EngineeringMathematics
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)} 
R_{1} ={(1,2), (1,8), (3,6), (5,4), (7,2), (7,8)}
where x+y is divisible by 3
R_{2} = {(2,2), (4,4), (6,2), (6,4), (8,2)}
where x+y is not divisible by 3
Then the composition of R_{1} with R_{2} denotes R_{1}R_{2}, is the relation from A to C defined by property such as:
(x,z) ∈ R_{1}R_{2}, iff if there is a y ∈ B such that (x,y) ∈ R_{1} and (y,z) ∈ R_{2}.
Thus, R_{1}R_{2} = {(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 
= 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 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 
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.
So, answer is (C).
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 
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 = 91 = 8.
Put n=2 in (C), (D)
C) 7
D) 8
So, (D) is the answer.
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 
∠5(1 – 1/∠1 + 1/∠2 – 1/∠3 + 1/∠4 – 1/∠5)
= 44
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 
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 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 
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 
The number of reflexive relations is 2^(n^2n).
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^2n) /2^ (n^2)
= 2^( n^2n 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. x^{2}sin x
III. √(x^{3}+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 
I. e^{x}
II. f'(x) = e^{x}
f'(x)<0 on the interval [0,1] so this is not an increasing function.
II. x^{2}sinx
f'(x) = 2x – cosx
at x=0, f'(0) = 2(0) – 1 = 1 < 0
f(x) = x^{2} – sinx is decreasing over some interval, increasing over some interval as cosx is periodic.
As the question is asked about increasing everywhere II is false.
III. √(x^{3}+1) = (x^{3}+1)^{1/2}
f'(x) = 1/2(3x^{2}/√(x^{3}+1))>0
f(x) is increasing over [0,1].
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 
‘x’ is a vector chosen randomly from {0,1}^{n}
‘a’ can have 2(^{n}1) possibilities, x can have 2^{n} possibilities.
∑a_{i}x_{i} have (2^{n}1)(2^{n}) possibilities, which is an even number of outcomes.
The probability of https://solutionsadda.in/wpcontent/uploads/2020/02/41.jpg is odd is ½.
For example:
Take n=3
a = {001, 010, 100, 011, 101, 111}
x = {000, 001, 010, 011, 100, 101, 110, 111}
Computed as [001]×[000] = 0+0+0 = 0 Output = even
[001]×[001] = 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 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 
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 ( K_{3×4} 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 
~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 
― ― ― ― ―
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

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 2×2 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. 
(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 
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. 
(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 
A  36 
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(n1)/2 = 9*8/2 = 36
Question 25 
A  7 
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 
A  1/2 
When 0 is substituted, we get 0/0
Apply L Hospital rule
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 
Given,
(Assume x1,x2,x3 as a,b,c)
Question 30 
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.) 
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 
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. 
The entries aii show the number of 2length 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 3length paths between the nodes i and j.
In A^n1 + I n, we will have at least n1 length paths, so there is no possibility of zero entires
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} 
(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 
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 
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 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 
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 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} 
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. 
→ 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,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 
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) 
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 
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 
E_{2} : Last card being ace
Note that E_{1} and E_{2} 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(E_{2} / E_{1}) = 3/51
∴ P(E_{1} and E_{2}) = P(E_{1}) ⋅ P(E_{2} / E_{1}) = 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 {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. 