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Computational Mathematics — MCQ (Multiple Choice Questions)
Module 1: Logic
1. Which of the following is a proposition? a) "Close the door." b) "What time is it?" c) "The sum of 2 and 2 is 4." d) "x + 2 = 5"
2. Which statement is logically equivalent to p → q? a) p ∧ ¬q b) ¬p ∧ q c) ¬p ∨ q d) p ∨ q
3. The negation of ∀x P(x) is: a) ∀x ¬P(x) b) ∃x ¬P(x) c) ¬∀x ¬P(x) d) ∃x P(x)
4. How many rows does a truth table for a formula with 3 distinct propositions have? a) 3 b) 6 c) 8 d) 9
5. Which rule is used to prove p → q by assuming p and deriving q? a) Modus ponens b) Modus tollens c) Implication introduction d) Reductio ad absurdum
Module 2: Well-Formed Formulae
6. Which of the following strings is NOT well-formed in propositional logic? a) (p ∧ q) → r b) p ∧ → q c) (¬p) ∨ (q ∧ r) d) ((p → q) ↔ ¬r)
7. A type system that rejects some safe programs but admits no unsafe ones is called: a) Complete b) Sound c) Decidable d) Expressive
8. In the formula ∀x (P(x, y) → Q(z)), which variable is free? a) x b) y c) z d) both y and z
Module 3: Graphs and Trees
9. A tree on n vertices has how many edges? a) n b) n − 1 c) n + 1 d) 2n
10. For which graph does an Eulerian circuit exist? a) A graph with exactly two odd-degree vertices b) A connected graph where all vertices have even degree c) Any connected graph d) A graph with no edges
11. In Kruskal's algorithm, an edge is rejected when it: a) Is the heaviest remaining edge b) Forms a cycle c) Has a greater weight than already-selected edges d) Connects two already-connected components
12. The Handshaking Lemma implies that any graph has an even number of vertices of: a) Even degree b) Odd degree c) Total degree d) No fixed parity
Module 4: Matrices
13. Matrix multiplication is: a) Always commutative b) Associative but not commutative c) Commutative but not associative d) Neither associative nor commutative
14. det(AB) equals: a) det(A) + det(B) b) det(A) · det(B) c) det(A) · det(B) − 1 d) det(A + B)
15. A 3×3 matrix A is invertible only if: a) All diagonal entries are 1 b) det(A) ≠ 0 c) A is symmetric d) All eigenvalues are positive
16. What does (A²)[i][j] count if A is the adjacency matrix of a graph? a) Number of edges from i to j b) Number of length-2 walks from i to j c) Number of vertices reachable in 2 steps d) Degree of vertex i times degree of vertex j
Module 5: Vector Spaces
17. Which condition is NOT required for a subset to be a subspace? a) Contains the zero vector b) Closed under addition c) Closed under scalar multiplication d) Contains all basis vectors
18. A basis of a vector space is: a) Any spanning set b) Any linearly independent set c) A linearly independent spanning set d) A set containing the zero vector
19. The dimension of P₃ (polynomials of degree ≤ 3) is: a) 3 b) 4 c) 5 d) Infinite
20. For orthogonal vectors u and v, which is true? a) u · v = 1 b) u · v = 0 c) u + v = 0 d) u × v = 0
Module 6: Eigenvalues
21. If λ is an eigenvalue of A, then: a) det(A − λI) = 0 b) A + λI is invertible c) A − λI has full rank d) λ is a root of the minimal polynomial only
22. The product of the eigenvalues of a matrix equals: a) Its trace b) Its determinant c) Its rank d) Its nullity
23. A matrix A is diagonalisable if: a) It is square b) det(A) ≠ 0 c) It has n linearly independent eigenvectors d) All eigenvalues are distinct
24. If A is similar to B (B = P⁻¹AP), then: a) det(A) = det(B) and trace(A) = trace(B) b) A = B c) A and B have different eigenvalues d) A is symmetric iff B is symmetric
25. Eigenvalues give the: a) Position of the kernel b) Scale factors along the eigenvectors c) Number of pivots in Gaussian elimination d) Entries of the inverse matrix
Practice questions (26–30)
26. In natural deduction, the rule that discharges an assumption to derive p → q is: a) Modus ponens b) Implication introduction c) Reductio ad absurdum d) Double-negation elimination
27. A tree on n ≥ 2 vertices has at least how many pendant (leaf) vertices? a) 1 b) 2 c) n d) 0
28. If A is an n×n invertible matrix with det(A) = 6, then det(A⁻¹) is: a) 6 b) 1/6 c) 36 d) −6
29. The characteristic equation used to find eigenvalues is: a) A·v = λ·v b) det(A − λI) = 0 c) A + λI = 0 d) det(A) = 0
30. Vectors in an orthogonal basis (but not orthonormal) satisfy: a) Each pair has dot product 0 b) Each vector has length 1 c) The vectors sum to 0 d) Each pair has dot product 1
Answers (key only)
Reveal answers after attempting all questions
1-c, 2-c, 3-b, 4-c, 5-c, 6-b, 7-b, 8-d, 9-b, 10-b, 11-b, 12-b, 13-b, 14-b, 15-b, 16-b, 17-d, 18-c, 19-b, 20-b, 21-a, 22-b, 23-c, 24-a, 25-b, 26-b, 27-b, 28-b, 29-b, 30-a.
Extended practice: Questions 31–70
| No. | Question | A | B | C | D |
|---|---|---|---|---|---|
| 31 | A tautology is true under: | One valuation | Every valuation | No valuation | Only false premises |
| 32 | p ↔ q is true when p and q: | Differ | Have the same truth value | Are both propositions only | Are both false only |
| 33 | The contrapositive of p → q is: | q → p | ¬p → ¬q | ¬q → ¬p | p → ¬q |
| 34 | De Morgan's negation of p ∧ q is: | ¬p ∧ ¬q | ¬p ∨ ¬q | p ∨ q | ¬p → q |
| 35 | From p → q and p, infer q by: | Modus ponens | Modus tollens | Addition | Resolution |
| 36 | ∃x P(x) is disproved by showing: | P(a) once | ∀x ¬P(x) | ∃x ¬P(x) | ∀x P(x) |
| 37 | A formula's main connective is the: | First symbol | Outermost operator | Last variable | Deepest negation |
| 38 | In ∀x(P(x) ∨ Q(y)), y is: | Bound | Free | A constant | Quantified by x |
| 39 | Type-checker soundness means accepted programs do not exhibit the modeled: | Formatting | Type errors | Algorithms | Comments |
| 40 | Undecidability implies a checker cannot be both complete and terminating for: | Every nontrivial semantic property | Integer addition | Tokenization | Finite tables only |
| 41 | The sum of degrees in an undirected graph is: | ` | E | ` | `2 |
| 42 | A graph has an Euler trail but no Euler circuit when it is connected and has: | 0 odd vertices | Exactly 2 odd vertices | 4 odd vertices | 1 odd vertex |
| 43 | A path repeats: | No vertices | No edges but may repeat vertices | Every edge | Its start always |
| 44 | The complete graph K_n has: | n edges | n−1 edges | n(n−1)/2 edges | n² edges |
| 45 | A spanning tree of a connected graph contains: | Every edge | Every vertex | Every cycle | Only odd vertices |
| 46 | Kruskal chooses the next safe edge by: | Lowest weight globally | Highest degree | Lowest vertex label | DFS order |
| 47 | Prim grows: | A forest without a start | One tree from a chosen start | An Euler circuit | An adjacency matrix |
| 48 | An adjacency matrix of an undirected simple graph is: | Skew-symmetric | Symmetric | Always diagonal | Rectangular only |
| 49 | Matrices A(m×n) and B(p×q) can multiply as AB when: | m=q | n=p | m=p | n=q only |
| 50 | (AB)^T equals: | A^T B^T | B^T A^T | AB | A+B |
| 51 | A square matrix is invertible exactly when its determinant is: | 0 | Nonzero | 1 only | Positive only |
| 52 | The determinant of a triangular matrix is the: | Sum of diagonal entries | Product of diagonal entries | Rank | Trace squared |
| 53 | Row swapping changes a determinant's: | Sign | Magnitude only | Dimension | Rank to zero |
| 54 | The identity matrix satisfies: | AI=IA=A | AI=0 | A+I=A | I=A⁻¹ always |
| 55 | Matrix multiplication is generally: | Commutative | Noncommutative | Undefined for square matrices | Element-wise |
| 56 | Rank is the number of: | Entries | Linearly independent rows/columns | Zero rows before reduction | Eigenvalues only |
| 57 | A subset W is a subspace only if it contains: | Any basis of V | The zero vector and is closed | Exactly two vectors | No dependent vectors |
| 58 | A set containing the zero vector is necessarily: | Independent | Dependent | A basis | Orthogonal |
| 59 | If n vectors in an n-dimensional space are independent, they: | Cannot span | Form a basis | Include zero | Have equal length |
| 60 | Dimension is the number of vectors in: | Any spanning set | Any basis | The whole space | Any dependent set |
| 61 | Orthogonal nonzero vectors are automatically: | Dependent | Independent | Equal | Unit length |
| 62 | An orthonormal set is orthogonal and each vector has norm: | 0 | 1 | n | −1 |
| 63 | A linear transformation satisfies T(u+v) =: | T(u)+T(v) | T(u)T(v) | u+v | 0 always |
| 64 | Zero is an eigenvalue of A exactly when A is: | Invertible | Singular | Orthogonal | Diagonal only |
| 65 | The sum of eigenvalues, counting multiplicity, equals the: | Determinant | Trace | Rank | Nullity |
| 66 | Eigenvectors must be: | Zero vectors | Nonzero vectors | Unit vectors only | Row vectors only |
| 67 | Distinct eigenvalues have eigenvectors that are: | Linearly independent | Always orthogonal | Equal | Zero |
| 68 | A real symmetric matrix has: | Complex eigenvalues only | Real eigenvalues and orthogonal eigenvectors | No diagonalization | Zero trace |
| 69 | Similar matrices have the same: | Entries | Characteristic polynomial | Eigenvectors always | Row operations only |
| 70 | For eigenvalue λ, an eigenvector lies in the null space of: | A+λI | A−λI | A only | I only |
Extended answer key
31-B, 32-B, 33-C, 34-B, 35-A, 36-B, 37-B, 38-B, 39-B, 40-A, 41-B, 42-B, 43-A, 44-C, 45-B, 46-A, 47-B, 48-B, 49-B, 50-B, 51-B, 52-B, 53-A, 54-A, 55-B, 56-B, 57-B, 58-B, 59-B, 60-B, 61-B, 62-B, 63-A, 64-B, 65-B, 66-B, 67-A, 68-B, 69-B, 70-B.