solutions adda
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Question 1 |
Let a(x, y), b(x, y,) and c(x, y) be three statements with variables x and y chosen from some universe. Consider the following statement:
(∃x)(∀y)[(a(x, y) ∧ b(x, y)) ∧ ¬c(x, y)]Which one of the following is its equivalent?
A | (∀x)(∃y)[(a(x, y) ∨ b(x, y)) → c(x, y)] |
B | (∃x)(∀y)[(a(x, y) ∨ b(x, y)) ∧¬ c(x, y)] |
C | ¬(∀x)(∃y)[(a(x, y) ∧ b(x, y)) → c(x, y)] |
D | ¬(∀x)(∃y)[(a(x, y) ∨ b(x, y)) → c(x, y)] |
Question 2 |
Let R1 be a relation from A = {1, 3, 5, 7} to B = {2, 4, 6, 8} and R2 be another relation from B to C = {1, 2, 3, 4} as defined below:
1. An element x in A is related to an element y in B (under R1) if x + y is divisible by 3.
2. An element x in B is related to an element y in C (under R2) if x + y is even but not divisible by 3.
Which is the composite relation R1R2 from A to C?
A | R1R2 = {(1, 2), (1, 4), (3, 3), (5, 4), (7, 3)} |
B | R1R2 = {(1, 2), (1, 3), (3, 2), (5, 2), (7, 3)} |
C | R1R2 = {(1, 2), (3, 2), (3, 4), (5, 4), (7, 2)} |
D | R1R2 = {(3, 2), (3, 4), (5, 1), (5, 3), (7, 1)} |
Question 3 |
What is the maximum number of edges in an acyclic undirected graph with n vertices?
A | n - 1 |
B | n |
C | n + 1 |
D | 2n - 1 |
Question 4 |
What values of x, y and z satisfy the following system of linear equations?
A | x = 6, y = 3, z = 2 |
B | x = 12, y = 3, z = -4 |
C | x = 6, y = 6, z = -4 |
D | x = 12, y = -3, z = 0 |
Question 5 |
Let p, q and s be four primitive statements. Consider the following arguments:
P: [(¬p ∨ q) ∧ (r → s) ∧ (p ∨ r)] → (¬s → q)
Q: [(¬p ∧ q) ∧ [q → (p → r)] → ¬r
R: [[(q ∧ r) → p] ∧ (¬q ∨ p)] → r
S: [p ∧ (p → r) ∧ (q ∨ ¬r)] → q
Which of the above arguments are valid?
A | P and Q only |
B | P and R only |
C | P and S only |
D | P, Q, R and S |
Question 6 |
Let A be an Let A be an n × n matrix of the following form.
What is the value of the determinant of A?
A | ![]() |
B | ![]() |
C | ![]() |
D | ![]() |
Question 7 |
Let X and Y be two exponentially distributed and independent random variables with mean α and β, respectively. If Z = min(X,Y), then the mean of Z is given by
A | 1/α+β |
B | min(α, β) |
C | αβ/α + β |
D | α + β |
Question 8 |
Let H1, H2, H3, ... be harmonic numbers. Then, for n ∈ Z+, can be expressed as
A | nHn+1 – (n + 1) |
B | (n + 1)Hn – n |
C | (n + 1)Hn – n |
D | (n+1)Hn+1 – (n+1) |
Question 9 |
In how many ways can we distribute 5 distinct balls, B1, B2, …, B5 in 5 distinct cells, C1, C2, …, C5 such that Ball B, is not in cell Ci, ∀i = 1, 2, …, 5 and each cell contains exactly one ball?
A | 44 |
B | 96 |
C | 120 |
D | 3125 |
Question 10 |
If matrix and X2 - X + I = 0 (I is the Identity matrix and 0 is the zero matrix), then the inverse of X is:
A | ![]() |
B | ![]() |
C | ![]() |
D | ![]() |
Question 11 |
What is the number of vertices in an undirected connected graph with 27 edges, 6 vertices of degree 2, 3 vertices of degree 4 and remaining of degree 3?
A | 10 |
B | 11 |
C | 18 |
D | 19 |
Question 12 |
If f(1) = 2, f(2) = 4 and f(4) = 16, what is the value of f(3) using Lagrange’s interpolation formula?
A | 8 |
B | 8(1/3) |
C | 8(2/3) |
D | 9 |
Question 13 |
Consider the following iterative root finding methods and convergence properties:
Iterative root finding Convergence properties methods (Q) False Position (I) Order of convergence = 1.62 (R) Newton Raphson (II) Order of convergence = 2 (S) Secant (III) Order of convergence = 1 with guarantee of convergence (T) Successive Approximation (IV) Order of convergence = 1 with no guarantee of convergence
A | Q-II, R-IV, S-II, T-I |
B | Q-III, R-II, S-I, T-IV |
C | Q-II, R-I, S-IV, T-III |
D | Q-I, R-IV, S-II, T-III |
Question 14 |
Let G be a group of 35 elements. Then the largest possible size of a subgroup of G other than G itself is ______.
A | 7 |
Question 15 |
Let R be the set of all binary relations on the set {1,2,3}. Suppose a relation is chosen from R at random. The probability that the chosen relation is reflexive (round off to 3 decimal places) is _____.
A | 0.125 |
Question 16 |
Consider the functions
Which of the above functions is/are increasing everywhere in [0,1]?
A | II and III only |
B | III only |
C | II only |
D | I and III only |
Question 17 |
For n>2, let a ∈ {0,1}n be a non-zero vector. Suppose that x is chosen uniformly at random from {0,1}n. Then, the probability that is an odd number is _____.
A | 0.5 |
Question 18 |
Graph G is obtained by adding vertex s to K3,4 and making s adjacent to every vertex of K3,4. The minimum number of colours required to edge-colour G is _____.
A | 7 |
Question 19 |
Which one of the following predicate formulae is NOT logically valid?
Note that W is a predicate formula without any free occurrence of x.
A | ∃x(p(x) → W) ≡ ∀x p(x) → W |
B | ∀x(p(x) → W) ≡ ∀x p(x) → W
|
C | ∃x(p(x) ∧ W) ≡ ∃x p(x) ∧ W |
D | ∀x(p(x) ∨ W) ≡ ∀x p(x) ∨ W |
Question 20 |
The number of permutations of the characters in LILAC so that no character appears in its original position, if the two L’s are indistinguishable, is _______.
A | 12 |
Question 21 |
Let A and B be two n×n matrices over real numbers. Let rank(M) and det(M) denote the rank and determinant of a matrix M, respectively. Consider the following statements,
Which of the above statements are TRUE?
A | I and II only
|
B | I and IV only |
C | III and IV only |
D | II and III only
|
Question 22 |
A | Statement 1 is correct and Statement 2 is wrong. |
B | Statement 1 is wrong and Statement 2 is correct. |
C | Both Statement 1 and Statement 2 are correct. |
D | Both Statement 1 and Statement 2 are wrong. |
Question 23 |
A | ![]() |
B | ![]() |
C | If the order of G is 2 , then G is commutative. |
D | If G is commutative, then a subgroup of G need not be commutative. |
Question 24 |
A | 36 |
Question 25 |
A | 7 |
Question 26 |
A | 1/2 |
Question 27 |
A | ![]() |
B | ![]() |
C | ![]() |
D | ![]() |
Question 28 |
A | A ^3 |
B | A^ 3 divided by 2 |
C | A ^3 divided by 3 |
D | A ^3 divided by 6 |
Question 29 |
A | ![]() |
B | ![]() |
C | ![]() |
D | ![]() |
Question 30 |
A | The chromatic number of the graph is 3. |
B | The graph has a Hamiltonian path |
C | The following graph is isomorphic to the Peterson graph. ![]() |
D | The size of the largest independent set of the given graph is 3. (A subset of vertices of a graph form an independent set if no two vertices of the subset are adjacent.) |
Question 31 |
A | The diagonal entries of A 2 are the degrees of the vertices of the graph. |
B | If the graph is connected, then none of the entries of A^ n + 1 + I n can be zero. |
C | If the sum of all the elements of A is at most 2( n- 1), then the graph must be acyclic. |
D | If there is at least a 1 in each of A ’s rows and columns, then the graph must be Connected. |
Question 32 |
A | ![]() |
B | ![]() |
C | ![]() |
D | ![]() |
Question 33 |
Let A and B be sets and let Ac and Bc denote the complements of the sets A and B. The set (A – B) ∪ (B - A) ∪ (A∩B) is equal to
A | A ∪ B |
B | Ac ∪ Bc |
C | A ∩ B |
D | Ac ∩ Bc |
Question 34 |
Let X = {2,3,6,12,24}, Let ≤ be the partial order defined by X ≤ Y if x divides y. Number of edge as in the Hasse diagram of (X,≤) is
A | 3 |
B | 4 |
C | 9 |
D | None of the above |
Question 35 |
Suppose X and Y are sets and X Y and are their respective cardinalities. It is given that there are exactly 97 functions from X to Y. from this one can conclude that
A | |X| = 1, |Y| = 97 |
B | |X| = 97, |Y| = 1 |
C | |X| = 97, |Y| = 97 |
D | None of the above |
Question 36 |
Which of the following statements is false?
A | The set of rational numbers is an abelian group under addition. |
B | The set of integers in an abelian group under addition. |
C | The set of rational numbers form an abelian group under multiplication. |
D | The set of real numbers excluding zero in an abelian group under multiplication. |
Question 37 |
Two dice are thrown simultaneously. The probability that at least one of them will have 6 facing up is
A | 1/36 |
B | 1/3 |
C | 25/36 |
D | 11/36 |
Question 38 |
The formula used to compute an approximation for the second derivative of a function f at a point X0 is
A | f(x0+h) + f(x0-h)/2 |
B | f(x0+h) - f(x0-h)/2h |
C | f(x0+h) + 2f(x0) + f(x0-h)/h2 |
D | f(x0+h) - 2f(x0) + f(x0-h)/h2 |
Question 39 |
Let Ax = b be a system of linear equations where A is an m × n matrix and b is a m × 1 column vector and X is a n × 1 column vector of unknowns. Which of the following is false?
A | The system has a solution if and only if, both A and the augmented matrix [A b] have the same rank.
|
B | If m < n and b is the zero vector, then the system has infinitely many solutions. |
C | If m = n and b is non-zero vector, then the system has a unique solution. |
D | The system will have only a trivial solution when m = n, b is the zero vector and rank (A) = n. |
Question 40 |
Let R denotes the set of real numbers. Let f: R×R → R×R be a bijective function defined by f(x,y) = (x+y,x-y), The inverse function of f is given by
A | ![]() |
B | ![]() |
C | ![]() |
D | ![]() |
Question 41 |
Let R be a non-empty relation on a collection of sets defined by A R B if and only if A ∩ B = ф. Then, (pick the true statement)
A | R is reflexive and transitive |
B | R is symmetric and not transitive |
C | R is an equivalence relation |
D | R is not reflexive and not symmetric |
Question 42 |
Which of the following is false? Read ∧ as AND, ∨ as OR, ~ as NOT, → as one way implication and ↔ as two way implication.
A | ((x → y) ∧ x) → y |
B | ((x → y) ∧ (x ∧ y)) → x |
C | (x → (x ∨ ψ)) |
D | ((x ∨ y) ↔ (x → y) |
Question 43 |
Which one of the following is false?
A | The set of all bijective functions on a finite set forms a group under function composition.
|
B | The set {1, 2, ……., p–1} forms a group under multiplication mod p where p is a prime number. |
C | The set of all strings over a finite alphabet forms a group under concatenation. |
D | A subset s ≠ ф of G is a subgroup of the group |
Question 44 |
Newton-Raphson iteration formula for finding 3√c, where c > 0 is,
A | ![]() |
B | ![]() |
C | ![]() |
D | ![]() |
Question 45 |
The matrices and
commute under multiplication
A | if a = b or θ = nπ, is an integer |
B | always |
C | never |
D | if a cos θ ≠ b sin θ |
Question 46 |
The probability that top and bottom cards of a randomly shuffled deck are both aces in
A | 4/52×4/52 |
B | 4/52×3/52 |
C | 4/52×3/51 |
D | 4/52×4/51 |
Question 47 |
Let f be a function defined by
Find the values for the constants a, b, c and d so that f is continuous and differentiable every where on the real line.
A | Theory Explanation. |
Question 48 |
Let F be the collection of all functions f: {1,2,3} → {1,2,3}. If f and g ∈ F, define an equivalence relation ~ by f ~ g if and only if f(3) = g(3).
a) Find the number of equivalence classes defined by ~.
b) Find the number of elements in each equivalence class.
A | Theory Explanation. |
Question 49 |
The Fibonacci sequence {f1,f2,f3,...,fn} is defined by the following recurrence:
fn+2 = fn+1 + fn, n ≥ 1; f2=1 : f1=1
Prove by induction that every third element of the sequence is even.
A | Theory Explanation. |
Question 50 |
Let and
be two matrices such that AB = I. Let
and CD = 1. Express the elements of D in terms of the elements of B.
A | Theory Explanation. |