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1 ASSIGNMENT ON SETS LEVEL 1 (CBSE/NCERT/STATE BOARDS) 1 What the differene between a olletion and a set? Give reasons to support your answer? If A = {0,1,,3,4,5,6, 7,8,9,10}, then insert the appropriate symbol or in eah of the following blank spaes: (i) 4...A (ii) -4...A (iii) 1...A (iv) 9...A (v) 0...A (vi) -...A 3 Write the set of all integers whose ube it an even integer. 4 Write the set of all real numbers whih annot be written as the quotient of two integers in the set-builder form. 5 Desribe eah of the following sets in Roster form (i) { : a positive integer and a divor of 9} (ii) { : Z and } (iii) { : a letter of the word 'PROPORTION'} n (iv) : and 1 n 3, where n N n Write the set,,,,,,,, in the set-builder form Write the set X = ,,,,,... in the set -builder form Let A and B be two sets having 3 and 6 elements respetively. Write the minimum number of elements that A B an have. 9 Let A = { : N, a multiple of 3) and B = { : N and a multiple of 5}. Write A B. 10 If A = {(, y) : y = 1, 0 R} and B = {(, y) : y = -, R}, then write A B. 11 If A = {(, y) : y = e, R} and B = {(, y): y = e -, R), then write A B. 1 Write the following sets in Roster form: (i) A = {a n : n N,a n+1 = 3a n and a 1 = } (ii) B = {a n : n N, a n+ = a n+1 + a n, a 1 = a = 1} 13 Desribe the following sets in set-builder form: By: Sumit luthra (M.S. B.Ed. M.Phill.) Ph: sumit.luthra1981@gmail.om Page 1

2 (i) A = {1,,3,4,5, 6}; (ii) B = {1,1/, 1/3, 1/4, 1/5,...}; (iii) C = {0,3, 6, 9,1,...}; (iv) D = {10,11,1,13,14,15}; 14 State whih of the following sets are finite and whih are infinite: (i) A = { : Z and = 0} (ii) B = { : Z and even} (iii) C = { : Z and = 36} (iv) D = { : Z and > -10} 15 Consider the following sets:, A = {1, ), B = {1,4,8}, C = {1,,4,6, 8} Insert the orret symbol or between eah of the following pair of sets: (i)...b (ii) A...B (iii) A... C (iv) B... C 16 In eah of the following, determine whether the statement true or false. If it true, prove it. If it false, give an eample. (i) If A and A B, then B (ii) If A B and B C, then A C (iii) If A B and B C, then A C (iv) If A B and B C, then A C (v) If A and A B, then B (vi) If A B and B, then A 17 Write the following subsets of R as intervals: (i) { : R,-4 < 6} (ii) { : R,- 1 < < -10} (iii) { : R, 0 < 7} (iv) { : R,3 4}. 18 Whih of the following statements are orret? Write a orret form of eah of the inorret statements. (i) a {a, b, } (ii) {a} {a, b, } (iii) a {{a}, b} (iv) {a} {{a),b} (v) {b, } {a, {b, }} (vi) {a, b} {a, {b, }} (vii) {a, b} (viii) {a, b, } (i) { : + 3 = 3} = 19 Let A = {a, b, {, d},e}. Whih of the following statements are false and why? (i) {. d} A (ii) {, d} A (iii) {{, d}} A (iv) a A (v) a A (vi) {a, b, e} A (vii) {a, b, e} A (viii) {a, b, } A (i) A () {} A 0 Let A = {, {}, 1, {1, 0}, }, Whih of the following are true? (i) A (ii) {} A (iii) {1} A (iv) {, } A (v) A (vi) {, {1}} A (vii) {{}, {1}} A (viii) {, {}, {1, }} A (i) {{}} A. By: Sumit luthra (M.S. B.Ed. M.Phill.) Ph: sumit.luthra1981@gmail.om Page

3 1 If A = {1, 3, 5, 7,11,13, 15, 17}, B = {, 4, 6,..., 18} and N the universal set, then find A' ((A B'). Suppose A 1, A,... A 30 are thirty sets eah with five elements and B 1, B,.,., B n are n sets eah with three elements. Let eatly ten of the A i 's and eatly 9 f B j 's. Find n. 30 n Ai Bj S. Assume that eah element of S belongs to i1 j1 3 Find sets A, B and C suh that A B,A C and B C are non-empty sets and A B C =. 4 For any two sets A and B, show that the following statements are equivalent: (i) A B (ii) A - B = 0 (iii) A B = B (iv) A B = A. 5 For three sets A, B and C, show that (i) A B = A C need not imply B = C (ii) A B C - B C - A 6 In a group of 50 people, 35 speak Hindi, 5 speak both Englh and Hindi and all the people speak at least one of the two languages. How many people speak only Englh and not Hindi 1 How many people speak Englh? 7 There are 00 individuals with a skin dorder, 10 has been eposed to hemial C 1, 50 to hemial C and 30 to both the hemials C 1 and C. Find the number of individuals eposed to (i) hemial C 1 or hemial C (ii) hemial C 1 but not hemial C (iii) hemial C but not hemial C 1. 8 A market researh group onduted a survey of 000 onsumers and reported that 170 onsumers liked produt P 1 and 1450 onsumers liked produt P. What the least number that must have liked both the produts? 9 In a survey of 700 students in a ollege, 180 were lted as drinking Lima, 75 as drinking Miranda and 95 were lted as both drinking Lima as well as Miranda. Find how many students were drinking neither Lima nor Miranda. 30 In a lass of 35 students, 17 have taken mathemat, 10 have taken mathemat but not eonom. Find the number of students who have taken both mathemat and eonom and the number of students who have taken eonom but not mathemat, if it given that eah student has taken either mathemat or eonom or both. By: Sumit luthra (M.S. B.Ed. M.Phill.) Ph: sumit.luthra1981@gmail.om Page 3

4 31 In a town of '10,000 families it was found that 40% families buy newspaper A, 0% families buy newspaper B and 10% families buy newspaper C. 5% families buy A and B, 3% buy B and C and 4% buy A and C. If % families buy all the three news papers, find the number of families whih buy (i) A only (ii) B only (in) none of A, B and C. 3 A ollege awarded 38 medals in Football, 15 in Basketball and 0 to Criket. If, these medals went to a total of 58 men and only three men got medals in all the three sports, how many reeived medals in eatly two of the three sports? 33 In a survey of 5 students, it was found that 15 had taken mathemat, 1 had taken phys and 11 had taken hemtry, 5 had taken mathemat and hemtry, 9 had taken mathemat and phys, 4 had taken phys and hemtry and 3 had taken all the three subjets.find the number of students that had (i) only hemtry. (ii) only mathemat. (iii) only phys. (iv) phys and hemtry but not mathemat. (v) mathemat and phys but not hemtry. (vi) only one of the subjets. (vii) at least one of the three subjets. (viii) none of the subjets. 34 Of the members of three athleti teams in a ertain shool, 1 are in the basketball team, 6 in hokey team and 9 in the football team. 14 play hokey and basket ball, 15 play hokey and football, 1 play football and basketball and 8 play all the three games. How many members are there in all? 35 A survey of 500 televion viewers produed the following information; 85 wath football, 195 wath hokey, 115 wath basketball, 45 wath football and basketball, 70 wath football and hokey, 50 wath hokey and basketball, 50 do not wath any of the three games. How many wath all the three games? How many wath eatly one of the three games? 36 In a survey it was found that 1 persons liked produt P 1, 6 liked produt P and 9 liked produt P 3. If 14 persons liked produts P 1 and P ; 1 persons liked produt P 3 and P 1 ; 14 persons liked produts P and P 3 and 8 liked all the three produts. Find how many liked produt P 3 only. By: Sumit luthra (M.S. B.Ed. M.Phill.) Ph: sumit.luthra1981@gmail.om Page 4

5 ASSIGNMENT ON SETS LEVEL (OBJECTIVE QUESTIONS FOR IIT-MAIN AND OTHER ENTRANCES) 1. The set of intelligent students in a lass [AMU 1998] A null set (b) A singleton set A finite set (d) Not a well defined olletion. Whih of the following the empty set [Karnataka CET 1990] { : a real number and 1 0} (b) { : a real number and 1 0} { : a real number and 9 0} (d) { : a real number and } 6. Given the sets A { 1,,3}, B {3,4}, C = {4, 5, 6}, then A ( B C) [MNR 1988; Kurukshetra CEE 1996] {3} (b) {1,, 3, 4} {1,, 4, 5} (d) {1,, 3, 4, 5, 6} 7. If A and B are any two sets, then A A [Karnataka CET 1996] A A (b) B 8. If A and B are two given sets, then A ( A [AMU 1998; Kurukshetra CEE 1999] A (d) (d) 9. If the sets A and B are defined as (b) B B A B 1 A {(, y) : y, 0 R} B {(, y) : y, R}, then A B A (b) A B B A B 10. Let A [ : R, 1] ; B [ : R, 1 1] and A B R D, then the set D [ :1 ] (b) [ :1 ] [ :1 ] 11. If the sets A and B are defined as A {(, y): y e, R} ; 3. The set A { : R, 16 and 6} B {(, y): y, R}, then [UPSEAT 1994, 99, 00] equals B A (b) A B [Karnataka CET 1995] A B (d) A B A (b) {14, 3, 4} n 1. If X { 4 3n 1 : n N} and {3} (d) {4} Y { 9( n 1) : n N}, then X Y 4. If a set A has n elements, then the total number of X (b) Y subsets of A [Roorkee 1991; Karnataka CET 199, 000] N n (b) n 13. Let n ( U) 700, n( A) 00, n( 300 and n (d) n n ( A 100, then n ( A B ) 5. The number of proper subsets of the set {1,, 3} [JMIEE 000] 400 (b) (b) (d) 00 6 (d) In a town of 10,000 families it was found that 40% family buy newspaper A, 0% buy newspaper B and 10% families buy newspaper C, 5% families buy A and B, 3% buy B and C and 4% buy A and C. If % families buy all the three newspapers, then number of families whih buy A only [Roorkee 1997] 3100 (b) (d) In a ity 0 perent of the population travels by ar, 50 perent travels by bus and 10 perent travels by both ar and bus. Then persons travelling by ar or bus [Kerala (Engg.) 00] 80 perent 60 perent (b) 40 perent (d) 70 perent 16. In a lass of 55 students, the number of students studying different subjets are 3 in Mathemat, 4 in Phys, 19 in Chemtry, 1 in Mathemat By: Sumit luthra (M.S. B.Ed. M.Phill.) Ph: sumit.luthra1981@gmail.om Page 5

6 and Phys, 9 in Mathemat and Chemtry, 7 in Phys and Chemtry and 4 in all the three subjets. The number of students who have taken eatly one subjet [UPSEAT 1990] 6 (b) 9 7 (d) All of these 17. If A, B and C are any three sets, then A (B C) 7. Whih set the subset of all given sets [Pb. CET 001] {1,, 3, 4,...} (b) {1} (A (A C) (b) (A (A {0} (d) {} C) (A (A C) 18. If A, B and C are any three sets, then A (B C) (A (A C) (b) (A (A C) (A C (d) (A C 19. If A, B and C are non-empty sets, then (A (B 15 (b) 14 A) equals [AMU 199, 1998; DCE 1998] 16 (d) 17 (A B (b) A (A 30. The smallest set A suh that A {1, } = {1,, 3, 5, (A (A (d) (A (A 9} {, 3, 5} (b) {3, 5, 9} 0. If A {, 4,5}, B {7, 8,9}, then n( A equal to 6 (b) 9 3 (d) 0 1. If the set A has p elements, B has q elements, then 3. If A and B are two sets, then A B A B iff the number of elements in A B [Karnataka CET 1999] A B (b) B A p q (b) p q 1 A B (d) None of pq (d) p these. If A { a, b}, B {, d}, C { d, e}, then {( a, ),( a, d),( a, e),( b, ),( b, d),( b, e)} [AMU 1999; Him. CET 00] A (B C) (b) A (B C) A (B C) (d) A (B C) 3. If P, Q and R are subsets of a set A, then R (P Q ) = [Karnataka CET 1993] (R P) (R Q) ( R Q) R P) (b) ( R P) R Q) 4. In rule method the null set represented by [Karnataka CET 1998] {} (b) { : } (d) { : } 5. A { : } represents [Kurukshetra CEE 1998] {0} (b) {} {1} (d) {} 1 6. If Q :, where y N, then y 0 Q (b) 1 Q Q (d) Q 3 8. Let S {0,1,5,4,7 }. Then the total number of subsets of S 64 (b) 3 40 (d) 0 9. The number of non-empty subsets of the set {1,, 3, 4} [Karnataka CET 1997; AMU 1998] {1,, 5, 9} 31. If A B = B, then [JMIEE 000] A B (b) B A A (d) B 33. Let A and B be two sets. Then A B A B A B = A B 34. Let A {(, y): y e, R}, B {(, y): y e, R}. Then (b) A B A B A B (b) A B A B R 35. If A = {, 3, 4, 8, 10}, B = {3, 4, 5, 10, 1}, C = {4, 5, 6, 1, 14} then (A (A C) equal to {3, 4, 10} (b) {, 8, 10} {4, 5, 6} (d) {3, 5, 14} 36. If A and B are any two sets, then A (A A (b) B A 37. If A, B, C be three sets suh that A B = A C and A B = A C, then [Roorkee 1991] A = B (b) B = C A = C (d) B (d) A = B = C By: Sumit luthra (M.S. B.Ed. M.Phill.) Ph: sumit.luthra1981@gmail.om Page 6

7 38. Let A = {a, b, }, B = {b,, d}, C = {a, b, d, e}, then A (B C) [Kurukshetra CEE 1997] {a, b, } (b) {b,, d} {a, b, d, e} (d) {e} 39. If A and B are sets, then A (B A) (b) A B 40. If A and B are two sets, then A A equal to A (b) B 41. Let U {1,, 3, 4, 5, 6, 7, 8, 9,10}, A { 1,,5}, B {6,7}, then A B B (b) A A (d) B 4. If A any set, then A A (b) A A U A A U 43. If [ an: n N}, then N N a N 7 5 N 7 [Kerala (Engg.) 005] (b) N N 35 (d) N 5 (e) N If an { a : N}, then the set 3N 7N 1 N 4 N (b) 10 N 45. The shaded region in the given figure [NDA 000] A (B C) A (b) A (B C) A (B C) (d) A (B C) 46. If A and B are two sets then (A (B A) (A A B (b) A B A (d) B 47. Let A and B be two sets then ( A A A (b) A B 48. Let U be the universal set and A B C U. Then {( A B C) C A) } A B C (b) A ( B C) A B C (d) A ( B C) 49. If n ( A) 3, n ( 6 and A B. Then the number of elements in C A B 3 (b) 9 B Let A and B be two sets suh that n ( A) 0.16, n( 0.14, n( A 0.5. Then n( A [JMIEE 001] 0.3 (b) If A and B are djoint, then n( A n (A) (b) n ( n( A) (d) n ( A). n( 5. If A and B are not djoint sets, then n( A n( A) (b) n( A) n( A [Kerala (Engg.) 001] n( A) A (d) n ( A) n( (e) n( A) 53. In a battle 70% of the ombatants lost one eye, 80% an ear, 75% an arm, 85% a leg, % lost all the four limbs. The minimum value of 10 (b) Out of 800 boys in a shool, 4 played riket, 40 played hokey and 336 played basketball. Of the total, 64 played both basketball and hokey; 80 played riket and basketball and 40 played riket and hokey; 4 played all the three games. The number of boys who did not play any game [DCE 1995; MP PET 1996] 18 (b) (d) A survey shows that 63% of the Amerians like heese whereas 76% like apples. If % of the Amerians like both heese and apples, then 39 (b) teahers of a shool either teah mathemat or phys. 1 of them teah mathemat while 4 teah both the subjets. Then the number of teahers teahing phys only 1 (b) Of the members of three athleti teams in a shool 1 are in the riket team, 6 are in the hokey team and 9 are in the football team. Among them, 14 play hokey and riket, 15 play hokey and football, and 1 play football and riket. Eight play all the three games. The total number of members in the three athleti teams By: Sumit luthra (M.S. B.Ed. M.Phill.) Ph: sumit.luthra1981@gmail.om Page 7

8 43 (b) In a lass of 100 students, 55 students have passed in Mathemat and 67 students have passed in Phys. Then the number of students who have passed in Phys only [DCE 1993; ISM Dhanbad 1994] (b) (d) If A and B are two sets, then A B = B A iff A B (b) B A A B 60. If A and B be any two sets, then ( A A Β (b) A B A B (d) A B 61. Let A and B be subsets of a set X. Then A B A B (b) A B A B A B A B (d) A B A B 6. Let A and B be two sets in the universal set. Then A B equals A B (b) A B A B 63. If A, B and C are any three sets, then A ( B C) ( A A C) (b) ( A A C) ( A C (d) ( A C 64. If A, B, C are three sets, then A (B C) (A (A C) (b) (A (A C) (A (A C) 65. If A = {1,, 4}, B = {, 4, 5}, C = {, 5}, then (A (B C) {(1, ), (1, 5), (, 5)} (b) {(1, 4)} (1, 4) 66. If (1, 3), (, 5) and (3, 3) are three elements of A B and the total number of elements in A B 6, then the remaining elements of A B are (1, 5); (, 3); (3, 5) (b) (5, 1); (3, ); (5, 3) (1, 5); (, 3); (5, 3) 67. A = {1,, 3} and B = {3, 8}, then (A (A {(3, 1), (3, ), (3, 3), (3, 8)} (b) {(1, 3), (, 3), (3, 3), (8, 3)} {(1, ), (, ), (3, 3), (8, 8)} (d) {(8, 3), (8, ), (8, 1), (8, 8)} 68. If A = {, 3, 5}, B = {, 5, 6}, then (A (A {(3, ), (3, 3), (3, 5)} (b) {(3, ), (3, 5), (3, 6)} {(3, ), (3, 5)} 69. In a lass of 30 pupils, 1 take needle work, 16 take phys and 18 take htory. If all the 30 students take at least one subjet and no one takes all three then the number of pupils taking subjets 16 (b) 6 8 (d) If n ( A) 4, n ( 3, n ( A B C) 4, then n (C) 88 (b) 1 1 (d) 17 (e) 71. The number of elements in the set {( a, b) : a 3b [Kerala (Engg.) 005] 35, a, b Z}, where Z the set of all integers, [Kerala (Engg.) 005] (b) 4 8 (d) 1 (e) If A {1,, 3, 4} ; B { a, b} and f a mapping suh that {(a, 1), (3, b)} (b) {(a, ), (4, b)} f : A B, then A B {(1, a), (1, b), (, a), (, b), (3, a), (3, b), (4, a), (4, b)} 73. If A = {1,, 3, 4, 5}, B = {, 4, 6}, C = {3, 4, 6}, then ( A C [Orsa JEE 004] {3, 4, 6} (b) {1,, 3} {1, 4, 3} 74. If A = {, y} then the power set of A y y {, } (b) {,, y} {, {}, {y}} (d) {, {}, {y}, {, y}} [Pb. CET 004, UPSEAT 000] 75. A set ontains n 1 elements. The number of subsets of th set ontaining more than n elements n 1 n 1 (b) (d) [UPSEAT 001, 04] n n 76. Whih of the following a true statement [UPSEAT 005] {a} {a, b, } (b) {a} {a, b, } By: Sumit luthra (M.S. B.Ed. M.Phill.) Ph: sumit.luthra1981@gmail.om Page 8

9 {a, b, } 77. If A = { : a multiple of 4} and B = { : a multiple of 6} then A B onsts of all multiples of[upseat 000] 16 (b) 1 8 (d) A lass has 175 students. The following data shows the number of students obtaining one or more subjets. Mathemat 100, Phys 70, Chemtry 40; Mathemat and Phys 30, Mathemat and Chemtry 8, Phys and Chemtry 3; Mathemat, Phys and Chemtry 18. How many students have offered Mathemat alone 35 (b) (d) (e) Consider the following relations : (1) A B A A () A ( A A (3) A ( B C) ( A A C) [Kerala (Engg.) 003] whih of these /are orret [NDA 003] 1 and 3 (b) only and 3 (d) 1 and 80. If two sets A and B are having 99 elements in ommon, then the number of elements ommon to a b 37 4 b b a a a d eah of the sets A B and B A are 46 a 47 [Kerala a (Engg.) ] (b) (d) 18 (e) Given n ( U) 0, n ( A) 1, n ( 9, n ( A 4, where U the universal set, A and B C are subsets of U, then n (( A ) 71 7 [Kerala (Engg.) ] a 74 d 75 d 17 (b) 9 11 (d) 3 (e) 16 1 d b 3 a b 7 a 8 d 9 10 b 11 1 b b d 17 a 18 b 19 0 b 1 3 a,b 4 d 5 b 6 b 7 d 8 b 9 a 30 b 31 b 3 33 b 34 b 35 a 51 5 b 53 a 54 d a 57 a 58 d b 61 d 6 a 63 a 64 b 65 b 66 a 67 b a 70 e 76 a 77 b d 80 b 81 d By: Sumit luthra (M.S. B.Ed. M.Phill.) Ph: sumit.luthra1981@gmail.om Page 9

Inverse Trigonometrical Functions 1. Properties of Inverse Trigonometrical Function. 1. The domain of sin x is [Roorkee Screening 1993] (a) (d)

Inverse Trigonometrical Functions Basic Level Properties of Inverse Trigonometrical Function. The domain of [Roorkee Screening 99] ( ) [ ] ( 0 ) ( ). The range of [DCE 00] ( ) ( 0 ). cos equal to [Pb.