### SET 1

Question 1

**In the following LU decomposition of the matrix , if the diagonal elements of U are both 1, then the lower diagonal entry 1**_{22} of L is

Answer Discuss it!

.

Correct answer is :5

Question 2

**lim**_{x→ ∞} x ^{1/x}

A : ∞

B : 0

C : 1

D : Not defined

Answer Discuss it!

.

Correct answer is :C

Question 3

**For a set A, the power set of A is denoted by 2**^{A}. If A = {5, {6}, {7}}, which of the following options are True?

1. ∅ ∈ 2^{A}

2. ∅ ⊆ 2^{A}

3. {5,(6)} ∈ 2 ^{A}

4. {5,(6)} ⊆ 2^{A}

A : 1 and 3 only

B : 2 and 3 only

C : 1,2 and 3 only

D : 1,2 and 4 only

Answer Discuss it!

.

Correct answer is :C

Question 4

**Which one of the following is Not equivalent to p ↔ q?**

A : (¬p ∨ q) ∧(p ∨ ¬q)

B : (¬p ∨ q) ∧ (q -> p)

C : (¬p ∧ q) ∨ (p ∧ ¬q)

D : (¬p ∧ ¬q) ∨ (p ∧ q)

Answer Discuss it!

.

Correct answer is :C

Question 5

A : h(x)/g(x)

B : -1/x

C : g(x)/h(x)

D : x/(1-x)^{2}

Answer Discuss it!

.

Correct answer is :A

Question 6

**Suppose L = {p, q, r, s, t} is a lattice represented by the following Hasse diagram.For any xy?L, not necessarily distinct, x ? y and x ? y are join and meet of x, y respectively. Let L**^{3}= {(x,y,z): x, y, z ? L} be the set of all ordered triplets of the elements of L. Let pr be the probability that an element (x,y,z) ∈ | L^{3} chosen equiprobably satisfies x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z). Then

A : pr = 0

B : pr = 1

C : 0 < pr <= 1/5

D : 1/5 < pr <1

Answer Discuss it!

.

Correct answer is :D

Question 7

**Consider the following 2 × 2 matrix A where two elements are unknown and are marked by a and b. The eigen values of this matrix are –1 and 7. What are the values of a and b?**

A : a = 6,b = 4

B : a = 4,b = 6

C : a = 3,b = 5

D : a = 5,b = 3

Answer Discuss it!

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Question 8

Answer Discuss it!

.

Correct answer is :-1

Question 9

**Let G be a connected planar graph with 10 vertices. If the number of edges on each face is three, then the number of edges in G is _______________.**

Answer Discuss it!

.

Correct answer is :24

Solution :

By euler's formula :

|V|+|R| = |E|+2 _________(1) where |V|, |E|, |R| are respectively number of vertices, edges and faces (regions)

Given |V| = 10 _______(2) and number of edges on each face is three

3|R| = 2|E| => |R| = (2/3) |E| __________ (3)

substituting 2 , 3 in 1 ,we get

10 + (2/3)|E| = |E| +2 => (|E| /3) = 8 => |E| =24

Question 10

Answer Discuss it!

.

Correct answer is :0.99

Question 11

**Let R be the relation on the set of positive integers such that a aRb if and only if a and b are distinct and have a common divisor other than 1. Which one of the following statements about R is true?**

A : R is symmetric and reflexive but not transitive

B : R is reflexive but not symmetric and not transitive

C : R is transitive but not reflexive and not symmetric

D : R is symmetric but not reflexive and not transitive

Answer Discuss it!

.

Correct answer is :D

Solution :

R is not reflexive as each element can?t be related to itself.

R is symmetric

Let a = 3, b = 6 and c = 10 then 3 and 6 have a common division other than 1

6 and 10 have a common division other than 1

but 3 &10 have no common division other than 1

3R6 and 6R10 but 3not related10

=> R is not transitive

Question 12

**Consider the following two statements.**

S1 : if a candidate is known to be corrupt, then he will not be elected

S2 : if a candidate is kind, he will be elected

Which one of the following statements follows from S1 and S2 per sound interference rules of logic?

A : If a person is known to corrupt, he is kind

B : If a person is not known to be corrupt, he is not kind

C : If a person is kind, he is not known to be corrupt

D : If a person is not kind, he is not known to be corrupt

Answer Discuss it!

.

Correct answer is :C

Solution :

Let P: candidate known to be corrupt

q: candidate will be elected

r: candidate is kind

then S_{1} = p -> ~q

=q -> ~p (contrapoitive rule)

and s2 : r -> q => r -> ~p (transitive rule )

i.e., If a person is kind, he is not known to be corrupt

Option is C

Question 13

**The larger of the two eigenvalues of the matrix is ____________.**

Answer Discuss it!

.

Question 14

**The cardinality of the power set of {0, 1, 2,.......10} is _________**

Answer Discuss it!

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Correct answer is :2048

Solution :

cardinality of the power set of {0, 1, 2, … , 10} is 11^{2} i.e.., 2048

Question 15

**The number of divisors of 2100 is _______.**

Answer Discuss it!

.

Correct answer is :36

Solution :

Let N =2100

2^{2} * 3*5^{2} * 7 (i.e.,product of primes)

Then the number of division of 2100 is

(2+1).(1+1)e. 36.(2+1).(1+1) i.e. (3)(2)(3)(2) i.e. 36

Question 16

**In a connected graph, a bridge is an edge whose removal disconnects a graph. Which one of the following statements is true?**

A : A tree has no bridges

B : A bridge cannot be part of a simple cycle

C : Every edge of a clique with size >= 3 is a bridge (A clique is any compete sub graph of a graph)

D : A graph with bridges cannot have a cycle

Answer Discuss it!

.

Correct answer is :B

Solution :

Since, every edge in a tree is bridge (A) is false

Since, every edge in a complete graph k _{n}(n>= 3) is not a bridge =>c is false

Since, in a cycle every edge is not a bridge => (B) is true

Question 17

**Let f(X) = x**^{-(1/3)} and A denote the area of the region bounded by f(x) and the X-axis, when x varies from – 1 to 1. Which of the following statements is/are TRUE?

(I) f is continuous in [-1,1]

(II) f is not bounded in [-1,1]

(III) A is nonzero and finite

A : II only

B : III only

C : II and III only

D : I, II and III

Answer Discuss it!

.

Correct answer is :C

Question 18

**The number of onto function (surjective function) from set X = {1,2,3,4}to set Y ={a,b,c} is ______.**

Answer Discuss it!

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Correct answer is :36

Question 19

**Consider two relations R1(A,B) with the tuples (1.5), (3,7) and R2 (A,C) = (1,7), (4,9). Assume that R(A,B,C) is the full natural outer join of R1 and R2 . Consider the following tuples of the form (A,B,C): a =(1.5,null), b=(1,null,7) c=(3,null,9), d=(4,7,null), e =(1,5,7), f=(3,7,null), g =(4,null,9).Which one of the following statements is correct?**

A : R contains a,b,e,f,g but not c, d.

B : R contains all of a,b,c,d,e,f,g

C : R contains e,f,g but not a,b

D : R contains e but not f,g.

Answer Discuss it!

.

Correct answer is :C

Question 20

**A young tableau is a 2D array of integers increasing from left to right and from top to bottom. Any unfilled entries are marked with ∞ , and hence there cannot be any entry to the right of , or below a ∞ . The following Young tableau consists of unique entries.**

1 2 5 14 3 4 6 23 10 12 18 25 31 ∞ ∞ ∞

When an element is removed from a Young tableau, other elements should be moved into its place so that the resulting table is still a Young tableau (unfilled entries may be filled in with a ∞ ). The minimum number of entries (other than 1) to be shifted, to remove 1 from the given Young tableau is _________

Answer Discuss it!

.

Correct answer is :5

Question 21

**Perform the following operations on the matrix **

(i) Add the third row to the second row

(ii) Subtract the third column from the first column.

The determinant of the resultant matrix is __________.

Answer Discuss it!

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Correct answer is :0

Solution :

Determinant is unaltered by the operations (i) and (ii)

=> Determinant of the resultant matrix = Determinant of the given matrix

Question 22

**Which one of the following well formed formulae is tautology?**

A : A only

B : B only

C : C only

D : D only

Answer Discuss it!

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Correct answer is :C

Question 23

**A graph is self-complementary if it is isomorphic to its complement For all selfcomplementary graphs on n vertices, n is**

A : A multiple of 4

B : Even

C : Odd

D : Congruent to 0 mod 4, or, 1 mod 4

Answer Discuss it!

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Correct answer is :D

Solution :

An n vertex self complementary graph has exactly half number of edges of the complete graph i.e., n(n-1)/4 edges . Since n(n-1) must be divisible by 4 , n must be congruent to 0 or 1 module 4.

Question 24

**The secant method is used to find the root of an equation f(x) = 0 . It is started from two distinct estimates, a b x and x for the root. It is an iterative procedure involving linear interpolation to a root. The iteration stops if f(xb) is very small and then xb is the solution. The procedure is given below. Observe that there is an expression which is missing and is marked by? Which is the suitable expression that is to be put in place of ? so that it follows all steps of the secant method?**

Secant

Initialize: xa, xb, ,N // ε = convergence indicator and N = maximum no. of iterations

fb= -f(xb)

i=0

while(i ( ε) do

i = i + 1 //update counter

xt = ? //missing expression for

xa=xb // intermediate value and reset xa

xb=xt //reset xb

fb = f(xb) // function value at new xb

end while

if |fb| > ε then //loop is terminated with i=N

write “Non-convergence”

else

A : x_{b} - (f_{b} - f(x_{a} ) f_{b} / (x_{b} - x_{a})

B : x_{a} - (f_{b} - f(x_{a} ) f_{a} / (x_{b} - x_{a})

C : x_{b} - (x_{b} - x_{a})f(x_{b} ) / f_{b} / (f_{b} -

D : x_{a} - (x_{b} - x_{a})f(x_{a} ) / f_{b} / (f_{b} -

Answer Discuss it!

.

Correct answer is :D

Solution :

Secant method direct formula

Question 25

**Let X and Y denote the sets containing 2and 20 distinct objects respectively and F denote the set of all possible functions defined from X to Y. let f be randomly chosen from F .The probability of f being one-to-one is _______.**

Answer Discuss it!

.

Correct answer is :0.95

Solution :

|X| =2 , |Y| =20

Number of functions from X to Y is 20^{2}i.e., 400 and number of one-one functions from X to Y is 20P2 i.e 20 * 19 =380

Probability of a function f being one-one is 380/400 = 0.95

Question 26

**In a room there are only two types of people, namely Type 1 and Type 2. Type 1 people always tell the truth and Type 2 people always lie. Your give a fair coin to a person in that room, without knowing which type he is from and tell him to loss it and hide the result from you till you ask for it. Upon asking, the person replies the following.**

“The result of the toss is head if and only if I am telling the truth.”

Which of the following options are correct?

A : The result is head

B : The result is tail

C : If the person is of Type 2, then the result is tail

D : If the person is of Type 1, then the result is tail

Answer Discuss it!

.

Correct answer is :C

Question 27

**Suppose U is the power set of the set S = {1,2,3,4,5,6}. For any T # U, let |T| denote the number of element in T and T' denote the complement of T. For any T,R # U, let T\R be the set of all elements in T which are not in R. Which one of the following is true?**

A : ∀ X ∈ U ( |X| = |X|' )

B : ∃ X isin;U ∃ Y∈U ( |X| =5, |Y|=5 and X ∩ Y = ∅

C : ∀X ∈ U ∀ Y∈ U ( |X| =2, |Y| =3 and 3 and X \ Y =∅

D : ∀X ∈ U ∀Y ∈ U(X \ Y = Y'\ X')

Answer Discuss it!

.

Correct answer is :D

Question 28

**The value of given equation is **

A : 0

B : 1/2

C : 1

D : ∞

Answer Discuss it!

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Correct answer is :A

Question 29

**The number of 4 digit numbers having their digits in non-decreasing order (from left to right) constructed by using the digits belonging to the set {1,2,3} is _____ .**

Answer Discuss it!

.

Correct answer is :15

Solution :

4-digit numbers with first digit ‘1’

1111, 1112, 1113, 1122, 1123, 1133, 1222, 1223, 1233, 1333 i.e., 10

4 digit numbers with first digit 2 : 2222, 2223, 2233, 2333 i.e, 4

4 digit numbers with first digit 3: 3333 i.e, 1

Question 30

**In the given matrix one of the eigen values is 1. The eigenvectors corresponding to the eigen value 1 are**

A : B :

C : D :

Answer Discuss it!

.

Correct answer is :B

Question 31

**Consider the equality Σ **_{i=0}^{n} = X and the following choices for X

I. θ (n^{4})

II. θ ( n^{5})

III.O(n^{5})

IV. Ω (n^{3})

The equality above remains correct if X is replaced by

A : only I

B : only II

C : I or III or IV but not II

D : II or III or IV but not I

Answer Discuss it!

.

Correct answer is :C

Solution :

X= sum of the cubes of first n natural numbers = (n^{2}(n+1)^{2})/4

Question 32

**In for non-zero x, af(X) + bf(1/x) = 1/x - 25 where a ≠ then ∫**_{1}^{2} f(x)dx is

A : (1/a^{2} - b^{2}) [a(ln 2-25) + 47b/2 ]

B : (1/a^{2} - b^{2}) [a(2ln 2-25) - 47b/2 ]

C : (1/a^{2} - b^{2}) [a(2ln 2-25) + 47b/2 ]

D : (1/a^{2} - b^{2}) [a(ln 2-25) + 47b/2 ]

Answer Discuss it!

.

Correct answer is :A

Question 33

**Suppose Xi for i=1,2,3 ar independent and identically distributed random variables whose probability mass functions are Pr [Xi = 0] = Pr [Xi =1] =1/ 2 for i =1,2,3. Define another random variable Y = X1 X2 ⊕ X , where ⊕ denotes XOR.Then Pr [Y = 0| X3 = 0] = _____ .**

Answer Discuss it!

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Correct answer is :0.75

Question 34

**The velocity v (in kilometer/minute) of a motorbike which start from rest, is given at fixed intervals of time t (in minutes as follows.**

t 2 4 6 8 10 12 14 16 18 20 10 v 18 25 29 32 20 11 5 2 0

The approximate distance (in kilometers) rounded to two places of decimals covered in 20 minutes using Simpson’s 1/3rd rule is ________.

Answer Discuss it!

.

Correct answer is :309.33

Solution :

Let ‘S’ be the distance covered in 20 minutes, then by simpson’s 1/3rd rule

= 2/3[(0+0)+4(10+25+32+11+2)+2(18+29+20+5)]

=309.33km (Here length of each of the subinterval is h = 2)

Question 35

**Let G be a connected undirected graph of 100 vertices and 300 edges. The weight of a minimum spanning tree of G is 500. When the weight of each edge of G is increased by five, the weight of a minimum spanning tree becomes __________.**

Answer Discuss it!

.

Correct answer is :995

Solution :

G has 100 verticles => spanning tree contain 99 edges given, weight of a minimum spanning tree of G is 500 since, each edge of G is increased by five

Weight of a minimum spanning tree becomes 500 + 5 × 99 = 995

Question 36

**If the following system has non-trivial solution. **

px + qy + rz =0

qx + ry + pz = 0

rx + py + qz =0

Then which one of the following options is TRUE?

A : p ? q + r = 0 or p = q = ?r

B : p + q ? r = 0 or p = ?q = r

C : p + q + r = 0 or p = q = r

D : p ? q + r = 0 or p = ?q = ?r

Answer Discuss it!

.

Correct answer is :A

Question 37

**Let R be a relation on the set of ordered pairs of positive integers such that ( ( p , q ) , ( r , s ) ) ∈ R if and only if p - s = q - r. Which one of the following is true about R?**

A : Both reflexive and symmetric

B : Reflexive but not symmetric

C : Not reflexive but symmetric

D : Neither reflexive nor symmetric

Answer Discuss it!

.

Correct answer is :C

Solution :

Since p- q≠ q-p

(p,q)R(p,q)

R is not reflective

Let (p,q) R (r,s) then p- s= q- r

r-q=s-p

(r,s) R (p,q)

R is symmetric

Question 1

_{22}of L is

.

Correct answer is :5

Question 2

_{x→ ∞}x

^{1/x}

.

Correct answer is :C

Question 3

^{A}. If A = {5, {6}, {7}}, which of the following options are True?

1. ∅ ∈ 2

^{A}

2. ∅ ⊆ 2

^{A}

3. {5,(6)} ∈ 2

^{A}

4. {5,(6)} ⊆ 2

^{A}

.

Correct answer is :C

Question 4

.

Correct answer is :C

Question 5

.

Correct answer is :A

Question 6

^{3}= {(x,y,z): x, y, z ? L} be the set of all ordered triplets of the elements of L. Let pr be the probability that an element (x,y,z) ∈ | L

^{3}chosen equiprobably satisfies x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z). Then

.

Correct answer is :D

Question 7

.

Question 8

.

Correct answer is :-1

Question 9

.

Correct answer is :24

Solution :

By euler's formula :

|V|+|R| = |E|+2 _________(1) where |V|, |E|, |R| are respectively number of vertices, edges and faces (regions)

Given |V| = 10 _______(2) and number of edges on each face is three

3|R| = 2|E| => |R| = (2/3) |E| __________ (3)

substituting 2 , 3 in 1 ,we get

10 + (2/3)|E| = |E| +2 => (|E| /3) = 8 => |E| =24

Question 10

.

Correct answer is :0.99

Question 11

.

Correct answer is :D

Solution :

R is not reflexive as each element can?t be related to itself.

R is symmetric

Let a = 3, b = 6 and c = 10 then 3 and 6 have a common division other than 1

6 and 10 have a common division other than 1

but 3 &10 have no common division other than 1

3R6 and 6R10 but 3not related10

=> R is not transitive

Question 12

S1 : if a candidate is known to be corrupt, then he will not be elected

S2 : if a candidate is kind, he will be elected

Which one of the following statements follows from S1 and S2 per sound interference rules of logic?

.

Correct answer is :C

Solution :

Let P: candidate known to be corrupt

q: candidate will be elected

r: candidate is kind

then S

_{1}= p -> ~q

=q -> ~p (contrapoitive rule)

and s2 : r -> q => r -> ~p (transitive rule )

i.e., If a person is kind, he is not known to be corrupt

Option is C

Question 13

.

Question 14

.

Correct answer is :2048

Solution :

cardinality of the power set of {0, 1, 2, … , 10} is 11

^{2}i.e.., 2048

Question 15

.

Correct answer is :36

Solution :

Let N =2100

2

^{2}* 3*5

^{2}* 7 (i.e.,product of primes)

Then the number of division of 2100 is

(2+1).(1+1)e. 36.(2+1).(1+1) i.e. (3)(2)(3)(2) i.e. 36

Question 16

.

Correct answer is :B

Solution :

Since, every edge in a tree is bridge (A) is false

Since, every edge in a complete graph k

_{n}(n>= 3) is not a bridge =>c is false

Since, in a cycle every edge is not a bridge => (B) is true

Question 17

^{-(1/3)}and A denote the area of the region bounded by f(x) and the X-axis, when x varies from – 1 to 1. Which of the following statements is/are TRUE?

(I) f is continuous in [-1,1]

(II) f is not bounded in [-1,1]

(III) A is nonzero and finite

.

Correct answer is :C

Question 18

.

Correct answer is :36

Question 19

.

Correct answer is :C

Question 20

1 | 2 | 5 | 14 |

3 | 4 | 6 | 23 |

10 | 12 | 18 | 25 |

31 | ∞ | ∞ | ∞ |

When an element is removed from a Young tableau, other elements should be moved into its place so that the resulting table is still a Young tableau (unfilled entries may be filled in with a ∞ ). The minimum number of entries (other than 1) to be shifted, to remove 1 from the given Young tableau is _________

.

Correct answer is :5

Question 21

(i) Add the third row to the second row

(ii) Subtract the third column from the first column.

The determinant of the resultant matrix is __________.

.

Correct answer is :0

Solution :

Determinant is unaltered by the operations (i) and (ii)

=> Determinant of the resultant matrix = Determinant of the given matrix

Question 22

.

Correct answer is :C

Question 23

.

Correct answer is :D

Solution :

An n vertex self complementary graph has exactly half number of edges of the complete graph i.e., n(n-1)/4 edges . Since n(n-1) must be divisible by 4 , n must be congruent to 0 or 1 module 4.

Question 24

Secant

Initialize: xa, xb, ,N // ε = convergence indicator and N = maximum no. of iterations

fb= -f(xb)

i=0

while(i

i = i + 1 //update counter

xt = ? //missing expression for

xa=xb // intermediate value and reset xa

xb=xt //reset xb

fb = f(xb) // function value at new xb

end while

if |fb| > ε then //loop is terminated with i=N

write “Non-convergence”

else

A : x

_{b}- (f

_{b}- f(x

_{a}) f

_{b}/ (x

_{b}- x

_{a})

B : x

_{a}- (f

_{b}- f(x

_{a}) f

_{a}/ (x

_{b}- x

_{a})

C : x

_{b}- (x

_{b}- x

_{a})f(x

_{b}) / f

_{b}/ (f

_{b}-

D : x

_{a}- (x

_{b}- x

_{a})f(x

_{a}) / f

_{b}/ (f

_{b}-

.

Correct answer is :D

Solution :

Secant method direct formula

Question 25

**Let X and Y denote the sets containing 2and 20 distinct objects respectively and F denote the set of all possible functions defined from X to Y. let f be randomly chosen from F .The probability of f being one-to-one is _______.**

.

Correct answer is :0.95

Solution :

|X| =2 , |Y| =20

Number of functions from X to Y is 20

^{2}i.e., 400 and number of one-one functions from X to Y is 20P2 i.e 20 * 19 =380

Probability of a function f being one-one is 380/400 = 0.95

Question 26

**In a room there are only two types of people, namely Type 1 and Type 2. Type 1 people always tell the truth and Type 2 people always lie. Your give a fair coin to a person in that room, without knowing which type he is from and tell him to loss it and hide the result from you till you ask for it. Upon asking, the person replies the following.**

“The result of the toss is head if and only if I am telling the truth.”

Which of the following options are correct?

“The result of the toss is head if and only if I am telling the truth.”

Which of the following options are correct?

A : The result is head

B : The result is tail

C : If the person is of Type 2, then the result is tail

D : If the person is of Type 1, then the result is tail

.

Correct answer is :C

Question 27

**Suppose U is the power set of the set S = {1,2,3,4,5,6}. For any T # U, let |T| denote the number of element in T and T' denote the complement of T. For any T,R # U, let T\R be the set of all elements in T which are not in R. Which one of the following is true?**

A : ∀ X ∈ U ( |X| = |X|' )

B : ∃ X isin;U ∃ Y∈U ( |X| =5, |Y|=5 and X ∩ Y = ∅

C : ∀X ∈ U ∀ Y∈ U ( |X| =2, |Y| =3 and 3 and X \ Y =∅

D : ∀X ∈ U ∀Y ∈ U(X \ Y = Y'\ X')

.

Correct answer is :D

Question 28

**The value of given equation is**

A : 0

B : 1/2

C : 1

D : ∞

.

Correct answer is :A

Question 29

**The number of 4 digit numbers having their digits in non-decreasing order (from left to right) constructed by using the digits belonging to the set {1,2,3} is _____ .**

.

Correct answer is :15

Solution :

4-digit numbers with first digit ‘1’

1111, 1112, 1113, 1122, 1123, 1133, 1222, 1223, 1233, 1333 i.e., 10

4 digit numbers with first digit 2 : 2222, 2223, 2233, 2333 i.e, 4

4 digit numbers with first digit 3: 3333 i.e, 1

Question 30

**In the given matrix one of the eigen values is 1. The eigenvectors corresponding to the eigen value 1 are**

A : B :

C : D :

.

Correct answer is :B

Question 31

**Consider the equality Σ**

I. θ (n

II. θ ( n

III.O(n

IV. Ω (n

The equality above remains correct if X is replaced by

_{i=0}^{n}= X and the following choices for XI. θ (n

^{4})II. θ ( n

^{5})III.O(n

^{5})IV. Ω (n

^{3})The equality above remains correct if X is replaced by

A : only I

B : only II

C : I or III or IV but not II

D : II or III or IV but not I

.

Correct answer is :C

Solution :

X= sum of the cubes of first n natural numbers = (n

^{2}(n+1)

^{2})/4

Question 32

**In for non-zero x, af(X) + bf(1/x) = 1/x - 25 where a ≠ then ∫**

_{1}^{2}f(x)dx isA : (1/a

^{2}- b

^{2}) [a(ln 2-25) + 47b/2 ]

B : (1/a

^{2}- b

^{2}) [a(2ln 2-25) - 47b/2 ]

C : (1/a

^{2}- b

^{2}) [a(2ln 2-25) + 47b/2 ]

D : (1/a

^{2}- b

^{2}) [a(ln 2-25) + 47b/2 ]

.

Correct answer is :A

Question 33

**Suppose Xi for i=1,2,3 ar independent and identically distributed random variables whose probability mass functions are Pr [Xi = 0] = Pr [Xi =1] =1/ 2 for i =1,2,3. Define another random variable Y = X1 X2 ⊕ X , where ⊕ denotes XOR.Then Pr [Y = 0| X3 = 0] = _____ .**

.

Correct answer is :0.75

Question 34

**The velocity v (in kilometer/minute) of a motorbike which start from rest, is given at fixed intervals of time t (in minutes as follows.**

The approximate distance (in kilometers) rounded to two places of decimals covered in 20 minutes using Simpson’s 1/3rd rule is ________.

t | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 |

10 | v | 18 | 25 | 29 | 32 | 20 | 11 | 5 | 2 | 0 |

The approximate distance (in kilometers) rounded to two places of decimals covered in 20 minutes using Simpson’s 1/3rd rule is ________.

.

Correct answer is :309.33

Solution :

Let ‘S’ be the distance covered in 20 minutes, then by simpson’s 1/3rd rule

= 2/3[(0+0)+4(10+25+32+11+2)+2(18+29+20+5)]

=309.33km (Here length of each of the subinterval is h = 2)

Question 35

**Let G be a connected undirected graph of 100 vertices and 300 edges. The weight of a minimum spanning tree of G is 500. When the weight of each edge of G is increased by five, the weight of a minimum spanning tree becomes __________.**

.

Correct answer is :995

Solution :

G has 100 verticles => spanning tree contain 99 edges given, weight of a minimum spanning tree of G is 500 since, each edge of G is increased by five

Weight of a minimum spanning tree becomes 500 + 5 × 99 = 995

Question 36

**If the following system has non-trivial solution.**

px + qy + rz =0

qx + ry + pz = 0

rx + py + qz =0

Then which one of the following options is TRUE?

px + qy + rz =0

qx + ry + pz = 0

rx + py + qz =0

Then which one of the following options is TRUE?

A : p ? q + r = 0 or p = q = ?r

B : p + q ? r = 0 or p = ?q = r

C : p + q + r = 0 or p = q = r

D : p ? q + r = 0 or p = ?q = ?r

.

Correct answer is :A

Question 37

**Let R be a relation on the set of ordered pairs of positive integers such that ( ( p , q ) , ( r , s ) ) ∈ R if and only if p - s = q - r. Which one of the following is true about R?**

A : Both reflexive and symmetric

B : Reflexive but not symmetric

C : Not reflexive but symmetric

D : Neither reflexive nor symmetric

.

Correct answer is :C

Solution :

Since p- q≠ q-p

(p,q)R(p,q)

R is not reflective

Let (p,q) R (r,s) then p- s= q- r

r-q=s-p

(r,s) R (p,q)

R is symmetric