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I Semester B.C.A. (Full Stack Development) (AI&ML) (Data Science)

Examination, January 2025

Subject: COMPUTER SCIENCE
Course Code: 24BCA12 : Mathematical Foundations
Max. Marks: 80
Time: 3 Hours

Instruction

Answer all the Sections.

SECTION – A

Answer any eight questions. Each question carries two marks:
(8 × 2 = 16)

1. If $A = \{3, 5, 7\}, B = \{2, 4, 6\}$, find $A \cup B$.
2. Find the symmetric difference of $A$ and $B$, where $A = \{1, 2, 3, 4\}$ and $B = \{3, 4, 5, 6\}$.
3. Construct the truth table for $(P \lor Q) \implies R$.
4. Evaluate $\begin{vmatrix} 3 & 2 \\ 1 & 4 \end{vmatrix}$.
5. Define a diagonal matrix with an example.
6. Differentiate between a tautology and a contingency.
7. Prove that $(A^T)^T = A$ for $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$.
8. Define combinations with an example.
9. Define a weighted graph.
10. Explain the concept of graph connectivity with an example.

SECTION – B

Answer any four questions. Each question carries six marks:
(4 × 6 = 24)

11. In a group of 40 students, 22 like mathematics and 18 like science. If 10 students like both, how many like only one subject?
12. Show that $P \land \lnot (Q \lor R) \equiv (P \land \lnot Q) \land (P \land \lnot R)$.
13. If $A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}, B = \begin{bmatrix} 1 & 2 \\ 3 & 5 \end{bmatrix}$, find $AB$.
14. Solve the system of equations using the matrix method:
    $x + 2y + z = 5, \quad 2x - y + 3z = 10, \quad 3x + y - z = 6.$
15. Explain the concept of Eulerian and Hamiltonian paths in graphs.

SECTION – C

Answer any five questions. Each question carries eight marks:
(5 × 8 = 40)

16. Define walk, trail, and circuit in a graph, and provide examples of each.
17. Let $A = \{2, 4, 6, 8\}$ and define a relation $R$ on $A$ such that $R = \{(a, b) \mid a \leq b\}$. Do the following:
    1. Represent $R$ as a matrix.
    2. Find the domain of $R$.
    3. Find the range of $R$.
    4. Determine if $R$ is reflexive, symmetric, or transitive.
18. Prove using mathematical induction that $1^3 + 2^3 + \dots + n^3 = \left( \frac{n(n+1)}{2} \right)^2$.
19. Solve using Cramer’s rule: $2x + 3y = 8$, $4x - 5y = -2$.
20. Discuss the differences between null matrix, identity matrix, and skew-symmetric matrix with examples.
21. Prove that $P \lor (Q \land R) \equiv (P \lor Q) \land (P \lor R)$ using logical equivalence laws.
22. Obtain the minimum spanning tree for the graph below using Prim's algorithm:
    (Provide a new graph here).
23. a) In how many ways can the letters of the word "SCIENCE" be arranged such that vowels always appear together?
    b) Write the converse, inverse, and contrapositive of the statement:
    "If $n$ is even, then $n^2$ is even."

End of Question Paper

Would you like details on creating the graph for Q22, or explanations for specific questions?

Follow-Up Questions:

1. How does a tautology differ from a contingency logically?
2. What is the significance of the determinant in solving linear equations?
3. Can you provide a step-by-step guide for Cramer’s rule with examples?
4. How do you identify Eulerian circuits in complex graphs?
5. What is the difference between reflexive and symmetric relations?

Tip: Always verify your calculations for matrices by re-checking row and column operations!