### SET 3

Question 1

**K4 and Q3 are graphs with the following structures **

Which one of the following statements is TRUE in relation to these graphs?

A : K4 is planar while Q3 is not

B : Both K4 and Q3 are planar

C : Q3 is planar while K4 is not

D : Neither K4 not Q3 is planar

Answer Discuss it!

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Correct answer is :B

Question 2

**If the difference between the expectation of the square of random variable E[X**^{2} ] and the square of the expectation of the random variable ( E[X])^{2} is denoted by R then

A : R=0

B : R<0

C : R ≥ 0

D : R>0

Answer Discuss it!

.

Correct answer is :C

Question 3

**If two fair coins are flipped and at least one of the outcomes is known to be a head, what is the probability that both outcomes are heads?**

A : 1/3

B : 1/4

C : 1/2

D : 2/3

Answer Discuss it!

.

Correct answer is :A

Solution :

Sample space = {HH, HT, TH} Required probability = 1/3

Question 4

**Consider the matrix as given below.**

Which one of the following provides the CORRECT values of eigenvalues of the matrix?

A : 1,4,3

B : 3,7,3

C : 7,3,2

D : 1,2,3

Answer Discuss it!

.

Correct answer is :A

Solution :

Given matrix is upper triangular matrix and its diagonal elements are its eigen values = 1, 4, 3

Question 5

**Which one of the following options is CORRECT given three positive integers x, y and z, and a predicate**

P(x) = ¬ ( x = 1 ) ∧ ∀ y ( ∃ z ( x = y * z ) => ( y = x ) ∨ ( y = 1 ) )

A : P(x) being true means that x is a prime number

B : P(x) being true means that x is a number other than 1

C : P(x) is always true irrespective of the value of x

D : P(x) being true means that x has exactly two factors other than 1 and x

Answer Discuss it!

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Correct answer is :A

Question 6

**Consider a finite sequence of random values X = [x1,x2,...xn ]. Let μx be the mean and x σ be the standard deviation of X .Let another finite sequence Y of equal length be derived from this as yi = a* xi + b, where a and b are positive constants. Let μy be the mean and σy be the standard deviation of this sequence. Which one of the following statements is INCORRECT?**

A : Index position of mode of X in X is the same as the index position of mode of Y in Y.

B : Index position of median of X in X is the same as the index position of median of Y in Y.

C : μ_{y} = aμ_{x} + b

D : σ_{y} = aσ_{x} + b

Answer Discuss it!

.

Correct answer is :D

Question 7

**A deck of 5 cards (each carrying a distinct number from 1 to 5) is shuffled thoroughly. Two cards are then removed one at a time from the deck. What is the probability that the two cards are selected with the number on the first card being one higher than the number on the second card?**

A : 1/5

B : 4/25

C : 1/4

D : 2/5

Answer Discuss it!

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Correct answer is :A

Solution :

(2,1), (3,2), (4,3), (5,4) Required probability =4/(5*4) = 4/20 = 1/5

Question 8

**Given i=√-1, what will be the evaluation of the integral **

A : 0

B : 2

C : -i

D : i

Answer Discuss it!

.

Correct answer is :D

Question 9

**Consider a random variable X that takes values + 1 and -1 with probability 0.5 each. The values of the cumulative distribution function F(x) at x = -1 and +1 are**

A : 0 and 0.5

B : 0 and 1

C : 0.5 and 1

D : 0.25 and 0.75

Answer Discuss it!

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Correct answer is :C

Solution :

The cumulative distribution function F(x)=P(X<=x)

F(-1)=P(X=-1) =0.5

F(+1)=P(X<=+1) =P(X=-1) = 0.5+0.5 = 1

Question 10

**Let G be a simple undirected planar graph on 10 vertices with 15edges. If G is a connected graph, then the number of bounded faces in any embedding of G on the plane is equal to**

A : 3

B : 4

C : 5

D : 6

Answer Discuss it!

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Correct answer is :D

Solution :

We have the relation V-E+F=2, by this we will get the total number of faces,F = 7. Out of 7 faces one is an unbounded face, so total 6 bounded faces.

Question 11

**What is the correct translation of the following statement into mathematical logic? **

“Some real numbers are rational”

A : ∃ x (real(x) v rational (x))

B : ∀x ( real(x) -> rational (x) )

C : ∃ x ( real(x) ∧ rational (x) )

D : ∃ x ( rational(x) -> real (x) )

Answer Discuss it!

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Correct answer is :C

Solution :

Option A: There exists x which is either real or rational and can be both.

Option B: All real numbers are rational

Option C: There exists a real number which is rational.

Option D: There exists some number which is not rational or which is real.

Question 12

**Let A be the 2 x 2 matrix with elements a11 = a12 = a21= + 1 and a22 =-1. Then the eigen values of the matrix A**^{19} are

A : 1024 and -1024

B : 1024 √2 and -1024 √2

C : 4 √2 and - 4 √2

D : 512 √2 and - 512 √2

Answer Discuss it!

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Correct answer is :D

Question 13

**Consider the function f(x) = sin(x) in the interval [π/4, 7π/4]. The number and location(s) of the local minima of this function are**

A : One, at π/2

B : One, at 3π / 2

C : Two, at π / 2 and 3π / 2

D : Two, at π / 4 and 3π / 2

Answer Discuss it!

.

Correct answer is :D

Solution :

Sin x has a maximum value of 1 at , π/2 and a minimum value of –1 at 3π/2 and at all angles conterminal with them.

∴ In the interval [ π/4, 7π/4 ] , it has one local minimum at x= 3π/2

Question 14

**Consider the following logical inferences. **

I1 : If it rains then the cricket match will not be played.

The cricket match was played.

Inference: There was no rain.

I2 : If it rains then the cricket match will not be played.

It did not rain.

Inference: The cricket match was played.

Which of the following is TRUE?

A : Both I1 and I2 are correct inferences

B : I1 is correct but I2 is not a correct inference

C : I1 is not correct but I2 is a correct inference

D : Both I1 and I2 are not correct inferences

Answer Discuss it!

.

Correct answer is :B

Question 15

**Let G be a complete undirected graph on 6 vertices. If vertices of G are labeled, then the number of distinct cycles of length 4 in G is equal to**

A : 15

B : 30

C : 90

D : 45

Answer Discuss it!

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Correct answer is :D

Solution :

4 vertices from 6 vertices can be chosen in in 6C4. Number of cycles of length 4 that can be formed from those selected vertices is (4-1)!/2 (left or right/ up or down does not matter), so total number of 4 length cycles are (6C4.3!)/2 = 45.

Question 16

**How many onto (or surjective) functions are there from an n-element (n >= 2) set to a 2- element set?**

A : 2^{n}

B : 2^{n} - 1

C : 2^{n} - 2

D : 2 ( 2^{n} - 2 )

Answer Discuss it!

.

Correct answer is :C

Solution :

Total number of functions is 2^{n}, out of which there will be exactly two functions where all elements map to exactly one element, so total number of onto functions is 2^{n}-2

Question 17

**Suppose a fair six-sided die is rolled once. If the value on the die is 1, 2, or 3, the die is rolled a second time. What is the probability that the sum total of values that turn up is at least 6?**

A : 10/21

B : 5/12

C : 2/3

D : 1/6

Answer Discuss it!

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Correct answer is :B

Solution :

Required probability= 1/6 * 2/6 + 1/6* 3/6 + 1/6 * 4/6 + 1/6 = 15/36 = 5/12

Question 18

**The bisection method is applied to compute a zero of the function f(x) = x^4 -x^3 - x^2 – 4 in the interval [1,9]. The method converges to a solution after ______ iterations.**

A : 1

B : 3

C : 5

D : 7

Answer Discuss it!

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Correct answer is :B

Question 19

**Which of the following graph is isomorphic to**

A : B :

C : D :

Answer Discuss it!

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Correct answer is :B

Solution :

The graph in option (A) has a 3 length cycle whereas the original graph does not have a 3 length cycle

The graph in option (C) has vertex with degree 3 whereas the original graph does not have a vertex with degree 3

The graph in option (D) has a 4 length cycle whereas the original graph does not have a 4 length cycle

Question 20

**Consider an undirected random graph of eight vertices. The probability that there is an edge between a pair of vertices is ½. What is the expected number of unordered cycles of length three?**

A : 1/8

B : 1

C : 7

D : 8

Answer Discuss it!

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Correct answer is :C

Solution :

P(edge) = 1/2

Number of ways we can choose the vertices out of 8 is 8C3

(Three edges in each cycle)

Expected number of unordered cycles of length 3 = 8C3 * (1/2)^{3} = 7

Question 21

**Which of the following statements is/are TRUE for undirected graphs? **

P: Number of odd degree vertices is even.

Q: Sum of degrees of all vertices is even.

A : P only

B : Q only

C : Both P and Q

D : Neither P nor Q

Answer Discuss it!

.

Correct answer is :C

Solution :

Q: Sum of degrees of all vertices = 2×(number of edges)

Question 22

**Function f is known at the following points:**

A : 8.983

B : 9.003

C : 9.017

D : 9.045

Answer Discuss it!

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Correct answer is :D

Question 23

**Which one of the following functions is continuous at x = 3 ?**

A : B :

C : D :

Answer Discuss it!

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Correct answer is :A

Question 24

**Which one of the following does NOT equal to**

A : B :

C : D :

Answer Discuss it!

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Correct answer is :A

Solution :

If matrix B is obtained from matrix A by replacing the lth row by itself plus k times the mth row, for 1 ≠ m then det(B)=det(A). With this property given matrix is equal to the matrices given in options (B),(C) and (D).

Question 25

**Suppose p is number of cars per minute passing through a certain road junction between 5 PM and 6PM, and p has a Poisson distribution with mean 3. What is the probability of observing fewer than 3 cars during any given minute in this interval?**

A : 8 / ( 2e^{3} )

B : 9 / ( 2e^{3} )

C : 17 / ( 2e^{3} )

D : 26 / ( 2e^{3} )

Answer Discuss it!

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Correct answer is :C

Solution :

P(p < 3) = P(p = 0) + P(p =1) + P(p = 2)

substituting values = e^{-3} (1+3+9/2) = 17/2e^{3}

Question 26

**Which one of the following is NOT logically equivalent to ¬∃ x (∀ y(a) ∧ ∀z (b))?**

(A)∀ x ( ∃ z (¬b) -> ∀ y(a))

(B)∀ x ( ∀ z (b) -> ∃ y(¬a))

(C)∀ x ( ∀ y (a) -> ∃ z(¬b))

(D)∀ x ( ∃ y (¬a) -> ∃ z(b))

A : A

B : B

C : C

D : A and D

Answer Discuss it!

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Correct answer is :A

Question 27

**The line graph L(G) of a simple graph G is defined as follows: **

• There is exactly one vertex v(e) in L(G) for each edge e in G.

• For any two edges e and e’ in G, L(G) has an edge between v(e) and v(e’), if and only if e and e’ are incident with the same vertex in G.

Which of the following statements is/are TRUE?

(P) The line graph of a cycle is a cycle.

(Q) The line graph of a clique is a clique.

(R) The line graph of a planar graph is planar.

(S) The line graph of a tree is a tree.

A : P only

B : P and R only

C : R only

D : P,Q and S only

Answer Discuss it!

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Correct answer is :A

Question 28

**What is the logical translation of the following statement? **

“None of my friends are perfect.”

A : ∃ x ( F(x) ∧ ¬P(x) )

B : ∃ x ( ¬ F(x) ∧ P(x) )

C : ∃ x ( ¬ F(x) ∧ ¬ P(x) )

D : ¬∃ x ( F(x) ∧ P(x) )

Answer Discuss it!

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Correct answer is :D

Solution :

“None of my friends are perfect”

= ∀ (F(x)-> ¬P(x))

= ∀x( ¬F(x) ∨ ¬P(x))

= ¬∃ x ( F(x) ∧ P(x) )

Question 1

Which one of the following statements is TRUE in relation to these graphs?

.

Correct answer is :B

Question 2

^{2}] and the square of the expectation of the random variable ( E[X])

^{2}is denoted by R then

.

Correct answer is :C

Question 3

.

Correct answer is :A

Solution :

Sample space = {HH, HT, TH} Required probability = 1/3

Question 4

Which one of the following provides the CORRECT values of eigenvalues of the matrix?

.

Correct answer is :A

Solution :

Given matrix is upper triangular matrix and its diagonal elements are its eigen values = 1, 4, 3

Question 5

P(x) = ¬ ( x = 1 ) ∧ ∀ y ( ∃ z ( x = y * z ) => ( y = x ) ∨ ( y = 1 ) )

.

Correct answer is :A

Question 6

.

Correct answer is :D

Question 7

.

Correct answer is :A

Solution :

(2,1), (3,2), (4,3), (5,4) Required probability =4/(5*4) = 4/20 = 1/5

Question 8

.

Correct answer is :D

Question 9

.

Correct answer is :C

Solution :

The cumulative distribution function F(x)=P(X<=x)

F(-1)=P(X=-1) =0.5

F(+1)=P(X<=+1) =P(X=-1) = 0.5+0.5 = 1

Question 10

.

Correct answer is :D

Solution :

We have the relation V-E+F=2, by this we will get the total number of faces,F = 7. Out of 7 faces one is an unbounded face, so total 6 bounded faces.

Question 11

“Some real numbers are rational”

.

Correct answer is :C

Solution :

Option A: There exists x which is either real or rational and can be both.

Option B: All real numbers are rational

Option C: There exists a real number which is rational.

Option D: There exists some number which is not rational or which is real.

Question 12

^{19}are

.

Correct answer is :D

Question 13

.

Correct answer is :D

Solution :

Sin x has a maximum value of 1 at , π/2 and a minimum value of –1 at 3π/2 and at all angles conterminal with them.

∴ In the interval [ π/4, 7π/4 ] , it has one local minimum at x= 3π/2

Question 14

I1 : If it rains then the cricket match will not be played.

The cricket match was played.

Inference: There was no rain.

I2 : If it rains then the cricket match will not be played.

It did not rain.

Inference: The cricket match was played.

Which of the following is TRUE?

.

Correct answer is :B

Question 15

.

Correct answer is :D

Solution :

4 vertices from 6 vertices can be chosen in in 6C4. Number of cycles of length 4 that can be formed from those selected vertices is (4-1)!/2 (left or right/ up or down does not matter), so total number of 4 length cycles are (6C4.3!)/2 = 45.

Question 16

.

Correct answer is :C

Solution :

Total number of functions is 2

^{n}, out of which there will be exactly two functions where all elements map to exactly one element, so total number of onto functions is 2

^{n}-2

Question 17

.

Correct answer is :B

Solution :

Required probability= 1/6 * 2/6 + 1/6* 3/6 + 1/6 * 4/6 + 1/6 = 15/36 = 5/12

Question 18

.

Correct answer is :B

Question 19

.

Correct answer is :B

Solution :

The graph in option (A) has a 3 length cycle whereas the original graph does not have a 3 length cycle

The graph in option (C) has vertex with degree 3 whereas the original graph does not have a vertex with degree 3

The graph in option (D) has a 4 length cycle whereas the original graph does not have a 4 length cycle

Question 20

.

Correct answer is :C

Solution :

P(edge) = 1/2

Number of ways we can choose the vertices out of 8 is 8C3

(Three edges in each cycle)

Expected number of unordered cycles of length 3 = 8C3 * (1/2)

^{3}= 7

Question 21

P: Number of odd degree vertices is even.

Q: Sum of degrees of all vertices is even.

.

Correct answer is :C

Solution :

Q: Sum of degrees of all vertices = 2×(number of edges)

Question 22

.

Correct answer is :D

Question 23

.

Correct answer is :A

Question 24

.

Correct answer is :A

Solution :

If matrix B is obtained from matrix A by replacing the lth row by itself plus k times the mth row, for 1 ≠ m then det(B)=det(A). With this property given matrix is equal to the matrices given in options (B),(C) and (D).

Question 25

.

Correct answer is :C

Solution :

P(p < 3) = P(p = 0) + P(p =1) + P(p = 2)

substituting values = e

^{-3}(1+3+9/2) = 17/2e

^{3}

Question 26

(A)∀ x ( ∃ z (¬b) -> ∀ y(a))

(B)∀ x ( ∀ z (b) -> ∃ y(¬a))

(C)∀ x ( ∀ y (a) -> ∃ z(¬b))

(D)∀ x ( ∃ y (¬a) -> ∃ z(b))

.

Correct answer is :A

Question 27

• There is exactly one vertex v(e) in L(G) for each edge e in G.

• For any two edges e and e’ in G, L(G) has an edge between v(e) and v(e’), if and only if e and e’ are incident with the same vertex in G.

Which of the following statements is/are TRUE?

(P) The line graph of a cycle is a cycle.

(Q) The line graph of a clique is a clique.

(R) The line graph of a planar graph is planar.

(S) The line graph of a tree is a tree.

.

Correct answer is :A

Question 28

“None of my friends are perfect.”

.

Correct answer is :D

Solution :

“None of my friends are perfect”

= ∀ (F(x)-> ¬P(x))

= ∀x( ¬F(x) ∨ ¬P(x))

= ¬∃ x ( F(x) ∧ P(x) )