GATE - CS - Engineering Mathematics

GATE CS Topic wise QuestionsEngineering Mathematics

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YEAR 2003

Question. 1

Let ( )P E denote the probability of the event E .

Given ( ) , ( ) / ,P A P B1 1 2= = the values of ( / )P A B and ( / )P B A respectively are

(A) 1/4, 1/2 (B) 1/2, 1/4

(C) 1/2, 1 (D) 1, 1/2

Question. 2

Consider the set */ of all strings over the alphabet /={0,1}. */ with the concatenation operator for strings

(A) does not form a group

(B) forms a non-commutative group

(C) does not have a right identity element

(D) forms a group if the empty string is removed from */

Question. 3

Let A be a sequence of 8 distinct integers sorted in ascending order.

How many distinct pairs of sequences. B and C are there such that (i) each is sorted in ascending order,

(ii) B has 5 and C 3 element, and

(iii) the result of merging B and C gives A?

CS Topicwise 2003-2010Engineering Mathematics

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(A) 2 (B) 30

(C) 56 (D) 256

Question. 4

n couples are invited to a party with the condition that every husband should be accompanied by his wife. However, a wife need not be accompanied by her husband. The number of different gatherings possible at the party is

(A) * 2nn2 n

b l (B) 3n

(C) ( )n

22 1

n (D) nn2

b l

Question. 5

Let G be an arbitrary graph with n nodes and k components. If a vertex is removed from G , the number of components in the resultant graph must necessarily lie between.

(A) k and n (B) k 1− and k 1+

(C) k 1− and n 1− (D) k 1+ and n k−

Question. 6

Let ( , )S # be a partial order with two minimal elements a and b , and a maximum elementc . Let :P S "{True, False} be a predicate defined on S . Suppose that ( )P a = True, ( )P b =False and ( ) ( )P x p y& for all ,x y S! satisfying ,x y# where & stands for logical implication.

Which of the following statements CANNOT be true?

(A) ( )P x =True for all x S! such that x ! such that x b!

(B) ( )P x =False for all x S! such that x a! and x c!

(C) ( )P x =False for all x S! such that b x# such that x c!

(D) ( )P x =False for all x S! such that a x# such that b x#

Question. 7

Which of the following is a valid first order formula?

(Here α and β are first order formulae with x as their only free

variable)


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(A) ( )[ ] ( )[ ] ( )[ ]x x x& & &6 6 6α β α β

(B) ( )[ ] [ ][ ]x x& /6 7α α β

(C) ( )[ ] (( )[ ] ( )[ ]x x x& &06 7 6α β α α

(D) ( )[ ] (( )[ ] ( )[ ])x x x& & &6 6 6α β α β

Question. 8

Consider the following formula α and its two interpretations

I1 and I2 α:( )[ ( )[ ]] ( )[ ]x P y Q Q x Px xy yy x+ + &6 6 J 6 J

I :1 Domain : the set of natural numbers

' 'P xx / is a prime number’

'Q yxy / divides 'x

I2; same as I1 except that 'P xx = is a composite number.’

Which of the following statements is true?

(A) I1 satisfies ,I2α does not

(B) I2 satisfies α,I1 does not

(C) Neither I2 nor I2 satisfies α

(D) Both I1 and I2 satisfy α

Question. 9

Consider the following logic program P

( ) ( , ), ( )

( , )

A x B x y C y

B x x

!

!

Which of the following first order sentences is equivalent to P?

(A) ( )[ ][ ( , ) ( )] ( )] ( )[ ( )]x y B x y C y A x x B xx&/ /6 7 J 7

(B) ( )[ ][ ( , ) ( )] ( )] ( )[ ( )]x y B x y C y A x x B xx&/ /6 6 J 7

(C) ( )[ ][ ( , ) ( )] ( )] ( )[ ( )]x y B x y C y A x x B xx&/ 06 7 J 7

(D) ( )[ ][ ( , ) ( )] ( )] ( )[ ( )]x y B x y C y A x x B xx&/ /6 6 J 7

Question. 10

The following resolution rule is used in logic programming:

Derive clause( )P Q0 from clauses ( ),( )P R Q R0 0 J

Which of the following statements related to this rule is FASLE?

(A) ( ) ( ) ( )P R Q R P Q&0 / 0 0J is logically valid


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(B) ( ) ( ) ( )P Q P R Q R&0 0 / 0 J is logically valid

(C) ( )P Q0 is satisfiable if and only if ( ) ( )P R Q R0 / 0 J is satisfiable

(D) ( )P Q &0 FALSE if and only if both P and Q are unclassifiable

Question. 11

A program consists of two modules executed sequentially. Let ( )f t1 and ( )f t2 respectively denote the probability density functions of time taken to execute the two modules. The probability density function of the overall time taken to execute the program is given by

(A) ( ) ( )f t f t1 2+ (B) ( ) ( )f x f x dxt

1 20

#

(C) ( ) ( )f x f t x dxt

1 20

−# (D) max { ( ), ( )}f t f t1 2

Question. 12

Let : A B" be injective (one-to-one) function. Define :g 2 2B"

/ as:

( ) { ( ) ),g C f x x C!= for all subsets C of A.

Define :h 2 2B A" as : ( ) { | , ( ) },h D x x A f x D! != for all subsets D of

B .

Which of the following statements is always true?

(A) ( )( ))g h D D3 (B) ( )( ))g h D D4

(C) ( )( ))g h D D+ φ= (D) ( )( )) ( )g h D B D+ = φ−

Question. 13

Consider the set { , , }a b c with binary operators + and # defined as follows:

+ a b c # a b c

a b a c a a b c

b a b c b b c a

c a c b c c c b

For example, , ,a c c c a a c b c#+ = + = = and b c a# = . Given the following set of equations;

( ) ( )

( ) ( )

a x a y c

b x c y c

# #

# #

+ =

+ =


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The number of solution ( )( . .,s i e pair ( )( , )s x y that satisfies the equations) is

(A) 0 (B) 1

(C) 2 (D) 3

Question. 14

Let { ,, , , , }a b c d e/ = be an alphabet, We define an encoding scheme as follows ( ) , ( ) , ( ) , ( ) , ( ) .g a g b g c g d g e3 5 7 9 11= = = = =

Let pi denote the i th− prime number( 1 2)p = .

For a non-empty string ..... ,s a an1= where each ,ai ! / define ( )f s p ( )n

i ig a

1iΠ= = . For a non-empty sequence ....s s< >n1 of strings

from */ , define ( ..... )h s s p< > ( ).n

ni i

f s1 1

iΠ= = .

Which of the following numbers is the encoding, h of a non-empty sequence of strigs?

(A) 2 3 57 7 7 (B) 2 3 58 8 8

(C) 2 3 59 9 9 (D) 2 3 510 10 10

Question. 15

m identical balls are to be placed in n distinct bags. You are given that ,m kn$ where k is a natural number 1$ . In how many ways can the balls be placed in the bags if each bag must contain at least k balls?

(A) 1m kn

−−f p (B)

11

m kn nn

− + −−f p

(C) 1m

n k−−f p (D)

2m kn n kn k

− + + −−f p

Question. 16

Consider the following recurrence relation

( ) ( )T n T n n1 1+ = + +6 @ for all n 1$

The value of ( )T m2 for m 1$ is

(A) ( )m m6 21 39 4− +

(B) ( 3 )m m m6 4 52+−


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(C) ( )m m m2 3 11 20 525+− −

(D) ( )m m m m6 5 34 137 104 653 2− + − +

Question. 17

How many perfect matching are there in a complete graph of 6 vertices?

(A) 15 (B) 24

(C) 30 (D) 60

Question. 18

A Graph ( , )G V E= satisfies | | | |E V3 6# − . The min-degree of G is defined as min

v V! {degree (v)}. Therefore, min-degree of G cannot be

(A) 3 (B) 4

(C) 5 (D) 6

Question. 19

Consider the following system of linear equations

2 1 4

4 3 12 5

x

y

z1 2 8 7

α−−−

=

R

T

SSSS

R

T

SSSS

R

T

SSSS

V

X

WWWW

V

X

WWWW

V

X

WWWW

Notice that the second and the third columns of the coefficient matrix are linearly dependent. For how many values of α. does this system

of equations have infinitely many solutions?

(A) 0 (B) 1

(C) 2 (D) infinitely many

Question. 20

A piecewise linear function ( )f x is plotted using thick solid lines in the figure below (the plot is drawn to scale).


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If we use the Newton-Raphson method to find the roots of ( )f x 0= using ,x x0 1 and x2 respectively as initial guesses, the roots obtained would be

(A) 1.3, 0.6 and 0.6 respectively

(B) 0.6, 0.6 and 1.3 respectively

(C) 1.3, 1.3 and 0.6 respectively

(D) 1.3, 0.6 and 1.3 respectively

YEAR 2004

Question. 21

Identify the correct translation into logical notation of the following assertion. Some boys in the class are taller than all the girls

Note: taller( , )x y is true if x is taller than y .

(A) ( )x7 (boy( ) ( )x y" 6 (girl( )y / taller ( , )))x y

(B) ( )x7 (boy( ) ( )x y/ 6 (girl( )y / taller ( , )))x y

(C) ( )x7 (boy( ) ( )x y" 6 (girl( )y " taller ( , )))x y

(D) ( )x7 (boy( ) ( )x y/ 6 (girl( )y " taller ( , )))x y

Question. 22

If a fair coin is tossed four times. What is the probability that two heads and two tails will result?

(A) 3/8 (B) 1/2

(C) 5/8 (D) 3/4


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Question. 23

Consider the binary relation:

{( , )|S x y y x 1= = + and , { , , ,....}}x y 0 1 2!

The reflexive transitive closure of S is

(A) {( , )|x y y x> and , { , , ,....}}x y 0 1 2!

(B) {( , )|x y y x$ and , { , , ,....}}x y 0 1 2!

(C) {( , )|x y y x< and , { , , ,....}}x y 0 1 2!

(D) {( , )|x y y x# and , { , , ,....}}x y 0 1 2!

Question. 24

The number of different n n# symmetric matrices with each element being either 0 or 1 (Note : power ( , )x2 is same as 2x )

(A) power( , )n2 (B) power( , )n2 2

(C) power( ,( )/ )n n2 22 + (D) power( ,( )/ )n n2 22 −

Question. 25

Let , , ,A B C D be n n# matrices, each with non-zero determinant, If ABCD I= , then B 1− is

(A) ,D C A1 1 1− − − (B) CDA

(C) ADC (D) does not necessarily exist

Question. 26

The following propositional statement is

( ( )) (( ) )P Q R P Q R" " "0 /

(A) satisfiable but not valid (B) valid

(C) a contradictions (D) none of the above

Question. 27

An examination paper has 150 multiple-choice questions of one mark each, with each question having four choices. Each incorrect answer fetches .0 25− mark. Suppose 1000 students choose all their answers randomly with uniform probability. The sum total of the expected marks obtained by all these students is

(A) 0 (B) 2550

(C) 7525 (D) 9375


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Question. 28

Two n bit binary strings,S1 and S2 are chosen randomly with uniform probability. The probability that the hamming distance between these strings (the number of bit positions where the two strings differ) is equal to d is

(A) 2Cn

d

n/ (B) 2Cn

d

d/

(C) /2d n (D) /1 2d

Question. 29

A point is randomly selected with uniform probability in the x Y− . plane within the rectangle with corners at (0,0), (1,0),(1,2) and (0, 2). If p is the length of the position vector of the point, the expected value of p2 is

(A) 2/3 (B) 1

(C) 4/3 (D) 5/3

Question. 30

The following is the incomplete operation table of a 4-element group.

* e a b c

e e a b c

a a b c e

b

c

The last row of the table is

(A) c a e b (B) c b a e

(C) c b e a (D) c e a b

Question. 31

The inclusion of which of the following sets into

S ={{1, 2},{1,2,3},{1,3,5},{1,2,4},{1,2,3,4,5}} is necessary and sufficient to make S a complete lattice under the partial order defined by set containment?

(A) {1} (B) {1},{2,3}

(C) {1},{1,3} (D) {1},{1,3},{1,2,3,4},{1,2,3,5}


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Question. 32

Mala has a colouring book in which each English letter is drawn two times. She wants to paint each of these 52 prints with one of k colours, such that the colour-pairs used to colour any two letters are different. Both prints of a letter can also be coloured with the same colour. What is the minimum value of k that satisfies this requirement?

(A) 9 (B) 8

(C) 7 (D) 6

Question. 33

In an M N# matrix such that all non-zero entries are covered in a rows and b columns. Then the maximum number of non-zero entries, such that no two are on the same row or column, is

(A) a b# + (B) # max ( , )a b

(C) [ , ]min M a N b# − − (D) { , }min a b#

Question. 34

The minimum number of colour required to colour the following graph, such that no two adjacent vertices are assigned the same colour, is

(A) 2 (B) 3

(C) 4 (D) 5

Question. 35

How many graphs on n labeled vertices exist which have at least

( )/n n3 22 − edges?

(A) C( ^ )/ ( ^ )/n n n n2 2 2 3 2− − (B) ( ^ )( ^ )/

n n C

k

n n2

0

2 3 2k−

=

−

/

(C) C( ^ )/n n n2 2− (D) C( ^ )

k

nn n

k0

22

=

−

/


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Question. 36

Let G1 ( , )V E1= and ( , )G V E2 2= be connected graphs on the same vertex set V with more than two vertices. if ( , )G G V E E1 2 1 2+ += is not a connected graph, then the graph ( , )G G V E E1 2 1 2, ,=(A) cannot have cut vertex

(B) must have a cycle

(C) must have a cut-edge (bridge)

(D) has chromatic number strictly greater than those of G1 and G2

Question. 37

How many solutions does the following system of linear equations have?

5x y− + 1=− x y− 2= 3x y+ 3=(A) infinitely many (B) two distinct solutions

(C) unique (D) none

YEAR 2005

Question. 38

Let ( )f x be the continuous probability density function of a random variable X . The probability that ,a X b< # is

(A) ( )f b a− (B) ( ) ( )f b f a−

(C) ( )f x dxa

b

# (D) ( )xf x dxa

b

#

Question. 39

Let ,A B and C be non-empty sets and let ( )X A B C= − − and ( ) ( )Y A C B C= − − −

Which one of the following is TRUE?

(A) X Y= (B) X Y1

(C) Y X1 (D) none of these

Question. 40

The following is the Hasse diagram of the poset [{ , , , , }, ]a b c d e #


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The poset is

(A) not a lattice

(B) a lattice but not a distributive lattice

(C) a distributive lattice but not a Boolean algebra

(D) a Boolean algebra

Question. 41

The set {1,2,4,7,8,11,13,14} is a group under multiplication modulo 15. The inverses of 4 and 7 are respectively

(A) 3 and 13 (B) 2 and 11

(C) 4 and 13 (D) 8 and 14

Question. 42

Let G be a simple connected planar graph with 13 vertices and 19 edges. Then, the number of faces in the planar embedding of the graph is

(A) 0 (B) 8

(C) 9 (D) 13

Question. 43

Let G be a simple graph will 20 vertices and 100 edges. The size of the minimum vertex cover of G is 8. Then, the size of the maximum independent set of G is

(A) 12 (B) 8

(C) less than 8 (D) more than 12

Question. 44

Let ,P Q and R be three atomic prepositional assertions. Let X denote ( )P Q R"0 and Y denote ( ) ( ).P R Q R" "0 Which one of


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the following is a tautology?

(A) X Y/ (B) X Y"

(C) Y X" (D) Y X"J

Question. 45

What is the first order predicate calculus statement equivalent to the following? Every teacher is liked by some student

(A) ( )x6 {teacher ( ) ( )x y" 7 [student ( )y " likes ( , )]]y x

(B) ( )x6 {teacher ( ) ( )x y" 7 [student ( )y / likes ( , )]]y x

(C) ( )y7 ( )x6 {teacher ( )x "[student ( )y / likes ( , )]]y x

(D) ( )x6 [teacher ( ) ( )x y/ 7 [student ( )y " likes ( , )]]x y

Question. 46

Let R and S be any two equivalence relations on a non-empty set A. Which one of the following statements is TRUE?

(A) ,R S R S+ , are both equivalence relations

(B) R S, is an equivalence relation

(C) R S+ is an equivalence relations

(D) Neither R S, nor R S+ is an equivalence relation

Question. 47

Let :f B C" and :g A B" be two function and let .h f o g= Given that h is an onto function. Which one of the following is TRUE?

(A) f and g should both be onto functions

(B) f should be but g need not be onto

(C) g should be onto but f not be onto

(D) both f and g need not be onto

Question. 48

What is the minimum number of ordered pairs of non-negative numbers that should be chosen to ensure that there are two pairs ( , )a b and ( , )c d in the chosen set such that a c/ mod 3 and b d/ mod 5

(A) 4 (B) 6

(C) 16 (D) 24


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Question. 49

Let ( ) /( ) ( )G x x g i x1 1 i

i

2

0= − =

3

=/ where | | |x < . What is ( )g i ?

(A) i (B) i 1+

(C) i2 (D) 2i

Question. 50

Which one of the following graphs if NOT plannar?

Data for Q. 51 & 52 are given below.

Solve the problems and choose the correct answers.

Let s and t be two vertices in a undirected graph ( , )G V E= having distinct positive edge weights. Let [ , ]X Y be a partition of V such that s X! and t Y! . Consider the edge having the minimum weight amongst all those edges that have one vertex in X and one vertex in Y .

Question. 51

The edge e must definitely belong to:

(A) the minimum weighted spanning tree of G

(B) the weighted shortest path from s to t

(C) each path from s to t


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(D) the weighted longest path from s to t

Question. 52

Let the weight of an edge e denote the congestion on that edge. The congestion on a path is defined to be the maximum of the congestions on the edges of the path. We wish to find the path from s to t having minimum congestion. Which one of the following paths is always such a path of minimum congestion?

(A) a path from s to t in the minimum weighted spanning tree

(B) a weighted shortest path from s to t

(C) an Euler walk from s to t

(D) a Hamiltonian path from s to t

Question. 53

Consider the set H of all 3 3# matrices of the type

0

a f e

b d

c0 0

R

T

SSSS

V

X

WWWW

Where , , , ,a b c d e and f are real numbers and .abc 0! Under the matrix multiplication operation, the set H is

(A) a group

(B) a monoid but not group

(C) a semigroup but not a monoid

(D) neither a group nor a semigroup

Question. 54

Consider the following system of equations in three real variables

,x x1 2 and x3

2 3x x x1 2 3= + 1= 3 2 5x x x1 2 3− + 2= 4x x x1 2 3− − + 3=

This system of equation has

(A) no solution

(B) a unique solution

(C) more than one but a finite number of solutions


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(D) an infinite number of solutions

Question. 55

What are the eigenvalues of the following 2 2# matrix?

2 1

4 5

−−= G

(A) 1− and 1 (B) 1 and 6

(C) 2 and 5 (D) 4 and 1−

YEAR 2006

Question. 56

Let , , ,X Y Z be sets of sizes ,x y and z respectively. Let W X Y#= and E be the set of all subsets of W . The number of functions from Z to E is

(A) z (B) z 2xy#

(C) z (D) 2xyz

Question. 57

The set {1,2,3,5,7,8,9} under multiplication modulo 10 is not a group. Given below are four plausible reasons. Which one of them is false?

(A) It is not closed

(B) 2 does not have an inverse

(C) 3 does not have an inverse

(D) 8 does not have an inverse

Question. 58

A relation R is defined on ordered pairs of integers as follows

( , ) ( , )x y R u v if x u< and y v> . Then R is

(A) Neither a Partial Order nor an Equivalence Relation

(B) A Partial Order but not a Total Order

(C) A Total Order

(D) An Equivalence Relation


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Question. 59

Which one of the first order predicate calculus statements given below correctly expresses the following English statement?

(A) [(x6 tiger ( )x / lion ( )) {(x " hungry( )x 0 threatened ( ))x " attacks ( )}]x

(B) [(x6 tiger ( )x 0 lion ( )) {(x " hungry( )x 0 threatened ( ))x / attacks ( )}]x

(C) [(x6 tiger ( )x 0 lion ( )) {(x " attacks( )x " (hungry( )x / Threatened ( )}]x

(D) [(x6 tiger ( )x 0 lion ( )) {(x " hungry( )x / Threatened ( ))x "

attacks( )}]x

Question. 60

Consider the following propositional statements:

1:(( ) )) (( ) ( ))

;(( ) )) (( ) ( ))

P A B C A C B C

P A B C A C B C2

" " "

" " "

/ /

0 0

/

/

Which one of the following is true?

(A) P1 is a tautology, but not P2

(B) P2 is a tautology, but not P1

(C) P1 and P2 are both tautologies

(D) Both P1 and P2 are not tautologies

Question. 61

For each element in set of size n2 , an unbiased coin is tossed. The n2 coin tossed are independent. An element is chosen if the corresponding coin toss were head. The probability that exactly n elements are chosen is

(A) n

n

24nd n (B)

2n

n2nd n

(C) 2n

n1 d n (D) 12

Question. 62

Let ,E F and G be finite sets.

Let ( ) ( )X E F F G+ += − and ( ( )) ( ).Y E E G E F+= − − − Which


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one of the following is true?

(A) X Y1 (B) X Y2

(C) X Y= (D) X Y Q!− and Y X Q!−

Question. 63

Let S={1,2,3............,m} .m 3> Let ,......,X Xn1 be subsets of S each of size 3 Define a function f from S to the set of natural numbers as, ( )f i is the number of sets Xj that contain the element i .

That is ( ) |{ | }|.f i j i Xj!=

Then ( )f ii

m

1=/ is

(A) m3 (B) n3

(C) sm 1+ (D) n2 1+

Question. 64

40. A logical binary relation 9, is defined as follows

A B A 9 B

True True True

True False True

False True False

False False True

Let~be the unary negation (NOT) operator, with higher precedence, than 9. Which one of the following is equivalent to A B/ ?

(A) (~ )A B9 (B) ~( ~ )A B9

(C) ~(~ ~ )A B9 (D) ~(~ )A B9

Question. 65

Given a set of elements { , ,...... }N n1 2= and two arbitrary subsets A N3 andB N3 , how many of the !n permutations π from N to N

satisfy min [ ( )]Aπ =min { ( )]Bπ , where min ( )S is the smallest integer

in the set of integers S . and ( )Sπ is the set of integers obtained by

applying permutation π to each element of S ?


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(A) ( | |)| || |n A B A B,−

(B) (| | | | )A B n2 2 2+

(C) !| || |

nA BA B,+

(D)

| |

| |n

A B

A B 2

,

+

f p

Data for Q. 66, 67 & 68 are given below.


The 2n vertices of graph G correspond to all subsets of a set of size n , for 6$ . Two vertices of G are adjacent if and only if the corresponding sets intersect in exactly two elements.

Question. 66

The number of vertices of degree zero in G is

(A) 1 (B) n

(C) n 1+ (D) 2n

Question. 67

The maximum degree of a vertex in G is

(A) /2n

22 /n 2d n (B) 2n 2−

(C) 2 3n 3 #− (D) 2n 1−

Question. 68

The number of connected components in G is

(A) n (B) n 2+

(C) 2 /n 2 (D) n2n

Question. 69

F is an n n# real matrix. b is an n 1# real vector. Suppose there are


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two n 1# vectors, u and v such that u v! , and , .Fu b Fv b= =

Which one of the following statements is false?

(A) Determinant of F is zero

(B) There are an iffinite number of solutions to Fx b=

(C) There is an x 0! such that Fx 0=

(D) F must have two identical rows

YEAR 2007

Question. 70

Let S be a set of n elements. The number of ordered pairs in the largest and the smallest equivalence relations on S are

(A) n and n (B) n2 and n

(C) n2 and 0 (D) n and 1

Question. 71

Let G be the non-planar graph with the minimum possible number of edges. Then G has.

(A) 9 edges and 5 vertices (B) 9 edges and 6 vertices

(C) 10 edges and 5 vertices (D) 10 edges and 6 vertices

Question. 72

Consider the DAG with { , , , , . }V 1 2 3 4 5 6= , shown below

Which of the following is NOT a topological ordering?

(A) 1 2 3 4 5 6 (B) 1 3 2 4 5 6

(C) 1 3 2 4 5 6 (D) 3 2 4 1 6 5


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Question. 73

Consider the following two statements about the function ( | |f x x=^ h

: ( )P f x is continuous for all real values of x

: ( )Q f x is differentiable for all real values of x

Which of the following is TRUE?

(A) P is true and Q is false (B) P is false and Q is true

(C) Both P and Q are true (D) Both P and Q are false

Question. 74

Let Graph ( )x be a predicate which denotes that x is a graph. Let connected ( )x be a predicate which denotes that x is connected. Which of the following first order logic sentences DOES NOT represent the statement; ‘Not every graph is connected”?

(A) xJ6 (Graph( )x & Connected ( ))x

(B) x7 (Graph( )x /J Connected ( ))x

(C) xJ6 (JGraph( )x 0 Connected ( ))x

(D) x6 (Graph( )x & J Connected ( ))x

Question. 75

Which of the following is TRUE about formulae in Conjunctive Normal Form

(A) For any formula, there is a truth assignment for which at least half the clauses evaluate to true.

(B) For any formula, there is a truth assignment for a which all the clauses evaluate to true.

(C) For is a formula such that for each truth assignment at most one-rourth of the clauses evaluate to true.

(D) None of the above.

Question. 76

Suppose we uniformly and randomly select a permutation from the 20! permutations of 1, 2, 3.......20. What is the probability that 2 appears at an earlier position that any other even number in the selected permutation?

(A) 21 (B) 10

1


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(C) !!

209 (D) None of these

Question. 77

How many different non-iscomorphic Abelian groups of order 4 are there?

(A) 2 (B) 3

(C) 4 (D) 5

Question. 78

Consider the set { , , , }S a b c d= . Consider the following 4 partitions , , , ,1 2 3 4π π π π on

: { }, { , }, { , },S abcd ab cd abc d1 2 3π π π= =

{ , , , }a b c d4π =

Let ' ve the partial

order on the set of partitions ' ( )S , . ,1 2 3 4π π π π= defined as follows:

i j'π π if and only if iπ refines jπ . The poset digram for ( ', )S ' is



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Suppose that a robot is placed on the Cartesian plane. At each step it is allowed to move either one unit up or one unit right, i.e., if it is at ( , )i j then it an move to either ( ), )i j1+ or ( , )i j 1+ .

Question. 79

How many distinct paths are there for the robot to reach the point (10, 10) starting from the initial position (0, 0)?

(A) 20

10d n (B) 220

(C) 210 (D) None of these

Question. 80

Suppose that the robot is not allowed to traverse the line segment from (4, 4) to (5, 4). With this constraint, how many distinct paths are there for the robot to reach (10, 10) starting from (0, 0)?

(A) 29 (B) 219

(C) 8 11

4 5#d dn n (D)

20 8 11

10 4 5#−d d dn n n

Question. 81

Which of the following graphs has an Eulerian circuit?

(A) Any k-regular graph where k is an even number

(B) A complete graph on 90 vertices

(C) The complement of a cycle on 25 vertices

(D) None of the above

Question. 82

Let A be a4#4 matrix with eigenvalues , , ,5 2 1 4= − . Which of the following is an eigenvalue of

A I

I A= G

Where I is the 4 4# identity matrix?

(A) 5− (B) 7−

(C) 2 (D) 1


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Question. 83

Consider the set of (column) vectors defined tyX { ' |x R!= 0x x x1 2 3+ + = where , , ] }x x x xT T

1 2 3= .

Which of the following is TRUE?

(A) {[1, 1,0] ,T− [ , , ] }1 0 1 T− is a basis for the subspace X.

(B) {[ , , ] ,[ , , ] }1 1 0 1 0 1T T− − is a linearly independent set, but it does not span X and therefore is not a basis of X

(C) X is not a subspace for R3

(D) None of the above

Question. 84

Consider the series , 0.5x xx x2 89

nn

n1 0= + =+ obtained from the

Newton-Raphson method. The series converges to

(A) 1.5 (B) 2

(C) 1.6 (D) 1.4

YEAR 2008

Question. 85

Which of the following tuple relational calculus expressions) is/are equivalent to ( ( ))t r P t6 !

1. ( ( ))t r P tJ7 ! 2. ( ( ))t r P t7 g

3. ( ( ))t r P tJ7 Jg 4. ( ( ))t r P t7 Jg

(A) 1 only (B) 2 only

(C) 3 only (D) 3 and 4 only

Question. 86

If , ,P Q R are subsets of the universal set U , then ( ) ( )P Q R P Q R Q Rc c c+ + , + + , , is

(A) Q Rc c, (B) P Q Rc c, ,

(C) P Q Rc c c, , (D) U

Question. 87

The following system of equations


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2x x x1 2 3+ + 1= 2 3x X X1 3 3+ + 2= 4X X ax1 2 3+ + 4=

has a unique solution. The only possible value (s) for a is/are

(A) 0 (B) either 0 or 1

(C) one of 0, 1 or 1− (D) any real number other than 5

Question. 88

lim cossin

x xx x

x +−

"3 equals

(A) 1 (B) 1−

(C) 3 (D) 3−

Question. 89

Let fsa and pda be two predicates such that fsa( )x means x is a finite state automation, and pda( )y means, that y is a pushdown automation. Let equivalent be another predicate such that equivalent ( , )a b means aa and b are equivalent. Which of the following first order logic statement represents the following:

Each finite state automation has an equivalent pushdown automation.

(A) ( ( )) ( pda(y)x x yfsa & /6 7 equivalent( , ))x y

(B) ( ) ( pda(y)y x yfsa & /6 7− equivalent( , ))x y

(C) ( (fsa( ) (y))x y x pda/ /6 7 equivalent( , ))x y

(D) ( (fsa( ) pda(x)x y x / /6 7 equivalent( , ))x y

Question. 90

P and Q are two propositions. Which of the following logical expression are equivalent?

1. ~P Q0

2. ~(~ )P Q/

3. ( ) ( ~ ) (~ ~ )P Q P Q P Q/ 0 / 0 /

4. ( ) ( ~ ) (~ )P Q P Q P Q/ 0 / 0 /

(A) Only 1 and 2 (B) Only 1, 2 and 3

(C) Only 1, 2 and 4 (D) All of 1, 2, 3, and 4


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Question. 91

Aishwarya studies either computer science or mathematics everyday. if she studies computer science on a day, then the probability that she studies mathematics the next day is 0.6. If she studies mathematics on a day, then the probability that she studies computer science the next day is 0.4. Given that Aishwarya studies computer science on Monday, what is the probability that she studies computer science on Wednesday?

(A) 0.24 (B) 0.36

(C) 0.40 (D) 0.60

Question. 92

Let X be a random variable following normal distribution with mean + land variance 4. Let Y be another normal variable with mean 1− and variance unknown, If ( ) ( )P X P Y1 2# $− = the standard deviation of Y is

(A) 3 (B) 2

(C) 2 (D) 1

Question. 93

Let Q ,p i iand1 2 1 2i ki

i kiodd evene

==# # # #

/ /

where k is positive integer. Then

(A) P Q k= − (B) P Q k= +

(C) P Q= (D) P Q k2= +



Let xn denote the number of binary strings of length n that contain no consecutive so.

Question. 94

Which of the following recurrences does xn satisfy?

(A) x x2n n 1= − (B) x x 1[ / ]n n 2= +

(C) x x n[ / ]n n 2= + (D) x x xn n n1 2= +− −


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Question. 95

The value of x5 is

(A) 5 (B) 7

(C) 8 (D) 16

Question. 96

Which of the following statements is true for every planar graph on n vertices

(A) The graph is connected

(B) The graph is Eulerian

(C) The graph has a vertex-cover of size at most /n3 4

(D) The graph has an independent set of size at least /n 3

Question. 97

How many of the following matrices have an eigenvalue 1?

1 0.0 1

.1 1 1 0

0 0 0 0 1 1 1 1and

− −−

= = = =G G G G

(A) one (B) two

(C) three (D) four

Question. 98

The minimum Number of equal length subintervals needed to

approximater xe dxx

1

2

# to an accuracy of at least 31 10 6# − using the

trapezoidal rule is

(A) 1000e (B) 1000

(C) 100e (D) 100

Question. 99

The Newton-Raphson iteration x xxR

21

n nn

1 = ++ −c m can be used to

compute the

(A) square of R (B) reciprocal of R

(C) square root of R (D) logarithm of R


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Question. 100

A point on a curve is said to be an extremum if it is a local minimum or a local maximum. The number of distinct extreme for the curve x x x3 16 24 374 3 2− − + is

(A) 0 (B) 1

(C) 2 (D) 3

YEAR 2009

Question. 101

Which one of the following is NOT necessarily a property of a Group ?

(A) Commutativity

(B) Associativity

(C) Existence of inverse of every element

(D) Existence of identity

Question. 102

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

Question. 103

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 tow vertices have the same degree

(C) At least three vertices have the same degree

(D) All vertices have the same degree.

Question. 104

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


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(B) R is NOT symmetric but antisymmetric

(C) R is both symmetric and antisymmetric

(D) R is neither symmetric nor antisymmetric

Question. 105

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) 0.492

Question. 106

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

Question. 107

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


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( ):P X X is precious

(A) ( ( )) ( ( ) ( ))x P x G x S x" /6

(B) (( ( ) ( )) ( ))x G x S x P x"/6

(C) (( ( ) ( )) ( ))x G x S x P x"/7

(D) (( ( ) ( )) ( ))x G x S x P x"06

Question. 108

The binary operation > is defined as follows:

P Q PXQT T TT F TF T FF F T

Which one of the following is equivalent to P0Q ?

(A) JQ J4 P (B) P J4 Q

(C) JP4Q (D) JP J4 Q

Question. 109

( )/( )tan tanx x dx1 10

4− +

π

# evaluates to

(A) 0 (B) 1

(C) In 2 (D) 1/2 In 2

Question. 110

Consider the following well-formed formulae :

I ( ( ))x P xJ6

II ( ( ))P xJ7

III ( ( ))P xJ7 J

IV ( ( ))x P x7 J

Which of the above are equivalent ?

(A) I and III (B) I and IV

(C) II and III (D) II and IV


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YEAR 2010

Question. 111

Let ( , )G V E= be a graph. Define ( ) ,G i ddd

#ξ = / where id is the

number of vertices of degree d in G. If S and T are two different trees with ( ) ( )S Tξ ξ= , then

(A) S T2= (B) S T 1= −

(C) S T= (D) S T 1= +

Question. 112

Newton-Raphson method is used to compare a root of the equation x 13 02 − = with 3.5 as the initial value. The approximation after one iteration is

(A) 3.575 (B) 3.677

(C) 3.667 (D) 3.607

Question. 113

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

(A) 210 (B) 215

(C) 220 (D) 225

Question. 114

Consider the set { , , }S 1 2ω ω= , where ω and 2ω 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

Question. 115

What is the value of lim n1 1n

n2−

"3b l ?

(A) 0 (B) e 2−

(C) ( )/n 1 2− (D) 1


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Question. 116

In a binary tree with n nodes, every node has an odd number of descendants. Every node is considered to be its own descendant. What is the number of nodes in the tree that have exactly one child ?

(A) 0 (B) 1

(C) ( )/n 1 2− (D) n 1−

Question. 117

Consider a company that assembles computers. The probability of a faulty assembly of any computer is p . The company therefor 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 p q1 1+ − − (B) ( )q p1 −

(C) ( )p q1 − (D) pq

Question. 118

What is the probability that a divisor of 1099 is a multiple of 1096 ?

(A) 1/625 (B) 4/625

(C) 12/625 (D) 16/625

Question. 119

The degrees sequence of a simple graph is the sequence of the degrees of the nodes in the graph in decreasing order. Which of the following sequence 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

Question. 120

Consider the following matrix.

A X Y2 3

= > H


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If the eigenvalues of A are 4 and 8, then

(A) ,x y4 10= = (B) ,x y5 8= =

(C) ,x y3 9=− = (D) ,x y4 10=− =

Question. 121

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 t6 7 7 J ?

(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.

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GATE - CS - Engineering Mathematics