Arjuna JEE 2025 Board Pattern Test - 02 PDF

Arjuna JEE (2025) BOARD PATTERN TEST - 02 DATE : 10/11/2024 M. MARKS : 220 Topics Covered Physics :...

Arjuna JEE (2025) BOARD PATTERN TEST - 02 DATE : 10/11/2024 M. MARKS : 220 Topics Covered Physics : Units and Measurements (Complete Chapter), Mathematical Tools (Complete Chapter), Motion in a Straight Line (Complete Chapter), Motion in a Plane (Complete Chapter), Laws of Motion (Complete Chapter), Circular Motion (Complete Chapter), Work, Energy and Power (Complete Chapter), Centre of Mass & System of Particles (Complete Chapter) Chemistry : Atomic Structure (Full Chapter), Mole Concept (Full Chapter), States of Matter (Full Chapter), Thermodynamics (Full Chapter), Chemical equilibrium (Full Chapter), Ionic Equilibrium (Full Chapter), Redox reaction (Full Chapter), Classification of Elements and Periodicity in Properties (Full Chapter), Chemical Bonding and Molecular Structure (Full Chapter), P-block Elements (Group 13 and 14) (Full Chapter) Mathematics : Basic Mathematics (Complete Chapter), Sets (Complete Chapter), Relation Function (Complete Chapter), Trigonometric Functions (Complete Chapter), Trigonometric Equation (Complete Chapter), Quadratic Equations (Complete Chapter), Complex Number-I (Complete Chapter), Sequence and Series (Complete Chapter), Permutations and Combinations (Complete Chapter), Binomial theorem (Complete Chapter), Straight Lines (Complete Chapter), Circles (Various forms of equation of circle) General Instructions: 1. Immediately fill in the particulars on this page of the test booklet. 2. The test is of 3 hours for each subject. 3. The test booklet consists of 104 questions. The maximum marks are 220. 4. This paper is divided Subject-wise in which Physics consists 5 Sections, Chemistry consists 4 Sections, Maths consists 5 Sections. Each type is mentioned in the paper. 5. Marking is mentioned with each type in the paper. 6. No student is allowed to carry any textual material, printed or written, bits of papers, pager, mobile phone, any electronic device, etc. inside the examination room/hall. 7. On completion of the test, the candidate must hand over the Answer Sheet to the Invigilator on duty in the Room/Hall. However, the candidates are allowed to take away this Test Booklet with them. 8. Do not fold or make any stray mark on the Answer Sheet (OMR). OMR Instructions: 1. Use blue/black dark ballpoint pens. 2. Darken the bubbles completely. Don't put a tick mark or a cross mark where it is specified that you fill the bubbles completely. Half-filled or over-filled bubbles will not be read by the software. 3. Never use pencils to mark your answers. 4. Never use whiteners to rectify filling errors as they may disrupt the scanning and evaluation process. 5. Writing on the OMR Sheet is permitted on the specified area only and even small marks other than the specified area may create problems during the evaluation. 6. Multiple markings will be treated as invalid responses. 7. Do not fold or make any stray mark on the Answer Sheet (OMR). Name of the Student (In CAPITALS) : _______________________________________________________________ Roll Number : _____________________________________________________________________________________________ OMR Bar Code Number : ________________________________________________________________________________ Candidate’s Signature : _______________________________ Invigilator’s Signature _____________________  SECTION-I (PHYSICS) SECTION - A 7. The position of a particle moving along x-axis is Very Short Answer Type Questions (1-10) (10 × 1 = 10) given by x = 10t – 2t2. Then the time (t) at which it 1. Which of the following four statements is true? will momentarily comes to rest is (1) A body can have zero velocity and still be (1) 0 (2) 2.5 s accelerated (3) 5 s (4) 10 s (2) A body can have a constant velocity and still have a varying speed 8. When a 4 kg rifle is fired, the 10 g bullet receives (3) A body cannot have a constant speed if it has an acceleration of 3 × 106 cm/s2. The magnitude of a varying velocity the force acting on the rifle (in newton) is (4) The direction of the velocity of a body cannot (1) Zero change when its acceleration is constant. (2) 120 (3) 300 2. In a projectile motion, the velocity: (4) 3000 (1) is always perpendicular to the acceleration (2) is never perpendicular to the acceleration 9. A person standing at the top of a building of 45 m (3) is perpendicular to the acceleration for one height drops a ball of mass 250 g. If acceleration instant only due to gravity, g = 10 ms–2, then the velocity with (4) is perpendicular to the acceleration for two which it hits the ground is instants: (1) 5.0 m/s (2) 10.0 m/s 3. A man is at rest in the middle of a pond on (3) 20.0 m/s perfectly smooth ice. He can get himself to the (4) 30.0 m/s shore by making use of newton's (1) First law 10. A heavy box is pushed across a rough floor with (2) Second law an initial speed of 4 m/s. It stops moving after (3) Third law 8 seconds. If the average resisting force of friction (4) Law of gravitation is 10 N, the mass of the box (in kg) is (1) 40 (2) 20 4.   A force F  3iˆ  5 ˆj  7kˆ N is applied over a (3) 5 (4) 2.5 particle which displaces it from origin to a point Assertion & Reason Type Questions (11–14) (4 × 1 = 4)   11. Assertion: Work and energy have same r  3iˆ  ˆj m. The work done on the particle is dimensional formula. (1) 9 J (2) 6 J Reason: Dimensional formula for work is (3) 4 J (4) 1 J [ML2T–2] (1) Both, A and R, are true and R is the correct 5. A body of mass 2 kg is released from the top of a explanation of A cliff. After travelling a vertical distance of 20 m, (2) Both, A and R, are true but R is not the its velocity becomes 15 m/s. Then the magnitude correct explanation of A of work done by the resistance of air will be (3) If A is true but R is false [Consider g = 10 m/s2] (4) If A is false but R is true (1) 400 J (2) 225 J 12. Assertion: The slope of momentum versus time (3) 175 J curve gives us the net force acting on the object. (4) 100 J Reason: Acceleration is given by the rate of 6. A force of 10 N acts upon an object of mass 2 kg change of velocity. for 5 s. If the object was initially at rest then what (1) Both, A and R, are true and R is the correct is the final velocity of the object? explanation of A (1) 25 m/s (2) Both, A and R, are true but R is not the (2) 40 m/s correct explanation of A (3) 75 m/s (3) If A is true but R is false (4) 90 m/s (4) If A is false but R is true 13. Assertion: Two objects of different masses (1 kg Energy is basically the capacity of doing work and 1.5 kg) are projected with same speed at same (depends upon the force applied and displacement angle of projection. The maximum height attained produced in the direction of the force applied) and by both the objects will be same. this energy can neither be created nor be destroyed Reason: The maximum height attained by the but it can be changed from one form to another projected object is independent of the mass of the such that the appearing energy is in the form of object. disappearing energy. The most common types of (1) Both, A and R, are true and R is the correct energy are kinetic energy and potential energy. explanation of A Kinetic energy is basically the energy possessed (2) Both, A and R, are true but R is not the by the body by virtue of its velocity. The relation correct explanation of A between kinetic energy and momentum can also (3) If A is true but R is false be established of an object which is under motion. (4) If A is false but R is true The change in kinetic energy is equal to the work done on it by the net force. 14. Assertion: When the momentum of an object is (a) If the momentum of an object is tripled, then doubled, its kinetic energy becomes 4 times. what will be the change in its kinetic energy? Reason: Kinetic energy of an object is directly (b) How much work is done by the net force during proportional to the square of its momentum. the displacement from x = 0 to 2 m if a particle (1) Both, A and R, are true and R is the correct of mass 1 kg travels in a straight line with a explanation of A velocity v = 5x2? (2) Both, A and R, are true but R is not the (c) A rectangular block of mass 16 kg moves correct explanation of A horizontally on a frictionless horizontal surface (3) If A is true but R is false with a velocity of 8m/s. What is the work done (4) If A is false but R is true by normal reaction on the block? (d) Calculate the change in kinetic energy, when SECTION - B the velocity of a body of mass 1 kg moving Case Study Based Questions (15–16) (2 × 4 = 8) with an initial velocity of 4iˆ  3 ˆj m/s 15. Read the passage and answer the following questions: changes into 6iˆ  2 ˆj m/s. An object that is in flight after being thrown or projected is known as a projectile. Such projectile SECTION - C can be a football, a cricket ball, a baseball, or any Short Answer Type Questions (17–25) (9 × 2 = 18) other object. However, the motion of a projectile 17. Why do we have different units for the same may be thought of as the result of two separate, physical quantity? simultaneously occurring components of motions one along the horizontal direction without any 18. A body falls towards earth in air. Will its total acceleration and the other along the vertical mechanical energy be conserved during the fall? direction with constant acceleration due to the Justify. force of gravity. It was Galileo who first started this independence of the horizontal and the 19. A body is moved along a closed loop. Is the work vertical components of projectile motion. done in moving the body necessarily zero? If not, Consider a particle projected from the surface of state the condition under which work done over a the Earth with a speed of 30 m/s at an angle of 300 closed path is always zero. with the horizontal (Take, g = 9.8 m/s2) (a) What will be the time of flight of the particle? 20. A nucleus is at rest in the laboratory frame of (b) What will be the range of this particle? reference. Show that if it disintegrates into two (c) What will be the maximum height of the smaller nuclei the products must move in opposite particle? directions. (d) What is the equation of trajectory for this particle? 21. A uniformly moving cricket ball is turned back by hitting it with a bat for a very short time interval. 16. Read the passage and answer the following Show the variation of its acceleration with time questions: (Take acceleration in the backward direction as positive). 22. Explain why 30. Two balls of masses 12 kg and 14 kg approach (a) a horse cannot pull a cart and run in empty each other with velocities 12 m/s and 14 m/s and space, collide elastically. What will be their velocities (b) passengers are thrown forward from their after the collision? seats when a speeding bus stops suddenly SECTION - E 23. Show that for a projectile the angle between the Long Answer Type Questions (31 – 33) (3 × 5 = 15) velocity and the x-axis as a function of time is 31. What will be the acceleration of masses and given by tension in the string when two masses m1 = 16 kg  voy  gt  (t )  tan 1   and m2 = 12 kg are connected at the two ends of  vox  the light inextensible string that passes over a frictionless pulley? 24. A calorie is a unit of heat or energy and it equals about 4.2 J where 1 J = 1 kgm2 s–2. Suppose we employ a system of units in which the unit of mass equals  kg, the unit of length equals  m, the unit of time is  s. Show that a calorie has a magnitude 4.212  2 in terms of the new units. 25. Two masses of 10 kg and 20 kg respectively are connected by a massless spring as shown in figure. A force of 200 N acts on the 20 kg mass. At the instant shown the 10 kg mass has acceleration 12 m/s2 towards right. Find the acceleration of 20 kg mass at this instant. 32. Check by the method of dimensional analysis whether the following relation are correct. P (i) v  where v = velocity of sound and D SECTION - D P = pressure, D = density of medium. Short Answer Type Questions (26 – 30) (5 × 3 = 15) 1 F 26. Find v and a of the particle if the position of a (ii) n  , where n = frequency of vibration 2l m particle is given by r  iˆ  4t 2 ˆj  2tkˆ l = length of the string Also, calculate the magnitude of velocity and F = stretching force acceleration of the particle at t = 1s? m = mass per unit length of the string. 27. (a) Define scalar and vector quantities along with 33. (a) Show that the centre of mass of two particles three examples of each. divides the line joining the two particles in the (b) Consider a biker riding on an inclined circular inverse ratio of their masses. track uniformly. Mention all the forces that act on the bike. (b) Locate the centre of mass of the system consisting of three particles having mass m, 28. The resultant of two forces A and B which have a 3m and 2m that are placed at the corners of ratio 2 : 5 is 35N. If the angle between the two right-angled triangle of sides 3 cm, 4 cm and forces is 60o, then calculate the value of A and B. 5 cm. 29. A ball of mass 90 g falls from a height of 3 m and rebounds to a height of 1 m. What will be the value of impulse and average force between them if the time duration for which they are in contact is 0.1s? (Take g = 9.8 m/s2).  SECTION-II (CHEMISTRY) SECTION - A (i) If NA is Avogadro's number, then the Passage Type Questions (34–35) (2 × 4 = 8) number of valence electrons in 4.2 g of Passage-I nitride ions (N3–) is: 34. Read the passage given below and answer the (1) 4.2 NA (2) 2.4 NA following questions: (3) 1.6 NA (4) 3.2 NA Shape of the compound depend on type and number of electron pair around central atom. (ii) The vapour density of a gas is 11.2. The These electron pair repel each other and stay as volume occupied by 11.2 g of gas at NTP far as possible. The repulsion sequence is as will be Lone pair- Lone Pair > Lone Pair- Bond Pair > (1) 22.4 L (2) 11.2 L Bond Pair- Bond Pair (3) 1 L (4) 44.8 L (i) Choose the incorrect match Compound Structure (iii) The number of molecules in 16 g of (1) SnCl2 Linear methane is: (2) CO2 Linear 16 (1) 3.0 × 1023 (2)  10 23 (3) I3 Linear 6.022 (4) N 3 Linear (3) 6.022 × 1023 (4) 16  1023 3.0 (ii) Which of the given compound is planar? (1) XeF6 (2) XeF4 (iv) If 3.01 × 1020 molecules are removed from 98 mg of H2SO4, then the number of moles (3) SO 24  (4) All of these of H2SO4 left will be (1) 0.1 × 10–3 (2) 1.66 × 10–3 –2 (iii) In which of the following d z2 orbital take (3) 9.95 × 10 (4) 0.5 × 10–3 part in hybridisation? (1) dsp3 Objective Type Questions (36–49) (14 × 1 = 14) (2) sp3d2 36. The maximum number of molecules are in: (3) d2sp3 (1) 28 g of CO (4) All of these (2) 46 g of C2H5OH (3) 36 g of H2O (iv) The type of hybrid orbitals used by chlorine (4) 54 g of N2O5 atom in ClO 2 is 37. The shape of ClF3 molecule is (1) sp3 (2) sp2 (1) Triangular (2) Pyramidal (3) sp (4) sp3d (3) T-shape (4) Linear Passage-II 38. The wavelength of a spectral line for an 35. Read the passage given below and answer the electronic transition is inversely related to: following questions: (1) Number of electrons undergoing transition Atoms and molecules are so small in size that it (2) The nuclear charge of the atom is neither possible to count them individually nor (3) Velocity of an electron undergoing possible to determine their mass. These are transition counted collectively in terms of Avogadro's (4) The difference in the energy of energy number. The mass of Avogadro's number of levels involved in the transition atoms and molecules is known as gram atomic mass and gram molecular mass respectively. The 39. Which of the following elements do not belong volume occupied by Avogadro's number of to the family indicated? molecules of a gas or vapours is known as molar (1) Cu – Coinage metal volume. (2) Ba – Alkaline earth metal The following questions are multiple choice (3) Zn – Alkaline earth metal questions. Choose the most appropriate answer: (4) Xe – Noble gas 40. Which of the following equation was suggested 47. For the reaction by de Broglie? N 2 (g)  O 2 (g) 2NO (g) p If pressure is increased by reducing the volume (1) 2r = n (2)   h of the container then: nh (1) Total pressure at equilibrium will remain (3) r2 = n (4) 2 r  same  (2) Concentration of all the component at equilibrium will change 41. The first ionization potentials of Na, Mg, Al and (3) Concentration of all the component at Si are in the order equilibrium will remain same (1) Na < Mg > Al < Si (4) Equilibrium will shift in the backward (2) Na < Mg < Al > Si direction (3) Na > Mg > Al > Si (4) Na > Mg > Al < Si 48. Total number of orbitals associated with third shell will be 42. Which of the following has zero dipole moment? (1) 2 (2) 4 (3) 9 (4) 3 (1) CO2 (2) NH3 (3) NF3 (4) H2O 49. Which of the following has S = +ve? 43. Which of the following is not correct? (1) 2H H2 (g) (2) Boiling of egg (1) G is zero for a reversible reaction. (3) Crystallization of sugar (2) G is positive for a spontaneous reaction. (4) Formation of complex compound (3) G is negative for a spontaneous reaction. (4) G is positive for a non-spontaneous SECTION - B reaction. (Very Short Answer Questions) (50 – 58) (9 × 2 = 18) 44. Assertion: Electronic energy for hydrogen atom 50. What is the difference between molality and of different orbitals follow the sequence: molarity? 1 s < 2s = 2p < 3s = 3p = 3d. Reason: Electronic energy for hydrogen atom 51. Which of the following has the maximum bond depends only on 'n' and is independent of 'l' and angle? Why? 'm' values. H2O, CO2, NH3 CH4 (1) Both assertion and reason are correct and reason is the correct explanation for 52. Calculate w, q and U, when 0.75 mol of an assertion. ideal gas expands isothermally and reversibly at (2) Both assertion and reason are correct, but 27°C from a volume of 15 L to 25 L. reason is not the correct explanation for (log 5 = 0.699, log 3 = 0.477) assertion. (3) Assertion is true, but reason is false. 53. MnO 24  undergoes a disproportionation reaction (4) Assertion is false, but reason is true. in an acidic medium but MnO 4 does not. Give a reason. 45. Entropy change in a reversible adiabatic process is: 54. Discuss the pattern of variation in the oxidation (1) Zero states of (2) Always positive (i) B to Tl (3) Always negative (ii) C to Pb (4) Sometimes positive and sometimes negative 55. Using s, p, d, f notations, describe the orbital with the following quantum numbers 46. A base according to Bronsted concept is a (1) n  2,  1 substance which can: (2) n  2,  0 (1) lose pair of electron (3) n  5,  3 (2) donate protons (3) gain a pair of electrons (4) n  3,  2 (4) accept protons 56. A gas that follows Boyle’s law, Charle’s law and 62. Among the elements B, Al, C and Si Avogadro’s law is called an ideal gas. (a) Which has the highest first ionization Under what conditions a real gas would behave enthalpy? ideally? (b) Which has the largest atomic radius? 63. Calculate the average atomic mass of hydrogen 57. The species: H2O, HCO3 , HSO4 and NH3 can using the following data: act both as Bronsted acids and bases. For each % Natural case give the corresponding conjugate acid and Isotope Molar mass abundance conjugate base. 1 H 99.985 1 2 H 0.015 2 58. Compressibility factor, Z of a gas is given as Z = (pV/nRT). What is the value of Z for an ideal SECTION - D gas? Long Answer Type Questions (64 – 66) (3 × 5 = 15) 64. Find out the oxidation number of chlorine in the SECTION - C following compounds and arrange them in Short Answer Type Questions (59 – 63) (5 × 3 = 15) increasing order of oxidation number of chlorine. 59. Calculate the oxidation number of each sulphur NaClO4, NaClO3, NaClO, KClO2, Cl2O7, ClO3, atom in the following compounds: Cl2O, NaCl, Cl2, ClO2 (a) Na2S2O3 Which oxidation state is not present in any of the (b) Na2S4O6 above compounds? (c) Na2SO3 (d) Na2SO4 65. On the basis of the Le Chatelier principle explain how temperature and pressure can be adjusted to 60. What do you understand by: increase the yield of ammonia in the following (a) Inert pair effect reaction. (b) Allotropy and (c) Catenation N 2 (g)  3H 2 (g) 2NH3 (g) H = –92.38 kJ/mol 61. How can you predict the following stages of a What will be the effect of addition of argon to reaction by comparing the value of Kc and Qc? the above reaction mixture at constant volume? (i) Net reaction proceeds in the forward direction. 66. Describe hybridisation in the case of PCl5 and (ii) Net reaction proceeds in the backward SF6. The axial bonds are longer as compared to direction. equatorial bonds in PCl5 whereas in SF6 both (iii) No net reaction occurs. axial bonds and equatorial bonds have the same bond length. Explain SECTION-III (MATHEMATICS) Part - A Which of the following pair of sets are disjoint SECTION - I sets? (1) A and B Short Answer Type Questions (67-82) (16×1=16) (2) B and C 67. If set A = {1, 2, 3, 4}, then it can be written in set (3) C and A builder form as (4) None of these (1) A = {x : x  N and x  5} (2) A = {x : x  N and x < 5} 69. Two finite sets have m and n elements. The total (3) A = {x : x  N and 1 < x < 5} number of subsets of the first set is 56 more than (4) A = {x : x  W and x < 5} total number of subsets of second set. The values of m and n are (1) 7 and 6 68. Given that, (2) 6 and 3 A = {1, 2, 3, 4, 5, 6} (3) 5 and 1 B = {7, 8, 9, 10, 11} (4) 8 and 7 C = {6, 8, 10, 12, 14}.  x 2 5 1 78. Solution set of inequation 70. If   1, y     ,  , find the values of x and 3 3  3 3 ( x  1) 2 ( x 2  4)( x 2  5 x  6)  0 is y. ( x3  6 x 2 ) (1) 2 and 1 (2) 2 and 2 (1) x  (–6, –2]  {1, 2}  [3, ) (3) –2 and 1 (4) 1 and 1 (2) x  (–6, 2]  [3, ) (3) x  (–2, 6]  [8, ) 71. If A = {1, 2, 3}, B = {3, 4} and C = {4, 5, 6}, then (4) x  (–6, 2]  {1, 2}  [8, ) A × (B  C) is (1) {(1, 3), (2, 3), (3, 3)} (2) {(1, 4), (2, 4), (3, 4)}  sin 4 t  cos 4 t  1  Value of 3  6  sin t  cos6 t  1  79. is equal to (3) {(4, 1), (4, 2), (4, 3)}   (4) {(3, 4), (3, 5), (3, 6)} (1) 2 (2) 4 (3) 6 (4) 8 72. Let A = {1, 2, 3} and B = {–1, 0, 1}, then which of the following is not a relation from A to B? 80. The value of i2014 is equal to (1) R = {(1, –1), (2, 0), (3, 1)} (1) i (2) –i (2) R = {(1, –1), (1, 0), (2, 0), (2, 1)} (3) 1 (4) –1 (3) R = {(2, 0), (2, 1), (1, –1), (3, 1), (3, 0)} (4) R = {(1, –1), (2, 0), (1, 3)} 81. The multiplicative inverse of 4 – 3i is equal to 1 4 3i 4 3i 73. The domain of the real function f ( x)  is (1)  (2)  4  x2 25 25 25 25 (1) the set of all real numbers 4 3i (3)  (4) None of these (2) the set of all positive real numbers 16 25 (3) (–2, 2) (4) [–2, 2] 5  3 82. The 5th term in the expansion of  2 x 2   is  x  1 1 74. Let f  x    x 2  2 , x  R – {0}, then f(x) is equal to  x x (1) 810.x–2 (2) 810.x–4 equal to (3) 810 (4) 810.x–5 (1) x2 (2) x2 – 1 (3) x2 – 2, when |x|  2 SECTION - II (4) None of these Case Study Questions (83-84) (2×4=8) 83. A sequence whose terms increases or decreases by a fixed number, is called an Arithmetic Progression 75. The range of f(x) = 1 + 3cos2x is equal to (AP). In other words, we can say that, a sequence is (1) [2, 3] called an arithmetic progression if the difference of (2) [2, 4] a term and the previous term is always same i.e. (3) [–2, 4] an+1 – an = constant for all n. (4) None of the above This constant or same difference is called the common difference of an AP and it is denoted by d. 76. If α and β are roots of ax2 + bx + c = 0 then value of In an AP, we usually denote the first term by a, common difference by d and the nth term by an or 1 1  is equal to Tn defined as a  b a  b Tn = an = a + (n – 1)d a c Also, l = a + (n – 1)d, where l is the last term of the (1) (2) sequence. bc ab The sum of n terms, Sn of this AP is given by b (3) (4) None of these n ac S n  [2a  ( n  1) d ]. 2 Also, if l be the last term, then the sum of n terms 2x  4 77. Solution set of inequation  5 is of this AP is S n  n (a  l ). x 1 2 (1) x  (–, 2) (2) x  [7, ) Based on above information, answer the (3) x  (1, 3] (4) x  [4, 6] following questions.  (i) If nth term of an AP is given by an = 2n2 + 1, zz (iv) If z = 3 + 4i, then is equal to then its 10th term is equal to 2 (1) 200 (1) 1 (2) 2 (2) 301 (3) 3 (4) 4 (3) 400 (4) Given sequence is not an AP Part - B SECTION - III (ii) 11th term of an AP 11, 18, 25, … is equal to Short Answer Type Questions (85-94) (10×2=20) (1) 80 (2) 81 85. In a class of 35 students, 24 like to play cricket and (3) 71 (4) 70 16 like to play football. Also, each student likes to play at least one of the two games. How many (iii) If the sum of n terms of an AP is given by students like to play both cricket and football? S n = 3n + 2n 2 , then the common difference of the AP is 86. If f(x) = 3x4 – 5x2 + 9, find f(x – 1) (1) 3 (2) 2 (3) 6 (4) 4 87. A horse is tied to a post by a rope. If the horse moves along a circular path always keeping the rope (iv) Let Sn denote the sum of the first n terms of an tight and describes 88 metres when it has traced out AP, if S2n = 3Sn, then S3n : Sn is equal to 72° at the centre, find the length of the rope. (1) 4 (2) 6 (3) 8 (4) 10 88. Prove that, sin(–420°) cos(390°) + cos(–660°) (sin330°) = –1 84. The conjugate of a complex number z, is the complex number, obtained by changing the sign of 89. Prove that, imaginary part of z. It is denoted by z. sin( B  C ) sin(C  A) sin( A  B)   0 The modulus (or absolute value) of a complex cos B cos C cos C cos A cos A cos B number, z = a + ib is defined as the non-negative cos 2 33  cos 2 57 real number a  b. It is denoted by |z|, i.e., 2 2 90. Prove that,  2 2 21 2 69 sin  sin 2 2 z  a 2  b2 Multiplicative inverse of z is z. It is also called 91. Show that 3 cosec 20  sec 20  4 2 z 92. Find the least positive value of n, for which reciprocal of z. n 1 i  zz  z 2   1. 1 i  On the basis of above information, answer the following questions. 93. By using binomial theorem, expand (1 + x + x2)3 (i) If (x – iy)(3 + 5i) is the conjugate of – 6 – 24i, 94. Write the general term in the expansion of (x2 – y)6. then the value of x + y is equal to (1) 0 (2) 1 SECTION - IV Short Answer Type Questions (95-101) (7×3=21) (3) 2 (4) 3 95. The ratio of the sum of n terms of two A.P.'s is (7n + 1) : (4n + 27). Find the ratio of their mth (ii) The value of ( z  3)( z  3) is equivalent to terms. (1) |z + 3|2 (2) |z – 3| (3) z2 + 3 (4) None of these 96. Find the sum of n terms of the series 1.2.3 + 2.3.4 + 3.4.5 + …. (iii) If z1 = 1 – 3i and z2 = –2 + 4i, then |z1 + z2| is 5 equal to 1  97. If the third term in the expansion of   xlog10 x  (1) 2 (2) 2 x  (3) 3 (4) 1 is 1000, then find x. 98. Find the square root of 7 – 24i SECTION - V Long Answer Type Questions (102-104) (3×5=15) 1  2i 102. Find the equations of the median of a triangle 99. Convert the complex number , in the polar 1  3i formed by the lines x + y – 6 = 0, x – 3y – 2 = 0 and form 5x – 3y + 2 = 0. 100. Find general solution of the equation, 103. If y = 2x is a chord of the circle x2 + y2 – 10x = 0,    2  find the equation of circle with this chord as tan   tan      tan      3 diameter.  3  3  104. Find the sum of the series cos 5 x  cos 4 x 101. Prove that,   cos 2 x  cos x 5 + 55 + 555 + ……. upto n terms. 1  2 cos 3 x PW Web/App - https://smart.link/7wwosivoicgd4 Library- https://smart.link/sdfez8ejd80if