## Engineering-Mathematics

 Question 1
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 1 Explanation:
 Question 2
 A 3
Question 2 Explanation:
 Question 3
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 3 Explanation:
 Question 4
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 4 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 5
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 5 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 6
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 6 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 7
In an undirected connected planar graph G, there are eight vertices and five faces. The number of edges in G is ______
 A 11
Question 7 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 8
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 8 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)

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)

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 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 9
Consider the following expression.

The value of the above expression (rounded to 2 decimal places) is _______
 A 0.25
Question 9 Explanation:
 Question 10
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 10 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 11

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 12

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 13

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 13 Explanation:

So, "imply" is False making "does not imply" True.
 Question 14

(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 15

Find the inverse of the matrix

 A B C D
Question 15 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 16

Match the following items

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

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

 A
Question 17 Explanation:
For 1 element:
We can't draw lattice with 1 element.
For 2 element:

For 3 element:

For 4 element:
 Question 18

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 18 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 19

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

 A 2n-1
Question 19 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 20

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

 A d*n/2
Question 20 Explanation:
Sum of degree of vertices = 2 × no. of edges
d * n = 2 * |E|
∴ |E| = d*n/2
 Question 21

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 21 Explanation:
P(A∩B') = P(A) - P(A∩B)
P3 = P(A) - P2
P(A) = P2 + P3
 Question 22

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

 A fog
Question 22 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 23

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

 A symmetry
Question 23 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.
 Question 24

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 24 Explanation:
No. of substrings of length
n = 1
(n-1) = 2
(n-2) = 3
So, Total = n(n+1)/2
 Question 25

The tank of matrix is:

 A 0 B 1 C 2 D 3
Question 25 Explanation:
 Question 26

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 26 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 27

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 27 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 28

The number of distinct simple graphs with upto three nodes is

 A 15 B 10 C 7 D 9
Question 28 Explanation:
 Question 29

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 29 Explanation:
 Question 30

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 30 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 31

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 31 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 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 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 34

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

 A Theory Explanation.
 Question 35

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

 A Theory Explanation.
 Question 36

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 37

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

 A Theory Explanation.
 Question 38

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 38 Explanation:
Note: Out of syllabus.
 Question 39

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 39 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 40

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 40 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 41

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 41 Explanation:
Note: Out of syllabus.
 Question 42

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

Probability of first ball black and second one white is,
 Question 43

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 43 Explanation:
Dot product of two perpendicular vectors must be zero.
 Question 44

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
 A ac B bc C ab D cc
Question 44 Explanation:
concat (a, head (tail (tail (acbc))))
concat (a, b)
ab
 Question 45

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 45 Explanation:
Just put values of (C) in place of x. It will satisfy the equation.
 Question 46

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 46 Explanation:
 Question 47

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 47 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 48

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 48 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 49

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 49 Explanation:

x & cos(x) are intersecting at only one point.
 Question 50

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 50 Explanation:
Note: Out of syllabus.
There are 50 questions to complete.

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