse of the distance from the source for a spherical wave (C) The wave intensity decreases as the inverse square of the distance from the source for a spherical wave (D) Total power of the spherical wave over the spherical surface centered at the source decreases with distance. 9. A block of mass kg is hanging over a smooth and light pulley through a light string. The other end of the string is pulled by a constant force F = 40 N. At t = 0 the system is at rest as shown. Then in the time interval from t = 0 to t = 10 seconds, pick up the correct statement (s) : (g = 10 m/s ) (A) tension in the string is 40 N (B) work done by gravity is 0 J (C) work done by tension on block is 80 J (D) None of these kg F=40N 10. A solid cube of mass 5 kg is placed on a rough horizontal surface, in xy-plane as shown. The friction coefficient between the surface and the cube is 0.4. An external force F 3i ˆ 4j ˆ 0kˆ N is applied on the cube. (use g = 10 m/s (A) The block starts slipping over the surface z (B) The friction force on the cube by the surface is 5 N. y (C) The friction force acts in xy-plane at angle 17 with the positive x-axis in clockwise direction. (D) The contact force exerted by the surface on the cube is 95 N x 11. An ideal monoatomic gas undergoes a processes in which P = 3V 3. Choose the correct statements. (A) If volume increases then temperature will also increase (B) If temperature of the gas decreases, then pressure must also decrease. (C) For expansion, heat will have to be supplied to the gas. (D) If temperature increases then work done by gas is positive.

6 AIITS-HCT-VII (Paper-1)-PCM-JEE(Advanced)/16 6 Comprehension Type This section contains paragraphs. Based upon one of paragraphs multiple choice questions and based on the other paragraph 3 multiple choice questions have to be answered. Each of these questions has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct Paragraph for Questions 1 and 13 In an organ pipe (may be closed or open) of 99 cm length standing wave is setup, whose equation is given by longitudinal displacement = (0.1 mm) cos (y + 1 cm) cos (400) t 0.8 where y is measured from the top of the tube in meters and t in second. Here 1 cm is the end correction. 1. The air column is vibrating in (A) First overtone (C) Third harmonic (B) fifth harmonic (D) Fundamental mode 13. Equation of the standing wave in terms of excess pressure is (Bulk modulus of air B = N/m ) (A) P ex = (15 N/m ) sin (y + 1 cm) cos (400t) 0.8 (B) P ex = (15 N/m ) cos (y + 1 cm) sin (400t) 0.8 (C) P ex = (5 N/m ) sin (y + 1 cm) cos (00t) 0.8 (D) P ex = (5 N/m ) cos 0.8 (y + 1 cm) sin (00t)

7 7 AIITS-HCT-VII (Paper-1)-PCM-JEE(Advanced)/16 Paragraph for Question Nos. 14 to 16 A rod of mass m and length l is free to rotate in a vertical plane about fixed horizontal axis O passing through one end as shown in the figure. A sphere of mass m 4 hits the rod elastically with horizontal speed v 0 at a distance from O. Before collision, the rod is at rest in vertical position. 3 O l 3 m 4 If we consider the sphere and the rod as a system, then gravity and force due to axis O are external forces. For the rod alone, force exerted by O, gravity and impact force are all external forces. Similar conclusion can be drawn for the sphere. Net external force on any system is equal to the product of the mass of the system and the acceleration of the center of the mass of the system. 14. The maximum angular deflection of the rod from the vertical after the collision is (A) (C) 1 5 v0 cos 1 16 g 1 3 v0 cos 1 16 g 15. The velocity of the sphere just after collision is v0 v0 (A) (B) 7 v0 (C) 7 (D) v 0 (B) (D) 1 3 v0 sin 1 16 g 1 5 v0 sin 1 16 g 16. Magnitude of the impulse on the rod exerted by the axis O during the collision is 3mv (A) zero (B) 0 8 3mv (C) 0 3 (D) mv

8 AIITS-HCT-VII (Paper-1)-PCM-JEE(Advanced)/16 8 SECTION B (Matrix-Match Type) This section contains questions. Each question contains statements given in two columns which have to be matched. Statements (A, B, C, D) in column I have to be matched with statements (p, q, r, s, t) in column II. The answers to these questions have to be appropriately bubbled as illustrated in the following example. If the correct match are A-p, A-s, B-q, B-r, C-p, C-q and D-s, then the correctly bubbled 4 4 matrix should be as follows: A B C D p q r s p q r s p q r s p q r s p q r s t t t t t 1. Match the following : A block P of mass m placed on smooth horizontal surface and connected with a spring. Another block Q of same mass is placed on the block P. The two blocks are pulled by a distance A and released. The blocks oscillates without slipping on each other. Then match the matrix. Q P Column I (A) Friction force on the block Q, When both block coming towards equilibrium position form left. (exclude the equilibrium position) (B) Friction force at the block Q when both block going right from equilibrium position. (exclude the equilibrium position) (C) Friction force will be maximum on Q. (D) Friction force on P when both block is coming towards the equilibrium position from right. (exclude the equilibrium position) Column II (p) towards right (q) towards left (r) at equilibrium position (s) at extreme position (t) none of these

9 9 AIITS-HCT-VII (Paper-1)-PCM-JEE(Advanced)/16. Match the following columns. Column I (A) A B C K 0 3K 0 6K 0 100C T 10C Three homogenous thermal resistors A, B and C, each of length, (p) Column II (Temperature T in steady state) 50C cross-sectional area A and thermal conductivity K 0, 3K 0 and 6K 0 respectively are joined in series such that there is one dimensional heat flow from left end (temperature 100C) to the right end (temperature 10C). T is the temperature of the interface between resistor B and C in the steady state. (B) 5R 3R 100C R T 10C A spherical annular thermal resistor of thermal conductivity K and inner and outer radii R and 5R respectively, is in steady state with its inner surface maintained at 100C and its outer surface maintained at 10C(heat flow is radially outwards). The temperature of the concentric spherical interface of radius 3R in steady state is T. (q) 70C (C) 10C 8R 100C R R T A cylindrical and thermal resistor of inner radius R and outer radius 8R is in steady state with its inner surface maintained at 100C and its outer surface maintained at 10C(heat flow is radially outwards). The temperature of the co-axial cylindrical interface of radius R in steady state is T. (r) 55C (D) A L L (s) 5C 6R 100C b B 3R 10C T A homogeneous thermal resistor whose outer surface is the frustom of a solid cone, has the two base radii as 6R and 3R respectively and the perpendicular distance between its, two bases is 3L. The two bases are maintained at 100C and 10C. In the steady state(heat flows from left end to the right end) the temperature of the interface at a distance of L from the left end is T. Assume that L is very large in comparison to R and the lateral surface of the cylinder is thermally insulated from the surroundings. Find T. (t) none of these

10 AIITS-HCT-VII (Paper-1)-PCM-JEE(Advanced)/16 10 SECTION C Integer Answer Type This section contains 6 questions. The answer to each of the questions is a single-digit integer, ranging from 0 to 9. The appropriate bubbles below the respective question numbers in the ORS have to be darkened. For example, if the correct answers to question numbers X, Y, Z and W (say) are 6, 0, 9 and, respectively, then the correct darkening of bubbles will look like the as shown X Y Z W Water at temperature 0º C flows from a tap T into a heated container C. The container has a heating element (a resistor R) which is generating heat at the rate of P, that may be varied. The rate of water in flow from trap is F = 0.16 cc/minute. The heat generated is sufficient so that the water in the container is boiling and getting converted into steam at a steady rate. What is the minimum power P (in watt) that must be generated as heat in the steady state in resistor R so that the amount of liquid water in the container neither increases nor decreases with time? (Neglect other losses of heat, such as conduction from the container to the air and heat capacity of container) For water, specific heat c = 4. kj.kg 1 K 1, latent heat of vaporisation L vap =.3 MJ.kg 1, density p = 1000 kg.m 3.. A cylinder of height h, diameter h/ and mass M and with a homogeneous mass distribution is placed on a horizontal table. One end of a string running over a pulley is fastened to the top of the cylinder, a body of mass m is hung from the other end and the system is released. Friction is negligible everywhere. At what minimum ratio m/m will the cylinder tilt? h/ h M m

11 11 AIITS-HCT-VII (Paper-1)-PCM-JEE(Advanced)/16 3. Shown in the figure is a system of three particles connected by two springs. The accelerations of A, B and C at any instant are 1 m/sec, m/sec and (1/) m/sec, respectively, directed as shown in the figure. External force acting on the system is (m 1 = 1kg, m = kg and m 3 = 4kg) a 1 a a 3 A B C m 1 m m 3 4. In a lawn a water sprinkler is sprinkling water in all the possible directions while rotating. What is maximum area (in m ) on ground that can be sprinkled? The speed of water form nozzle is 400 1/ 4 m / s (g = 10 m/s ) 5. A horizontal board is being moved with a constant velocity v 0 on a smooth horizontal plane. A small block is projected on the block with velocity v 1 = 5 m/s at an angle = 37 relative to the ground. The coefficient of friction between the block and the board is = 0.3. If velocity of block (in m/s) relative to ground after time t = 10 sec. is v than v/10 is equal to.. 6. The 1/v versus displacement graph of a particle is shown in the figure, where v is the speed of the particle. The particle is moving in a straight line along positive x-axis. Calculate the time taken (in seconds) by the particle to reach from the point A to B. 1/v (s/m) 4 B A 0 x (m)

12 AIITS-HCT-VII (Paper-1)-PCM-JEE(Advanced)/16 Chemistry 1 PART II SECTION A Straight Objective Type This section contains 7 multiple choice questions numbered 1 to 7. Each question has 4 choices (A), (B), (C) and (D), out of which ONLY ONE is correct 1. Find the equivalent weight of H 3 PO 4 in the following reaction: Ca OH H PO Ca H PO H O (A) 98 (B) 49 (C) 3.7 (D) 37.. Which one of these does have greatest second ionization energy? (A) B (B) C (C) Sr (D) Ba 3. Ag NH aq 3aq AgNH3 aq What is the relationship between K c and K sp (AgCl)? 1 K NH sp 3 (A) K c AgNH3 Cl K Cl sp (C) K c AgNH3 NH3 (B) (D) K K K sp 3 c AgNH3 Cl NH K Ag NH sp 3 NH3 Cl c 4. The shape of XeOF is similar to (A) T (B) (C) (D)

13 13 AIITS-HCT-VII (Paper-1)-PCM-JEE(Advanced)/16 5. Find de Broglie wavelength of the electron in the first Bohr s orbit: 10 (A) 3 10 m 10 (B) 4 10 m 11 (C) 3 10 m 11 (D) 4 10 m 6. CH 3 H 3 C CH C CH (A) CH3 CH 3 CH HBr /HO (B) H 3 C CH C CH3 CH 3 H 3 C CH C CH3 (C) Br H 3 C CH CH CH CH 3 Br CH 3 (D) Br CH 3 H 3 C CH CH C CH 3 Br 7. For the following reaction: 1 COg O g CO g at 98K The standard entropy change is kj mol 1. The values of standard Gibbs free energy of formation of CO and CO are and 137. kj mol 1 respectively. The reaction is (A) non-spontaneous, endothermic (B) non spontaneous, exothermic (C) spontaneous, endothermic (D) spontaneous, exothermic Multiple Correct Choice Type This section contains 4 multiple choice questions. Each question has 4 choices (A), (B), (C) and (D) for its answer, out of which ONE OR MORE is/are correct 8. Which of the following carbocation will be relatively more stable than benzyl carbocation? (A) (B) CH Ph CH (C) (D) CH 3 C Ph CH 3

14 AIITS-HCT-VII (Paper-1)-PCM-JEE(Advanced)/ Which of the following molecules/species contains atleast one atom with sp 3 hybridization? (A) ClO (B) SO 3 3 (C) BO 3 (D) H 3 PO Which of the following is correct about the given reaction sequence? Br 3 /h alc. KOH Hg OCOCH /THF, H O NaOI NaBH, OH A B C D E F 4 (A) Molecular weight of (F) is nth multiple of weight of a methyl radical (where n = positive integer) (B) Compound (C) on heating with conc. H SO 4 gives (B) (C) Compound (B) can not exhibit position isomerism (D) Compound (A) can exhibit position isomerism 11. Chlorine gas is prepared in laboratory by the reaction: K Cr O HCl KCl CrCl Cl H O 7 3 If a 61.3 g sample of 96% K Cr O 7 is allowed to react with 30 ml of hydrochloric acid solution having density 1.15 g/ml and containing 30% HCl by mass. Now select the correct statement (s) among the following: (A) the mass of Cl generated is 4.63 g (B) equivalent weight of K Cr O 7 is 49 g (C) equivalent weight of HCl is g (D) the stoichiometric coefficient of HCl is 14 Comprehension Type This section contains paragraphs. Based upon one of paragraphs multiple choice questions and based on the other paragraph 3 multiple choice questions have to be answered. Each of these questions has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct Paragraph for Question Nos. 1 to 13 The decomposition of N O 5 is according to the equation: NO 5 g 4NO O g is a first order reaction. After 30 minutes from the start of the decomposition in a closed vessel, the total pressure is found to be 94.5 mm Hg. On complete decomposition the total pressure is mm Hg. An inert gas with partial pressure 10 mm Hg has always been present in vessel during decomposition of N O What is the pressure of the vessel before the beginning of decomposition of N O 5? (A) 3.8 mm (B) 33.8 mm (C) 43.8 mm (D) 53.8 mm H3O 13. What is the pressure of NO after 30 minutes? (A) 67.6 mm (C) 65.6 mm (B) 66.6 mm (D) 64.6 mm

15 15 AIITS-HCT-VII (Paper-1)-PCM-JEE(Advanced)/16 Paragraph for Question Nos. 14 to 16 Study following molecules before answering questions: CH 3 H CH H 3 H H 3 C CH 3 H H 3 C CH 3 H H H 3 C CH What is the molecular formula of isomer of any of these molecules? (A) C 5 H 10 (B) C 5 H 1 (C) C 5 H 14 (D) C 4 H The number of pairs of diastereomers are (A) 1 (B) (C) 3 (D) One of the seven cyclic isomers is missing. What is the number of C atom constituting the cyclic ring? (A) 3 member ring, hence 3 C atoms (B) 4 members ring, hence 4 C atoms (C) 5 member ring, hence 5 C atoms (D) 6 member ring, hence 6 C atoms

16 AIITS-HCT-VII (Paper-1)-PCM-JEE(Advanced)/16 16 SECTION B (Matrix-Match Type) This section contains questions. Each question contains statements given in two columns which have to be matched. Statements (A, B, C, D) in column I have to be matched with statements (p, q, r, s, t) in column II. The answers to these questions have to be appropriately bubbled as illustrated in the following example. If the correct match are A-p, A-s, B-q, B-r, C-p, C-q and D-s, then the correctly bubbled 4 4 matrix should be as follows: A B C D p q r s p q r s p q r s p q r s p q r s t t t t t 1. Match the following column I and column II: Column I (bond Order) (A) 0 (p) + N (B) 1.5 (q) N (C).5 (r) CO (D) 3 (s) O (t) He Column II (Molecule). Match the following: Column I (Quantity) (A) Degree of dissociation (p) T 1 (B) Pre-exponential factor (Arrhenius (q) L 0 T 0 equation) (C) Rate of reaction (r) M 0 L 0 (D) Activation energy (s) M 1 L 3 T 1 Column II (Dimensions) (T = Time, L = length, M = mol, W = mass) (t) W 1 L T

17 17 AIITS-HCT-VII (Paper-1)-PCM-JEE(Advanced)/16 SECTION C Integer Answer Type This section contains 6 questions. The answer to each of the questions is a single-digit integer, ranging from 0 to 9. The appropriate bubbles below the respective question numbers in the ORS have to be darkened. For example, if the correct answers to question numbers X, Y, Z and W (say) are 6, 0, 9 and, respectively, then the correct darkening of bubbles will look like the as shown X Y Z W Number of possible cyclic alcohols with formula C 5 H 10 O (excluding stereoisomer) is x than what is x/?. The number of pure sigma bonds in Cr O 7 will be. 3. Which number of hydrogen will be having least bond energy with carbon in given below compound? H() H(1) H(3) H (5) H(4) 4. Calculate the number of protons present in an acid whose 0.45 g requires 40 ml of N/ KOH solution. (MW = 90) 5. The ph of blood stream is maintained by a proper balance of H CO 3 and NaHCO 3 concentration. The volume of 10 M NaHCO 3 solution which is mixed with 10 ml sample of blood which is 4 M in H CO 3 in order to maintain a ph of 7.4 is x ml. K a for H CO 3 in blood is What will be the value of x/10? 6. If a mixture of 3 mol of H and 1 mole of N is completely converted in to NH 3, what would be the ratio of the initial and final volume at same temperature and pressure?

18 AIITS-HCT-VII (Paper-1)-PCM-JEE(Advanced)/16 18 Mathematics PART III SECTION A Straight Objective Type This section contains 7 multiple choice questions numbered 1 to 7. Each question has 4 choices (A), (B), (C) and (D), out of which ONLY ONE is correct 1. Two consecutive sides of a parallelogram are 4x + 5y = 0 & 7x + y = 0. If one diagonal is 11x + 7y = 9, then equation of other diagonal is (A) x + y = 0 (B) x + y = 0 (C) x y = 0 (D) none of these.. The curve x 3 + y 3 = 3axy is symmetric about (A) x-axis (C) y = x (B) y-axis (D) none of these If m i, i = 1,, 3, 4 are con-cyclic points, then the value of m 1 m m 3 m 4 is m i (A) 1 (B) 1 (C) 0 (D) none of these. 4. Coefficient of x r in the expansion of (1 x) 1/ is r! (A) (B) r! (C) r! r r! r! r r! (D) none of these.

19 19 AIITS-HCT-VII (Paper-1)-PCM-JEE(Advanced)/ If w is a complex cube root of unity then w + w (A) 1 (B) 0 (C) 9 (D) i. 6. If 1 + a + a + a a x = (1 + a)(1 + a )(1 + a 4 ) then x is (A) 3 (B) 5 (C) 7 (D) Ten persons, amongst whom are A, B and C to speak at a function. The number of ways in which it can be done if A wants to speak before B and B wants to speak before C is 10! (A) (B) 3! 7! 6 (C) 10P 3. 7! (D) none of these. is Multiple Correct Choice Type This section contains 4 multiple choice questions. Each question has 4 choices (A), (B), (C) and (D) for its answer, out of which ONE OR MORE is/are correct 8. If (1+ x + x ) n = a 0 + a 1 x + a x + +a n x n, then the value of a 0 + a 3 + a is (A) a 1 + a 4 +a (B) a + a 5 +a (C) 3 n-1 (D) none of these. 9. One vertex of the triangle of maximum area that can be inscribed in the curve z i =,is +i, remaining vertices is / are (A) -1+ i( + 3 ) (B) 1 i( + 3 ) (C) -1+ i( 3 ) (D) 1 i( 3 ) 10. If a, b, c be three unequal positive quantities in H.P. then (A) a c 100 > b 100 (B) a 3 +c 3 > b 3 (C) a 5 + c 5 > b 5 (D) a + c > b 11. If 1 sin A = sin A cos A, then A could lie in quadrant 4 (A) first (B) second (C) third (D) fourth

20 AIITS-HCT-VII (Paper-1)-PCM-JEE(Advanced)/16 0 Comprehension Type This section contains paragraphs. Based upon one of paragraphs multiple choice questions and based on the other paragraph 3 multiple choice questions have to be answered. Each of these questions has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct Read the following comprehension carefully: Paragraph for Question Nos. 1 to 13 Let f(n) denotes the number of different ways the positive integer n can be expressed as the sum of 1 s and s. For example, f(3) = 3, since 3 = = 1 + = + 1 f(4) = 5, since 4 = = = = = +. Note that order of the appearance of 1 s and s is important. Now, answer the following questions: 1. The value of f(6) is (A) 6 (B) 11 (C) 13 (D) f{f(6)} is equal to (A) 377 (B) 30 (C) 31 (D) 350

21 1 AIITS-HCT-VII (Paper-1)-PCM-JEE(Advanced)/16 Read the following comprehension carefully: Paragraph for Question Nos. 14 to 16 A coplanar beam of light emerging from a point source have equation x y + (1 + ) = 0, R. the rays of the beam strike an elliptical surface and get reflected. The reflected rays form another convergent beam having equation x y + (1 ) = 0, R. Further it is found that the foot of the perpendicular from the point (, ) upon any tangent to the ellipse lies on the circle x + y 4y 5 = 0. Now answer the following questions: 14. The eccentricity of the ellipse is equal to (A) 1 3 (B) 1 3 (C) 3 (D) The area of the largest triangle that an incident ray and the corresponding reflected ray can enclose with the axis of the ellipse is equal to (A) 4 5 (B) 5 (C) 5 (D) none of these 16. Total distance travelled by an incident ray and the corresponding reflected ray is the least if the point of incidence coincides with (A) an end of the minor axis (B) an end of the major axis (C) an end of the latus rectum (D) none of these

22 AIITS-HCT-VII (Paper-1)-PCM-JEE(Advanced)/16 SECTION B (Matching List Type) This section contains multiple choice questions. Each question has matching Column(s). The codes for the Column(s) have choices (A), (B), (C) and (D) out of which only ONE OR MORE is/are correct 1. Match the following Column-I with Column-II (A) Column I f(z) is a complex valued function f(z) = (a + ib)z where a, b R and a + ib = 1. It has the property that f(z) is always equidistant from 0 and z, then a b = (B) The number of all positive integers n = a 3 b (a, b 0) such that n 6 does not divide 6 n is (C) A is the region of the complex plane {z: z/4 and 4/ z have real and imaginary part in (0, 1)}, then [p] (where p is the area of the region A and [.] denotes the greatest integer function) is (D) If 3x + 4y + z = 5, where x, y, z R, then minimum value of6(x + y + z )is Column II (p) 5 (q) 0 (r) 6 (s) 5 (t) 10. Match the following Column-I with Column-II Column I Column II (A) sincos x (p) x R cos sin x (q) R n (B) 1 6 (C) tan( sinx) (r) x n, n (D) ln(tanx) (s) x n, n (t) x R n 8

23 3 AIITS-HCT-VII (Paper-1)-PCM-JEE(Advanced)/16 SECTION C Integer Answer Type This section contains 6 questions. The answer to each of the questions is a single-digit integer, ranging from 0 to 9. The appropriate bubbles below the respective question numbers in the ORS have to be darkened. For example, if the correct answers to question numbers X, Y, Z and W (say) are 6, 0, 9 and, respectively, then the correct darkening of bubbles will look like the as shown X Y Z W logb x 1. The value of a when a = 0. b = 5, x... is The value of c for which 7, where and are the roots of x + 7x + c = 0 is If coefficient of x 7 in ax bx 1 11 is equal to coefficient of x 7 in 1 ax bx 11, then ab =.. 4. In a ABC, a 3 cos(b C) + b 3 cos(c A) + c 3 cos (A B) = (abc) k then k is 5. The line L passes through the points of intersection of the circles x + y = 5 and x + y 8x + 7 = 0. The length of perpendicular from centre of second circle onto the line L is. 6. If the lines ax + y + 1 = 0, x + by + 1 = 0 and x + y + c = 0 are concurrent then (given a, b, c are distinct and different from 1) a 1 b 1 c