UGC NET · PAPER 2
UGC NET Paper 2 Computer Science – Unit 1: Discrete Structures and Optimization PYQ (December 2023)
Q1. What is the probability that a positive integer selected at random from the set of positive integers not exceeding 100 is divisible by either 2 or 5?
- 10/5
- 3/5
- 2/5
- 1/5
Q41. 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 ≠ 0) ↔ (xy ≠ 0))
Choose the correct answer from the options given below:
(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 ≠ 0) ↔ (xy ≠ 0))
Choose the correct answer from the options given below:
- (A) and (B) Only
- (A), (C) and (D) Only
- (A), (B) and (D) Only
- (A), (B), (C) and (D)
Q42. 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)
Choose the correct answer from the options given below:
(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)
Choose the correct answer from the options given below:
- (A), (B) and (C) Only
- (B) and (C) Only
- (C), (D) and (E) Only
- (C) and (E) Only
Q43. If N² = N × N, where N is the set of natural numbers, and R is a relation on N² such that ⟨x, y⟩ R ⟨u, v⟩ ⇔ xv = yu, which of the following are TRUE?
(A) Reflexive
(B) Symmetric
(C) Transitive
(D) Asymmetric
Choose the correct answer from the options given below:
(A) Reflexive
(B) Symmetric
(C) Transitive
(D) Asymmetric
Choose the correct answer from the options given below:
- (A) and (B) Only
- (B) and (C) Only
- (A), (C) and (D) Only
- (A), (B) and (C) Only
Q46. Which of the following are tautologies?
(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))
Choose the correct answer from the options given below:
(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))
Choose the correct answer from the options given below:
- (A) Only
- (B) Only
- (A) and (B) Only
- (C) and (D) Only
Q56. Let P(x) be the statement "x = x²". If the universe of discourse is the integers, which of the following have truth value TRUE?
(A) P(0)
(B) P(1)
(C) P(2)
(D) ∃x P(x)
(E) ∀x P(x)
Choose the correct answer from the options given below:
(A) P(0)
(B) P(1)
(C) P(2)
(D) ∃x P(x)
(E) ∀x P(x)
Choose the correct answer from the options given below:
- (A), (B) and (E) Only
- (A), (B) and (C) Only
- (A), (B) and (D) Only
- (B), (C) and (D) Only
Q57. 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)
Choose the correct answer from the options given below:
(A) ∀n (n² ≥ 0)
(B) ∃n (n² = 2)
(C) ∀n (n² ≥ n)
(D) ∃n (n² < 0)
Choose the correct answer from the options given below:
- (A) and (B) Only
- (B) and (C) Only
- (C) and (D) Only
- (A) and (C) Only
Q58. Which of the following graphs (on vertices a–f) are trees?
(A) edges: ac, bc, cd, cf, ef
(B) edges: ac, af, df, ef, be
(C) edges: ab, ac, ad, be, cf, de
(D) edges: af, be, bd, ce
Choose the correct answer from the options given below:
(A) edges: ac, bc, cd, cf, ef
(B) edges: ac, af, df, ef, be
(C) edges: ab, ac, ad, be, cf, de
(D) edges: af, be, bd, ce
Choose the correct answer from the options given below:
- (A) and (B) only
- (A), (B) and (D) only
- (A) and (D) only
- (A), (B), (C) and (D) only
Q66. Match List I with List II
| List I | List II |
|---|---|
| (A) φ ∩ {φ} | (I) φ |
| (B) {φ} ∩ {φ} | (II) {φ} |
| (C) {φ, {φ}} − φ | (III) {{φ}} |
| (D) φ ∪ {{φ}} | (IV) {φ, {φ}} |
- (A)-(I), (B)-(II), (C)-(III), (D)-(IV)
- (A)-(II), (B)-(I), (C)-(III), (D)-(IV)
- (A)-(II), (B)-(I), (C)-(IV), (D)-(III)
- (A)-(I), (B)-(II), (C)-(IV), (D)-(III)
Q72. Match List I with List II (logically equivalent expressions)
| List I | List II |
|---|---|
| (a) P ∧ (P → Q) | (I) P ∨ Q |
| (b) ¬(P ∨ Q) → (P ∧ Q) | (II) (P ∧ ¬P) ∨ (P ∧ Q) |
| (c) P → Q | (III) (¬P) ∨ Q |
| (d) P ∨ (Q ∧ R) | (IV) (P ∧ P) ∨ (P ∧ Q) ∨ (P ∧ R) ∨ (Q ∧ R) |
- (a)-(I), (b)-(II), (c)-(III), (d)-(IV)
- (a)-(II), (b)-(I), (c)-(III), (d)-(IV)
- (a)-(III), (b)-(I), (c)-(II), (d)-(IV)
- (a)-(IV), (b)-(III), (c)-(II), (d)-(I)
Q76. Arrange the following in ascending order:
(a) Remainder of 4916 when divided by 17
(b) Remainder of 2446 when divided by 9
(c) Remainder of 15517 when divided by 17
(d) Last digit of the number 745
Choose the correct answer from the options given below:
(a) Remainder of 4916 when divided by 17
(b) Remainder of 2446 when divided by 9
(c) Remainder of 15517 when divided by 17
(d) Last digit of the number 745
Choose the correct answer from the options given below:
- (a), (b), (c), (d)
- (a), (b), (d), (c)
- (a), (c), (b), (d)
- (d), (c), (b), (a)
Q86. Statement (I): If H is a non-empty finite subset of a group G and ab ∈ H for all a, b ∈ H, then H is also a group.
Statement (II): There is no homomorphism from (Z, +) to (Q, +), where Z is the set of integers and Q the set of rationals.
In the light of the above statements, choose the most appropriate answer from the options given below:
Statement (II): There is no homomorphism from (Z, +) to (Q, +), where Z is the set of integers and Q the set of rationals.
In the light of the above statements, choose the most appropriate answer from the options given below:
- Both Statement I and Statement II are correct
- Both Statement I and Statement II are incorrect
- Statement I is correct but Statement II is incorrect
- Statement I is incorrect but Statement II is correct
Directions (Q91–95): 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 E are 100 units and 120 units respectively. Let x be the quantity (in grams) of food X and y the quantity (in grams) of food Y.
Q91. The cost function of the total food is:
- Z = 6x + 7y
- Z = 8x + 12y
- Z = 12x + 20y
- Z = 20x + 12y
Q92. Which of the following are the constraints when formulating the LPP?
- 6x + 7y ≤ 100, 8x + 12y ≤ 120, x, y ≥ 0
- 6x + 8y ≤ 100, 7x + 12y ≤ 120, x, y ≥ 0
- 6x + 7y ≥ 100, 8x + 12y ≥ 120, x, y ≥ 0
- 6x + 8y ≥ 100, 7x + 12y ≥ 120, x, y ≥ 0
Q93. What are the quantities (in grams) of food X and Y respectively when the cost of food is minimum?
- 0 and 12½
- 15 and 5/4
- 120/7 and 0
- 0 and 10
Q94. The dual of the formulated LPP is:
- Max Z = 100u + 120v s.t. 6u + 7v ≤ 12, 8u + 12v ≤ 20, u, v ≥ 0
- Max Z = 12u + 20v s.t. 6u + 7v ≤ 100, 8u + 12v ≤ 120, u, v ≥ 0
- Max Z = 100u + 120v s.t. 6u + 7v ≤ 12, 8u + 7v ≤ 20, u, v unrestricted
- Max Z = 100u + 120v s.t. 6u + 7v ≥ 12, 8u + 12v ≥ 20, u, v ≥ 0