Engineering-Mathematics

Question 1
Consider the following expression.

The value of the above expression (rounded to 2 decimal places) is _______
A
0.25
Question 1 Explanation: 
Question 2
Let p and q be two propositions. Consider the following two formulae in propositional logic.

Which one of the following choices is correct?
A
Both S1and S2 are tautologies.
B
Neither S1and S2 are tautology.
C
S1is not a tautology but S2is a tautology.
D
S1is a tautology but S2is not a tautology.
Question 2 Explanation: 

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
A sender (S) transmits a signal, which can be one of the two kinds: H and L with probabilities 0.1 and 0.9 respectively, to a receiver (R). In the graph below, the weight of edge (u, v) is the probability of receiving v when u is transmitted, where u, v ∈ {H, L}. For example, the probability that the received signal is L given the transmitted signal was H, is 0.7.

If the received signal is H, the probability that the transmitted signal was H (rounded to 2 decimal places) is _______
A
0.04
Question 3 Explanation: 

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(H-r) = 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
 In an undirected connected planar graph G, there are eight vertices and five faces. The number of edges in G is ______
A
11
Question 4 Explanation: 

v - e + f = 2

v is number of vertices

e is number of edges

f is number of faces including bounded and unbounded

8-e+5=2

=> 13-2 =e

The number of edges  are =11

Question 5
Consider the two statements.
           S1: There exist random variables X and Y such that
                           EX-E(X)Y-E(Y)2>Var[X] Var[Y]
           S2: For all random variables X and Y,
                            CovX,Y=E|X-E[X]| |Y-E[Y]| 
Which one of the following choices is correct?
A
S1is false, but S2is true.
B
Both S1and S2are true.
C
S1is true, but S2is false.
D
Both S1and S2 are false.
Question 5 Explanation: 
Variance(X) = Var[X]= E((X-E(X))^2)
For a dataset with single values, we have variance 0. EX-E(X)Y-E(Y)2>Var[X] Var[Y]
This leads to inequance of 0>0 which is incorrect.

Its not |x-E(x)|. Thus S2 is also incorrect.
Question 6
Let G be a group of order 6, and H be a subgroup of G such that 1 < |H| < 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.
Question 6 Explanation: 

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 relation R is said to be circular of aRb and bRc together imply cRa. Which of the following options is/are correct?
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.
Question 7 Explanation: 

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 8 Explanation: 
Question 9
The lifetime of a component of a certain type is a random variable whose probability density function is exponentially distributed with parameter 2. For a randomly picked component of this type, the probability that its lifetime exceeds the expected lifetime (rounded to 2 decimal places) is _______.
A
0.37
Question 9 Explanation: 
Question 10
Let G = (V, E) be an undirected unweighted connected graph. The diameter of G is defined as:

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 10 Explanation: 
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 11 Explanation: 
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
Question 12 Explanation: 
If a relation is equivalence then it must be
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
Question 13 Explanation: 
S = {(φ), 1, (2, 3)}
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
Question 14 Explanation: 

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 15 Explanation: 
Note: Out of syllabus.
Question 16

The value of k for which 4x2 - 8xy + ky2 = 0 does not represent a pair of straight lines (both passing through the origin) is:

A
0
B
2
C
9
D
3
Question 16 Explanation: 
Note: Out of syllabus.
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 17 Explanation: 
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
Question 18 Explanation: 
In a normal graph number of edges required for n vertices is n-1, and in cyclic graph it is n.
In cyclic graph:
No. of edges = No. of vertices
⇒ n = n
Question 19

If the cube roots of unity are 1, ω and ω2, then the roots of the following equation are (x - 1)3 + 8 = 0

A
-1, 1 + 2ω, 1 + 2ω2
B
1, 1 - 2ω, 1 - 2ω2
C
-1, 1 - 2ω, 1 - 2ω2
D
-1, 1 + 2ω, -1 + 2ω2
Question 19 Explanation: 
Just put values of (C) in place of x. It will satisfy the equation.
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
Question 20 Explanation: 
concat (a, head (tail (tail (acbc))))
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+j-k)
C
1/3 (1-j-k)
D
1/√3 (1+j-k)
E
None of the above.
Question 21 Explanation: 
Dot product of two perpendicular vectors must be zero.
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
Question 22 Explanation: 
Probability of first ball white and second one black is,

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 23 Explanation: 
Note: Out of syllabus.
Question 24

Let A be the set of all non-singular matrices over real number and let* be the matrix multiplication operation. Then

A
A is closed under* but < A, *> is not a semigroup
B
is a semigroup but not a monoid
C
is a monoid but not a group
D
is a group but not an abelian group
Question 24 Explanation: 
As the matrices are non-singular so their determinant ±0. Hence, the inverse matrix always exist. But for a group to be Abelian it should follow commutative property. As matrix multiplication is not commutative, is a group but not an abelian group.
Question 25

The solution of differential equation y'' + 3y' + 2y = 0 is of the form

A
C1ex + C2e2x
B
C1e-x + C2e3x
C
C1e-x + C2e-2x
D
C1e-2x + C22-x
Question 25 Explanation: 
Note: Out of syllabus.
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
Question 26 Explanation: 
From the axiom ¬p → q, we can conclude that p ∨ q.
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 G1 and G2 be subgroups of a group G.
(a) Show that G1 ∩ G2 is also a group of G.
(b) Is G1 ∪ G2 always a subgroup of G?

A
Theory Explanation.
Question 30

Prove using mathematical induction for n≥5, 2n > n2

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 + 2x2.
(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
Question 33 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 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
Question 34 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 35

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 35 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.
As the question is asked about increasing everywhere II is false.
III. √(x3+1) = (x3+1)1/2
f'(x) = 1/2(3x2/√(x3+1))>0
f(x) is increasing over [0,1].
Question 36

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 36 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.
The probability of https://solutionsadda.in/wp-content/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 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 37 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 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
Question 38 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 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
Question 39 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 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
Question 40 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 41

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 41 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 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
Question 42 Explanation: 

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
Question 43 Explanation: 
From the given information we can write,
|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 44 Explanation: 
Rational number consists of number '0'. If 0 is present in a set inverse is not possible 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
Question 45 Explanation: 
1 - no. 6 on both dice
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 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 46 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 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 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 47 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 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,x-y), The inverse function of f is given by

A
B
C
D
Question 48 Explanation: 
Question 49

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 49 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 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)
Question 50 Explanation: 
When x = F and y = F
then option (D) will be False.
There are 50 questions to complete.

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