 ## Engineering-Mathematics

 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 1 Explanation: 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 2 Explanation:
From the given information,
R1 ={(1,2), (1,8), (3,6), (5,4), (7,2), (7,8)}
where x+y is divisible by 3
R2 = {(2,2), (4,4), (6,2), (6,4), (8,2)}
where x+y is not divisible by 3
Then the composition of R1 with R2 denotes R1R2, is the relation from A to C defined by property such as:
(x,z) ∈ R1R2, iff if there is a y ∈ B such that (x,y) ∈ R1 and (y,z) ∈ R2.
Thus, R1R2 = {(1,2), (3,2), (3,4), (5,4), (7,2)}
 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 3 Explanation:
Maximum number of edges in an acyclic undirected graph = No. of vertices - 1
= n - 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 4 Explanation: 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 5 Explanation:
Let p→q is conditional proposition here. p and q are compound propositions itself. Arguments to be valid if all combinations have to be tautology (like T→T, F→T, F→F) and its invalid if it have fallacy (T→F).
If we somehow get this fallacy (T→F) then an argument is invalid.
For options P and S you don't get any such combinations for T→F, so P and S are valid.
For option Q: If we put p=F, q=T, r=T then we get T→F. So its INVALID.
For option R: If we put p=F, q=F, r=F then we get T→F. So it is INVALID.
 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 6 Explanation:
Put n=1, you will get a matrix like .
Find its determinant, Determinant = 3.
Now check options, by putting n=1, I am getting following results,
A) 5
B) 7
C) 3
D) 3
(A), (B) can't be the answer.
Now, check for n=2, Determinant = 9-1 = 8.
Put n=2 in (C), (D)
C) 7
D) 8
 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 9 Explanation:
Just apply inclusion exclusion principle,
∠5(1 - 1/∠1 + 1/∠2 - 1/∠3 + 1/∠4 - 1/∠5)
= 44
 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 10 Explanation: 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 11 Explanation:
Let x = Total no. of vertices
By Handshaking Lemma,
6 * 2 + 3 * 4 + (x - 9) * 3 = 27 * 2
24 + (x - 9) * 3 = 54
x = 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 12 Explanation:
Note: Out of syllabus.
 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 13 Explanation:
Note: Out of syllabus.
 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 14 Explanation:
Lagrange’s Theorem:
If ‘H” is a subgroup of finite group (G,*) then O(H) is the divisor of O(G).
Given that the order of group is 35. Its divisors are 1,5,7,35.
It is asked that the size of largest possible subgroup other than G itself will be 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 15 Explanation:
For a set with n elements,
The number of reflexive relations is 2^(n^2-n).
The total number of relations on a set with n elements is 2^ (n^2).
The probability of choosing the reflexive relation out of set of relations is
= 2^(n^2-n) /2^ (n^2)
= 2^( n^2-n- n^2)
= 2^(-n)
Given n=3, the probability will be 2-n = ⅛ = 0.125
 Question 16

Consider the functions

I. e-x
II. x2-sin x
III. √(x3+1)

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 16 Explanation:
A function f(x) is said to be increasing if f'(x)>0 at each point in an interval.
I. e-x
II. f'(x) = -e-x
f'(x)<0 on the interval [0,1] so this is not an increasing function.
II. x2-sinx
f'(x) = 2x - cosx
at x=0, f'(0) = 2(0) - 1 = -1 < 0
f(x) = x2 - sinx is decreasing over some interval, increasing over some interval as cosx is periodic.
III. √(x3+1) = (x3+1)1/2
f'(x) = 1/2(3x2/√(x3+1))>0
f(x) is increasing over [0,1].
 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 17 Explanation:
‘a’ is a non-zero vector such that a∈{0,1}n
‘x’ is a vector chosen randomly from {0,1}n
‘a’ can have 2(n-1) possibilities, x can have 2n possibilities.
∑aixi have (2n-1)(2n) possibilities, which is an even number of outcomes.
For example:
Take n=3
a = {001, 010, 100, 011, 101, 111}
x = {000, 001, 010, 011, 100, 101, 110, 111}
Computed as × = 0+0+0 = 0 Output = even
× = 0+0+1 = 0 Output = odd
Similarly, there could be 28 even, 28 odd outputs for the a(size=7), x(size=8) of total 56 outputs.
 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 18 Explanation:
In k3x4 there are two sets with sizes 3,4. (it is a complete bipartite graph).
The vertex in the set of size 3 has 4 edges connected to 4 vertices on other set. So, edge color of G is max(3,4) i.e. 4.
When a vertex is added to the graph with 7 vertices ( K3x4 has 7 vertices), there would be 7 edges associated to that new vertex. As per the edge coloring “no two adjacent edges have same color).
As the new vertex with 7 edges need to be colored with 7 colors, the edge color of graph G is 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 19 Explanation:
Basic Rules:
~p→q ≡ ~p∨q
Demorgan laws:
~(∀x(a(x)) ≡ ∃x~a(x)
~(∃x(a(x)) ≡ ∀x~a(x)
(A) ∃x(p(x)→w) ≡ ∀x p(x)→w
LHS: ∃x(p(x)→w) ≡ ∃x(~p(x)∨w)
≡ ∃x(~p(x))∨w
Demorgan’s law:
~(∀x(a(x)) = ∃x ~ a(x)
≡ ~(∀x P(x)) ∨ w
≡ (∀x) P(x) → w ≡ RHS
It’s valid.
(B) ∀x(P(x) → w) ≡ ∀x(~P(x) ∨ w)
≡ ∀x(~P(x)) ∨ w
≡ ~(∃x P(x)) ∨ w
≡ ∃x P(x) → w
This is not equal to RHS.
(C) ∃x(P(x) ∧ w) ≡ ∃x P(x) ∧ w
‘w’ is not a term which contains x.
So the quantifier does not have any impact on ‘w’.
Thus it can be written as
∃x(P(x)) ∧ w) ≡ ∃x P(x) ∧ w
(D) ∀(x)(P(x) ∨ w) ≡ ∀x P(x) ∨ w
‘w’ is not a term which contains ‘x’.
So the quantifier does not have an impact on ‘w’.
Thus ∀(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 20 Explanation:
There are 5 places.
― ― ― ― ―
Given: L I L A C
The derangement formula ⎣n!/e⎦ cannot be directly performed as there are repeated characters.
Let’s proceed in regular manner:
The L, L can be placed in other ‘3’ places as (1) Can be arranged such that A, I, C be placed in three positions excluding ‘C’ being placed at its own position, which we get only 2×2×1 = 4 ways.
Similarly (2) can be filled as A, I, C being placed such that 4th position is not filled by A, so we have 2×2×1 = 4 ways. Similarly with (3).
Totally, we get 4+4+4 = 12 ways.
 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,

I. rank(AB) = rank(A) rank(B)
II. det(AB) = det(A) det(B)
III. rank(A + B) ≤ rank(A) + rank(B)
IV. det(A + B) ≤ det(A) + det(B)

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 21 Explanation:
Rank(AB) ≥ Rank(A) + Rank(B) − n. So option I is wrong.
Rank is the number of independent rows(vectors) of a matrix. On product of two matrices, the combined rank is more than the sum of individual matrices (subtracted with the order n)
det(AB) = det(A)∙det(B) as the magnitude remains same for the matrices after multiplication.
Note: We can just take a 2x2 matrix and check the options.
 Question 22 where tr() represents the trace of a matrix. Which one of the following holds?
 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 22 Explanation:
the matrices AB and BA have the same characteristic polynomial, so in particular the same trace, and the same determinant.
(In exam point of view, you can just consider small example and conclude).
So trace of AB = trace of BA
As well
Trace of CD = trace of DC
 Question 23
Which of the following statements is/are TRUE for a group G ?
 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 23 Explanation:
(A) : True. Circular wait is a necessary condition for the formation of a deadlock.
(B) : False. In a multiple instance graph, a cycle always does not indicate deadlock.
(C) : False. Unsafe state may or may not lead to deadlock.
(D) : True. Since every edge is allocated, that means there are no requests. Hence, cycle is not possible.
 Question 24
Consider a simple undirected graph of 10 vertices. If the graph is disconnected, then the maximum number of edges it can have is ____________.
 A 36
Question 24 Explanation:
We obtain maximum number of edges in a disconnected graph, with one isolated vertex and remaining completely connected.
Given 10 vertices,
Then we can have a complete graph with 9 vertices and one isolated verted.
Number of edges in complete graph with 9 edges is n(n-1)/2 = 9*8/2 = 36
 Question 25
The number of arrangements of six identical balls in three identical bins is______.
 A 7
Question 25 Explanation:
Given distribution of n identical items into r identical boxes.
As no other condition is given,
We need to consider that distinct distribution will be based on the count of balls.
6 identical balls into 3 identical bins.
That can be done in the combination of
[6,0,0], [5,1,0], [ 4,2,0], [3,3,0], [4,1,1],[3,2,1],[2,2,2]
I.e. in 7 ways
 Question 26
The value of the following limit is _____________. A 1/2
Question 26 Explanation: When 0 is substituted, we get 0/0
Apply L- Hospital rule  -1/2
 Question 27
Which one of the following is the decimal form of generating function of the sequences {an} n >=0 defined below A B C D Question 27 Explanation:  Question 28
Consider a simple undirected unweighted graph with at least three vertices. If A is the adjacency matrix of the graph, then the number of 3-cycles in the graph is given by the trace of
 A A ^3 B A^ 3 divided by 2 C A ^3 divided by 3 D A ^3 divided by 6
 Question 29
Consider solving the following system of simultaneous equations using LU decomposition. A B C D Question 29 Explanation:
The LU decomposition of the given matrix is Given, (Assume x1,x2,x3 as a,b,c)       Question 30
The following simple undirected graph is referred to as the Peterson graph. Which of the following statements is/are TRUE?
 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 30 Explanation:
Chromatic number is 3. Peterson graph ihas hamiltonian path but not hamiltonian cycle
Given graph in option C is isomorphic as the given has same number of vertice, edges, degree sequence and cycles.
LArgest independent set can be more than 3 Question 31
Which of the properties hold for the adjacency matrix A of a simple undirected unweighted graph having n vertices?
 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 31 Explanation:
IF A is adjacency matrix, then in the matrix A*A = A^2 we have
The entries aii show the number of 2-length paths between the nodes i and j. Why this happens is easy to see: if there is an edge ij and an edge jk, then there will be a path ik through j. The entries ii are the degrees of the nodes i.
Similarly in A^3 we have the entries aii that show the number of 3-length paths between the nodes i and j.
In A^n-1 + I n, we will have at least n-1 length paths, so there is no possibility of zero entires
 Question 32
Which of the following is/are the eigenvector(s) for the matrix given below? A B C D Question 32 Explanation:  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 33 Explanation:
(A – B) ∪ (B - A) ∪ (A∩B) (A - B) = 1
(B - A) = 2
(A∩B) = 3
A∪B = (1∪2∪3)
(A – B) ∪ (B - A) ∪ (A∩B) = 1∪2∪3 = (A∪B)
 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 34 Explanation: No. of edges = 4
 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 35 Explanation:
From the given information we can write,
|Y||X| = 97
→ Option A only satisfies.
 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 36 Explanation:
Rational number consists of number '0'. If 0 is present in a set inverse is not possible 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 37 Explanation:
1 - no. 6 on both dice
1 - (5/6 × 5/6) = 1 - (25/36) = 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 38 Explanation:
The formula which is used to compute the second derivation of a function f at point X is
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 39 Explanation:
→ It belongs to linear non-homogeneous equations. So by having m=n, we can't say that it will have unique solution.
→ Solution can be depends on rank of matrix A and matrix [A B].
→ If rank[A] = rank[A B] then it can have solution otherwise no solution.
 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 40 Explanation: 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 41 Explanation:
Let A = {1, 2, 3} and B = {4, 5} and C = {1, 6, 7}
Now,
A ∩ B = ф
& B ∩ C = ф
But A ∩ B ≠ ф
So, R is not transitive.
A ∩ B = A, so R is not reflexive.
If A ∩ B = ф
then definitely B ∩ A = ф.
Hence, R is symmetric.
So, option (B) is true.
 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 42 Explanation:
When x = F and y = F
then option (D) will be False.
 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 if and only if for any pair of elements a, b ∈ s, a* b-1 ∈ s.
Question 43 Explanation:
Option (C) is False because string concatenation operation is monoid (doesn't have inverse to become a group).
 Question 44

Newton-Raphson iteration formula for finding 3√c, where c > 0 is,

 A B C D Question 44 Explanation:
Note: Out of syllabus.
 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 45 Explanation: 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 46 Explanation:
E1 : First card being ace
E2 : Last card being ace
Note that E1 and E2 are dependent events, i.e., probability of last card being ace if first is ace will be lesser than the probability of last card being ace if first card is not ace.
So, probability of first card being ace = 4/52
Probability of last card being ace given that first card is ace is,
P(E2 / E1) = 3/51
∴ P(E1 and E2) = P(E1) ⋅ P(E2 / E1) = 4/52 × 3/51
 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.
There are 50 questions to complete.

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