M.C.A. (Online Mode) Semester I Examination

Paper Code: OCA-503 MJ
Title: Computational Mathematics–I
Time: 3 Hours
Maximum Marks: 70

Q1) Multiple Choice Questions with reasoning — answer any 25 of 30 [25 × 2 = 50]

For every attempted question, select one option and give a brief reason.

No.	Question	A	B	C	D
1	Which is a proposition?	Close the door	What time is it?	2 + 2 = 4	x + 2 = 5
2	p → q is equivalent to:	p ∧ ¬q	¬p ∧ q	¬p ∨ q	p ∨ q
3	The negation of ∀x P(x) is:	∀x ¬P(x)	∃x ¬P(x)	¬∃x P(x)	∃x P(x)
4	A truth table for three propositions has:	3 rows	6 rows	8 rows	9 rows
5	Assuming p and deriving q to prove p → q uses:	Modus ponens	Modus tollens	Implication introduction	Addition
6	Which string is not well formed?	(p ∧ q) → r	p ∧ → q	¬p ∨ (q ∧ r)	(p → q) ↔ ¬r
7	A type system that admits no modelled unsafe programs is:	Complete	Sound	Undecidable	Ambiguous
8	In ∀x(P(x,y) → Q(z)), the free variables are:	x only	y only	z only	y and z
9	A tree with n vertices has:	n edges	n−1 edges	n+1 edges	2n edges
10	A connected graph has an Euler circuit when:	Exactly two vertices are odd	Every vertex has even degree	Any vertex is odd	It has no cycles
11	Kruskal rejects a candidate edge when it:	Is currently heaviest	Forms a cycle	Joins different components	Has a unique weight
12	The handshaking lemma implies an even number of vertices have:	Even degree	Odd degree	Zero degree	Maximum degree
13	Matrix multiplication is:	Always commutative	Associative but not generally commutative	Commutative but not associative	Neither
14	det(AB) equals:	det(A)+det(B)	det(A)det(B)	det(A)-det(B)	det(A+B)
15	A square matrix is invertible exactly when:	Its diagonal is all 1	Its determinant is nonzero	It is symmetric	Its trace is positive
16	For a graph adjacency matrix A, (A²)ij counts:	Edges from i to j	Length-two walks from i to j	Reachable vertices only	Degree products
17	Which is not required of a subspace?	Contains zero	Closed under addition	Closed under scalar multiplication	Contains every basis vector of the parent space
18	A basis is:	Any spanning set	Any independent set	An independent spanning set	A set containing zero
19	The dimension of polynomials of degree at most 3 is:	3	4	5	Infinite
20	Orthogonal vectors u and v satisfy:	u·v=1	u·v=0	u+v=0	u=v
21	If λ is an eigenvalue of A, then:	det(A−λI)=0	A−λI has full rank	λ must be positive	A must be symmetric
22	The product of a matrix's eigenvalues equals its:	Trace	Determinant	Rank	Nullity
23	An n×n matrix is diagonalisable when it has:	Nonzero determinant	n independent eigenvectors	Only positive eigenvalues	Equal row sums
24	Similar matrices necessarily have equal:	Entries	Determinant and trace	Eigenvectors	Symmetry
25	Eigenvalues describe:	Kernel position	Scale factors along eigenvectors	Pivot locations	Inverse entries
26	Which rule discharges an assumption when deriving p → q?	Modus ponens	Implication introduction	Double negation	Resolution
27	A tree with at least two vertices has at least:	One leaf	Two leaves	n leaves	No leaves
28	If det(A)=6, then det(A⁻¹) is:	6	1/6	36	−6
29	The characteristic equation is:	Av=λv	det(A−λI)=0	A+λI=0	det(A)=0
30	Vectors in an orthogonal, not necessarily orthonormal, basis:	Have pairwise dot product 0	All have length 1	Sum to 0	Have pairwise dot product 1

Q2) Answer any five in approximately two sentences [5 × 2 = 10]

1. Define a tautology and a contradiction.
2. Distinguish free and bound variables.
3. State the handshaking lemma.
4. Distinguish a path from a trail.
5. State the condition for multiplying an m×n matrix by a p×q matrix.
6. Define linear independence.
7. What does it mean for two matrices to be similar?

Q3) Answer any two in approximately ten lines [2 × 5 = 10]

1. Construct the truth table for (p → q) ↔ (¬q → ¬p) and classify the formula.
2. Apply Kruskal's algorithm to edges AB=1, BC=2, AC=3, BD=4, CD=5; give the minimum spanning tree and total weight.
3. For A=[[1,2],[3,4]], find det(A), A⁻¹, and verify one entry of AA⁻¹=I.
4. Find the eigenvalues and eigenvectors of A=[[2,0],[0,3]], and state whether A is diagonalisable.