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INFOMATHS PUNE-2015 27 then find 13. The digit at unit place of the number (13) 0 (a) 1 (b) 7 (c) 9 (d) 3 na -|x| 14. The function f(x) = e is 0 f x ? (a) continuous everywhere but not differentiable at x (a) nk (b) (n+1)k (c) (n-1)k (d) 0 =0 2. Three square are chosen at random on the chess (b) continuous and differentiable everywhere board. The chance that they are in diagonal line is (c) only continuous at x = 0 7 3 7 3 (d) None of these (a) (b) (c) (d) 1 84 64 744 744 15. lim 1 ax x ? x 0 dy 3. Consider the differential equation 2y 0 (a) 0 (b) e-a (c) ea (d) None of these dx 2 16. If (3n +690) = (492k04) where n is a number and k (a) Every solution of equation is identically zero is a digit. Then what is the possible value of n? (b) All solutions of equation are unbounded (a) greater than – 1 and less than 3 (c) All solutions of equation approaches to zero (b) greater than 0 and less than 10 when x (c) greater than 9 and less than 18 (d) No solution of equation approaches to zero when (d) N.O.T x 17. Find the angle between the tangents to the curve y = 4. For what value of k the line y = 9x be the tangent to x2 – 5x + 6 at the points (3, 0) and (2, 0) is k x 1 ke the curve y at some point on the xy-plane (a) (b) (c) (d) x 1 3 6 2 4 with constraint that x > - 1 18. S is a set and P(S) denotes the power set of S then (a) k < 0 (b) k > 0 and k < 1 (a) P[P(S)] = P(S) (b) P[P(S)] P(S) = P(S) (c) k > 1 and k < 3 (d) k > 3 (c) P[P(S)] P(S) = {} (d) S P(S) 5. If m things are distributed among ‘a’ men and ‘b’ dy y x-y at (1, 1)? women, then the chance that men get odd number of 19. If x = e then find dx things? (a) 1 (b) 2 (c) 0 (d) 4 m m m m 2 1 b a b a 1 b a b a 20. Find the area bounded by x = 8y and the line x – 2y (a) . (b) . m m + 8 = 0 2 2 b a b a (a) 25 (b) 48 (c) 36 (d) 32 m m 1 b a b a 21. There are four machines and it is known that exactly (c) . (d) None of these m two of them are faulty. They are tested one by one, 2 b a in a random order till both the faulty machines are 6. Find the total number of distinct relations on the set identified. Then, the probability that only two tests containing 3 elements? are needed is (a) 6 (b) 128 (c) 18 (d) 512 (a) 1/3 (b) 1/6 (c) 1/2 (d) 1/4 x 4 7. Find the equation of tangent to the curve y 4e at 22. If v " u uv ' u ' v k then find the value of k? the point where it crosses Y-axis is (a) v ' u (b) u " v (c) u ' v ' (d) uv " (a) 4x + y = 4 (b) 4x – y = 16 23. If three numbers are selected from (2n + 1) (c) x – y = 4 (d) x + y = 4 3 2 consecutive integers then find the probability that the 8. The curve y = x – 6x + 9x + 1 has symmetric numbers from and A.P.? rotation about the point 3n 3 (a) (0, 0) (b) (3, 2) (c) (2, 3) (d) None of these (a) (b) 2 bx 4n 1 4n 1 9. Find the differential equation of the curve y ae 3n after eliminating arbitrary constants. (c) (d) None of these 2 2 4n 2 1 d 2 y dy d 2 y dy (a) y 2 0 (b) x 2 0 24. The error is measuring radius of a circle is 0.5%. dx dx dx dx Then what will be the percentage change in the area 2 d 2 y dy d2y dy of the circle is? y 0 (c) y 2 (d) 0 dx 2 (a) 0.1% (b) 0.0025% dx dx dx (c) 0.01% (d) 0.5% 10. If f(x + y) = f(x) . f(y) and f(6) = 3, f'(0) = then f'(6) 25. Five person entered in a lift cabin on the ground =? floor of 8 floor house. Suppose each of them with (a) 15 (b) 22 (c) 28 (d) 30 1/3 equal probability and independently can leave the 11. Find the maximum value of the function (x + 1) – 1/3 cabin at any floor beginning with the first. Then find (x – 1) is the probability that all five person leaving at (a) 0 (b) 1 (c) 2 (d) 3 different floors. 12. If f(9) = 9 and f'(9) = 3 then find the value of 7 7 7 P P P f x 3 (a) 55 (b) 75 (c) 55 (b) N.O.T lim 7 7 7 x 9 x 3 (a) 3 (b) 0 (c) 1 (d) 9 1. If f(x) = f(x + ) and f x k, a 1 INFOMATHS/MCA/MATHS/ INFOMATHS 26. The probability that a student passes in Mathematics, B: I. I never stolen the horse Physics and Chemistry are m, p and c respectively of II. G stolen the horse these subjects, the student has 75% chance of (a) B stolen the horse (b) G stolen the horse passing in at least once, a 50% chance of passing in (c) R stolen the horse (d) Data insufficient atleast, two and a 40% chance of passing in exactly 34. A cricket match is played and you want to know the two, which of the following relations are true. result of the match immediately you can know the result only by asking your friend. Your friend tries to 19 27 (a) p m c (b) p m c have some fun and tally you that you can ask only 20 20 questions which he answer as ‘yes’ or ‘no’. Your 1 1 friend tells you that he will lie but at most once. A (c) pmc (d) pmc 20 4 cricket match took place in which one team lost and 27. A is a set containing n elements. A subjects P of A is so another won. You want to know the result. What chosen at random. Then, returning back the elements is the minimum number of questions you need to ask of P the set A is completed. A subset Q is again your friend so you can know the result? chosen at random. Find the probability that have no (a) 2 (b) 3 (c) 4 (d) None element common. 35. P is a moving point and tangents from P meet a n n n circle. Whose centre is at the origin and radius is 1 2 3 (a) (b) (c) (d) N.O.T unity, at A and B [where, A and B are point of 2 3 4 contact] and angle AOB = 60. Then, locus of P is 28. If A has (n + 1) coins and B has n coins. Coins are 2 flipped simultaneously, then what is the probability (a) circle of radius (b) circle of radius 3 3 that a gets more heads than B? (c) circle of radius 2 (d) None 1 1 1 (a) (b) (c) (d) None 36. A and B are two candidates seeking admission for an 2 2 2 interview. The probability that A is getting selected 29. The maximum number of cricket ball, which are is 0.5 and the probability that both A and B are placed, such that touches the remaining is 4. Then selected is atmost 0.3. The probability of B getting the maximum number of coins that are placed such selected is that touches the remaining is (a) P(B) 0.8 (b) P(B) 0.8 (a) less than 5 (c) P(B) = 0.9 (d) Data Insufficient (b) greater than 4 but less than 8 37. 50 defective bulbs and 50 non-defective bulbs are (c) Greater than 7 but less than 11 selected at random, one by one with replacement. If (d) None X, Y and Z are the events defined as 30. f(x) is a differentiable function satisfying f(x3) = 4x4, X : Chance of getting 1st bulb defective then find the value of f'(8)? Y : Chance of getting 2nd bulb non – defective 64 32 16 16 2 Z : Chance of getting both defective or both non(a) (b) (c) (d) 3 3 3 3 defective 31. If f and g are two positive functions having relation Then, which of the following is true. (a) X, Y, Z are independent f x R such that f – g, it is given that lim 1, (b) X, Y, Z are pairwise independent x g x (c) X, Y, Z are mutually exclusive which one of the following is not consequence? (d) None of these (a) f2 ~ g2 (b) f ~ g 38. Numbers is selected from the numbers 00, 01, ……, 99 and the product obtained of the 2 digits of the (c) ef ~ eg (d) (f + g) ~ 2f numbers is 18. If 4 numbers are selected, then what 32. A chess match starts between 6 pm – 7 pm and ends is the chance that atleast 3 numbers are such that between 9 pm – 10 pm. Observing in 12 hours wall their digits have products 18? clock, when the match starts the minute hand points at position m and hour hand points at position h. 97 7 3 97 (a) (b) (c) (d) 4 4 4 When the match ends, the hour hand points at m and 100 25 25 25 minute hand at h, then what is the minimum time the 39. Let a z+ is element of A and a relation a R b game is played? defined on A such that a = bk where k is a positive (a) less than 130 minutes integer then, which of the following belongs to this (b) Greater than 131 minutes and less than 181 relation? minutes (a) (8, 128) (b) (16, 256) (c) Greater than 180 minutes and less than 210 (c) (169, 13) (d) (11, 3) minutes. 40. A ring of diameters 20 cm is suspended horizontally (d) none by 6 strings of equal length, such that the height 33. One day a horse is stolen and G, R, B are rounded up between the centre of the ring and top of the string is for questioning. Each of them made two statement 24 cm, and the strings are attached in such a way that and none of them made more than one false they are equidistant from each other then, what is the statement cosine of the angle between two adjacent strings? R : I. Did not stole the horse II. The one who has stolen the horse is arab. 313 335 13 (a) (b) (c) (d) N.O.T G : I. R never stole the horse 338 338 7 II. The one who has stolen the horse is Jewish 2 INFOMATHS/MCA/MATHS/ INFOMATHS 41. A coin is tossed 10 times and a bag is filled with 49. How many NDD of length 2 can be formed from white and black ball accordingly by getting heads or NDD of lengths 20. tails respectively 10 balls are drawn from the bag (a) less than 201 with replacement and we get all white balls then, (b) More than 200 what is probability that all balls in the bag are white? (c) more than 300 and less than 701 (d) None of these 10 9 2 1 (a) (b) 9 (c) 9 (d) 9 50. How many NDD of length 3 can be formed from 9 2 2 2 2 NDD of length 30? 42. In the equation SHOO + SHOO = BRAIN, if all (a) less than 201 digits represent distinct numbers, then the sum of (b) more than 200 digits of minimum possible number of SHOO is (c) More than 300 & less than 701 (a) less than 18 (d) None (b) Number lying between 13 – 18 51. How many NDD of length 4 can be formed from (c) Number lying between 17 – 22 NDD of length 30? (d) None of these (a) less than 201 (b) more than 200 Directions : [43 – 45] Group questions (c) more than 300 & less than 701 NEWTON + ALWAYS = TOTALED (d) None of these Here, each letters belongs to a single digit integer. On the 52. If z is represented by abcde and x = 1abcde and y = basis of the given statement, answer the following abcde 1 and y = 3x, then z is questions: (a) greater than 30001 but less than 40001 43. In the above statement, NEWTON belong to (b) less than 30000 (a) less than 500001 (c) greater than 40000 but less than 50001 (b) greater than 500000 or less than 600001 (d) None (c) greater than 600000 or less than 700001 (d) greater than 700000 Directions (Q. Nos. 53-56) These questions are on the 44. In the above statement, ALWAYS belongs to following information. (a) less than 500001 From time to time the managing director of a company (b) greater than 500000 or less than 600001 appoints planning committee, each consisting of exactly (c) greater than 600000 or less than 700001 three members. Eligible for appointment are three (d) greater than 700000 executives from finance; B, C and D and three executives 45. In the above statement, D belongs to which integer from operations E, F and M. or single digit Any given committee is subject to the following (a) 0 (b) 1 (c) 3 (d) None of these restrictions on appointments. (i) At least one member must be from finance, and at least Directions : [46-48]: Group Questions one member must be from operations. Consider an infinite non-decreasing series in which every (ii) If B is appointed, C cannot be appointed. natural numbers repeats for n times. The first 12 digit of (iii) Neither D nor F can be appointed unless the other is the series is [122333444455]. also appointed. 46. The sum of first 200 elements is (iv) If E is appointed, M must be appointed. (a) less than 2000 53. Which of the following is an acceptable committee? (b) greater than 2001 and less than 2301 (a) E, F and M (b) D, E and F (c) greater than 2301 and less than 2701 (c) D, F and M (d) B, D and M (d) greater than 2701 54. If appointees from operations are in majority in a 47. The smallest value of n, when the sum of numbers committee, that committee must include exceed by 1 million (a) M (b) F (c) E (d) C (a) less than 40 (b) > 41 and < 50 55. If the restrictions on appointments apply also to a (c) > 51 and < 60 (d) None of these four-member committee appointed from the same 48. The sum of all the elements of the sequence starting group of executives, which of the following will be from 1001th to 10000th element: true? (a) less than 500000 (a) If B is appointed, M must be appointed (b) greater than 800000 and less than 1300001 (b) If F is appointed, C must be appointed (c) greater than 500000 and less than 800001 (c) If C is appointed, E must be appointed (d) none (d) None of the above will be true 56. If B is appointed to the same committee as M, which Directions [Q. No. 49 to 51] Group questions of the following will be true of that committee? If NDD is described as finite non-decreasing distinct (a) E is not a committee member sequence (b) F is a committee member Eg S1 = {1, 2, 4, 7, 31, 100} is a NDD. (c) Appointees from finance are in majority And S2 = {1, 2, 2, 5} is not NDD because of 2 is (d) Appointees from operations are in majority repeated. Directions (Q. Nos. 57-60) Read the following New NDDS can be formed from a given sequence of information to answer these questions. NDD example from above sequence S1, NDD {1, Ankit is decorating his room and trying to arrange six 2, 7} and {1, 7, 31, 100} can be formed. paintings on the East and West walls in his room. The paintings are each multi colour representations of one of 3 INFOMATHS/MCA/MATHS/ INFOMATHS the letters of the alphabets E, H, M, O, R, T. Ankit does not want the three letters on each wall to make any common English word. Also, the colours of the O and E do not look good next to each other, nor do the T and O go well together. 3. 57. If Ankit puts E, H and M on the East wall, which of the following must he true? (a) R and M cannot face each other. (b) O cannot be in the centre of the West wall. (c) E cannot be in the centre of the East wall. (d) T and M cannot face each other. 58. If Ankit’s mother is coming to visit and he decides to celebrate the visit by having his paintings spell ‘MOTHER’, starting with the leftmost painting on the East wall and going around the room, which of the following will be false? (a) O is opposite to E (b) H is next to E (c) T is next to O 4. (d) T is opposite to R 59. Which of the following is not possible? (a) T and O to be opposite to each other (b) T, H and R to be on the same wall (c) H, M and R to the on the same wall (d) M and O to be opposite to each other 60. If Ankit trades his M painting for another O painting just like the one he has now, which of the following must be false ? (a) O can be on opposite walls in the middle 6. (b) Either R or H will be next to an O (c) Either R or H will be next to either T or E (d) T will be opposite to either O or E 7. PUNE-2015 1 A 11 C 21 B 31 C 41 51 D 2 A 12 A 22 B 32 B 42 C 52 C 3 C 13 B 23 C 33 A 43 B 53 C 4 D 14 A 24 A 34 B 44 C 54 C 5 A 15 B 25 A 35 A 45 A 55 C 6 D 16 D 26 B 36 A 46 C 56 D 7 D 17 A 27 C 37 B 47 B 57 B 8 C 18 D 28 A 38 D 48 B 58 D 9 A 19 C 29 A 39 C 49 A 59 C 10 D 20 C 30 C 40 A 50 D 60 A na 0 8. a 0 [ f(x) is a periodic with a] 0 2. f x dx nk [ dy 2y 0 dx dy 2 y dx dy 2dx y log y = - 2x + c y = Ae-2x lim y lim Ae2 x 0 x x Hence, all solution of equation approaches to zero when x ke We have, y x 1 k x 1 k k .e x 1 2 x 1 2 x 1 k k dy dx x 1.e k x 1 k x 1 1 x 1 2 x 1 dy ke 3/ 2 dx 2 x 1 k Ans. (d) Set containing 3 elements 2 Total number of relation is 23 29 512 Ans. (d) Equation of curve is y = 4e-x/4 Put, x = 0 in y = 4e-x/4, we get y = 4e0 y = 4 [ e0 = 1] dy 1 4e x / 4 e x / 4 dx 4 dy e 0 dx 0,4 =-1 Hence, equation of tangent at (0, 4) y – 4 = - x(x – 0) x+y=4 f x dx n f x dx na Ans. (c) We have, 392 7 41664 744 Now, ANSWER WITH EXPLANATIONS 1. Ans. (a) We have, f(x) = f(x + a) i.e. f(x) is a periodic function with period a Required probability f x dx k ] a 0 Ans. (a) Total number of square on chess board = 64 Total number of ways of chosen three square from 64! 64 63 62 41664 64 square is 64 C3 61!3! 3 2 1 The square are diagonal line Total number of favourable out comes is 4(3C3 + 4C3 + 5C3 + 6C3 + 7C3) + 2 8C3 = 4(1 + 3 + 10 + 20 + 35) + 2 56 = 280 + 112 = 392 4 Ans. (c) We have, y = x3 – 6x2 + 9x + 1 dy 3 x 2 12 x 9 dx dy 0, dx Then, 3x2 – 12x + 9 = 0 (x – 3) (x – 1) = 0 f(x) is increasing x (- , 1) and (3, ) f(x) is decreasing x (1, 3) Graph of f(x) is INFOMATHS/MCA/MATHS/ INFOMATHS f ' 9 9 f 9 3 3 3 [ 3 f(9) = 9, f ' (9) = 3] 13. Ans. (b) We have, (13)27 (134)6 133 (28561)6 2197 Digit of unit place of (28561)6 is 1. And digit of unit place of (28561)6 2197 = 7 Hence, digit of unit place of (13)27 = 7. Symmetric rotation about the point of f(x) is 1 3 5 1 , 2,3 2 2 9. 14. Ans. (a) We have, f(x) = e-|x| e x , x 0 f x x e , x0 f(x) is continuous at x = 0 If lim f x lim f x f 0 Ans. (a) We have, y = aebx dy abebx dx dy by dx d 2 y bdy 2 dx dx 2 d 2 y dy y 2 dx dx x 0 x 0 x 0 f(x) is continuous at x = 0 e x x 0 Now, f ' x x x0 e f(x) is discontinuous at x = 0 If LHD of f(0) = RHD = 0 of f(0) LHD of f ' (0) = - 1 and RHD of f '(0) = 1 f(x) is not differentiable at x = 0. 2 y x 0 lim e x lim ex f 0 1 1 1 d 2 y dy 0 dx 2 dx 1 15. Ans. (b) We have, lim 1 ax x x 0 10. Ans. (d) We have, f(x + y) = f(x) . f(y) Differentiating w.r.t.x, y as constant, we get f ' (x + y) = f ' (x) . f(y) …(i) Put x = 0 and y = 6 in Eq. (i), we get f ' (0 + 6) = f '(0) . f(6) f '(6) = f '(0) . f(6) f '(6) = 10 3 = 30 [ f '(0) = 10, f(6) = 3] 1 lim 1 ax 1 e x 0 x g x lim a [ lim[ f x ] lim f x 1 g x ] x 0 xa e e-a x a e 16. Ans. (d) (3n + 690)2 = 492 k04 Taking square in both sides, we get 3n 690 492k 04 3n + 690 702.70 3n + 690 702.70 702.70 690 n 3 n4 3n + 690 701.4 3n 114 n 3.8 Hence, n = 4 11. Ans. (c) Let f(x) = f(x + 1)1/3 – (x – 1)1/3 1 1 1 f ' x 2/3 2/3 3 x 1 x 1 For maxima and minima, f ' (x) = 0 1 1 1 0 2/3 2/3 3 x 1 x 1 x=0 Now, f '(x) is increasing if, x (-, 0) and decreasing in x (0, ) 17. Ans. (a) We have, y = x2 – 5x + 6 f(x) is maximum at x = 0 dy 2x 5 Hence, f(0) = (1)1/3 – (-1)1/3 = 2 dx Maximum value of f(x) is 2. dy 65 1 dx 3,0 f x 3 12. Ans. (a) We have, lim x 9 dy x 3 And 4 5 1 dx 2,0 f ' x 0 dy dy 2 f x lim dx [ Apply L’Hospital’s rule] dx x 9 1 3,0 2,0 tan 0 dy dy 2 x 1 dx 3,0 dx 2,0 5 [if k = 9] [if k = 0] INFOMATHS/MCA/MATHS/ INFOMATHS tan 11 11 23. Ans. (c) Three numbers can be selected out of (2n + n 4n2 1 1) integer is 2n 1c3 3 Total number of AP with common difference with 1, 2, 3, …. n are (2n – 1), (2n – 3) …., respectively. Favourable outcomes = 2n – 1 + 2n – 3 + …. + 1 = n2 3n 2 3n 2 Required probability 2 4 n 1 n 4n 1 1 0 Hence, 2 18. Ans. (d) Let x be a element of s. xs Now, P(S) be the power set of s. {x} P(S) Hence, s P(S) dr 100 0.5% r Area of circle = r2 A = r2 dA 2 r dr dA = 2 rdr dA 2 rdr 100 100 A A dA 2 rdr 100 100 A r2 dr 2 100 r = 2 0.5 = 1% Hence, change in area of circle 1%. [ s = {x}] 24. Ans. () We have, 19. Ans. (c) We have, xy = ex-y Taking log both the sides, we get log xy = log ex-y y log x = (x – y) Differentiating w.r.t. x, we get y dy dy dy x y log x 1 log x 1 x dx dx dx x dy x y dy 1 1 0 dx x log x 1 dx 1,1 1 log1 1 20. Ans. (c) We have, x2 = 8y and x – 2y + 8 = 0 Solving these equation, we get x = - 4, or 8. 25. Ans. (a) The total number of ways in which each of the five persons can leave the cabin at any of the 7 floors = 75 Five persons can leave the cabin at 5 different floor in 7p5 ways. 7 p Required probability 5 5 7 Area of shaded region 26. Ans. (b) It is given that, probability of passing atleast once in 75% i.e. P(A B C) = P(A) + P(B) + P(C) - P(A B) – P(B C) – P(A C) + P (A B C) 75 P A P B P C P A B 100 - P(B C) – P(A C) + P(A B C) …(i) Probability of passing at least two is 50%. i.e. 50% = P(A B) + P(B C) + P(A C) - 2P (A B C) …(ii) And probability of passing exactly two is 40%. 40% = P(A B) + P(B C) + P(A C) - 3P (A B C) …(iii) Solving Eqs. (ii) and (iii), we get P(A B C) = 10% 1 i.e. pmc 10 Solving eqs. (i) and (ii), we get P(A) + P(B) + P(C) = 135% 135 27 i.e. p + m + c 100 20 8 x 8 x 2 x 2 8 x x3 dx 4 2 8 4 2 24 4 64 64 64 16 32 8 = 36 2 3 4 2 3 4 8 21. Ans. (b) The total number of ways in which two machine can be selected out of four machine is 4C2 = 6. If only two tests are required to identify faulty machine then, in first two tests faulty machines are identified. So, favourable outcomes = 1 1 Required probability 6 22. Ans. (b) We have, v ' u dx u v " dx u ' v " dx dx c uv ' u ' v ' dx c uv ' u v ' dx [u " v ' dx].dx c 27. Ans. (c) Since, set A contains n elements, So it has 2n subsets. Set P can be chosen in 2n ways. Similarly, set Q can be chosen in 2n ways. P and Q can be chosen in 2 n – 2n = 4n ways. uv ' u ' v u " v.dx c Hence, k u " v 6 INFOMATHS/MCA/MATHS/ INFOMATHS 3 3 Suppose, P contains r elements, where r varies from 97 4 96 4 1 97 O to n, then, P can be chosen in nCr ways for 0 to be 4 25 25 100 25 25 25 disjoint from A, it should be chosen from the set of all subsets of set consisting of remaining (n – r) elements. 39. Ans. (c) We have, a = bk, where k is positive integers This can be done in 2n-r ways. a>b P and Q can be chosen in nCr . 2n-r ways. 169 = 132 But, r can vary from 0 to n. Hence, option (c) is correct. Total number of disjoint sets P and Q n 53. Ans. (c) D and F should be appointed together, so n n cr 2n r 1 2 3n option (a) and (d) are ruled out. r 0 E and M must be appointed together, so option (b) is n 3n 3 ruled out. Option (c) is correct because it satisfies all So, required probability n the conditions 4 4 54. Ans. (c) This committee has two executives from operation and one executive from finance. Executives from operations are E, F and M. E and F cannot be selected because of (iv). So, the committee must include M. 30. Ans. (c) We have, f(x)3 = 4x2 f ' (x3) . 3x2 = 16x3 Put x = 2, we get f ' (23) . 3.22 = 16.23 16 2 32 f ' 8 3 3 35. Ans. (a) We have, In OPB, OBP = 90 and POB = 30 OB cos 30 OP 55. Ans. (c) A four member committee can be formed with B, D, and M. So, option (a) is correct. 56. Ans. (d) If B and M are appointed then, cannot be appointee D and F have to be appointed together. So, the third person appointed must be E. So, appointees from operations are in majority. 57. Ans. (b) O cannot be in the centre of the West wall. Because the only ways in which rest three letters O, R, T to the displayed in East wall, when O is in the centre are ROT and TOR. And in both the cases O and T are next to each other which is not permissible. 58. Ans. (d) Letter of the word MOTHER can be arrange in the following ways OP 2 3 [ OB = r = 1] Locus of p is circle of radius 2 . All option are correct except (d). 3 59. Ans. (c) If H, M and R on the wall, then three letters on the other wall will be T O E. and in all the possibilities either O and E will be next to each other or O and T will be next to each other and it will be violation of the condition. 36. Ans. (a) We have, P(A) = 0.5 P(A B) 0.3 P(A) + P(B) – P(A B) 1 P(B) 1 + P(A B) – P(A) P(B) 1 + 0.3 – 0.5 P(B) 0.8 60. Ans. (a) If painting M is exchanged from O and if, O is in the middle of each wall, then the positions of all 38. Ans. (d) Let P be the probability getting product of the six painting will be as under two-digit number is 18. i.e. such number are 29, 36, 63, 92 4 P 100 4 96 Now, any position of paintings E and T will violate q 1 100 100 the conditions. Required probability is P(X 3) = P(X=3)+P(x=4) = 4c3 p3q + 4c4 p4 3 4 4 96 4 4 100 100 100 7 INFOMATHS/MCA/MATHS/