Answer: A. Answer: C. 3. If (G,.) is a group such that a2 = e, a G, then G is A. abelian group B. non-abelian group C. semi group D.

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1 1. The set of all real numbers under the usual multiplication operation is not a group since A. zero has no inverse B. identity element does not exist C. multiplication is not associative D. multiplication is not a binary operation 2. If (G,.) is a group such that (ab)- 1 = a-1b-1, a, b G, then G is a/an A. commutative semi group B. non-abelian group C. abelian group D. None of these 3. If (G,.) is a group such that a2 = e, a G, then G is A. abelian group B. non-abelian group C. semi group D. none of these 4. The inverse of - i in the multiplicative group, {1, - 1, i, - i} is A. -1 B. 1 C. -i D. i 5. The set of integers Z with the binary operation "*" defined as a*b =a +b+ 1 for a, b Z, is a group. The identity element of this group is A. -1 B. 0 C. 1 D If R = {(1, 2),(2, 3),(3, 3)} be a relation defined on A= {1, 2, 3} then R. R (= R2) is A. {(1, 2),(1, 3),(3, 3)} B. {(1, 3),(2, 3),(3, 3)} C. {(2, 1),(1, 3),(2, 3)} D. R itself

2 7. Which of the following statements is false? A. If R is relexive, then R R-1 φ B. R R-1 φ =>R is anti-symmetric. C. If R, R' are relexive relations in A, then R - R' is reflexive D. If R, R' are equivalence relations in a set A, then R R' is also an equivalence relation in A 8. If (G,.) is a group, such that (ab)2 = a2 b2 a, b G, then G is a/an A. abelian group B. non-abelian group C. commutative semi group D. none of these 9. (Z,*) is a group with a*b = a+b+1 a, b Z. The inverse of a is A. 0 B. -2 C. a-2 D. -a Let G denoted the set of all n x n non-singular matrices with rational numbers as entries. Then under multiplication G is a/an A. subgroup B. ininite, abelian C. finite abelian group D. infinite, non abelian group 11. Let A be the set of all non-singular matrices over real numbers and let * be the matrix multiplication operator. Then A. < A, * > is a monoid but not a group B. < A, * > is a group but not an abelian group C. < A, * > is a semi group but not a monoid D. A is closed under * but < A, * > is not a semi group

3 12. Let (Z, *) be an algebraic structure, where Z is the set of integers and the operation * is defined by n * m = maximum (n, m). Which of the following statements is TRUE for (Z, *)? A. (Z, *) is a group B. (Z, *) is a monoid C. (Z, *) is an abelian group D. None of these 13. Some group (G, 0) is known to be abelian. Then which one of the following is TRUE for G? A. G is of finite order B. g = g² for every g G C. g = g-1 for every g G D. (g o h)² = g²o h² for every g,h G 14. If the binary operation * is deined on a set of ordered pairs of real numbers as (a,b)*(c,d)=(ad+bc,bd) and is associative, then (1, 2)*(3, 5)*(3, 4) equals A. (7,11) B. (23,11) C. (32,40) D. (74,40) 15. If A = (1, 2, 3, 4). Let ~= {(1, 2), (1, 3), (4, 2)}. Then ~ is A. reflexive B. transitive C. symmetric D. not anti-symmetric 16. If a, b are positive integers, define a * b = a where ab = a (modulo 7), with this * operation, then inverse of 3 in group G (1, 2, 3, 4, 5, 6) is A. 1 B. 3 C. 5 D. 7

4 17. Which of the following is TRUE? A. Set of all matrices forms a group under multipication C. Set of all rational negative numbers forms a group under multiplication B. Set of all non-singular matrices forms a group under multiplication D. None of these 18. The set of all nth roots of unity under multiplication of complex numbers form a/an A. group B. abelian group C. semi group with identity D. commutative semigroups with identity 19. Which of the following statements is FALSE? A. The set of rational numbers form an abelian group under multiplication C. The set of rational integers is an abelian group under addition B. The set of rational numbers is an abelian group under addition D. None of these 20. In the group G = {2, 4, 6, 8) under multiplication modulo 10, the identity element is A. 2 B. 4 C. 6 D Let D30 = {1, 2, 3, 4, 5, 6, 10, 15, 30} and relation I be partial ordering on D30. The all lower bounds of 10 and 15 respectively are A. 1,5 B. 1,7 C. 1,3,5 D. None of these 22. Hasse diagrams are drawn for A. lattices B. boolean Algebra

5 C. partially ordered sets D. none of these 23. A self-complemented, distributive lattice is called A. Self dual lattice B. Complete lattice C. Modular lattice D. Boolean algebra 24. Let D30 = {1, 2, 3, 5, 6, 10, 15, 30} and relation I be a partial ordering on D30. The lub of 10 and 15 respectively is A. 1 B. 5 C. 15 D Let X = {2, 3, 6, 12, 24}, and be the partial order defined by X Y if X divides Y. Number of edges in the Hasse diagram of (X, ) is A. 1 B. 3 C. 4 D If lattice (C, ) is a complemented chain, then A. C 2 B. C 1 C. C >1 D. C doesn't exist 27. A self-complemented, distributive lattice is called A. Self dual lattice B. Modular lattice C. Complete lattice D. Boolean algebra

6 28. The less than relation, <, on reals is A. not a partial ordering because it is not antisymmetric and not reflexive. C. a partial ordering since it is anti-symmetric and reflexive. B. not a partial ordering because it is not asymmetric and not reflexive D. a partial ordering since it is asymmetric and reflexive. 29. Principle of duality is defined as A. all properties are unaltered when is replaced by other than 0 and 1 element. B. all properties are unaltered when is replaced by C. LUB becomes GLB D. is replaced by 30. Different partially ordered sets may be represented by the same Hasse diagram if they are A. same B. isomorphic C. order-isomorphic D. lattices with same order 31. The absorption law is defined as A. a * ( a b ) = a B. a * ( a * b ) = b C. a * ( a b ) = b D. a * ( a * b ) = a b 32. A partial order is deined on the set S = {x, a1, a2, a3,... an, y} as x a i for all i and ai y for all i, where n 1. Number of total orders on the set S which contain partial order A. n! B. 1 C. n D. n + 2

7 33. Let L be a set with a relation R which is transitive, antisymmetric and reflexive and for any two elements a, b L. Let least upper bound lub (a, b) and the greatest lower bound glb (a, b) exist. A. L is a Poset B. L is a lattice C. L is a boolean algebra D. none of these 34. On solving 2p - 3q - 4r + 6r - 2q + p, answer will be A. 8q -5r B. 7p + 5r C. 3p - 5q + 2r D. 10p + 3q - 5r 35. Is the equation 3(2 x 4) = 18 equivalent to 6x 12 = 18? A. Yes, the equations are equivalent by the Distributive Property of Multiplication over B. Yes, the equations are equivalent by the Commutative Property of Multiplication Addition. C. Yes, the equations are equivalent by the Associative Property of Multiplication. D. No, the equations are not equivalent = A. 2 B. 4 C. 6 D Which number does not have a reciprocal? A. -1 B. 0 C. 1 D. 1/1000

8 38. What is the multiplicative inverse of 1/2? A. -2 B. 2 C. -1/2 D. 1/2 39. What is the solution for this equation? 2x 3 = 5 A. x = 1 or x = 4 B. x = 1 or x = 3 C. x = 4 or x = 4 D. x = 4 or x = What is the solution set of the inequality 5 x + 4 3? A. 2 x 6 B. 12 x 4 C. x 2 or x 6 D. x 12 or x Which equation is equivalent to 5x 2 (7 x + = 1) 14 x? A. 9x + 2=14 x B. 9x + 1=14 x C. 9x 2 =14 x D. 12x 1 =14 x 42. Answer of factorization of expression 4z(3a + 2b - 4c) + (3a + 2b - 4c) is A. (4z - 1)(3a - 2b -4c) B. (4z + 1)(3a + 2b -4c) C. (4z + 1) - (3a + 2b -4c) D. (4z + 1) + (3a + 2b -4c)

9 43. By factorizing expression 2bx + 4by - 3ax -6ay, answer must be A. (2b - 3a)(x + 2y) B. (2b + 3a)(x - 2y) C. (2a- 3b)(3x - 2y) D. (2a + 3b)(2x - 4y) 44. If -4x + 5y is subtracted from 3x + 2y then answer will be A. x - 3y B. x + 3y C. 2x + 5y D. 3x + 6y 45. On solving algebraic expression -38b 2, answer will be A. 19b B. 19b C. 56b D. 56b 46. Simplify 15ax² 5x A. 3ax B. 3ax² C. 5ax D. 5ax² 47. Simplify x A. 2x 5 B. 5x 2 C. 5 2x D. 2 5x 48. Simplify a(c - b) - b(a - c) A. ac + bc B. ac - 2ab - bc C. ac - 2ab + bc D. ac + 2ab + bc

10 49. Expand and simplfy (x - 5)(x + 4) A. x² - x - 1 B. x² - x - 9 C. x² - x - 20 D. x² + 9x Expand and simplfy (x - y)(x + y) A. x² - y² B. x² + y² C. x²- 2xy + y² D. x² + 2xy + y² 51. (a - b)² = A. a² + b² B. a² - b² C. a² + 2ab + b² D. a² - 2ab + b² 52. Factorise x² + x - 72 A. (x + 8)(x - 9) B. (x - 8)(x + 9) C. (x -?72)² D. (x -?72)(x +?72) 53. Factorise -20x² - 9x + 20 A. (5-4x)(4-5x) B. (5-4x)(4 + 5x) C. (5 + 4x)(4-5x) D. (5 + 4x)(4 + 5x)

11 54. Expand and simplify (x + y)³ A. x³ - y³ B. x³ + y³ C. x³ + 3xy(x - y) + y³ D. x³ + 3xy(x + y) + y³ 55. Simplify (x - 9)(x + 10) (x² - 81) A. (x + 10) (x - 9) B. (x + 10) (x + 9) C. (x² + x - 90) (x² - 81) D. None of above 56. Simplify (3x 2y)(x 3y) A. x 2y B. x² (6y²) C. x² (2y²) D. (3x²) (2y²) 57. Simplify (1 (x² - 1)) (1 (x + 1)) A. 1 (x + 1) B. 1 (x - 1) C. 1 (x² - 1) D. None of above 58. Simplify (1 s) - (1 t) A. (t - s) st B. (s - t) st C. 1 (s - t) D Simplify (2 tana) + (4 tanb) A. 6 (tana + tanb) B. (2tanA + 4tanB) tanatanb C. (2tanB + 4tanA) tanatanb D. None of above

12 60. 1 (x² - 1) can be expressed as (A (x+1)) + (B (x-1)), which of following pairs of values of A and B is correct? A. A = 1, B = 0 B. A = 1, B = -1 C. A = -1 2, B = 1 2 D. A = 1 2, B = Simplify (a^(-1 3) a^(1 3)) a^(-1 2) A. a^(1 2) B. a^(-1 2) C. a^(-7 18) D. None of above 62. Evaluate (1 64)^(-1 3) A. 2 B. 4 C. 2^-1 D. 4^ What is value of M in (p/q)2m+2 = (q/p)9-m A. 6 B. 5 C. -7/2 D. -11