Discrete Mathematics (MTH202)

1. The functions f o g and g o f are always equal.

   1. TRUE
   2. FALSE

2. A non-zero integer d divides an integer n if and only if there exists an integer k such that _________.

   1. n = d / k
   2. n = d k
   3. n = d + k
   4. n = d - k

3. If p is false and q is false, then ∼p implies q is ________.

   1. True
   2. False

4. (-2)! = _________ ?

   1. -2
   2. 0
   3. 2
   4. Undefined

5. A tree is normally constructed from ________.

   1. right
   2. center
   3. left to right
   4. right to left

6. The converse of the conditional statement 'If I live in Quetta, then I live in Pakistan' is ________.

   1. If I live in Pakistan, then I live in Quetta.
   2. If I live in Pakistan, then I do Not live in Quetta.
   3. If I do Not live in Quetta, then I do Not live in Pakistan
   4. If I do Not live in Quetta, then I live in Pakistan

7. There are three bus lines between A and B, and two bus lines between B and C. Find the number of ways a person can travel round trip by bus from A to C by way of B?

   1. 5
   2. 6
   3. 10
   4. 36

8. An integer n is a perfect square if and only if ________ for some integer k.

   1. n = 2k
   2. n = k^2
   3. n = square-root of k
   4. n = k^3

9. A predicate becomes _________ when its variables are given specific values.

   1. sentence
   2. statement
   3. algorithm
   4. iteration

10. The disjunction p ∨ q is False when ________.

    1. p is False, q is True.
    2. p is True, q is False.
    3. p is True, q is True.
    4. p is False, q is False.

11. Which of the following statements is true according to the Division Algorithm?

    1. 17 = 5 x 1 + 12
    2. 17 = 5 x 3 + 2
    3. 17 = 5 x 4 - 3
    4. 17 = 5 x 5 - 8

12. The functions 'f' and 'g' are inverse of each other if and only if their composition gives _______.

    1. constant function
    2. identity function
    3. bijective function
    4. injective function

13. If A and B are disjoint finite sets then n(A ∪ B) = ______.

    1. n(A) − n(B)
    2. n(A) + n(B) − n(A ∩ B)
    3. n(A) + n(B)
    4. n(A) + n(B) + n(A ∩ B)

14. For the following relation to be a function, x can not be what values?
    R = {(2,4), (x,1), (4,2), (5,6)}

    1. x cannot be 2, 4 or 5
    2. x cannot be 4, 1 or 6
    3. x cannot be 2, 4 or 6
    4. x cannot be 1, 2 or 6

15. Determine values of x and y, where (2x, x + y) = (8, 6).

    1. x = 3 and y = 5
    2. x = 4 and y = 2
    3. x = 6 and y = 12
    4. x = 4 and y = 12

16. Let A and B be subsets of U with n(A) = 12, n(B) = 15, n(A') = 17, and n(A intersection B) = 8, then n(U)=______ .

    1. 27
    2. 29
    3. 20
    4. 35

17. The switches in parallel act just like ________.

    1. NOT gate
    2. AND gate
    3. OR gate
    4. XOR gate

18. Range of the relation {(0,1), (3,22), (90,34)} is __________ .

    1. {0, 3, 90}
    2. {1, 22, 34}
    3. {0, 1, 3}
    4. {0, 1, 3, 22, 90, 34}

19. If r is a positive real number, then the value of r in 3.r.r = −27r is ________.

    1. −9
    2. +9
    3. 0
    4. None of the given

20. Suppose there are 8 different tea flavors and 5 different biscuit brands. A guest wants to take one tea and one brand of biscuit. How many choices are there for this guest?

    1. 5
    2. 8
    3. 13
    4. 40

21. Let f(x) = x² + 1 define functions f from R to R and c = 2 be any scalar, then c.f(x) is ______.

    1. 2
    2. x² + 1
    3. 2x² - 1
    4. 2x² + 2

22. In how many ways a student can choose a course from 2 science courses,3 literature courses and 5 art courses.

    1. 30
    2. 10
    3. 1440
    4. 240

23. Let p1, p2, p3 be True premises in a given Truth Table. If the conjunctions of the Conclusion with each of p1, p2, p3 are True, then the argument is ________.

    1. False
    2. True
    3. Invalid
    4. Valid

24. Which of the followings is the product set A * B * C ? where A = {a}, B = {b}, and C = {c, d}.

    1. {(a, b, c), (a, b, d)}
    2. {(a, c, b), (a, d, b)}
    3. {(b, c, a), (b, d, a)}
    4. {(c, b, a), (d, b, a)}

25. The conjunction p ∧ q is True when _________.

    1. p is True, q is False
    2. p is False, q is True
    3. p is True, q is True
    4. p is False, q is False

26. 'p is equivalent to q' means ________.

    1. p is not necessary but p is sufficient for q.
    2. p is neither necessary nor sufficient for q.
    3. p is necessary and sufficient for q.
    4. p is necessary but not sufficient for q.

27. What is the minimum number of students in a class to be sure that two of them are born in the same month?

    1. 11
    2. 12
    3. 13
    4. 14

28. Let g be a function defined by g(x) = x + 1. Then the composition of (g o g)(x)is ______.

    1. x
    2. x + 1
    3. x + 2
    4. x² + 2x + 1

29. Find the number of distinct permutations that can be formed using the letters of the word ”BENZENE”

    1. 120
    2. 220
    3. 320
    4. 420

30. A Random variable is also called a _________.

    1. Chance Variable
    2. Constant

31. P(0, 0)=______?

    1. 0
    2. 1
    3. 2
    4. undefined

32. If A and B are any two sets, then A − B = B − A

    1. True
    2. False

33. If f and g are two one-to-one functions, then their composition that is gof is one-to-one.

    1. TRUE
    2. FALSE

34. A box contains 5 different colored light bulbs. Which of the followings is the number of ordered samples of size 3 with replacement?

    1. 8
    2. 15
    3. 125
    4. 243

35. Which of the followings is the product set A * B * C ? where A = {a}, B = {b}, and C = {c, d}.

    1. {(a, b, c), (a, b, d)}
    2. {(a, c, b), (a, d, b)}
    3. {(b, c, a), (b, d, a)}
    4. {(c, b, a), (d, b, a)}

36. A student can choose a computer project from one of the two lists. The two lists contain 12 and 18 possible projects, respectively. How many possible projects are there to choose from?

    1. 12
    2. 18
    3. 30
    4. 216

37. Let R be a relation on a set A. If R is reflexive then its compliment is ____________.

    1. Reflexive
    2. Irreflexive
    3. Symmetric
    4. Antisymmetric

38. A student is to answer five out of nine questions on exams. Find the number of ways that can choose the five questions.

    1. 216
    2. 316
    3. 126
    4. None of the given

39. There are 5 girls students and 20 boys students in a class. How many students are there in total ?

    1. 4
    2. 15
    3. 25
    4. 100

40. What is the truth value of the sentence?
    'It rains if and only if there are clouds.'

    1. True
    2. False

41. The method of loop invariants is used to prove __________ of a loop with respect to certain pre and post-conditions.

    1. falseness
    2. correctness

42. The statement p → q is logically equivalent to ∼q → ∼p

    1. True
    2. False

43. The set of prime numbers is _________.

    1. finite set
    2. infinite set
    3. continuous set
    4. None of the given

44. Let R be a relation on a set A. If R is reflexive then its compliment is ________ .

    1. Reflexive
    2. Irreflexive
    3. Symmetric
    4. Antisymmetric

45. Reductio and absurdum' is another name of _________.

    1. Direct Method of proof
    2. proof by contradiction
    3. proof by contapositive
    4. None of the given

46. In how many ways can 6 people be seated on 6 available seats?

    1. 120
    2. 6
    3. 12
    4. 720

47. The set Z of all integers is _____.

    1. uncountable
    2. countable

48. Let A = {1, 2, 3, 4} and B = {7} then the constant function from A to B is _________ .

    1. Onto
    2. One to one
    3. Both one to one and onto
    4. Neither one to one nor onto

49. Let p → q be a conditional statement, then the statement q → p is called ________.

    1. Inverse
    2. Converse
    3. Contrapositive
    4. Double conditional

50. If X and Y are independent random variables, then E(XY) is equal to

    1. E(XY)
    2. XE(Y)
    3. YE(X)
    4. E(x)E(y)

51. Let A = {2, 3, 5, 7}, B = {2, 3, 5, 7, 2}, C = Set of first five prime numbers. Then from the following which statement is true ?

    1. A = B
    2. A = C
    3. B = C
    4. All the three sets are equal.

52. Let f(x)=3x and g(x) = 3x − 2 define functions f and g from R to R. Then (f+g)(x) = ________.

    1. −2
    2. 6x + 2
    3. 6x − 2
    4. 6x.x − 2

53. If p is false and q is true, then ∼p ↔ q is ________.

    1. True
    2. False

54. Let p be True and q be True, then ( ∼p ∧ q ) is ________.

    1. t ( where t is tautology. )
    2. c ( where c is contradiction. )
    3. True
    4. False

55. If p ↔ q is True, then ________.

    1. Only p is True.
    2. Only q is True.
    3. p and q both are True.
    4. None of the given.

56. If A = Set of students of virtual university then A has been written in the _________.

    1. Tabular form
    2. Set builder form
    3. Descriptive form
    4. A is not a set

57. Real valued function is a function that assigns _______ to each member of its domain.

    1. negative real number
    2. positive real number
    3. only a real number
    4. any arbitrary real number

58. Find the number of the word that can be formed of the letters of the word “ELEVEN”.

    1. 120
    2. 110
    3. 220
    4. None of the given

59. The logical statement p ∧ q means ________.

    1. p OR q
    2. p NOT q
    3. p AND q
    4. p XOR q

60. Which relations below are functions?
    R1 = {(3,4), (4,5), (6,7), (8,9)}
    R2 = {(3,4), (4,5), (6,7), (3,9)}
    R3 = {(-3,4), (4,-5), (0,0), (8,9)}
    R4 = {(8,11), (34,5), (6,17), (8,19)}

    1. R1 and R3 are functions
    2. R1 and R2 are functions
    3. R2 and R4 are functions
    4. R3 and R2 are functions

61. R = {(a,1), (b,2), (c,3), (d,4)} then the inverse of this relation is _______.

    1. {(a,1), (b,2), (3,c), (4,d)}
    2. {(1,a), (2,b), (3,c), (4,d)}
    3. {(a,1), (2,b), (3,c), (4,d)}
    4. {(1,a), (b,2), (3,c), (4,d)}

62. Let A = {1, 2, 3} and B = {2, 4} then number of functions from A to B are _________.

    1. 6
    2. 8
    3. 16
    4. 64

63. The total number of terms in an arithmetic series 0 + 5 + 10 + 15 + .... + 50 are ________.

    1. 9
    2. 10
    3. 11
    4. infinite

64. If p = It is raining, q = She will go to college
    "It is raining and she will not go to college”
    will be denoted by

    1. p ∧ ∼q
    2. p ∧ q
    3. ∼(p ∧ q)
    4. ∼p ∧ q

65. The contrapositive of the conditional statement 'If it is Sunday, then I go for shopping' is ________.

    1. I do Not go for shopping, then it is Not Sunday.
    2. I go for shopping, then it is Sunday.
    3. I do Not go for shopping, then it is Sunday.
    4. I go for shopping, then it is Not Sunday.

66. If A and B be events with P(A) = 1/3, P(B) = 1/4 and P(A ∩ B) = 1/6, then P(A ∪ B) = ________ .

    1. 2/3
    2. 5/12
    3. 1/24
    4. 1/2

67. If X and Y are random variables, then E(aX) is equal to

    1. E(aX)
    2. aE(X)
    3. aX
    4. None of the given

68. One-to-One correspondence means the condition of ______.

    1. one-One
    2. identity
    3. onto
    4. one-One and onto

69. If X and Y are independent random variables and a and b are constants, then Var(aX + bY)is equal to

    1. aVar(X) + bVar(Y)
    2. (a + b)[Var(X) + Var(Y)]
    3. Var(aX) + Var(bY)
    4. a^2 Var(X) + b^2 Var(Y)

70. If order matters and repetition is allowed, then which counting method should be used in order to select 'k' elements from a total of 'n' elements?

    1. K-Selection
    2. K-Sample
    3. K-combination
    4. K-Permuatation

71. Among 20 people, 15 either swim or jog or both. If 5 swim and 6 swim and jog, how many jog?

    1. 6
    2. 16
    3. 24
    4. 46

72. Let R be the universal relation on a set A then which one of the following statement about R is true?

    1. R is not symmetric
    2. R is not reflexive
    3. R is not transitive
    4. R is reflexive, symmetric and transitive.

73. The negation of “Today is Friday” is

    1. Today is Saturday
    2. Today is not Friday
    3. Today is Thursday
    4. None of the given

74. The disjunction of p and q is written as ________.

    1. p ∨ q
    2. p ∧ q
    3. p XOR q
    4. None of the given

75. In how many ways a student can choose one of each of the courses when he is offered 3 mathematics courses, 4 literature courses and 2 history courses.

    1. 9
    2. 24
    3. 288
    4. 14

76. ∼(P → q) is logically equivalent to _________.

    1. p ∧ ∼q
    2. p ∨ ∼q
    3. ∼p ∧ q
    4. ∼p ∨ q

77. Which of the followings is the factorial form of 5 . 4?

    1. 5/3
    2. 5!/3
    3. 5!/3!
    4. 5/3!

78. A set is called finite, if and only if, it is the ________ or there is ________ .

    1. empty set, onto
    2. empty set, one-to-one
    3. one-to-one, onto
    4. empty set, bijective

79. Let X = {1, 2, 3}, then 2-combinations of the 3 elements of the set X are _________?

    1. {1, 2}, {1, 3} and {2, 3}
    2. {1, 2}, {2, 1}, {1, 3}, {3, 1}, {2, 3}, and {3, 2}
    3. {1, 2}, {2, 1}, {1, 3} and {2, 3}
    4. {1, 2}, {2, 1},{1, 3} and {3, 1}

80. ( p ∨ ∼p ) is the ________.

    1. Contradiction
    2. Conjunction
    3. Tautology
    4. Contingency

81. Let f(x) = 3x and g(x) = x + 2 define functions f and g from R to R, then (f.g)(x) is _____.

    1. 2x − 2
    2. 3x + 2
    3. 4x + 2
    4. 3x² + 6x

82. If a function (g o f)(x):X→Z is defined as (g o f)(x) = g(f(x)) for all x ∈ X. Then the function ________ is known as composition of f and g.

    1. (f o g)
    2. f^-1(g(x))
    3. (g o f)
    4. g^-1(f(x))

83. The number of the words that can be formed from the letters of the word,“COMMITTEE” are

    1. 9p9
    2. 9C9
    3. 9! / (2!2!2!)
    4. None of the given

84. Let R be a relation on a set A. If R is symmetric then its compliment is __________.

    1. Reflexive
    2. Irreflexive
    3. Symmetric
    4. Antisymmetric

85. The converse of the conditional statement p → q is

    1. q → p
    2. ∼q → ∼p
    3. ∼p → ∼q
    4. None of the given

86. Let X = {2, 4, 5} and Y= {1, 2, 4} and R be a relation from X to Y defined by R = {(2,4), (4,1), (a,2)}. For what value of ‘a‘ the relation R is a function ?

    1. 1
    2. 2
    3. 4
    4. 5

87. How many possible outcomes are there when a fair coin is tossed four times?

    1. 4
    2. 8
    3. 16
    4. 32

88. x belongs to A or x belongs to B, therefore x belongs to ________.

    1. A intersection B
    2. A union B
    3. A difference B
    4. A symmetric difference B

Select a course code for Objective Questions:

ACC311 ACC501 BIF101 BIF401 BIF501 BIO101 BIO201 BIO202 BIO203 BIO204 BIO301 BIO302 BIO303 BIO401 BIO503 BNK603 BT101 BT201 BT301 BT302 BT401 BT402 BT403 BT404 BT405 BT406 BT501 BT504 BT505 BT604 CHE201 CHE301 CS001 CS101 CS201 CS202 CS204 CS301 CS302 CS304 CS311 CS401 CS402 CS403 CS408 CS501 CS502 CS504 CS506 CS508 CS601 CS602 CS604 CS605 CS607 CS609 CS610 CS614 CS707 CS710 ECO401 ECO402 ECO403 ECO404 EDU302 EDU401 EDU402 EDU501 EDU601 ENG101 ENG201 ENG301 ENG501 FIN611 FIN624 FIN625 ISL201 IT430 MCM101 MCM301 MGMT623 MGT101 MGT211 MGT301 MGT401 MGT402 MGT501 MGT502 MGT503 MGT510 MGT601 MGT603 MKT501 MKT610 MKT630 MTH101 MTH202 MTH302 MTH501 MTH601 MTH634 PAK301 PHY101 PHY301 PSY101 PSY403 PSY502 PSY512 SOC101 SOC301 SOC401 STA301 STA630 STA641 ZOO102 ZOO401 ZOO501 ZOO502 ZOO503 ZOO504 ZOO505 ZOO506 ZOO510 ZOO630

Select a course code for Subjective Questions:

ACC311 CS001 CS101 CS201 CS301 CS607 CS701 CS702 CS703 CS704 CS707 CS708 CS710 CS711 CS718 CS721 CS724 ECO401 ECO403 ENG101 ENG301 MGT703 MTH501 STA301