UGC NET · PAPER 2
UGC NET Paper 2 Computer Science – Unit 1: Discrete Structures and Optimisation PYQ (June 2023)
Q3. Consider the universe of positive integers X = {n | 1 ≤ n ≤ 8}, proposition P = "n is an even integer", Q = "(3 ≤ n ≤ 7) ∧ (n ≠ 6)". The truth set of P ↔ Q is
- {1, 4}
- {2, 6}
- {3, 4, 5}
- {1}
Q4. Let (Z, +) denote the group of all integers under addition. Then the number of automorphisms of (Z, +) is
- 1
- 2
- 3
- 4
Q23. Let R = {x | x ∈ N, x is a multiple of 3 and x ≤ 100} and S = {x | x ∈ N, x is a multiple of 5 and x < 100}. What is the number of elements in (R ∩ S) × (S ∩ R)?
- 36
- 33
- 20
- 6
Q27. There are m points on one straight line AB and n points on another straight line AC, none of them being A. How many triangles can be formed with these points as vertices?
- mn(m + n − 2)
- ½ mn(m + n − 2)
- ½ mn(m + n − 1)
- mn(m + n − 1)
Q33. Let N be the set of natural numbers and R the relation on N × N defined by (a, b) R (c, d) if ad(b + c) = bc(a + d). Then R is
- Symmetric only
- Reflexive only
- Transitive only
- An equivalence relation
Q50. How many integral solutions are there to x + y + z + w = 29, where x ≥ 1, y ≥ 2, z ≥ 3 and w ≥ 0?
- 2400
- 2600
- 2800
- 3000
Q57. In a multi-user operating system, on average 20 requests are made to use a particular resource per hour. The probability that no request is made in 45 minutes is
- e⁻¹⁵
- e⁻⁵
- 1 − e⁻⁵
- 1 − e⁻¹⁰
Q67. Let G = R \ {0} (non-zero reals) and H = {−1, 1} be groups under multiplication. Then the map φ : G → H defined by φ(x) = x/|x| is
- Not a homomorphism
- A one-one homomorphism, which is not onto
- An onto homomorphism, which is not one-one
- A homomorphism
Q70. Find the sum of all four-digit numbers formed using the digits 1, 2, 4 and 6 (each used once).
- 86,658
- 88,858
- 91,958
- 93,358
Q76. Match List I with List II
| List I | List II |
|---|---|
| A. A △ B | I. (A − B) ∪ (A − C) |
| B. A − (B ∪ C) | II. (A − B) ∩ (A − C) |
| C. A − (B ∩ C) | III. (A − B) ∪ (B − A) |
| D. A ∩ (B − C) | IV. (A ∩ B) − (A ∩ C) |
- A-III, B-II, C-I, D-IV
- A-II, B-III, C-IV, D-I
- A-IV, B-III, C-I, D-II
- A-IV, B-I, C-III, D-II
Q82. Consider the following statements:
A. The identity is unique in any monoid.
B. A monoid is a group if every element of the monoid has an inverse.
C. A semigroup has closure, associative and identity properties.
D. A quasigroup has the closure property.
A. The identity is unique in any monoid.
B. A monoid is a group if every element of the monoid has an inverse.
C. A semigroup has closure, associative and identity properties.
D. A quasigroup has the closure property.
- A, B and D only
- B, C and D only
- A, B and C only
- A, C and D only
Q84. If A = {4n + 2 | n is a natural number} and B = {3n | n is a natural number}, which of the following is correct for A ∩ B?
- {12n² + 6n | n is a natural number}
- {24n − 12 | n is a natural number}
- {60n + 30 | n is a natural number}
- {12n − 6 | n is a natural number}