Discrete Structures and Optimisation – Previous Year Questions
272 questions from past UGC NET Paper 2 – Computer Science papers, each with the correct answer. Open any question for its explanation.
Topic-wise mock tests
- Which of the following statements about sets are true? A. 0 ∈ ∅ B. ∅ ∈ {0} C. ∅ ∈ {∅} D. {∅} ∈ {∅} E. {∅} ⊂ {∅, {∅}}
- Consider the following relations on {1, 2, 3, 4}. Which of them are reflexive? (A) R₁ = {(1,1), (1,2), (2,1), (2,2), (3,4), (4,1), (4,4)} (B) R₂ = {(1,1), (1,2), (2,1)} (C) R₃ = {(1,1), (1,2), (1,4), (2,1), (2,2), (3,3)…
- Let P be "It is hot day" and q be "The temperature is 48°C". Write in simple sentences the meaning of ¬p∧¬q.
- 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 . . . .
- 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)…
- If p and q are two statements, which of the following equivalences hold? (A) ~(p ∧ q) ≡ ~p ∨ ~q (B) ~(p ∨ q) ≡ ~p ∧ ~q (C) p ∧ ~p ≡ T (D) ~(p ⇒ q) ≡ p ∧ ~q (E) p ∨ q ≡ ~p ∨ ~q
- Match List - I with List - II. List - I (Queries) List - II (Probability) (A) A bag contains 6 (I) 3 / 68 white and 4 red balls. Two balls are drawn at random. What is the chance they will the same color?
- Translate into English: ∀x ∃y (x < y), where the domain of both variables is the real numbers.
- If f(n) = 5f(n/2) + 3 and f(1) = 7, where n = 2ᴷ and K is a positive integer, then f(2ᴷ) =
- Which of the following are true if the universe of discourse consists of all integers? A. ∀n∃m (n² < m) B. ∃n∀m (n < m²) C. ∃n∃m (n² + m² = 5) D. ∃n∃m (n² + m² = 6) E. ∃n∃m (m + n = 4 ∧ n − m = 1)
- Which of the following sequences aₙ are solutions of the recurrence aₙ = 8aₙ₋₁ − 16aₙ₋₂? A. aₙ = 1 B. aₙ = 2ⁿ C. aₙ = 4ⁿ D. aₙ = n·4ⁿ E. aₙ = (−4)ⁿ
- Arrange the following simple graphs based on a number of spanning trees they have : A. K [Complete Graph with 3 nodes] 3 B.K [Complete Graph with 4 nodes] 4 C.K (Complete bipartite graph of the 2 nodes 2,2 in each parti…
- Match List - I with List - II.
- Arrange the following sets in increasing order of cardinality: A. A₁ = {{1, 2}, {3}} B. A₂ = {{1}, {2}, {3}, {4}} C. A₃ = {{1, 2, 3, 4, 5, 6}} D. A₄ = {{1, 2}, {2, 3, 4}, {5}} E. A₅ = {{1}, {2}, {3}, {4}, {5}}
- The chairs of an auditorium are to be labelled with a letter followed by a positive two digits integer where labelling is starting with A01. How many maximum possible chairs could be labelled in this way?
- Z is set of integers, <Z,*> forms group if * in :
- Which of the following are tautologies? A. (p ∨ r) → p B. p → (p ∧ r) C. ~p → (p → r) D. (p ∧ r) → (p → r) E. ~(p → r) → p
- Which of the following is not the most common property in social network ?
- When we formulate the LPP from the above-given passage. Which of the following is an objective function for its dual problem?
- How many unit length used for cloth A when income is maximized ?
- If x unit length of type A cloths and y unit length of type B cloths, then contrains formulate for the LPP, is :
- On the basis of available material, maximum total income is :
- If the universe of discourse is all real numbers, which of the following are true? (A) ∃x∀y (x + y = y) (B) ∀x∀y (((x ≥ 0) ∧ (y < 0)) → (x − y > 0)) (C) ∃x∃y (((x ≤ 0) ∧ (y ≤ 0)) ∧ (x − y > 0)) (D) ∀x∀y (((x ≠ 0) ∧ (y ≠…
- If the universe of discourse is the set of integers, which of the following are TRUE? (A) ∀n∃m (n² < m) (B) ∃n∀m (n < m²) (C) ∃n∀m (nm = m) (D) ∃n∃m (n² + m² = 6) (E) ∃n∃m (n + m = 4 ∧ n − m = 1)
- Let N² = N × N, where N is the set of natural numbers, and let R be the relation on N² with ⟨x, y⟩ R ⟨u, v⟩ ⇔ xv = yu. Which of the following are TRUE? (A) Reflexive (B) Symmetric (C) Transitive (D) Asymmetric
- Which of the following are tautology? → → → → (A) (P (P Q)) (P Q) → → → ∨ (B) ((P Q) Q) (P Q) ∨¬ → → ∨¬ → (C) ((P P) Q) ((P P) R) → ∧¬ → → ∧¬ (D) (Q (P P)) (R (P P))
- Let P(x) be the statement "x = x²". If the universe of discourse is the integers, which of the following are true? (A) P(0) (B) P(1) (C) P(2) (D) ∃x P(x) (E) ∀x P(x)
- Which of the following statements are true if the universe of discourse is the set of integers? (A) ∀n (n² ≥ 0) (B) ∃n (n² = 2) (C) ∀n (n² ≥ n) (D) ∃n (n² < 0)
- Which of the following graphs are trees? (see figure)
- Match List I with List II. List I: (A) ∅ ∩ {∅} = (B) {∅} ∩ {∅} = (C) {∅, {∅}} − ∅ = (D) ∅ ∪ {{∅}} =. List II: (I) ∅ (II) {∅} (III) {{∅}} (IV) {∅, {∅}}