UGC NET Computer Science Discrete Structures and Optimization Previous Year Questions (PYQs)

All 49 Discrete Structures and Optimization PYQs

Ordered newest exam first. Each question links back to the full paper it came from.

1. Question 1

   UGC NET January 2025

   Arrange the following graphs by number of edges in increasing order, for n > 3.

   A. Kₙ (complete graph)
   B. Cₙ (cycle graph)
   C. Wₙ (wheel graph)
   D. Kₙ,ₙ (complete bipartite graph)
   E. Qₙ (n-cube graph)

   1. AA, B, C, D, E
   2. BB, A, C, D, E
   3. CA, B, C, E, D
   4. DE, D, C, A, B

   Answer: (B) B, A, C, D, E

   Explanation

   The cycle has the fewest edges. Applying the graph definitions and the paper’s official key gives B, A, C, D, E; the remaining sequences place the cycle or complete graph incorrectly. The correct option follows from the stated definition or calculation; the other options omit a necessary condition or describe a different concept.

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2. Question 2

   UGC NET January 2025

   Which of the following is the complement of the Boolean function AB + CD′ + A′B + CD′?

   1. AA′B + CD
   2. B(A′ + B)(C + D′)
   3. C(A + B′)(C′ + D)
   4. DAB′ + CD′

   Answer: (C) (A + B′)(C′ + D)

   Explanation

   First simplify AB + A′B to B, giving F = B + CD′. By De Morgan’s law, F′ = B′(C′ + D), which is represented by the keyed option. The correct option follows from the stated definition or calculation; the other options omit a necessary condition or describe a different concept.

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3. Question 3

   UGC NET January 2025

   Consider the LPP: maximise z = 30x − 18y subject to 3x + 4y ≤ 60, 5x − 3y ≥ 20, and x, y ≥ 0. Which listed point is the solution?

   A. (4, 0)
   B. (2, 0)
   C. (7, 5)
   D. (0, 15)
   E. (8, 5)

   1. AA and C only
   2. BB only
   3. CE only
   4. DD only

   Answer: (C) E only

   Explanation

   Feasibility must be checked first, then the objective compared. Point E satisfies both inequalities and gives z = 150, exceeding the other feasible listed points. The correct option follows from the stated definition or calculation; the other options omit a necessary condition or describe a different concept.

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4. Question 4

   UGC NET January 2025

   If x and y are elements of a group G, x⁵ = y³ = e, and e is the identity, then the inverse of x²yx⁴y² is

   1. Ay²xy²x⁴
   2. Byxy²x³
   3. Cyx⁶y⁶x³
   4. Dx⁴y²x²y

   Answer: (B) yxy²x³

   Explanation

   The inverse reverses the factor order: y⁻²x⁻⁴y⁻¹x⁻². Since x⁵ = e and y³ = e, this becomes yxy²x³, which is option B. The correct option follows from the stated definition or calculation; the other options omit a necessary condition or describe a different concept.

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5. Question 5

   UGC NET January 2025

   Which statements about the sets are true?

   A. ∅ ∈ ∅
   B. ∅ ∈ {∅}
   C. {∅} ⊂ {∅, {∅}}
   D. {∅} ∈ {∅}
   E. {∅} ⊂ (∅, {∅})

   1. AA, B, C, D and E
   2. BA, B, C and E only
   3. CA and C only
   4. DC and E only

   Answer: (D) C and E only

   Explanation

   The empty set is not an element of itself, but it is an element of {∅}. Careful distinction between membership (∈) and subset (⊂) makes C and E the correct statements. The correct option follows from the stated definition or calculation; the other options omit a necessary condition or describe a different concept.

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6. Question 6

   UGC NET January 2025

   Which one is not a basic solution of the system x₁ + 2x₂ + x₃ = 4 and 2x₁ + x₂ + 5x₃ = 5?

   1. Ax₁ = −1, x₂ = 2, x₃ = 1
   2. Bx₁ = 2, x₂ = 1
   3. Cx₁ = 5, x₃ = −1
   4. Dx₂ = 5/3, x₃ = 2/3

   Answer: (A) x₁ = −1, x₂ = 2, x₃ = 1

   Explanation

   A basic solution sets one variable to zero and solves the two equations for the remaining variables. Option A gives three nonzero assignments and does not satisfy that basic-solution construction.

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7. Question 7

   UGC NET January 2025

   Match Boolean algebra laws with axioms.

   List-I	List-II
   A. Absorption law B. Bounded law C. Identity law D. Distributive law	I. a + 1 = 1 II. a + 0 = a III. a(b + c) = ab + ac IV. a + ab = a

   1. AA-IV, B-I, C-II, D-III
   2. BA-IV, B-III, C-I, D-II
   3. CA-III, B-IV, C-II, D-I
   4. DA-II, B-III, C-IV, D-I

   Answer: (A) A-IV, B-I, C-II, D-III

   Explanation

   Absorption is a + ab = a; bounded law is a + 1 = 1; identity is a + 0 = a; and distributivity is a(b + c) = ab + ac. The correct option follows from the stated definition or calculation; the other options omit a necessary condition or describe a different concept.

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8. Question 8

   UGC NET January 2025

   Which is the simplified form of the Boolean function with minterms Σm(0, 1, 3, 7) over variables A, B and C?

   1. AA′B′C + A′B′C + A′BC + ABC
   2. BA′B′ + A′BC + ABC
   3. CA′B′ + BC
   4. DA′B′ + BC′

   Answer: (C) A′B′ + BC

   Explanation

   A′B′ covers minterms 0 and 1, while BC covers minterms 3 and 7. Together these terms cover exactly Σm(0,1,3,7). The correct option follows from the stated definition or calculation; the other options omit a necessary condition or describe a different concept.

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9. Question 9

   UGC NET January 2025

   The probability that A hits a target is 1/4 and the probability that B hits the target is 2/5. Both shoot. What is the probability that at least one hits the target?

   1. A3/5
   2. B11/9
   3. C2/20
   4. D11/20

   Answer: (D) 11/20

   Explanation

   Use the complement: P(at least one hit) = 1 − P(neither hits) = 1 − (3/4)(3/5) = 1 − 9/20 = 11/20. The correct option follows from the stated definition or calculation; the other options omit a necessary condition or describe a different concept.

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10. Question 10

    UGC NET January 2025

    Match List-I with List-II: Match the logical equivalence propositions.

    List-I	List-II
    A. p → q B. ¬(p ∨ (¬p ∧ q)) C. p ↔ q D. ¬(p ↔ q)	I. (p ∧ q) ∨ (¬p ∧ ¬q) II. ¬p ∨ q III. ¬(p ∨ q) IV. ¬p ↔ q

    1. AA-I, B-III, C-II, D-IV
    2. BA-I, B-II, C-III, D-IV
    3. CA-II, B-III, C-I, D-IV
    4. DA-II, B-III, C-IV, D-I

    Answer: (C) A-II, B-III, C-I, D-IV

    Explanation

    p → q is ¬p ∨ q; ¬(p ∨ (¬p ∧ q)) simplifies to ¬(p ∨ q); p ↔ q is (p ∧ q) ∨ (¬p ∧ ¬q); and its negation is ¬p ↔ q. The correct option follows from the stated definition or calculation; the other options omit a necessary condition or describe a different concept.

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11. Question 11

    UGC NET January 2025

    The mathematical notation to describe logical entailment of a sentence α entails another sentence β is

    1. Aα ⊨ β
    2. Bα ⊆ β
    3. Cβ ⊨ α
    4. Dβ ⊆ α

    Answer: (A) α ⊨ β

    Explanation

    Semantic entailment is written α ⊨ β: every interpretation that satisfies α also satisfies β. It is neither set inclusion nor entailment in the reverse direction. The correct option follows from the stated definition or calculation; the other options omit a necessary condition or describe a different concept.

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12. Question 12

    UGC NET January 2025

    Which of the three displayed graphs is/are planar?

    

    Graphs A, B and C for Question 133

    1. AA and C only
    2. BB only
    3. CA only
    4. DA and B only

    Answer: (C) A only

    Explanation

    Graph A is planar because it can be redrawn without any edge crossings. Graphs B and C contain non-planar configurations, so only A is planar and option C is correct. The correct option follows from the stated definition or calculation; the other options omit a necessary condition or describe a different concept.

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13. Question 13

    UGC NET August 2024

    Consider the given number (45) y where y is the
    base of the number. Some of the possible values
    of y are given below.
    A. 5
    B. 6
    C. 7
    D. 8

    1. A(A), (B) and (C) Only
    2. B(B), (C) and (D) Only
    3. C(A), (C) and (D) Only
    4. D(A), (B), (C) and (D)

    Answer: (B) (B), (C) and (D) Only

    Explanation

    In base y every digit used must be smaller than y. The numeral 45 uses the digit 5, so y must exceed 5. Base 5 is therefore invalid, while bases 6, 7 and 8 all admit the digits 4 and 5. The valid values are (B), (C) and (D) — option B.

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14. Question 14

    UGC NET August 2024

    Let P be "It is hot day" and q be "The
    temperature is 48°C". Write in simple sentences
    the meaning of ¬p∧¬q.

    1. AIt is hot day or temperature is 48°C
    2. BIt is cold day or temperature is 48°C
    3. CIt is neither a hot day nor temperature is 48°C
    4. DIt is not a hot day

    Answer: (C) It is neither a hot day nor temperature is 48°C

    Explanation

    ¬p reads “it is not a hot day” and ¬q reads “the temperature is not 48°C”. The conjunction asserts both at once, which in plain English is “it is neither a hot day nor is the temperature 48°C”. Option C says exactly that; options A and B use “or”, and option D drops the second conjunct.

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15. Question 15

    UGC NET August 2024

    A graph G with number of vertices greater and
    equal than three i.e. (n ≥ 3) is a Hamiltonian
    graph, if the degree of each vertex is greater and
    equal to . . . .

    1. AEqual to number of vertices
    2. BDouble of number of vertices
    3. CHalf of number of vertices
    4. DFour times of number of vertices

    Answer: (C) Half of number of vertices

    Explanation

    Dirac’s theorem states that a simple graph on n ≥ 3 vertices in which every vertex has degree at least n/2 is Hamiltonian. The threshold is therefore half the number of vertices, so option C is correct.

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16. Question 16

    UGC NET August 2024

    A coin is tossed successively three times. Find the
    Probability (P), Event (E), Sample space (S) of
    getting exactly one head or two heads, where n
    is number of occurrence.
    A. n(S) = 8 and n(E) = 4
    B. n(E) = 6 and n(S) = 8
    C. P(E) = 3/4
    D. P(E) = 1/2

    1. A(A), (B), (C) and (D)
    2. B(B) and (C) Only
    3. C(A) and (D) Only
    4. D(C) and (D) Only

    Answer: (B) (B) and (C) Only

    Explanation

    Three tosses give n(S) = 2³ = 8. “Exactly one head or exactly two heads” covers every outcome except HHH and TTT, so n(E) = 6 and P(E) = 6/8 = 3/4. Statements (B) and (C) hold while (A) and (D) do not, so option B is correct.

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17. Question 17

    UGC NET August 2024

    Out of the following steps in the proper
    sequence for simplifying a Boolean function
    using a Karnaugh map (K-map).
    A. Identify and group the largest possible
    cluster of 1's
    B. Draw the K-map for the given Boolean
    function
    C. Write the simplified Boolean expression from
    the grouped clusters
    D. Transfer the truth table values to the K-map

    1. A(B), (D), (A), (C)
    2. B(D), (B), (A), (C)
    3. C(B), (A), (D), (C)
    4. D(A), (B), (C), (D)

    Answer: (A) (B), (D), (A), (C)

    Explanation

    Simplifying with a Karnaugh map goes: draw the map for the function (B), transfer the truth-table values into its cells (D), group the largest possible clusters of 1s (A), then read the simplified expression off those groups (C). That order is option A.

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18. Question 18

    UGC NET August 2024

    Let L (x , y) be the statement "x loves y" where
    the domain for both x and y consists of all people
    in the world. Use quantifiers to express "Joy is
    loved by everyone".

    1. A∀x L(x,Joy)
    2. B∀y L(Joy,y)
    3. C∃y∀x L(x,y)
    4. D∃x¬ L(Joy,x)

    Answer: (A) ∀x L(x,Joy)

    Explanation

    “Joy is loved by everyone” says that for every person x, x loves Joy — that is ∀x L(x, Joy). Option B reverses the roles and says Joy loves everyone; option C only asserts that somebody is universally loved; option D is an existential negation. Option A is correct.

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19. Question 19

    UGC NET August 2024

    Match List-I with List-II.

    List-I (Queries)	List-II (Probability)
    (A) A bag contains 6 white and 4 red balls. Two balls are drawn at random. What is the chance they will be the same colour?	(I) 3/68
    (B) In a pack of 52 cards, one card is drawn at random. What is the probability that it is either a king or a queen?	(II) 14/68
    (C) A bag contains 6 red, 4 white and 8 blue balls. If three balls are drawn at random, find the probability of 1 red and 2 white balls.	(III) 2/13
    (D) A bag contains 6 red, 4 white and 8 blue balls. If three balls are drawn at random, find the probability of 2 blue and 1 red balls.	(IV) 7/15

    1. A(A)-(III), (B)-(IV), (C)-(I), (D)-(II)
    2. B(A)-(III), (B)-(IV), (C)-(II), (D)-(I)
    3. C(A)-(IV), (B)-(III), (C)-(II), (D)-(I)
    4. D(A)-(IV), (B)-(III), (C)-(I), (D)-(II)

    Answer: (D) (A)-(IV), (B)-(III), (C)-(I), (D)-(II)

    Explanation

    There are C(10,2) = 45 equally likely pairs and C(6,2) + C(4,2) = 21 same-colour pairs, so the first probability is 21/45 = 7/15 and (A)-(IV). A pack holds 4 kings and 4 queens, so 8/52 = 2/13 and (B)-(III). From 18 balls C(18,3) = 816; one red with two white gives C(6,1)C(4,2) = 36, so 36/816 = 3/68 and (C)-(I). Two blue with one red gives C(8,2)C(6,1) = 168, so 168/816 = 14/68 and (D)-(II). Option D is correct.

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20. Question 20

    UGC NET August 2024

    Match List-I with List-II.

    List-I	List-II
    (A) Dijkstra’s Algorithm	(I) Find the shortest path between all pairs of vertices in a graph with positive or negative edge weights.
    (B) Floyd-Warshall Algorithm	(II) Finds the shortest path in a weighted graph with non-negative edge weights.
    (C) Bellman-Ford Algorithm	(III) Finds single-source shortest paths with possible negative weights.
    (D) Prim’s Algorithm	(IV) Finds the Minimum Spanning Tree (MST).

    1. A(A)-(I), (B)-(II), (C)-(III), (D)-(IV)
    2. B(A)-(II), (B)-(I), (C)-(III), (D)-(IV)
    3. C(A)-(I), (B)-(II), (C)-(IV), (D)-(III)
    4. D(A)-(III), (B)-(II), (C)-(IV), (D)-(I)

    Answer: (B) (A)-(II), (B)-(I), (C)-(III), (D)-(IV)

    Explanation

    Dijkstra’s algorithm needs non-negative edge weights, so (A)-(II). Floyd-Warshall solves all-pairs shortest paths, giving (B)-(I). Bellman-Ford handles single-source shortest paths when negative weights are present, so (C)-(III). Prim’s algorithm builds a minimum spanning tree, so (D)-(IV). Option B is correct.

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21. Question 21

    UGC NET August 2024

    An undirected graph has vertex set V and edge set E. If it has l edges, what is the sum of the degrees of all vertices?

    1. A2l
    2. Bl/2
    3. Cl²
    4. D√l

    Answer: (A) 2l

    Explanation

    Each undirected edge contributes one to the degree of each of its two endpoints. By the handshaking lemma, the total degree is 2l, so option A is correct.

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22. Question 22

    UGC NET August 2024

    40 software professionals were interviewed for a
    job. 25 knew PYTHON 20 knew JAVA and 7 knew
    neither language. How many knew both
    languages ?

    1. A12
    2. B53
    3. C10
    4. D88

    Answer: (A) 12

    Explanation

    Seven knew neither language, so 33 of the 40 knew at least one. By inclusion-exclusion |P ∪ J| = |P| + |J| − |P ∩ J|, giving 33 = 25 + 20 − |P ∩ J|, so |P ∩ J| = 12. Option A is correct.

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23. Question 23

    UGC NET August 2024

    Translate ∀x ∃y (x < y) into English. Consider the domain to be the real numbers for both variables.

    1. AFor all real numbers x, there exists a real number y such that x is less than y
    2. BFor every real numbers y, there exists a real number x such that x is less than y
    3. CFor some real numbers x, there exists a real number y such that x is less than y
    4. DFor each and every real numbers x and y, x is less than y

    Answer: (A) For all real numbers x, there exists a real number y such that x is less than y

    Explanation

    ∀x binds every real number x, and for each of them ∃y asserts that some real y exists with x < y. Read literally: for all real numbers x there exists a real number y such that x is less than y. Option A states this; option B swaps the quantified variables, option C weakens ∀ to “for some”, and option D collapses both quantifiers into a single universal claim.

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24. Question 24

    UGC NET December 2023

    What is the probability that a positive integer selected at random from the set of positive integer not exceeding 100 is divisible by either 2 or 5 ?

    1. A10/5
    2. B3/5
    3. C2/5
    4. D1/5

    Answer: (B) 3/5

    Explanation

    The key idea is stated in the problem: What is the probability that a positive integer selected at random from the set of positive integer not exceeding 100 is divisible by either 2 or 5 ?. The correct answer is B: 3/5. It follows the governing definition, algorithm, or calculation in this discrete structures and optimization item; the other choices fail to satisfy one or more stated conditions.

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25. Question 25

    UGC NET December 2023

    In a feed forward neural network with the following specifications : Input layer has 4 neurons, hidden layer has 3 neurons and output layer has 2 neurons using the sigmoid activation function for given input values [0.5, 0.8, 0.2, 0.6] as well as the initial weights for the connections. WI : (0.1, 0.3, 0.5, 0.2] W2e [02,04 0.6,.0.2] Input layer to hidden layer weights W3 : [0.3, 0.5, 0.7, 0.2] W4 : (0.4, 0.1, 0.3] W5 : [0.5, 0.2, 0.4] Hidden layer to output layer weights What is the output of the output layer when the given input values are passed through neural network ? Round the answer to two decimal places :

    1. A[0.62, 0.68]
    2. B[0.72, 0.78]
    3. C[0.82, 0.88]
    4. D[0.92, 0.98]

    Answer: (A) [0.62, 0.68]

    Explanation

    The key idea is stated in the problem: In a feed forward neural network with the following specifications : Input layer has 4 neurons, hidden layer has 3 neurons and output layer has 2 neurons using the sigmoid activation function for given input values [0.5, 0.8, 0.2, 0.6] as well as the initial w. The correct answer is A: [0.62, 0.68]. It follows the governing definition, algorithm, or calculation in this discrete structures and optimization item; the other choices fail to satisfy one or more stated conditions.

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26. Practise Discrete Structures and Optimization in a timed set

27. Question 26

    UGC NET December 2023

    If universe of disclosure are all real numbers, then which of the following are true ?
    A. 3x ¥y (&+y=y)
    B. Wx Vy((K>0)\y<0)) xy) (©) 3x Sy (((X<0)(y<0))AGe—y>0))
    D. Vx Vy((x#0)A(y#0)>(xy#0)) Choose the correct answer from the options given below :

    1. A(A) and (B) Only
    2. B(A), © and (D) Only
    3. C(A), (B) and (D) Only
    4. D(A), (B), () and (D) Only

    Answer: (D) (A), (B), () and (D) Only

    Explanation

    The key idea is stated in the problem: If universe of disclosure are all real numbers, then which of the following are true ?
    A. 3x ¥y (&+y=y)
    B. Wx Vy((K>0)\y<0)) xy) (©) 3x Sy (((X<0)(y<0))AGe—y>0))
    D. Vx Vy((x#0)A(y#0)>(xy#0)) Choose the correct answer from the options given below :. The correct answer is D: (A), (B), () and (D) Only. It follows the governing definition, algorithm, or calculation in this discrete structures and optimization item; the other choices fail to satisfy one or more stated conditions.

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28. Question 27

    UGC NET December 2023

    If the universe of disclosure is set of integers, then which of the followings are TRUE ?
    A. Vn 3m(n2< m)
    B. 3n Vm(n < m2)
    C. Jn Vm(nm=m)
    D. Jn Sm(n?+m?=6)
    E. Jn dm(n+m=4 \ n—-m=1) Choose the correct answer from the options given below :

    1. A(A), (B) and (©) Only
    2. B(B) and (C) Only
    3. C(©), (D) and @&) Only
    4. D(© and (E) Only

    Answer: (B) (B) and (C) Only

    Explanation

    The key idea is stated in the problem: If the universe of disclosure is set of integers, then which of the followings are TRUE ?
    A. Vn 3m(n2< m)
    B. 3n Vm(n < m2)
    C. Jn Vm(nm=m)
    D. Jn Sm(n?+m?=6)
    E. Jn dm(n+m=4 \ n—-m=1) Choose the correct answer from the options given below :. The correct answer is B: (B) and (C) Only. It follows the governing definition, algorithm, or calculation in this discrete structures and optimization item; the other choices fail to satisfy one or more stated conditions.

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29. Question 28

    UGC NET December 2023

    Which of the following(s) are main memory ?
    A. Virtual memory
    B. Cache memory
    C. RAM
    D. SSD Choose the correct answer from the options given below :

    1. A(A) and (© Only
    2. B(B) and (C) Only
    3. C(C) and (D) Only
    4. D(A), (B) and (© Only

    Answer: (B) (B) and (C) Only

    Explanation

    The key idea is stated in the problem: Which of the following(s) are main memory ?
    A. Virtual memory
    B. Cache memory
    C. RAM
    D. SSD Choose the correct answer from the options given below :. The correct answer is B: (B) and (C) Only. It follows the governing definition, algorithm, or calculation in this discrete structures and optimization item; the other choices fail to satisfy one or more stated conditions.

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30. Question 29

    UGC NET December 2023

    The statement P(x) : "x=x?". If the universe of disclosure consists of integers, what are the following have truth values :
    A. PC)
    B. PC) () PQ)
    D. Ax P(x) (BE) Vx P(x) Choose the correct answer from the options given below :

    1. A(A), @) and (E) Only
    2. B(A), (B) and (C) Only
    3. C(A), (B) and (D) Only
    4. D(B), (C) and (D) Only

    Answer: (C) (A), (B) and (D) Only

    Explanation

    The key idea is stated in the problem: The statement P(x) : "x=x?". If the universe of disclosure consists of integers, what are the following have truth values :
    A. PC)
    B. PC) () PQ)
    D. Ax P(x) (BE) Vx P(x) Choose the correct answer from the options given below :. The correct answer is C: (A), (B) and (D) Only. It follows the governing definition, algorithm, or calculation in this discrete structures and optimization item; the other choices fail to satisfy one or more stated conditions.

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31. Question 30

    UGC NET December 2023

    Which of the following statement are truth statements if universe of disclosure is set of integers :
    A. Vn(n?20)
    B. An(n?=2)
    C. Vn(n?>n)
    D. n(n*<0) Choose the correct answer from the options given below :

    1. A(A) and (B) Only
    2. B(B) and (C) Only
    3. C(©) and (D) Only
    4. D(A) and (C) Only

    Answer: (D) (A) and (C) Only

    Explanation

    The key idea is stated in the problem: Which of the following statement are truth statements if universe of disclosure is set of integers :
    A. Vn(n?20)
    B. An(n?=2)
    C. Vn(n?>n)
    D. n(n*<0) Choose the correct answer from the options given below :. The correct answer is D: (A) and (C) Only. It follows the governing definition, algorithm, or calculation in this discrete structures and optimization item; the other choices fail to satisfy one or more stated conditions.

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32. Question 31

    UGC NET December 2023

    Arrange the following encoding strategies used in Genetic Algorithms (GAs) in the correct sequence starting from the initial step and ending with the final representation of solutions :
    A. Binary Encoding
    B. Real valued Encoding
    C. Permutation Encoding
    D. Gray coding Choose the correct answer from the options given below :

    1. A(D), (B), (A). (©)
    2. B(B), (D), (A), ©
    3. C(©), ), (A), @)
    4. D(B), (©, (A), (D)

    Answer: (C) (©), ), (A), @)

    Explanation

    The key idea is stated in the problem: Arrange the following encoding strategies used in Genetic Algorithms (GAs) in the correct sequence starting from the initial step and ending with the final representation of solutions :
    A. Binary Encoding
    B. Real valued Encoding
    C. Permutation Encoding
    D. . The correct answer is C: (©), ), (A), @). It follows the governing definition, algorithm, or calculation in this discrete structures and optimization item; the other choices fail to satisfy one or more stated conditions.

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33. Question 32

    UGC NET December 2023

    Arrange the following steps in the correct order for a DHCP Client to renew its IP lease with a DHCP server :
    A. DHCP client sends a DHCPREQUEST message
    B. DHCP server acknowledges the renewal with a DHCPACK message
    C. DHCP client checks the local lease timer and initiates renewal
    D. DHCP server updates its lease database Choose the correct answer from the options given below :

    1. A(A), (B), ©, ©)
    2. B(©), ©), (B), (A)
    3. C(©), (8), (A), @)
    4. D(©, (A), (), (D)

    Answer: (D) (©, (A), (), (D)

    Explanation

    The key idea is stated in the problem: Arrange the following steps in the correct order for a DHCP Client to renew its IP lease with a DHCP server :
    A. DHCP client sends a DHCPREQUEST message
    B. DHCP server acknowledges the renewal with a DHCPACK message
    C. DHCP client checks the local lease tim. The correct answer is D: (©, (A), (), (D). It follows the governing definition, algorithm, or calculation in this discrete structures and optimization item; the other choices fail to satisfy one or more stated conditions.

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34. Question 33

    UGC NET December 2023

    The minimum cost of food is :

    Food X contains 6 units of Vitamin D per gram and 7 units of Vitamin E per gram and costs Rs. 12 per gram. Food Y contains 8 units of Vitamin D per gram and 12 units of Vitamin E per gram and costs Rs. 20 per gram. The daily minimum requirements of vitamin D and vitamin E are 100 units and 120 units respectively. Let x and y be the quantities in grams of Food X and Food Y.

    1. A205
    2. B250
    3. C330
    4. D200

    Answer: (A) 205

    Explanation

    The key idea is stated in the problem: The minimum cost of food is :. The correct answer is A: 205. It follows the governing definition, algorithm, or calculation in this discrete structures and optimization item; the other choices fail to satisfy one or more stated conditions.

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35. Question 34

    UGC NET December 2023

    Number of tuples obtained by applying cartesian product over X and Y are :

    X (S, Si, C)	Y (S, P, D)
    (J, 1, M) (B, 2, N) (R, 3, H) (T, 4, G)	(J, S₁, CA) (B, P₁, AB) (R, D₁, DC) (A, H₁, MD)

    1. A16
    2. B12
    3. C04
    4. D32

    Answer: (A) 16

    Explanation

    The key idea is stated in the problem: Number of tuples obtained by applying cartesian product over X and Y are :. The correct answer is A: 16. It follows the governing definition, algorithm, or calculation in this discrete structures and optimization item; the other choices fail to satisfy one or more stated conditions.

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36. Question 35

    UGC NET December 2023

    Match List-I with List-II.

    Source matching prompt	Source values
    See question stem	See answer choices

    1. A(A)-(D, (B)-(), ©-€V), (D)-@)
    2. B(A)-(), (@)-C), ©-(, (B)-av)
    3. C(A)-(), (8)-(), ©-€V), (D)-@)
    4. D(A)-(ID), (B)-(IV), (C)-(H), (D)-()

    Answer: (C) (A)-(), (8)-(), ©-€V), (D)-@)

    Explanation

    The key idea is stated in the problem: Match List - I with List - II. List - I List - II
    A. Bresenham (I) Hidden surface removal
    B. Cohen-Sutherland (I) Line drawing algorithm
    C. Sutherland-Hodgeman (II) Line clipping algorithm
    D. Z-Buffer (IV) Polygon clipping algorithm Choose the correct answ. The correct answer is C: (A)-(), (8)-(), ©-€V), (D)-@). It follows the governing definition, algorithm, or calculation in this discrete structures and optimization item; the other choices fail to satisfy one or more stated conditions.

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37. Question 36

    UGC NET December 2023

    Match List-I with List-II.

    Source matching prompt	Source values
    See question stem	See answer choices

    1. A(A)-€0N, (B)-(), ()-@, ()-(IV)
    2. B(A)-CV), (B-(), (QC, (D)-(m)
    3. C(A)-C, @)-€), (C)-CV), (P)-
    4. D(A)-(IV), (B)-(D), (C)-(I1), (D)-()

    Answer: (B) (A)-CV), (B-(), (QC, (D)-(m)

    Explanation

    The key idea is stated in the problem: Match List - I with List - II. List - I List - II
    A. BCNF iff (I) every JD is implied by the candidate keys
    B. 5 NF iff (II) all underlying domains contain scalar values only
    C. 1 NFiff (III) every MVD is implied by the candidate keys
    D. 4 NF iff (IV) ever. The correct answer is B: (A)-CV), (B-(), (QC, (D)-(m). It follows the governing definition, algorithm, or calculation in this discrete structures and optimization item; the other choices fail to satisfy one or more stated conditions.

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38. Question 37

    UGC NET December 2023

    Match List-I with List-II.

    Source matching prompt	Source values
    See question stem	See answer choices

    1. A(A)-0), (BC), (CC, (D)-(IV)
    2. B(A), (B)-(), (CCD, (D)-(IV)
    3. C(A)-CD, (B)-
    4. D(A)-CV), (BCD, ()-M), (P)-

    Answer: (B) (A), (B)-(), (CCD, (D)-(IV)

    Explanation

    The key idea is stated in the problem: Match List - I with List - II. List - I List - IT Propositions Disjunctive Normal Form (DNF)
    A. PA(P3Q) @ Pva
    B. APVQ)(PAQ, —@)_ (PASP)V(PAQ) (CQ) PQ (I) (AP)vQ
    D. PV(QAR) (IV) (PAP)V(PAQ)V(PAR)V(QAR) Choose the correct answer from the options given below :. The correct answer is B: (A), (B)-(), (CCD, (D)-(IV). It follows the governing definition, algorithm, or calculation in this discrete structures and optimization item; the other choices fail to satisfy one or more stated conditions.

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39. Question 38

    UGC NET November 2020

    The number of positive integers not exceeding 100 that are either odd or the square of an integer is ____.

    1. A63
    2. B59
    3. C55
    4. D50

    Answer: (C) 55

    Explanation

    There are 50 odd integers through 100. Of the squares 1² through 10², the five even squares (2², 4², 6², 8² and 10²) are not already in the odd set, so the union has 50 + 5 = 55 integers. Counting all ten squares again would double-count the five odd squares.

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40. Question 39

    UGC NET November 2020

    How many ways are there to pack six copies of the same book into four identical boxes, where a box can contain as many as six books?

    1. A4
    2. B6
    3. C7
    4. D9

    Answer: (D) 9

    Explanation

    Because both books and boxes are identical, each arrangement is a partition of 6 into at most four positive parts. The nine partitions are 6; 5+1; 4+2; 4+1+1; 3+3; 3+2+1; 3+1+1+1; 2+2+2; and 2+2+1+1.

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41. Question 40

    UGC NET November 2020

    Which of the following pairs of propositions are not logically equivalent?

    1. A((p → r) ∧ (q → r)) and ((p ∨ q) → r)
    2. Bp ↔ q and (¬p ↔ ¬q)
    3. C((p ∧ q) ∨ (¬p ∧ ¬q)) and p ↔ q
    4. D((p ∧ q) → r) and ((p → r) ∧ (q → r))

    Answer: (D) ((p ∧ q) → r) and ((p → r) ∧ (q → r))

    Explanation

    The first expression in option D requires r only when both p and q are true. The second requires r whenever either p or q is true, so they differ when exactly one of p and q is true. The other pairs are standard equivalent rewritings.

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42. Question 41

    UGC NET November 2020

    Consider the linear programming problem:

    Maximise Z = 2x₁ + 3x₂
    Subject to 2x₁ + x₂ ≤ 4; x₁ + 2x₂ ≤ 5; x₁, x₂ ≥ 0.

    The optimum value of the LP is

    1. A23
    2. B9.5
    3. C13
    4. D8

    Answer: (D) 8

    Explanation

    A linear-programming optimum occurs at a feasible corner. The two binding constraints give x₁ = 1 and x₂ = 2, so Z = 2(1) + 3(2) = 8. The other corners give smaller values.

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43. Question 42

    UGC NET November 2020

    What is the radix of the numbers if the solution to x² − 10x + 26 = 0 is x = 4 and x = 7?

    1. A8
    2. B9
    3. C10
    4. D11

    Answer: (D) 11

    Explanation

    By Vieta’s rule, the sum of the roots is the value represented by 10 in that radix. Since 4 + 7 = 11 in decimal, 10 in the unknown base is 11, so the base is 11.

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44. Question 43

    UGC NET November 2020

    If f(x) means “x is my friend” and p(x) means “x is perfect”, translate: “Some of my friends are not perfect.”

    1. A∀x (f(x) ∧ ¬p(x))
    2. B∃x (f(x) ∧ ¬p(x))
    3. C∀x (f(x) ∧ p(x))
    4. D∃x (¬f(x) ∧ ¬p(x))

    Answer: (B) ∃x (f(x) ∧ ¬p(x))

    Explanation

    “Some” requires an existential quantifier, and “not perfect” is ¬p(x). The witness must also be a friend, giving ∃x(f(x) ∧ ¬p(x)); a universal quantifier would incorrectly claim this for every friend.

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45. Question 44

    UGC NET November 2020

    On A = {a,b,c,d,e,f,g}, R = {(a,a),(b,b),(c,d),(c,g),(d,g),(e,e),(f,f),(g,g)}. Which listed property is satisfied: reflexive, antisymmetric, symmetric?

    1. AOnly reflexive
    2. BOnly symmetric
    3. CBoth reflexive and antisymmetric
    4. DAntisymmetric but not reflexive

    Answer: (D) Antisymmetric but not reflexive

    Explanation

    R is not reflexive because (c,c) and (d,d) are absent. It is antisymmetric because no distinct pair occurs in both directions, but it is not symmetric because (c,d) occurs without (d,c).

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46. Question 45

    UGC NET November 2020

    For the premise ∀x(P(x) ∨ Q(x)), an argument tries to conclude (∀xP(x)) ∧ (∀xQ(x)) by instantiating P(c) ∨ Q(c) and then simplifying to P(c) and Q(c). Which assessment is correct?

    1. AThe argument is valid.
    2. BSteps deriving P(c) and Q(c) are not correct inferences.
    3. CThe universal-generalisation steps are not correct.
    4. DThe final conjunction is not correct.

    Answer: (B) Steps deriving P(c) and Q(c) are not correct inferences.

    Explanation

    From a disjunction P(c) ∨ Q(c), one cannot infer either disjunct by simplification. Therefore the derivations of P(c) and Q(c) are invalid, and the later universal conclusions have no support.

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47. Question 46

    UGC NET November 2020

    Which simplified expressions represent F(A,B,C,D) = Σ(0,1,2,3,6,12,13,14,15)?
    A. A′B′ + AB + A′C′D′
    B. A′B′ + AB + A′CD′
    C. A′B′ + AB + BC′D′
    D. A′B′ + AB + BCD′

    1. A(A) only
    2. B(B) only
    3. C(A) and (B) only
    4. D(B) and (D) only

    Answer: (D) (B) and (D) only

    Explanation

    A′B′ covers minterms 0–3 and AB covers 12–15. Expression B adds only minterm 6 through A′CD′, while D adds that same missing minterm through BCD′; the overlap at 14 is already covered by AB. In contrast, A and C also generate minterm 4, which is absent from the specified sum.

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48. Question 47

    UGC NET November 2020

    A radical person is electable if the person is conservative, and otherwise is not electable. Which listed logical formalisation correctly represents that condition?

    1. AOCR review required
    2. BOCR review required
    3. C(A) and (C) only (4) (B) and (D) only
    4. DOCR review required

    Answer: (C) (A) and (C) only (4) (B) and (D) only

    Explanation

    The intended proposition says that, for a radical person, electability holds exactly when conservatism holds. The matching formalisation preserves both the conditional case and the “otherwise not” case.

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49. Question 48

    UGC NET November 2020

    Match List I with List II

    Let R, = {(1,1), (2,2), (3,3)} and R2 = {(1,1), (1,2), (1,3), (1,4)}

    List I List IT
    A. R, UR: @ = {(1,1),.2).(1.3),2.4).(2.2).(3,3)}
    B. Ry —R, aq) {aD}
    C. RiARz MM) {(1.2),(2.3).(1.4}}
    @) R:-R; QV) {(2.2),(8,3)}

    List I	List II
    Items are listed in the question stem.	Match each item to its stated description or complexity.

    1. AALT. B-Il. C-IV. D-III (2) AI, B-IV, C-II, D-II
    2. BOCR review required
    3. CALI, B-II, C-I, D-IV (4) A-I, B-IV, C-II, D-IIT
    4. DOCR review required

    Answer: (A) ALT. B-Il. C-IV. D-III (2) AI, B-IV, C-II, D-II

    Explanation

    Compute union, difference, intersection, and composition directly from the listed ordered pairs; only option A matches all four resulting relations.

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50. Question 49

    UGC NET November 2020

    Given below are two statements:

    Statement I: 5 divides n5 — n whenever n is a nonnegative integer.
    Statement II: 6 divides n? — n whenever n is a nonnegative integer.
    In the light of the above statements. choose the correct answer from the options given below

    1. ABoth Statement I and Statement II are correct
    2. BBoth Statement I and Statement II are incorrect
    3. CStatement I is correct but Statement II is incorrect
    4. DStatement I is incorrect but Statement II is correct

    Answer: (A) Both Statement I and Statement II are correct

    Explanation

    Fermat’s little theorem gives n⁵ ≡ n (mod 5). Also n³ − n is the product of three consecutive integers, so it is divisible by 6; both statements are true.

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