### SET 4

Question 1

**Which one of the following in NOT necessarily a property of a Group?**

A : Commutativity

B : Associativity

C : Existence of inverse for every element

D : Existence of identity

Answer Discuss it!

.

Correct answer is :A

Solution :

A group is a set, G, together with an operation that combines any two elements a and b to form another element.

To qualify as a group, the set and operation, must satisfy four requirements known as the group axioms:

* Closure

* Associativity

* Identity element should exist

* Inverse element

Question 2

**What is the chromatic number of an n-vertex simple connected graph which does not contain any odd length cycle? Assume n >= 2.**

A : 2

B : 3

C : n-1

D : n

Answer Discuss it!

.

Correct answer is :A

Solution :

A simple graph with no odd cycles is bipartite graph and a Bipartite graph can be colored using 2 colors

Question 3

**Which one of the following is TRUE for any simple connected undirected graph with more than 2 vertices?**

A : No two vertices have the same degree

B : At least two vertices have the same degree

C : At least three vertices have the same degree

D : All vertices have the same degree

Answer Discuss it!

.

Correct answer is :B

Solution :

Since the graph is simple, there must not be any self loop and parallel edges. Since the graph is connected, the degree of any vertex cannot be 0. Therefore, degree of all vertices should be be from 1 to n-1. So the degree of at least two vertices must be same.

Question 4

**Consider the binary relation R = {(x, y), (x, z), (z, x), (z, y)} on the set {x, y, z}. Which one of the following is TRUE?**

A : R is symmetric but NOT antisymmetric

B : R is NOT symmetric but antisymmetric

C : R is both symmetric and antisymmetric

D : R is neither symmetric nor antisymmetric

Answer Discuss it!

.

Correct answer is :D

Solution :

R is not symmetric as (x, y) is present, but (y, x) is not present in R. R is also not antisymmetric as both (x, z) and (z, x) are present in R.

Question 5

**An unbalanced dice (with 6 faces, numbered from 1 to 6) is thrown. The probability that the face value is odd is 90% of the probability that the face value is even. The probability of getting any even numbered face is the same. If the probability that the face is even given that it is greater than 3 is 0.75, which one of the following options is closest to the probability that the face value exceeds 3?**

A : 0.453

B : 0.468

C : 0.485

D : 0.487

Answer Discuss it!

.

Correct answer is :B

Solution :

Let Xi be the probability that the face value is i.

We can say that X1 + X2 + X3 + X4 + X5 + X6 = 1

It is given that (X1 + X3 + X5) = 0.9*(X2 + X4 + X6)

It is also given that X2 = X4 = X6

Also given that (X4 + X6)/(X4 + X5 + X6) = 0.75

Solving the above equations, we can get X3 as 0.468

Question 6

**For the composition table of a cyclic group shown below**

Which one of the following choices is correct?

A : a, b are generators

B : b, c are generators

C : c, d are generators

D : d, a are generators

Answer Discuss it!

.

Correct answer is :C

Solution :

Check for all:-

a1 = a ,

a2 = a * a = a

a3 = a2 * a = a * a = a

a is not the generator since we are not able to express
other members of the group in powers of a

Check for c -

c1 = c

c2 = c * c = b

c3 = c2 * c = b * c = d

c4 = c2 * c2 = b * b = a

We are able to generate all the members of the group from c ,

Hence c is the generator
Similarly check for d

Question 7

**Which one of the following is the most appropriate logical formula to represent the statement? "Gold and silver ornaments are precious". The following notations are used: G(x): x is a gold ornament S(x): x is a silver ornament P(x): x is precious**

A : ∀ x( P(x) -> (G(x) ∧ S(x) ) )

B : ∀x ( (G(x) ∧ S(x)) -> P(x))

C : ∃x( (G(x) ∧ S(x)) -> P(x)

D : ∀x( (G(x) ∨ S(x)) -> P(x) )

Answer Discuss it!

.

Correct answer is :D

Solution :

D represents the correct form. For all x where x is gold ornament or silver ornament, x is precious.

Question 8

**Evalutae the following**

A : 0

B : 1

C : ln 2

D : 1/2 ln 2

Answer Discuss it!

.

Correct answer is :D

Solution :

(1-tanx)/(1+tanx) = (cosx - sinx)/(cosx + sinx)

Let cosx + sinx = t (-sinx + cosx)dx = dt (1/t)dt = ln t

=> ln(sinx + cosx) => ln(sin π/4 + cos π/4) => ln(1/√2 + 1/√2) => 1/2 ln 2

Question 9

**Consider the following well-formed formulae:**

I.¬∀ x(P(x))

II. ¬∃ x(P(x))

III. ¬∃ x(¬P(x))

IV. ¬∃x(¬P(x))

Which of the above are equivalent?

A : I and III

B : I and IV

C : II and III

D : II and IV

Answer Discuss it!

.

Correct answer is :B

Solution :

According to negation property of universal qualifier and existential quantifier

¬ ∀ ∈ X P(x) ≡ ∃x ∈ X ¬ P(x)

Question 10

**Newton-Raphson method is used to compute a root of the equation x**^{2} - 13 = 0 with 3.5 as the initial value. The approximation after one iteration is

A : 3.575

B : 3.676

C : 3.667

D : 3.607

Answer Discuss it!

.

Correct answer is :D

Solution :

According to Newton Raphson's method

f(x) = x^{2}-13

f`(x) = 2x

applying formula

x1 = 3.5 - (3.5*3.5 - 13)/2*3.5

x = 3.607

Question 11

**What is the possible number of reflexive relations on a set of 5 elements?**

A : 2^{10}

B : 2^{15}

C : 2^{20}

D : 2^{25}

Answer Discuss it!

.

Correct answer is :C

Solution :

No. of reflexive relations is 2^{n2-n}

substituting n=5 we get 2^{20}

Question 12

**Consider the set S = {1, ω , ω**^{2}}, where ω and ω^{2} are cube roots of unity. If * denotes the multiplication operation, the structure (S, *) forms

A : A group

B : A ring

C : An integral domain

D : A field

Answer Discuss it!

.

Correct answer is :A

Question 13

**What is the value of Lim**_{n->∞}(1-1/n)^{2n} ?

A : 0

B : e^{-2}

C : e^{-1/2}

D : 1

Answer Discuss it!

.

Correct answer is :B

Question 14

**Consider a company that assembles computers. The probability of a faulty assembly of any computer is p. The company therefore subjects each computer to a testing process.This testing process gives the correct result for any computer with a probability of q. What is the probability of a computer being declared faulty?**

A : pq + (1 - p)(1 - q)

B : (1-p)p

C : (1-p)q

D : pq

Answer Discuss it!

.

Correct answer is :A

Question 15

**What is the probability that divisor of 10**^{99} is a multiple of 10^{96}?

A : 1/625

B : 4/625

C : 12/625

D : 16/625

Answer Discuss it!

.

Correct answer is :A

Solution :

Divisors are of the form :

2^{k} * 5^{m}

So for 10^{99} we have (1,2^{1},2^{2}......2^{99})*(1,5^{1},5^{2}......5^{99}) divisors (one from both sets)

So there are total (100)*(100) = 10000 possibilities

Question 16

**The degree sequence of a simple graph is the sequence of the degrees of the nodes in the graph in decreasing order. Which of the following sequences can not be the degree sequence of any graph? **

I. 7, 6, 5, 4, 4, 3, 2, 1

II. 6, 6, 6, 6, 3, 3, 2, 2

III. 7, 6, 6, 4, 4, 3, 2, 2

IV. 8, 7, 7, 6, 4, 2, 1, 1

A : I and II

B : III and IV

C : IV only

D : II and IV

Answer Discuss it!

.

Correct answer is :D

Question 17

**Consider the following matrix**

If the eigenvalues of A are 4 and 8, then

A : x=4, y=10

B : x=5, y=8

C : x=3, y=9

D : x=-4, y=10

Answer Discuss it!

.

Correct answer is :D

Question 18

**Suppose the predicate F(x, y, t) is used to represent the statement that person x can fool person y at time t. which one of the statements below expresses best the meaning of the formula ∀x ∃y ∃t (¬F(x, y, t))?**

A : Everyone can fool some person at some time

B : No one can fool everyone all the time

C : Everyone cannot fool some person all the time

D : No one can fool some person at some time

Answer Discuss it!

.

Correct answer is :B

Question 1

.

Correct answer is :A

Solution :

A group is a set, G, together with an operation that combines any two elements a and b to form another element.

To qualify as a group, the set and operation, must satisfy four requirements known as the group axioms:

* Closure

* Associativity

* Identity element should exist

* Inverse element

Question 2

.

Correct answer is :A

Solution :

A simple graph with no odd cycles is bipartite graph and a Bipartite graph can be colored using 2 colors

Question 3

.

Correct answer is :B

Solution :

Since the graph is simple, there must not be any self loop and parallel edges. Since the graph is connected, the degree of any vertex cannot be 0. Therefore, degree of all vertices should be be from 1 to n-1. So the degree of at least two vertices must be same.

Question 4

.

Correct answer is :D

Solution :

R is not symmetric as (x, y) is present, but (y, x) is not present in R. R is also not antisymmetric as both (x, z) and (z, x) are present in R.

Question 5

.

Correct answer is :B

Solution :

Let Xi be the probability that the face value is i.

We can say that X1 + X2 + X3 + X4 + X5 + X6 = 1

It is given that (X1 + X3 + X5) = 0.9*(X2 + X4 + X6)

It is also given that X2 = X4 = X6

Also given that (X4 + X6)/(X4 + X5 + X6) = 0.75

Solving the above equations, we can get X3 as 0.468

Question 6

Which one of the following choices is correct?

.

Correct answer is :C

Solution :

Check for all:-

a1 = a ,

a2 = a * a = a

a3 = a2 * a = a * a = a

a is not the generator since we are not able to express other members of the group in powers of a

Check for c -

c1 = c

c2 = c * c = b

c3 = c2 * c = b * c = d

c4 = c2 * c2 = b * b = a

We are able to generate all the members of the group from c ,

Hence c is the generator Similarly check for d

Question 7

.

Correct answer is :D

Solution :

D represents the correct form. For all x where x is gold ornament or silver ornament, x is precious.

Question 8

.

Correct answer is :D

Solution :

(1-tanx)/(1+tanx) = (cosx - sinx)/(cosx + sinx)

Let cosx + sinx = t (-sinx + cosx)dx = dt (1/t)dt = ln t

=> ln(sinx + cosx) => ln(sin π/4 + cos π/4) => ln(1/√2 + 1/√2) => 1/2 ln 2

Question 9

I.¬∀ x(P(x))

II. ¬∃ x(P(x))

III. ¬∃ x(¬P(x))

IV. ¬∃x(¬P(x))

Which of the above are equivalent?

.

Correct answer is :B

Solution :

According to negation property of universal qualifier and existential quantifier

¬ ∀ ∈ X P(x) ≡ ∃x ∈ X ¬ P(x)

Question 10

^{2}- 13 = 0 with 3.5 as the initial value. The approximation after one iteration is

.

Correct answer is :D

Solution :

According to Newton Raphson's method

f(x) = x

^{2}-13

f`(x) = 2x

applying formula

x1 = 3.5 - (3.5*3.5 - 13)/2*3.5

x = 3.607

Question 11

.

Correct answer is :C

Solution :

No. of reflexive relations is 2

^{n2-n}

substituting n=5 we get 2

^{20}

Question 12

^{2}}, where ω and ω

^{2}are cube roots of unity. If * denotes the multiplication operation, the structure (S, *) forms

.

Correct answer is :A

Question 13

_{n->∞}(1-1/n)

^{2n}?

.

Correct answer is :B

Question 14

.

Correct answer is :A

Question 15

^{99}is a multiple of 10

^{96}?

.

Correct answer is :A

Solution :

Divisors are of the form :

2

^{k}* 5

^{m}

So for 10

^{99}we have (1,2

^{1},2

^{2}......2

^{99})*(1,5

^{1},5

^{2}......5

^{99}) divisors (one from both sets)

So there are total (100)*(100) = 10000 possibilities

Question 16

I. 7, 6, 5, 4, 4, 3, 2, 1

II. 6, 6, 6, 6, 3, 3, 2, 2

III. 7, 6, 6, 4, 4, 3, 2, 2

IV. 8, 7, 7, 6, 4, 2, 1, 1

.

Correct answer is :D

Question 17

If the eigenvalues of A are 4 and 8, then

.

Correct answer is :D

Question 18

.

Correct answer is :B