EngineeringMathematics
Question 1 
The value of the above expression (rounded to 2 decimal places) is _______
A  0.25 
Question 2 
Which one of the following choices is correct?
A  Both S_{1}and _{S2} are tautologies. 
B  Neither _{S1}and _{S2} are tautology. 
C  _{S1}is not a tautology but _{S2}is a tautology. 
D  _{S1}is a tautology but _{S2}is not a tautology. 
A tautology is a formula which is "always true" . That is, it is true for every assignment of truth values to its simple components.
Method 1:
S1: (~p ^ (p Vq)) →q
The implication is false only for T>F condition.
Let's consider q as false, then
(~p ^ (p Vq)) will be (~p ^ (p VF)) = (~p ^ (p)) =F.
It is always F>F which is true for implication. So there are no cases that return false, thus its always True i.e. its Tautology.
S2:
q>(~p (p Vq))
The false case for implication occurs at T>F case.
Let q=T then (~p (p Vq)) = (~p (p VT))= ~p. (It can be false for p=T).
So there is a case which yields T>F = F. Thus its not Valid or Not a Tautology.
Method 2:
Question 3 
If the received signal is H, the probability that the transmitted signal was H (rounded to 2 decimal places) is _______
A  0.04 
Bayes theorem:
Probability of event A happening given that event B has already happened is
P(A/B) = P(B/A)*P(A) / P(B)
Here, it is asked that P( H transmitted / H received).
S can send signal to H with 0.1 probability, S can send signal to L with 0.9 probability.
The complete diagram can be
Probability that H Transmitted (H_t) given that H received (H_r)is
P( H_t / H_r) = P( H_r/ H_t) * P(H_t) / P(H_r)
P(Hr) = probability that H received = P( H received from H)+ P(H received from L)
It can be observed from the graph that H can receive in two ways (S to H to H) and (S to L to H)
The P(H_r) = 0.1*0.3 + 0.9*0.8= 0.03+0.72 = 0.75
P(H_received given that H_transmitted) =0.3
P(H transmitted ) = 0.1 i.e.
P( H_t / H_r) = P( H_r/ H_t) * P(H_t) / P(H_r)
= 0.3*0.1 / 0.75 = 0.04
Question 4 
A  11 
v  e + f = 2
v is number of vertices
e is number of edges
f is number of faces including bounded and unbounded
8e+5=2
=> 132 =e
The number of edges are =11
Question 5 
S1: There exist random variables X and Y such that
EXE(X)YE(Y)2>Var[X] Var[Y]
S2: For all random variables X and Y,
CovX,Y=EXE[X] YE[Y]
Which one of the following choices is correct?
A  S_{1}is false, but S_{2}is true. 
B  Both S_{1}and S_{2}are true. 
C  S_{1}is true, but S_{2}is false. 
D  Both S_{1}and S_{2} are false. 
For a dataset with single values, we have variance 0. EXE(X)YE(Y)2>Var[X] Var[Y]
This leads to inequance of 0>0 which is incorrect.
Its not xE(x). Thus S2 is also incorrect.
Question 6 
Which one of the following options is correct?
A  G is always cyclic, but H may not be cyclic. 
B  Both G and H are always cyclic. 
C  G may not be cyclic, but H is always cyclic. 
D  Both G and H may not be cyclic. 
If ‘G’ is a group with sides 6, its subgroups can have orders 1, 2, 3, 6.
(The subgroup order must divide the order of the group)
Given ‘H’ can be 1 to 6, but 4, 5 cannot divide ‘6’.
Then ‘H’ is not a subgroup.
G can be cyclic only if it is abelian. Thus G may or may not be cyclic.
The H can be cyclic only for the divisors of 6 and H cannot be cyclic for any non divisors of 6.
Question 7 
A  If a relation S is reflexive and circular, then S is an equivalence relation. 
B  If a relation S is transitive and circular, then S is an equivalence relation. 
C  If a relation S is circular and symmetric, then S is an equivalence relation. 
D  If a relation S is reflexive and symmetric, then S is an equivalence relation. 
Theorem: A relation R on a set A is an equivalence relation if and only if it is reflexive and circular.
For symmetry, assume that x, y ∈ A so that xRy, lets check for yRx.
Since R is reflexive and y ∈ A, we know that yRy. Since R is circular and xRy and yRy, we know that yRx. Thus R is symmetric.
For transitivity, assume that x, y, z ∈ A so that xRy and yRz. Check for xRz. Since R is circular and xRy and yRz, we know that zRx. Since we already proved that R is symmetric, zRx implies that xRz. Thus R is transitive.
Question 8 
A  3 
Question 9 
A  0.37 
Question 10 
Let M be the adjacency matrix of G.
Define graph G2on the same set of vertices with adjacency matrix N, where
Which one of the following statements is true?
A  
B  
C  
D 
Question 11 
The probability that a number selected at random between 100 and 999 (both inclusive) will not contain the digit 7 is:
A  16/25 
B  (9/10)^{3} 
C  27/75 
D  18/25 
Question 12 
Let R be a symmetric and transitive relation on a set A. Then
A  R is reflexive and hence an equivalence relation 
B  R is reflexive and hence a partial order

C  R is reflexive and hence not an equivalence relation 
D  None of the above 
i) Symmetric
ii) Reflexive
iii) Transitive
If a relation is said to be symmetric and transitive then we can't say the relation is reflexive and equivalence.
Question 13 
The number of elements in he power set P (S) of the set S = {(φ), 1, (2, 3)} is:
A  2 
B  4 
C  8 
D  None of the above 
P(S) = {φ, {{φ}}, {1}, {{2, 3}}, {{φ}, 1}, {1, {2, 3}}, {{φ}, 1, {2, 3}}}
In P(S) it contains 8 elements.
Question 14 
In the interval [0, π] the equation x = cos x has
A  No solution 
B  Exactly one solution 
C  Exactly two solutions 
D  An infinite number of solutions 
x & cos(x) are intersecting at only one point.
Question 15 
If at every point of a certain curve, the slope of the tangent equals −2x/y the curve is
A  a straight line 
B  a parabola 
C  a circle 
D  an ellipse 
Question 16 
The value of k for which 4x^{2}  8xy + ky^{2} = 0 does not represent a pair of straight lines (both passing through the origin) is:
A  0 
B  2 
C  9 
D  3 
Question 17 
The rank of the following (n + 1)×(n+1) matrix, where a is a real number is
A  1 
B  2 
C  n 
D  Depends on the value of a 
Question 18 
The minimum number of edges in a connected cyclic graph on n vertices is:
A  n  1 
B  n 
C  n + 1 
D  None of the above 
In cyclic graph:
No. of edges = No. of vertices
⇒ n = n
Question 19 
If the cube roots of unity are 1, ω and ω^{2}, then the roots of the following equation are (x  1)^{3} + 8 = 0
A  1, 1 + 2ω, 1 + 2ω^{2} 
B  1, 1  2ω, 1  2ω^{2} 
C  1, 1  2ω, 1  2ω^{2} 
D  1, 1 + 2ω, 1 + 2ω^{2} 
Question 20 
A language with string manipulation facilities uses the following operations
head(s): first character of a string tail(s): all but the first character of a string concat(s1,s2):s1 s2 for the string acbc what will be the output of concat(head(s), head(tail(tail(s))))
A  ac 
B  bc 
C  ab 
D  cc 
concat (a, head (tail (cbc)))
concat (a, head (bc))
concat (a, b)
ab
Question 21 
A unit vector perpendicular to both the vectors a = 2i  2j + k and b = 1 + j  2k is:
A  1/√3 (1+j+k) 
B  1/3 (1+jk) 
C  1/3 (1jk) 
D  1/√3 (1+jk) 
E  None of the above. 
Question 22 
A bag contains 10 white balls and 15 black balls. Two balls are drawn in succession. The probability that one of them is black and the other is white is:
A  2/3 
B  4/5 
C  1/2 
D  2/1 
Probability of first ball black and second one white is,
Question 23 
The iteration formula to find the square root of a positive real number b using the Newton Raphson method is
A  
B  
C  
D  None of the above 
Question 24 
Let A be the set of all nonsingular matrices over real number and let* be the matrix multiplication operation. Then
A  A is closed under* but < A, *> is not a semigroup 
B  
C  
D 
Question 25 
The solution of differential equation y'' + 3y' + 2y = 0 is of the form
A  C_{1}e^{x} + C_{2}e^{2x} 
B  C_{1}e^{x} + C_{2}e^{3x} 
C  C_{1}e^{x} + C_{2}e^{2x} 
D  C_{1}e^{2x} + C_{2}2^{x} 
Question 26 
If the proposition ¬p ⇒ ν is true, then the truth value of the proposition ¬p ∨ (p ⇒ q), where ¬ is negation, ‘∨’ is inclusive or and ⇒ is implication, is
A  true 
B  multiple valued 
C  false 
D  cannot be determined 
So, either p or q must be True.
Now,
¬p ∨ (p → q)
= ¬p ∨ (¬p ∨ q)
= ¬p ∨ q
Since nothing c an be said about the truth values of p, it implies that ¬p ∨ q can also be True or False. Hence, the value cannot be determined.
Question 27 
(a) Determine the number of divisors of 600.
(b) Compute without using power series expansion
A  Theory Explanation. 
Question 28 
Obtain the principal (canonical) conjunctive normal form of the propositional formula
(p ∧ q) V (¬q ∧ r)
Where ‘∧’ is logical and, ‘v’ is inclusive or and ¬ is negation.
A  Theory Explanation. 
Question 29 
Let G_{1} and G_{2} be subgroups of a group G.
(a) Show that G_{1} ∩ G_{2} is also a group of G.
(b) Is G_{1} ∪ G_{2} always a subgroup of G?
A  Theory Explanation. 
Question 30 
Prove using mathematical induction for n≥5, 2^{n} > n^{2}
A  Theory Explanation. 
Question 31 
Prove that in finite graph, the number of vertices of odd degree is always even.
A  Theory Explanation. 
Question 32 
(a) Find the minimum value of 3  4x + 2x^{2}.
(b) Determine the number of positive integers (≤ 720) which are not divisible by
any of numbers 2, 3, and 5.
A  Theory Explanation. 
Question 33 
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 34 
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 35 
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 36 
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 37 
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_{3x4} 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 38 
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 39 
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 40 
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 2x2 matrix and check the options.
Question 41 
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 42 
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 43 
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 44 
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 45 
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 46 
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 47 
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 48 
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 49 
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 50 
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.