UGC NET · PAPER 2
UGC NET Paper 2 Computer Science – Unit 1: Discrete Structures and Optimisation PYQ (June 2024 (Cancelled Exam))
Q10. If f(n) = 5f(n/2) + 3 and f(1) = 7 is a recurrence relation, where k is a positive integer, then f(2ᵏ) =
- 5ᵏ(31/4) − 3/4
- 5ᵏ(29/4)
- 2ᵏ(31/4) − 3/4
- 2ᵏ(29/4)
Q22. 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)
Choose the correct answer from the options given below:
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)
Choose the correct answer from the options given below:
- A, B, C Only
- B, C, E Only
- C, D, E Only
- B, D Only
Q26. Which of the following sequences aₙ is a solution of the recurrence relation aₙ = 8aₙ₋₁ − 16aₙ₋₂?
A. aₙ = 1
B. aₙ = 2ⁿ
C. aₙ = 4ⁿ
D. aₙ = n·4ⁿ
E. aₙ = (−4)ⁿ
Choose the most appropriate answer from the options given below:
A. aₙ = 1
B. aₙ = 2ⁿ
C. aₙ = 4ⁿ
D. aₙ = n·4ⁿ
E. aₙ = (−4)ⁿ
Choose the most appropriate answer from the options given below:
- A, B, C Only
- B, C, D Only
- C, D Only
- C, D, E Only
Q38. Arrange the following simple graphs by the number of spanning trees they have (increasing):
A. K₃ (complete graph on 3 nodes)
B. K₄ (complete graph on 4 nodes)
C. K₂,₂ (complete bipartite graph with 2 nodes in each part)
D. C₅ (cycle graph on 5 nodes)
Choose the correct answer from the options given below:
A. K₃ (complete graph on 3 nodes)
B. K₄ (complete graph on 4 nodes)
C. K₂,₂ (complete bipartite graph with 2 nodes in each part)
D. C₅ (cycle graph on 5 nodes)
Choose the correct answer from the options given below:
- A, B, C, D
- A, C, B, D
- A, C, D, B
- D, B, C, A
Q43. A bit string of length four is generated at random. What is the probability that it contains at least two consecutive 0s, given that the first bit is 0?
- 1/2
- 5/16
- 5/8
- 1/4
Q46. Arrange the following sets in increasing order of their 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}}
Choose the correct answer from the options given below:
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}}
Choose the correct answer from the options given below:
- A, B, D, E, C
- A, B, E, D, C
- A, C, D, B, E
- C, A, D, B, E
Q49. The chairs of an auditorium are to be labelled with a letter followed by a positive two-digit integer, starting with A01. What is the maximum number of chairs that can be labelled this way?
- 2600
- 2574
- 2340
- 2366
Q55. Z is the set of integers. ⟨Z, *⟩ forms a group if * is:
- a * b = a + b + 2
- a * b = 2a + b
- a * b = (a + b)/2
- a * b = a² + b
Q63. Arrange the steps of mathematical modelling:
A. Solution fitting with data
B. Formulation mathematically
C. Solve mathematically
D. Repeat formulation if it fits poorly with the data
E. Interpret results
Choose the correct answer from the options given below:
A. Solution fitting with data
B. Formulation mathematically
C. Solve mathematically
D. Repeat formulation if it fits poorly with the data
E. Interpret results
Choose the correct answer from the options given below:
- B, C, A, D, E
- B, A, D, C, E
- D, A, B, C, E
- E, D, C, A, B
Q67. 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
Choose the correct answer from the options given below:
A. (p ∨ r) → p
B. p → (p ∧ r)
C. ~p → (p → r)
D. (p ∧ r) → (p → r)
E. ~(p → r) → p
Choose the correct answer from the options given below:
- A, B Only
- A, B, C Only
- B, C, D Only
- C, D, E Only
Q79. Which of the following is NOT a common property of social networks?
- The degree distribution follows a scale-free (power-law) distribution
- Average shortest path lengths are short in connected networks
- Most of them are directed networks
- They follow the small-world property
Directions (Q91–95): A firm can produce two types of cloth, A and B. Three kinds of wool are required: red, green and blue. One unit length of type A cloth needs 2 yards of red and 3 yards of green wool. One unit length of type B cloth needs 3 yards of red wool, 2 yards of green wool and 2 yards of blue wool. The firm has a stock of only 8 yards of red wool, 10 yards of green wool and 15 yards of blue wool. The income from one unit length of type A and type B cloth is ₹30 and ₹50 respectively. Let x and y be the unit lengths of type A and type B cloth produced.
Q91. When the LPP is formulated from the passage, which of the following is the objective function of its dual problem?
- Min P = 5u + 8v + 15w
- Min P = 8u + 10v + 15w
- Min P = 15u + 5v + 8w
- Max P = 10u + 8v + 15w
Q92. Which of the following is the appropriate objective function for the given problem?
- 2x + 3y
- 3x + 2y + 2z
- 30x + 50y
- 3x + 2y
Q93. How many unit lengths of cloth A are used when income is maximised (as per the keyed answer)?
- 0.8 unit
- 2.4 unit
- 2.8 unit
- 3.0 unit
Q94. If x unit lengths of type A and y unit lengths of type B cloth are produced, the constraints of the LPP are:
- y ≤ 7.5, 2x + 3y ≤ 8, 3y + 2x ≤ 10
- y ≤ 7.5, 2x + 3y ≤ 8, 3x + 2y ≤ 10
- 2x + 3y ≤ 8, 3x + 2y + 2z < 15
- 2x + 3y ≤ 10, 3x + 2y ≤ 10, 2y ≤ 15
Q95. On the basis of the available material, the maximum total income is:
- ₹133.3
- ₹124
- ₹100
- ₹153.3