### SET 2

Question 1

**Consider the statement: **

“Not all that glitters is gold”

Predicate glitters (x) is true if x glitters and predicate gold (x) is true if x is gold. Which one of the following logical formulae represents the above statement?

A : ∀x glitter (x) => ¬gold(x)

B : ∀xgold(x) => glitter(x)

C : ∃xgold(x) ∧ ¬ glitter(x)

D : ∃ glitter(x) ∧ ¬ gold(x)

Answer Discuss it!

.

Correct answer is :D

Solution :

It means “It is false that every glitter is gold” or “some glitters are not gold”. Then we can say “atleast one glitter object is not gold”.

Question 2

**Suppose you break a stick of unit length at a point chosen uniformly at random. Then the expected length of the shorter stick is ________ .**

Answer Discuss it!

.

Correct answer is :0.25

Solution :

The smaller sticks, therefore, will range in length from almost 0 meters up to a maximum of 0.5 meters, with each length equally possible.

Thus, the average length will be about 0.25 meters, or about a quarter of the stick.

Question 3

**Consider the following system of equations:**

3x + 2y= 1

4x +7z=1

x + y + z = 3

x - 2y + 7z = 0

The number of solutions for this system is __________________

Answer Discuss it!

.

Correct answer is :1

Question 4

**The value of the dot product of the eigenvectors corresponding to any pair of different eigen values of a 4-by-4 symmetric positive definite matrix is ______________.**

Answer Discuss it!

.

Correct answer is :0

Solution :

( The eigen vectors corresponding to distinct eigen values of real symmetric matrix are orthogonal)

Question 5

**Let the function given below where θ = ∈ [ π/6 , π/2] and f'(θ) denote the derivative of f with respect to θ . Which of the following statement is / are TRUE? **

(I) There exists θ ∈ (π/6 ,&pi/;3) such that f'(θ) = 0

(I) There exists θ ∈ (π/6 ,π/3) such that f'(θ) ≠ 0

A : I only

B : II only

C : Both I and II

D : Neither I and II

Answer Discuss it!

.

Correct answer is :C

Solution :

By mean value theorem

Question 6

**There are 5 bags labelled 1 to 5. All the coins in a given bag have the same weight. Some bags have coins of weight 10 gm, others have coins of weight 11 gm. I pick 1, 2, 4, 8, 16 coins respectively from bags 1 to 5. Their total weight comes out to 323 gm. Then the product of the labels of the bags having 11 gm coins is ___.**

Answer Discuss it!

.

Correct answer is :12

Solution :

Let the weight of coins in the respective bags (1 through 5) be a,b,c,d and e-each of which can take one of two values namely 10 or 11 (gm).

Now, the given information on total weight can be expressed as the following equation:

1.a+2.b+4.c+8.d+16.e = 323

a must be odd => a = 11

The equation then becomes: 11+2.b+4.c+8.d+16.e = 323

=>2.b+4.c+8.d+16.e = 312

=>b+2.c+4.d+8.e = 156

b must be even b = 10

The equation then becomes: 10+2.c+4.d+8.e = 156

=>2.c+4.d+8.e = 146

=> c+2.d+4.e = 73

c must be odd c = 11

The equation now becomes: 11+2.d+4.e = 73

=>2.d+4.e = 62

=>d+2.e = 31

e = 11 and e = 10

Therefore, bags labelled 1, 3 and 4 contain 11 gm coins => Required Product = 1*3*4* = 12.

Question 7

**The function f(x) = x sin x satisfies the following equation. f"(x) + f(x) +tcosx = 0. The value of t is______.**

Answer Discuss it!

.

Correct answer is :-2

Solution :

Given f "(x) +f (x)+ t cos x =0

and f(x)= xsin x

f '(x)= x cos x + sin x

f "(x)= x(- sin x) + cos x + cos x

= 2cos x - xsin x

= 2cos x - f(x)

2cos x - f(x)+ f(x) +t cos x = 0

=> 2cos x= -t cos x=> t= -2

Question 8

**A function f(x) is continuous the interval [0,2]. It is known that f(0) = f(2) = -1 and f(1) = 1. Which one of the following statements must be true?**

A : There exists a y in the interval (0,1) such that f(y) = f(y + 1)

B : For every y in the interval (0,1), f(y) = f(2 - y)

C : The maximum value of the function in the interval (0.2) is 1

D : There exists a y in the interval (0,1) such that f(y) =f(2 – y)

Answer Discuss it!

.

Correct answer is :A

Solution :

Define g(x) =f(x)-f(x+1) in [0,1]. g(0) is negative and g(1) is positive. By intermediate value

theorem there is y€(0,1) such that g(y)=0

That is f(y) =f(y+1).

Thus answer is (a)

Question 9

**Four fair six-sided dice are rolled. The probability that the sum of the results being 22is X/1296.The value of X is ______________.**

Answer Discuss it!

.

Correct answer is :10

Solution :

22 occurred in following ways

6 6 6 4 -> 4 ways

6 6 5 5 -> 6 ways

Required probability = 6+4/2296 = 10/2296 => x=10

Question 10

**A pennant is a sequence of numbers, each number being 1 or 2. An n-pennant is a sequence of numbers with sum equal to n. For example, (1,1,2) is a 4-pennant. The set of all possible 1- pennants is {(1)}, the set of all possible 2-pennants is {(2), (1,1)}and the set of all 3-pennants is {(2,1), (1,1,1), (1,2)}. Note that the pennant (1,2) is not the same as the pennant (2,1). The number of 10- pennants is ______________.**

Answer Discuss it!

.

Correct answer is :89

Solution :

No twos: 11111111111=> pennant

Single two: 211111111 => 9!/8!1! = 9 pennants

Two twos: 22111111 => 8!/6!.2! = 28

Three twos: 2221111 => 7!/3!.4! = 35

Four twos: 222211 => 6!/4!.2! = 15

Five twos: 22222 =>1

Total = 89 pennants.

Question 11

**Let S denote the set of all functions f:{0,1}**^{4} -> {01} . Denote by N the number of functions from S to the set {0,1}. The value of log_{2} log_{2} N is______.

Answer Discuss it!

.

Correct answer is :16

Solution :

The number of functions from A to B where size of A = |A| and size of B = |B| is |B||A|

{0,1}^{4} = {0,1} X {0,1} X {0,1}X {0.1} = 16

|S| = 2^{16}

N=2^{|S|}

loglogN= loglog 2^{|S|} = log |S| = log 2^{16} =16

Question 12

**An ordered n-tuple (d1,d2,…,dn) with d1 >= d2 >= .... >= dn is called graphic if there exists a simple undirected graph with n vertices having degrees d1,d2,.......dn respectively. Which of the following 6-tuples is NOT graphic?**

A : (1, 1, 1, 1, 1, 1)

B : (2, 2, 2, 2, 2, 2)

C : (3, 3, 3, 1, 0, 0)

D : ( 3, 2, 1, 1, 1, 0)

Answer Discuss it!

.

Correct answer is :C

Solution :

According to havel-hakimi theorem

(1,1,1,1,1,1) is graphic iff<1,1,1,1,0> is graphic

(0,1,1,1,1) is graphic iff (0,1,1,0) is graphic

(0,0,1,1) is graphic iff (0,0,0) is graphic

Since (0,0,0) is graphic (1,1,1,1,1,1) is also graphic.

(The process is always finding maximum degree and removing it from degree sequence, subtract 1 from each degree for d times from right to left where d is maximum degree)

(2,2,2,2,2,2) is graphic iff (2,2,22-1,2-1) = (2,2,2,1,1) is graphic

(1,1,2,2,2

Question 13

**Which one of the following propositional logic formulas is TRUE when exactly two of p, q, and r are TRUE?**

A : ((p <-> q) ∧ r) V (p ∧ q ∧ ~ r)

B : (~ (p <-> q) ∧ r) V (p ∧ q ∧ ~ r)

C : ((p -> q) ∧ r) V (p ∧ q ∧ ~ r)

D : (~ (p <-> q) ∧ r) ∧ (p ∧ q ∧ ~ r)

Answer Discuss it!

.

Correct answer is :B

Solution :

P = T q=F and r=T

Option A will become false

Option C will become false.

Option D is always false.

Question 14

**The security system at an IT office is composed of 10 computers of which exactly four are working. To check whether the system is functional, the officials inspect four of the computers picked at random (without replacement). The system is deemed functional if at least three of the four computers inspected are working. Let the probability that the system is deemed functional be denoted by p Then 100p= _____________.**

Answer Discuss it!

.

Correct answer is :11.85 - 11.95

Solution :

p= P [at least three computers are working]

=P (3 or 4 computers working)

= (4C3) * (6C1) / 10C4 + 4C4 / 10C4 = 5/42

100p=11.9

Question 15

**Each of the nine words in the sentence ”The quick brown fox jumps over the lazy dog” is written on a separate piece of paper. These nine pieces of paper are kept in a box. One of the pieces is drawn at random from the box. The expected length of the word drawn is _____________. (The answer should be rounded to one decimal place.**

Answer Discuss it!

.

Correct answer is :3.8889

Question 16

**The maximum number of edges in a bipartite graph on 12 vertices is __________________________.**

Answer Discuss it!

.

Correct answer is :36

Solution :

The number of edges in a bipartite graph on n-vertices is atmost n^{2}/4

The maximum number of edges in a bipartite graph on 12 –vertices is n^{2} /4 = 12*12 /4 = 36

Question 17

**If the matrix A is such that**

Then the determinant of A is equal to ________.

Answer Discuss it!

.

Correct answer is :0

Solution :

|A| = 0

Question 18

**A non-zero polynomial f(x) of degree 3 has roots at x = 1,x = 2 and x = 3. Which one of the following must be TRUE?**

A : f(0) f(4) < 0

B : f(0) f(4) > 0

C : f(0) + f(4) > 0

D : f(0) + f(4) < 0

Answer Discuss it!

.

Correct answer is :A

Solution :

Since, the roots of f(x) = 0 i.e., x = 1, 2, 3 lies between 0 and 4 and f(x) is of degree 3

f(0) and f(4) are of opposite signs

f(0).f(4)<0.

Question 19

**Suppose n and p are unsigned int variables in a C program. We wish to set p to nC3 . If n is large, which one of the following statements is most likely to set p correctly?**

A : p = n * (n – 1) * (n-2) / 6;

B : p = n * (n – 1) / 2 * (n-2) / 3;

C : p = n * (n – 1) / 3 * (n-2) / 2;

D : p = n * (n – 1) / 2 * (n-2) / 6.0;

Answer Discuss it!

.

Correct answer is :B

Solution :

n*(n-1) is an even number so we divide it by 2 and the rest by 3. The output will be same but overflow can be avoided.

Question 20

**In the Newton-Raphson method, an initial guess of x0 = 2 is made and the sequencex0,x1,x2 ... is obtained for the function**

0.75x^{3} - 2x^{2} - 2x +4 = 0

Consider the statements

(I) x3 = 0 .

(II) The method converges to a solution in a finite number of iterations.

Which of the following is TRUE?

A : Only I

B : Only II

C : Both I and II

D : Neither I and II

Answer Discuss it!

.

Correct answer is :A

Question 21

**The product of the non-zero eigenvalues of the matrix is**

Answer Discuss it!

.

Correct answer is :6

Question 22

**The probability that a given positive integer lying between 1 and 100 (both inclusive) is NOT divisible by 2, 3 or 5 is ______ .**

Answer Discuss it!

.

Correct answer is :0.259 - 0.261

Solution :

Let A = divisible by 2, B = divisible by 3 and C = divisible by 5, then

n(A) = 50, n(B) = 33, n(C) = 20

n(A ∩ B) = 16 (100/6)

n(A ∩ C) = 10 (100/10)

n(B ∩ C) = 6 (100/15)

n(A ∩ B ∩ C) = 3 (100/(2*3*5))

Now find n(A U B U C) = n(A) + n(B) + n(C) - n(A ∩ B) - n(A ∩ B) -n(B ∩ C) + n(A ∩ B ∩ C)

(100 - n(A U B U C))/100

Substituting the values we get answer as 0.26

Question 23

**The number of distinct positive integral factors of 2014 is _________________________ **

Answer Discuss it!

.

Correct answer is :8

Solution :

2014 = 2×19×53 i.e., product of prime factors

Number of distinct positive integral factors of 2014 is (2)×(2)×(2) = 8.

Question 24

**Consider the following relation on subsets of the set S of integers between 1 and 2014. For two distinct subsets U and V of S we say U < V if the minimum element in the symmetric difference of the two sets is in U.**

Consider the following two statements:

S1: There is a subset of S that is larger than every other subset.

S2: There is a subset of S that is smaller than every other subset.

Which one of the following is CORRECT?

A : Both S1 and S2 are true

B : S1 is true and S2 is false

C : S2 is true and S1 is false

D : Neither S1 nor S2 is true

Answer Discuss it!

.

Correct answer is :A

Solution :

From given data S1 is true ,since null set is larger than every other set ,and S2 is true since the universal set {1,2,...,2014} is smaller than every other set.

Both s1 and s2 are true.

Question 25

**A cycle on n vertices is isomorphic to its complement. The value of n is _____.**

Answer Discuss it!

.

Correct answer is :5

Solution :

Consider a cycle on five vertices C5

C5 and C5` are isomorphic

Question 26

**Which one of the following Boolean expressions is NOT a tautology?**

A : ((a -> b) ∧ (b -> c)) -> (a -> c)

B : (a <-> c) -> (~ b -> (a ∧ c))

C : (a ∧ b ∧ c) -> (c ∧ a )

D : a -> (b -> a)

Answer Discuss it!

.

Correct answer is :B

Question 27

**Let X and Y be finite sets and f : X -> Y be a function. Which one of the following statements is TRUE?**

A : For any subsets A and B of X, f (A U B) = |f (A)| + |f (B)|

B : For any subsets A and B of X, f (A ∧ B) =f (A) ∧ f (B)

C : For any subsets A and B of X, f (A ∧ B) =min{ f (A) , f (B) }

D : For any subsets S and T of Y, f^{-1} (S ∧ T) = f^{-1} (S) ∧ f^{-1}(T)

Answer Discuss it!

.

Correct answer is :D

Question 28

**Let G be a group with 15 elements. Let L be a subgroup of G. It is known that L ≠ G and that the size of L is at least 4. The size of L is _______.**

Answer Discuss it!

.

Correct answer is :5

Solution :

Order of subgroup divides order of group (Lagrange’s theorem).

3, 5 and 15 can be the order of subgroup. As subgroup has atleast 4 elements and it is not equal to the given group, order of subgroup can’t be 3 and 15. Hence it is 5.

Question 29

**Which one of the following statements is TRUE about every n × n matrix with only real eigenvalues?**

A : If the trace of the matrix is positive and the determinant of the matrix is negative, at least one of its eigenvalues is negative.

B : If the trace of the matrix is positive, all its eigenvalues are positive.

C : If the determinant of the matrix is positive, all its eigenvalues are positive.

D : If the product of the trace and determinant of the matrix is positive, all its eigenvalues are positive.

Answer Discuss it!

.

Correct answer is :A

Solution :

If the trace of the matrix is positive and the determinant of the matrix is negative then atleast one of its eigen values is negative.

Since determinant = product of eigen values.

Question 30

**If V1 and V2 are 4-dimensional subspaces of a 6-dimensional vector space V, then the smallest possible dimension of V1 ∧ v2 is _______.**

Answer Discuss it!

.

Correct answer is :2

Solution :

Let the basis of 6-dimensional vector space be {e1, e2, e3,e4, e5, e6}. In order for V1 ∧ V2 to have smallest possible dimension V1 and V2 could be, say, {e1, e2, e3,e4} and {e3, e4, e5, e6} respectively. The basis of V1 ∧ V2 would then be {e3, e4}. => Smallest possible dimension = 2.

Question 31

**Find the value of k**

Answer Discuss it!

.

Correct answer is :4

Question 32

**Let A be a square matrix size n × n. Consider the following pseudocode. What is the expected output? **

C = 100;

for i = 1 to n do

for j = 1 to n do

{

Temp = A[ i ] [ j ] + C ;

A [ i ] [ j ] = A [ j ] [ i ] ;

A [ j ] [ i ] = Temp – C ;

}

for i = 1 to n do

for j = 1 to n do

output (A[ i ] [ j ]);

A : The matrix A itself

B : Transpose of the matrix A

C : Adding 100 to the upper diagonal elements and subtracting 100 from lower diagonal elements of A

D : None of these

Answer Discuss it!

.

Correct answer is :A

Solution :

In the computation of given pseudo code for each row and column of Matrix A, each upper triangular element will be interchanged by its mirror image in the lower triangular and after that the same lower triangular element will be again re-interchanged by its mirror image in the upper triangular, resulting the final computed Matrix A same as input Matrix A.

Question 33

**With respect to the numerical evaluation of the definite integral, K= ∫**^{b}a x^{2} dx, where a and b are given, which of the following statements is/are TRUE?

(I) The value of K obtained using the trapezoidal rule is always greater than or equal to the exact value of the definite integral.

(II) The value of K obtained using the Simpson’s rule is always equal to the exact value of the definite integral.

A : I only

B : II only

C : Both I and II

D : Neither I and II

Answer Discuss it!

.

Correct answer is :C

Question 34

**The value of the integral given below is**

A : -2π

B : π

C : -π

D : 2π

Answer Discuss it!

.

Correct answer is :A

Question 35

**Let S be a sample space and two mutually exclusive events A and B be such that A ∪ B = S. If P(.) denotes the probability of the event, the maximum value of P(A)P(B) is ______ A**

Answer Discuss it!

.

Correct answer is :0.25

Question 36

**Consider the set of all functions f :{0,1,...,2014}->{0,1...,2014} such that f (f (i)) = i, for 0 <= i<= 2014 . **

Consider the following statements.

P. For each such function it must be the case that for every i, f(i) = i,

Q. For each such function it must be the case that for some i,f(i) = i,

R. Each such function must be onto.

Which one of the following is CORRECT?

A : P, Q and R are true

B : Only Q and R are true

C : Only P and Q are true

D : Only R is true

Answer Discuss it!

.

Correct answer is :B

Solution :

Let us consider a function (counter example) as

f(0)=1,f(1) =0,f(2)=3,f(3)=2,.........,f(2012)=2013,f(2013)=2012 andf(2014)=2014

Clearly f (f (i))= i for 0<= i <=2014

Here f(i) ≠ i for every i and f(i) = i for some i

Also f is onto

Hence , only Q and R are true

Question 37

**There are two elements x,y in a group (G,*) such that every element in the group can be written as a product of some number of x’s and y’s in some order. It is known that**

x * x = y * y = x * y *x * y = y* x * y *x = e

where e is the identity element. The maximum number of elements in such a group is _________________.

Answer Discuss it!

.

Correct answer is :4

Question 38

**If G is a forest with n vertices and k connected components, how many edges does G have? **

A : [n / k]

B : [n / k]

C : n-k

D : n-k+1

Answer Discuss it!

.

Correct answer is :C

Solution :

Let n1 ,n2 ,.....nk be the number of vertices respectively in K connected components of a forest G, then n1 ?1,n2 ?1,.....,nk ?1 be the number of edges respectively in K connected components and n1 + n2 + ..... + nk = n (number of vertices in G)

Hence, number of edges in G = number of edges in K connected components

(n1-1)+(n2-1)+..........+(nk-1) = n-k

Question 39

**Let d denote the minimum degree of a vertex in a graph. For all planar graphs on n vertices with d>=3, which one of the following is TRUE?**

A : In any planar embedding, the number of faces is at least n/2 +2

B : In any planar embedding, the number of faces is less than n/2 +2

C : There is a planar embedding in which the number of faces is less than n /2 +2

D : There is a planar embedding in which the number of faces is at most n /d +1

Answer Discuss it!

.

Correct answer is :A

Solution :

We know that v + r = e + 2e = n + r ? 2 ...(1)

Where V= n(number of vertices); r =number of faces and e =number of edges

Given d>=3 then 3n <=2e

e>=3n/2

n+r-2>=3n/2 (using(1))

r>=3n/2 -n +2 => r>= n/2 +2

No.of faces is atleast n/2 + 2

Question 40

**The CORECT formula for the sentence, “not all rainy days are cold” is**

A : ∀d (Rainy(d) ∧ ~Cold(d))

B : ∀d ( ~Rainy(d)->Cold(d))

C : ∃d(~Rainy(d) -> Cold(d))

D : ∃d (Rainy(d) ∧ Cold(d))

Answer Discuss it!

.

Correct answer is :D

Question 1

“Not all that glitters is gold”

Predicate glitters (x) is true if x glitters and predicate gold (x) is true if x is gold. Which one of the following logical formulae represents the above statement?

.

Correct answer is :D

Solution :

It means “It is false that every glitter is gold” or “some glitters are not gold”. Then we can say “atleast one glitter object is not gold”.

Question 2

.

Correct answer is :0.25

Solution :

The smaller sticks, therefore, will range in length from almost 0 meters up to a maximum of 0.5 meters, with each length equally possible.

Thus, the average length will be about 0.25 meters, or about a quarter of the stick.

Question 3

3x + 2y= 1

4x +7z=1

x + y + z = 3

x - 2y + 7z = 0

The number of solutions for this system is __________________

.

Correct answer is :1

Question 4

.

Correct answer is :0

Solution :

( The eigen vectors corresponding to distinct eigen values of real symmetric matrix are orthogonal)

Question 5

(I) There exists θ ∈ (π/6 ,&pi/;3) such that f'(θ) = 0

(I) There exists θ ∈ (π/6 ,π/3) such that f'(θ) ≠ 0

.

Correct answer is :C

Solution :

By mean value theorem

Question 6

.

Correct answer is :12

Solution :

Let the weight of coins in the respective bags (1 through 5) be a,b,c,d and e-each of which can take one of two values namely 10 or 11 (gm).

Now, the given information on total weight can be expressed as the following equation:

1.a+2.b+4.c+8.d+16.e = 323

a must be odd => a = 11

The equation then becomes: 11+2.b+4.c+8.d+16.e = 323

=>2.b+4.c+8.d+16.e = 312

=>b+2.c+4.d+8.e = 156

b must be even b = 10

The equation then becomes: 10+2.c+4.d+8.e = 156

=>2.c+4.d+8.e = 146

=> c+2.d+4.e = 73

c must be odd c = 11

The equation now becomes: 11+2.d+4.e = 73

=>2.d+4.e = 62

=>d+2.e = 31

e = 11 and e = 10

Therefore, bags labelled 1, 3 and 4 contain 11 gm coins => Required Product = 1*3*4* = 12.

Question 7

.

Correct answer is :-2

Solution :

Given f "(x) +f (x)+ t cos x =0

and f(x)= xsin x

f '(x)= x cos x + sin x

f "(x)= x(- sin x) + cos x + cos x

= 2cos x - xsin x

= 2cos x - f(x)

2cos x - f(x)+ f(x) +t cos x = 0

=> 2cos x= -t cos x=> t= -2

Question 8

.

Correct answer is :A

Solution :

Define g(x) =f(x)-f(x+1) in [0,1]. g(0) is negative and g(1) is positive. By intermediate value

theorem there is y€(0,1) such that g(y)=0

That is f(y) =f(y+1).

Thus answer is (a)

Question 9

.

Correct answer is :10

Solution :

22 occurred in following ways

6 6 6 4 -> 4 ways

6 6 5 5 -> 6 ways

Required probability = 6+4/2296 = 10/2296 => x=10

Question 10

.

Correct answer is :89

Solution :

No twos: 11111111111=> pennant

Single two: 211111111 => 9!/8!1! = 9 pennants

Two twos: 22111111 => 8!/6!.2! = 28

Three twos: 2221111 => 7!/3!.4! = 35

Four twos: 222211 => 6!/4!.2! = 15

Five twos: 22222 =>1

Total = 89 pennants.

Question 11

^{4}-> {01} . Denote by N the number of functions from S to the set {0,1}. The value of log

_{2}log

_{2}N is______.

.

Correct answer is :16

Solution :

The number of functions from A to B where size of A = |A| and size of B = |B| is |B||A|

{0,1}

^{4}= {0,1} X {0,1} X {0,1}X {0.1} = 16

|S| = 2

^{16}

N=2

^{|S|}

loglogN= loglog 2

^{|S|}= log |S| = log 2

^{16}=16

Question 12

.

Correct answer is :C

Solution :

According to havel-hakimi theorem

(1,1,1,1,1,1) is graphic iff<1,1,1,1,0> is graphic

(0,1,1,1,1) is graphic iff (0,1,1,0) is graphic

(0,0,1,1) is graphic iff (0,0,0) is graphic

Since (0,0,0) is graphic (1,1,1,1,1,1) is also graphic.

(The process is always finding maximum degree and removing it from degree sequence, subtract 1 from each degree for d times from right to left where d is maximum degree)

(2,2,2,2,2,2) is graphic iff (2,2,22-1,2-1) = (2,2,2,1,1) is graphic

(1,1,2,2,2

Question 13

.

Correct answer is :B

Solution :

P = T q=F and r=T

Option A will become false

Option C will become false.

Option D is always false.

Question 14

.

Correct answer is :11.85 - 11.95

Solution :

p= P [at least three computers are working]

=P (3 or 4 computers working)

= (4C3) * (6C1) / 10C4 + 4C4 / 10C4 = 5/42

100p=11.9

Question 15

.

Correct answer is :3.8889

Question 16

.

Correct answer is :36

Solution :

The number of edges in a bipartite graph on n-vertices is atmost n

^{2}/4

The maximum number of edges in a bipartite graph on 12 –vertices is n

^{2}/4 = 12*12 /4 = 36

Question 17

Then the determinant of A is equal to ________.

.

Correct answer is :0

Solution :

|A| = 0

Question 18

.

Correct answer is :A

Solution :

Since, the roots of f(x) = 0 i.e., x = 1, 2, 3 lies between 0 and 4 and f(x) is of degree 3

f(0) and f(4) are of opposite signs

f(0).f(4)<0.

Question 19

.

Correct answer is :B

Solution :

n*(n-1) is an even number so we divide it by 2 and the rest by 3. The output will be same but overflow can be avoided.

Question 20

0.75x

^{3}- 2x

^{2}- 2x +4 = 0

Consider the statements

(I) x3 = 0 .

(II) The method converges to a solution in a finite number of iterations.

Which of the following is TRUE?

.

Correct answer is :A

Question 21

.

Correct answer is :6

Question 22

.

Correct answer is :0.259 - 0.261

Solution :

Let A = divisible by 2, B = divisible by 3 and C = divisible by 5, then

n(A) = 50, n(B) = 33, n(C) = 20

n(A ∩ B) = 16 (100/6)

n(A ∩ C) = 10 (100/10)

n(B ∩ C) = 6 (100/15)

n(A ∩ B ∩ C) = 3 (100/(2*3*5))

Now find n(A U B U C) = n(A) + n(B) + n(C) - n(A ∩ B) - n(A ∩ B) -n(B ∩ C) + n(A ∩ B ∩ C)

(100 - n(A U B U C))/100

Substituting the values we get answer as 0.26

Question 23

.

Correct answer is :8

Solution :

2014 = 2×19×53 i.e., product of prime factors

Number of distinct positive integral factors of 2014 is (2)×(2)×(2) = 8.

Question 24

Consider the following two statements:

S1: There is a subset of S that is larger than every other subset.

S2: There is a subset of S that is smaller than every other subset.

Which one of the following is CORRECT?

.

Correct answer is :A

Solution :

From given data S1 is true ,since null set is larger than every other set ,and S2 is true since the universal set {1,2,...,2014} is smaller than every other set.

Both s1 and s2 are true.

Question 25

.

Correct answer is :5

Solution :

Consider a cycle on five vertices C5

C5 and C5` are isomorphic

Question 26

.

Correct answer is :B

Question 27

.

Correct answer is :D

Question 28

.

Correct answer is :5

Solution :

Order of subgroup divides order of group (Lagrange’s theorem).

3, 5 and 15 can be the order of subgroup. As subgroup has atleast 4 elements and it is not equal to the given group, order of subgroup can’t be 3 and 15. Hence it is 5.

Question 29

.

Correct answer is :A

Solution :

If the trace of the matrix is positive and the determinant of the matrix is negative then atleast one of its eigen values is negative.

Since determinant = product of eigen values.

Question 30

.

Correct answer is :2

Solution :

Let the basis of 6-dimensional vector space be {e1, e2, e3,e4, e5, e6}. In order for V1 ∧ V2 to have smallest possible dimension V1 and V2 could be, say, {e1, e2, e3,e4} and {e3, e4, e5, e6} respectively. The basis of V1 ∧ V2 would then be {e3, e4}. => Smallest possible dimension = 2.

Question 31

.

Correct answer is :4

Question 32

C = 100;

for i = 1 to n do

for j = 1 to n do

{

Temp = A[ i ] [ j ] + C ;

A [ i ] [ j ] = A [ j ] [ i ] ;

A [ j ] [ i ] = Temp – C ;

}

for i = 1 to n do

for j = 1 to n do

output (A[ i ] [ j ]);

.

Correct answer is :A

Solution :

In the computation of given pseudo code for each row and column of Matrix A, each upper triangular element will be interchanged by its mirror image in the lower triangular and after that the same lower triangular element will be again re-interchanged by its mirror image in the upper triangular, resulting the final computed Matrix A same as input Matrix A.

Question 33

^{b}a x

^{2}dx, where a and b are given, which of the following statements is/are TRUE?

(I) The value of K obtained using the trapezoidal rule is always greater than or equal to the exact value of the definite integral.

(II) The value of K obtained using the Simpson’s rule is always equal to the exact value of the definite integral.

.

Correct answer is :C

Question 34

.

Correct answer is :A

Question 35

.

Correct answer is :0.25

Question 36

Consider the following statements.

P. For each such function it must be the case that for every i, f(i) = i,

Q. For each such function it must be the case that for some i,f(i) = i,

R. Each such function must be onto.

Which one of the following is CORRECT?

.

Correct answer is :B

Solution :

Let us consider a function (counter example) as

f(0)=1,f(1) =0,f(2)=3,f(3)=2,.........,f(2012)=2013,f(2013)=2012 andf(2014)=2014

Clearly f (f (i))= i for 0<= i <=2014

Here f(i) ≠ i for every i and f(i) = i for some i

Also f is onto

Hence , only Q and R are true

Question 37

x * x = y * y = x * y *x * y = y* x * y *x = e

where e is the identity element. The maximum number of elements in such a group is _________________.

.

Correct answer is :4

Question 38

.

Correct answer is :C

Solution :

Let n1 ,n2 ,.....nk be the number of vertices respectively in K connected components of a forest G, then n1 ?1,n2 ?1,.....,nk ?1 be the number of edges respectively in K connected components and n1 + n2 + ..... + nk = n (number of vertices in G)

Hence, number of edges in G = number of edges in K connected components

(n1-1)+(n2-1)+..........+(nk-1) = n-k

Question 39

.

Correct answer is :A

Solution :

We know that v + r = e + 2e = n + r ? 2 ...(1)

Where V= n(number of vertices); r =number of faces and e =number of edges

Given d>=3 then 3n <=2e

e>=3n/2

n+r-2>=3n/2 (using(1))

r>=3n/2 -n +2 => r>= n/2 +2

No.of faces is atleast n/2 + 2

Question 40

.

Correct answer is :D