Engineering-Mathematics

Question 1

Let A and B be real symmetric matrices of size n × n. Then which one of the following is true?

A
AA′ = 1
B
A = A-1
C
AB = BA
D
(AB)' = BA
Question 1 Explanation: 
Question 2

Backward Euler method for solving the differential equation dy/dx = f(x,y) is specified by, (choose one of the following).

A
yn+1 = yn + hf(xn, yn)
B
yn+1 = yn + hf(xn+1, yn+1)
C
yn+1 = yn-1 + 2hf(xn, yn)
D
yn+1 = (1 + h) f(xn+1, yn+1)
Question 2 Explanation: 
dy/dx = f(x,y)
With initial value y(x0) = y0. Here the function f and the initial data x0 and y0 are known. The function y depends on the real variable x and is unknown. A numerical method produces a sequence y0, y1, y2, ....... such that yn approximates y(x0 + nh) where h is called the step size.
→ The backward Euler method is helpful to compute the approximations i.e.,
yn+1 = yn + hf(x n+1, yn+1)
Question 3

Let A and B be any two arbitrary events, then, which one of the following is true?

A
P(A∩B) = P(A)P(B)
B
P(A∪B) = P(A) + P(B)
C
P(A|B) = P(A∩B)P(B)
D
P(A∪B) ≤ P(A) + P(B)
Question 3 Explanation: 
(A) Happens when A and B are independent.
(B) Happens when A and B are mutually exclusive.
(C) Not happens.
(D) P(A∪B) ≤ P(A) + P(B) is true because P(A∪B) = P(A) + P(B) - P(A∩B).
Question 4

The number of distinct simple graphs with upto three nodes is

A
15
B
10
C
7
D
9
Question 4 Explanation: 
Question 5

The tank of matrix is:

A
0
B
1
C
2
D
3
Question 5 Explanation: 
Question 6

Some group (G,o) is known to be abelian. Then, which one of the following is true for G?

A
g = g-1 for every g ∈ G
B
g = g2 for every g ∈ G
C
(goh)2 = g2oh2 for every g,h ∈ G
D
G is of finite order
Question 6 Explanation: 
Associate property of a group (aob)oc = ao(boc)
For Abelian group, commutative also holds
i.e., (aob) = (boa)
Consider option (C):
(goh)2 = (goh)o(gog)
= (hog)o(goh)
= (ho(gog)oh)
= ((hog2)oh)
= (g2oh)oh
= g2o(hoh)
= g2oh2 [True]
Question 7

In a compact single dimensional array representation for lower triangular matrices (i.e all the elements above the diagonal are zero) of size n × n, non-zero elements (i.e elements of the lower triangle) of each row are stored one after another, starting from the first row, the index of the (i, j)th element of the lower triangular matrix in this new representation is:

A
i + j
B
i + j - 1
C
j + i(i-1)/2
D
i + j(j-1)/2
Question 7 Explanation: 
Though not mentioned in question, from options it is clear that array index starts from 1 and not 0.
If we assume array index starting from 1 then, ith row contains i number of non-zero elements. Before ith row there are (i-1) rows, (1 to i-1) and in total these rows has 1+2+3......+(i-1) = i(i-1)/2 elements.
Now at ith row, the jth element will be at j position.
So the index of (i, j)th element of lower triangular matrix in this new representation is
j = i(i-1)/2
Question 8

The number of substrings (of all lengths inclusive) that can be formed from a character string of length n is

A
n
B
n2
C
n(n-1)/2
D
n(n+1)/2
Question 8 Explanation: 
No. of substrings of length
n = 1
(n-1) = 2
(n-2) = 3
So, Total = n(n+1)/2
Question 9

On the set N of non-negative integers, the binary operation __________ is associative and non-commutative.

A
fog
Question 9 Explanation: 
The most important associative operation that is not commutative is function composition. If you have two functions f and g, their composition, usually denoted fog, is defined by
(fog)(x) = f(g(x))
It is associative, (fog)oh = fo(goh), but its usually not commutative. fog is usually not equal to gof.
Note that if fog exists then gof might not even exists.
Question 10

Amongst the properties {reflexivity, symmetry, anti-symmetry, transitivity} the relation R = {(x,y) ∈ N2 | x ≠ y } satisfies __________.

A
symmetry
Question 10 Explanation: 
It is not reflexive as xRx is not possible.
It is symmetric as if xRy then yRx.
It is not antisymmetric as xRy and yRx are possible and we can have x≠y.
It is not transitive as if xRy and yRz then xRz need not be true. This is violated when x=x.
So, symmetry is the answer.
Question 11

The number of subsets {1, 2, ... n} with odd cardinality is __________.

A
2n-1
Question 11 Explanation: 
Total no. of subsets with n elements is 2n.
And so, no. of subsets with odd cardinality is half of total no. of subsets = 2n /n = 2n-1
Question 12

The number of edges in a regular graph of degree d and n vertices is _________.

A
d*n/2
Question 12 Explanation: 
Sum of degree of vertices = 2 × no. of edges
d * n = 2 * |E|
∴ |E| = d*n/2
Question 13

The probability of an event B is P1. The probability that events A and B occur together is P2 while the probability that A and occur together is P3. The probability of the event A in terms of P1, P2 and P3 is __________.

A
P2 + P3
Question 13 Explanation: 
P(A∩B') = P(A) - P(A∩B)
P3 = P(A) - P2
P(A) = P2 + P3
Question 14

Let A, B and C be independent events which occur with probabilities 0.8, 0.5 and 0.3 respectively. The probability of occurrence of at least one of the event is __________

A
0.93
Question 14 Explanation: 
P(A∪B∪C) = P(A) + P(B) + P(C) - P(A∩B) - P(B∩C) - P(A∩C) + P(A∩B∩C)
Since all the events are independent, so we can write
P(A∪B∪C) = P(A) + P(B) + P(C) - P(A)P(B) - P(B)P(C) - P(A)P(C) + P(A)P(B) P(C)
= 0.8 + 0.5 + 0.3 - 0.4 - 0.5 - 0.24 + 0.12
= 0.93
Question 15

The Hasse diagrams of all the lattices with up to four elements are __________ (write all the relevant Hasse diagrams).

A
Question 15 Explanation: 
For 1 element:
We can't draw lattice with 1 element.
For 2 element:

For 3 element:

For 4 element:
Question 16

Match the following items

A
(i) - (b), (ii) - (c), (iii) - (d), (iv) - (a)
Question 16 Explanation: 
Note: Out of syllabus.
Question 17

Find the inverse of the matrix

A
B
C
D
Question 17 Explanation: 
Using eigen values, the characteristic equation we get is,
3 + 2λ2 - 2 = 0
Using Cayley-Hamiltonian theorem
-A3 + 2A2 - 2I = 0
So, A-1 = 1/2 (2A - A2)
Solving we get,
Question 18

Let p and q be propositions. Using only the truth table decide whether p ⇔ q does not imply p → q is true or false.

A
True
B
False
Question 18 Explanation: 

So, "imply" is False making "does not imply" True.
Question 19

(a) Let * be a Boolean operation defined as
If C = A * B then evaluate and fill in the blanks:
(i) A * A = _______
(ii) C * A = _______
(b) Solve the following boolean equations for the values of A, B and C:

A
Theory Explanation.
Question 20

A 3-ary tree is a tree in which every internal node has exactly three children. Use induction to prove that the number of leaves in a 3-ary tree with n interval nodes is 2(n-1)+3.

A
Theory Explanation.
Question 21

Every element a of some ring (R,+,0) satisfies the equation aoa = a.
Decide whether or not the ring is commutative.

A
Theory Explanation.
Question 22

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 22 Explanation: 
Question 23

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 23 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 24

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 24 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 25

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 25 Explanation: 

x & cos(x) are intersecting at only one point.
Question 26

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 26 Explanation: 
Note: Out of syllabus.
Question 27

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 27 Explanation: 
Note: Out of syllabus.
Question 28

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 28 Explanation: 
Question 29

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 29 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 30

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 30 Explanation: 
Just put values of (C) in place of x. It will satisfy the equation.
Question 31

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 31 Explanation: 
concat (a, head (tail (tail (acbc))))
concat (a, head (tail (cbc)))
concat (a, head (bc))
concat (a, b)
ab
Question 32

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 32 Explanation: 
Dot product of two perpendicular vectors must be zero.
Question 33

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 33 Explanation: 
Probability of first ball white and second one black is,

Probability of first ball black and second one white is,
Question 34

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 34 Explanation: 
Note: Out of syllabus.
Question 35

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 35 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 36

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 36 Explanation: 
Note: Out of syllabus.
Question 37

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 37 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 38

(a) Determine the number of divisors of 600.
(b) Compute without using power series expansion

A
Theory Explanation.
Question 39

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 40

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 41

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

A
Theory Explanation.
Question 42

Prove that in finite graph, the number of vertices of odd degree is always even.

A
Theory Explanation.
Question 43

(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 44

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 44 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 45

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 45 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 46

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 46 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 47

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 47 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 48

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 48 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 49

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 49 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 50

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 50 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.
There are 50 questions to complete.

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