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 R_{1} be a relation from A = {1, 3, 5, 7} to B = {2, 4, 6, 8} and R_{2} 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 R_{1}) if x + y is divisible by 3.
2. An element x in B is related to an element y in C (under R_{2}) if x + y is even but not divisible by 3.
Which is the composite relation R_{1}R_{2} from A to C?
A  R_{1}R_{2} = {(1, 2), (1, 4), (3, 3), (5, 4), (7, 3)} 
B  R_{1}R_{2} = {(1, 2), (1, 3), (3, 2), (5, 2), (7, 3)} 
C  R_{1}R_{2} = {(1, 2), (3, 2), (3, 4), (5, 4), (7, 2)} 
D  R_{1}R_{2} = {(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 H_{1}, H_{2}, H_{3}, ... be harmonic numbers. Then, for n ∈ Z^{+}, can be expressed as
A  nH_{n+1} – (n + 1) 
B  (n + 1)H_{n} – n 
C  (n + 1)H_{n} – n 
D  (n+1)H_{n+1} – (n+1) 
Question 9 
In how many ways can we distribute 5 distinct balls, B_{1}, B_{2}, …, B_{5} in 5 distinct cells, C_{1}, C_{2}, …, C_{5} such that Ball B, is not in cell C_{i}, ∀i = 1, 2, …, 5 and each cell contains exactly one ball?
A  44 
B  96 
C  120 
D  3125 
Question 10 
If matrix and X_{2}  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  QII, RIV, SII, TI 
B  QIII, RII, SI, TIV 
C  QII, RI, SIV, TIII 
D  QI, RIV, SII, TIII 
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 nonzero 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 K_{3,4} and making s adjacent to every vertex of K_{3,4}. The minimum number of colours required to edgecolour 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 A^{c} and B^{c} denote the complements of the sets A and B. The set (A – B) ∪ (B  A) ∪ (A∩B) is equal to
A  A ∪ B 
B  A^{c} ∪ B^{c} 
C  A ∩ B 
D  A^{c} ∩ B^{c} 
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 X_{0} is
A  f(x_{0}+h) + f(x_{0}h)/2 
B  f(x_{0}+h)  f(x_{0}h)/2h 
C  f(x_{0}+h) + 2f(x_{0}) + f(x_{0}h)/h^{2} 
D  f(x_{0}+h)  2f(x_{0}) + f(x_{0}h)/h^{2} 
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 nonzero 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,xy), The inverse function of f is given by
A  
B  
C  
D 
Question 41 
Let R be a nonempty 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 
NewtonRaphson 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 {f_{1},f_{2},f_{3},...,f_{n}} is defined by the following recurrence:
f_{n+2} = f_{n+1} + f_{n}, n ≥ 1; f_{2}=1 : f_{1}=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. 