Engineering Mathematics Questions

8/12/2019 Engineering Mathematics Questions

1/12

GATE EE Topic wise QuestionsENGINEERING MATHEMATICS

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YEAR 2010 ONE MARK

Question. 1

The value of the quantity P , where P xe dx x 0

1= # , is equal to

(A) 0 (B) 1

(C) e (D) 1/e

Question. 2

Divergence of the three-dimensional radial vector field r is

(A) 3 (B) / r 1

(C) i j k+ +t t t

(D) 3( )i j k+ +t t t

YEAR 2010 TWO MARKS

Question. 3

A box contains 4 white balls and 3 red balls. In succession, two ballsare randomly and removed form the box. Given that the first removedball is white, the probability that the second removed ball is red is

(A) 1/3 (B) 3/7

(C) 1/2 (D) 4/7


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EE Topic wise 2001-2010ENGINEERING MATHEMATICS

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Question. 4

At t 0= , the function ( ) sin f t t t = has

(A) a minimum (B) a discontinuity(C) a point of inflection (D) a maximum

Question. 5

An eigenvector of P 100

120

023

=

J

L

KKK

N

P

OOO is

(A) 1 1 1 T8 B (B) 1 2 1 T8 B(C) 1 1 2 T8 B (D) 2 1 1 T8 B

Question. 6

For the differential equationdt d x

dt dx x 6 8 02

2+ + = with initial conditions

( )x 0 1= and dt dx 0

t 0=

=

, the solution is

(A) ( ) 2x t e e t t 6 2= (B) ( ) 2x t e e t t 2 4=

(C) ( ) 2x t e e t t 6 4= + (D) ( ) 2x t e e t t 2 4= +

Question. 7For the set of equations, x x x x 2 4 21 2 3 4+ + + = and3 6 3 12 6x x x x 1 2 3 4+ + + = . The following statement is true.(A) Only the trivial solution 0x x x x 1 2 3 4= = = = exists

(B) There are no solutions(C) A unique non-trivial solution exists(D) Multiple non-trivial solutions exist

YEAR 2009 ONE MARK

Question. 8

The trace and determinant of a 2 2# matrix are known to be 2 and 35 respectively. Its eigenvalues are(A) 30 and 5 (B) 37 and 1

(C) 7 and 5 (D) 17.5 and 2


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YEAR 2009 TWO MARKS

Question. 9

( , ) f x y is a continuous function defined over ( , ) [ , ] [ , ]x y 0 1 0 1#!

. Given the two constraints, x y > 2 and y x > 2, the volume under( , ) f x y is

(A) ( , ) f x y dxdy x y

x y

y

y

0

12=

=

=

= # # (B) ( , ) f x y dxdy x y

x

y x

y 1122 =

=

=

= # #

(C) ( , ) f x y dxdy x

x

y

y

0

1

0

1

=

=

=

= # # (D) ( , ) f x y dxdy x

x y

y

y x

00 =

=

=

= # #

Question. 10

Assume for simplicity that N people, all born in April (a month of30 days), are collected in a room. Consider the event of at least twopeople in the room being born on the same date of the month, even ifin different years, e.g. 1980 and 1985. What is the smallest N so thatthe probability of this event exceeds 0.5 ?(A) 20 (B) 7

(C) 15 (D) 16

Question. 11

A cubic polynomial with real coefficients(A) Can possibly have no extrema and no zero crossings

(B) May have up to three extrema and upto 2 zero crossings(C) Cannot have more than two extrema and more than three zero

crossings

(D) Will always have an equal number of extrema and zerocrossings

Question. 12

Let x 117 02 = . The iterative steps for the solution using Newton-Raphons method is given by

(A) x x x 21 117

k k k

1 = ++ b l (B) x x x 117k k k 1 = +

(C) x x x 117k k k

1 = + (D) x x x x 21 117

k k k k

1 = ++ b l


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Question. 13

( , ) ( ) ( )x y x xy y xy F a ax y2 2= + + +t t

. Its line integral over the straightline from ( , ) ( , )x y 0 2= to ( , ) ( , )x y 2 0= evaluates to

(A) 8 (B) 4

(C) 8 (D) 0

YEAR 2008 ONE MARKS

Question. 14

X is a uniformly distributed random variable that takes values

between 0 and 1. The value of { }E X 3

will be(A) 0 (B) 1/8

(C) 1/4 (D) 1/2

Question. 15

The characteristic equation of a ( 3 3# ) matrix P is defined as

( )a I P 2 1 03 2 = = + + + =

If I denotes identity matrix, then the inverse of matrix P will be

(A) ( )P P I 22 + + (B) ( )P P I 2 + +

(C) ( )P P I 2 + + (D) ( )P P I 22 + +

Question. 16

If the rank of a ( )5 6# matrix Q is 4, then which one of the followingstatement is correct ?

(A) Q will have four linearly independent rows and four linearlyindependent columns

(B) Q will have four linearly independent rows and five linearlyindependent columns

(C) QQ T will be invertible

(D) Q Q T will be invertible


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YEAR 2008 TWO MARKS

Question. 17

Consider function ( ) ( 4) f x x 2 2=

where x is a real number. Thenthe function has(A) only one minimum (B) only tow minima(C) three minima (D) three maxima

Question. 18

Equation e 1 0x = is required to be solved using Newtons methodwith an initial guess x 10 = . Then, after one step of Newtonsmethod, estimate x 1 of the solution will be given by

(A) 0.71828 (B) 0.36784(C) 0.20587 (D) 0.00000

Question. 19

A is m n # full rank matrix with m n > and I is identity matrix. Letmatrix ' ( )A A A A1T T= - , Then, which one of the following statementis FALSE ?(A) 'AA A A= (B) ( ')AA 2

(C) 'A A I = (D) ' 'AA A A=

Question. 20

A differential equation / ( )dx dt e u t t 2= - , has to be solved usingtrapezoidal rule of integration with a step size .h 0 01= s. Function

( )u t indicates a unit step function. If ( )x 0 0=- , then value of x at.t 0 01= s will be given by

(A) 0.00099 (B) 0.00495(C) 0.0099 (D) 0.0198

Question. 21

Let P be a 2 2# real orthogonal matrix and x is a real vector [ ]x ,x1 2 T with length ( )x x x /12 22 1 2= + . Then, which one of the followingstatements is correct ?(A) P x x# where at least one vector satisfies P x x

(D) No relationship can be established between x and P x

YEAR 2007 ONE MARK

Question. 22

x x x x n 1 2 Tg= 8 B is an n-tuple nonzero vector. The n n # matrixV xxT=

(A) has rank zero (B) has rank 1

(C) is orthogonal (D) has rank n

YEAR 2007 TWO MARKS

Question. 23

The differential equation dt dx x 1= - is discretised using Eulers

numerical integration method with a time step T 0>3 . What is themaximum permissible value of T 3 to ensure stability of the solutionof the corresponding discrete time equation ?

(A) 1 (B) /2 (C) (D) 2

Question. 24

The value of( )z

dz 1

C 2+ # where C is the contour /z i 2 1 = is

(A) i 2 (B)

(C) tan z 1- (D) tani z 1 -

Question. 25

The integral ( )sin cost d 21

0

2

# equals(A) sin cost t (B) 0

(C) ( / )cos t 1 2 (D) (1/2)sin t


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Question. 26

A loaded dice has following probability distribution of occurrences

Dice Value 1 2 3 4 5 6Probability 1/4 1/8 1/8 1/8 1/8 1/4

If three identical dice as the above are thrown, the probability ofoccurrence of values 1, 5 and 6 on the three dice is

(A) same as that of occurrence of 3, 4, 5

(B) same as that of occurrence of 1, 2, 5

(C) 1/128

(D) 5/8

Question. 27

Let x and y be two vectors in a 3 dimensional space and x,y< > denote their dot product. Then the determinant

detx,xy,x

x,yy,y

< >

< >

< >

< >= G(A) is zero when x and y are linearly independent

(B) is positive when x and y are linearly independent

(C) is non-zero for all non-zero x and y

(D) is zero only when either x or y is zero

Question. 28

The linear operation ( )L x is defined by the cross product L(x) b x#= ,where b 0 1 0 T= 8 B and x x x x 1 2 3 T= 8 B are three dimensional vectors.The 3 3# matrix M of this operations satisfies ( ) M

x x x

L x1

2

3

=

R

T

SSSS

V

X

WWWW

Then the eigenvalues of M are

(A) , ,0 1 1+ (B) , ,1 1 1

(C) , ,i i 1 (D) , ,i i 0


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Statement for Linked Answer Question 29 & 30

Cayley-Hamilton Theorem states that a square matrix satisfies itsown characteristic equation. Consider a matrix

A32

20=

= GQuestion. 29

A satisfies the relation

(A) A I A3 2 01+ + =- (B) 2 2 0A A I 2 + + =

(C) ( ) ( )A I A I 2+ + (D) ( ) 0exp A =

Question. 30

A9 equals

(A) 511 510A I + (B) 309 104A I +

(C) 154 155A I + (D) ( )exp A9

YEAR 2006 TWO MARKS

Question. 31

The expression ( / )V R h H dh 1H 2 2

0= # for the volume of a cone

is equal to

(A) ( / )R h H dr 1R 2 2

0 # (B) ( / )R h H dh 1R 2 2

0 #

(C) ( / )rH r R dh 2 1H

0 # (D) rH Rr dr 2 1

R 2

0 ` j #

Question. 32

A surface ( , )S x y x y 2 5 3= + is integrated once over a pathconsisting of the points that satisfy ( ) ( )x y 1 2 1 2 2+ + = . Theintegral evaluates to

(A) 17 2 (B) 17 2

(C) /2 17 (D) 0


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Question. 33

Two fair dice are rolled and the sum r of the numbers turned up isconsidered

(A) ( )Pr r 6 61> =

(B) Pr ( /r 3 is an integer) 65=

(C) Pr ( /r r 8 4;= is an integer) 95=

(D) (Pr r 6 ;= r/5 is an integer) 181=

Statement for Linked Answer Question 34 & 35

, ,P Q R1013

25

9

27

12

T T T

=

=

=

R

T

SSSS

R

T

SSSS

R

T

SSSS

V

X

WWWW

V

X

WWWW

V

X

WWWW

are three vectors.

Question. 34

An orthogonal set of vectors having a span that contains P,Q,R is

(A)

6

36

4

23

R

T

SSSS

R

T

SSSS

V

X

WWWW

V

X

WWWW (B)

4

24

5

711

8

23

R

T

SSSS

R

T

SSSS

R

T

SSSS

V

X

WWWW

V

X

WWWW

V

X

WWWW

(C)671

322

394

R

T

SSSS

R

T

SSSS

R

T

SSSS

V

X

WWWW

V

X

WWWW

V

X

WWWW (D)

43

11

1313

534

R

T

SSSS

R

T

SSSS

R

T

SSSS

V

X

WWWW

V

X

WWWW

V

X

WWWW

Question. 35

The following vector is linearly dependent upon the solution to theprevious problem

(A)893

R

T

SSSS

V

X

WWWW (B)

217

30

R

T

SSSS

V

X

WWWW

(C)445

R

T

SSSS

V

X

WWWW (D)

1323

R

T

SSSS

V

X

WWWW


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YEAR 2005 ONE MARK

Question. 36

In the matrix equation x q P = , which of the following is a necessarycondition for the existence of at least on solution for the unknownvector x

(A) Augmented matrix [ ]q P must have the same rank as matrix P

(B) Vector q must have only non-zero elements

(C) Matrix P must be singular

(D) Matrix P must be square

Question. 37

If P and Q are two random events, then the following is TRUE

(A) Independence of P and Q implies that probability ( )P Q 0+ =

(B) Probability ( )P Q , $ Probability (P) + Probability (Q)

(C) If P and Q are mutually exclusive, then they must beindependent

(D) Probability ( )P Q + # Probability (P)

Question. 38

If S x dx 31

=3 - # , then S has the value

(A) 31 (B) 4

1

(C) 21 (D) 1

Question. 39

The solution of the first order differential equation '( ) ( )x t x t 3= ,(0)x x 0= is

(A) ( )x t x e t 0 3= - (B) ( )x t x e 0 3= -

(C) ( )x t x e /0 1 3= - (D) ( )x t x e 0 1= -


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YEAR 2005 TWO MARKS

Question. 40

For the matrix p 300

220

211

=

R

T

SSSS

V

X

WWWW, one of the eigen values is equal to 2

Which of the following is an eigen vector ?

(A)32

1

R

T

SSSS

V

X

WWWW (B)

321

R

T

SSSS

V

X

WWWW

(C)12

3

R

T

SS

SS

V

X

WW

WW

(D)25

0

R

T

SS

SS

V

X

WW

WWQuestion. 41

If R122

013

11

2=

R

T

SSSS

V

X

WWWW, then top row of R 1- is

(A) 5 6 48 B (B) 5 3 18 B(C) 2 0 18 B (D) /2 1 1 2

8 BQuestion. 42A fair coin is tossed three times in succession. If the first toss producesa head, then the probability of getting exactly two heads in threetosses is

(A) 81 (B) 2

1

(C)83 (D)

43

Question. 43

For the function ( ) f x x e x 2= - , the maximum occurs when x is equalto(A) 2 (B) 1

(C) 0 (D) 1


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Question. 44

For the scalar field u x y 2 32 2

= + , magnitude of the gradient at thepoint (1, 3) is

(A) 913 (B) 2

9

(C) 5 (D) 29

Question. 45

For the equation '' ( ) ' ( ) ( )x t x t x t 3 2 5+ + = ,the solution ( )x t approacheswhich of the following values as t " 3 ?

(A) 0 (B) 25

(C) 5 (D) 10

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Engineering Mathematics Questions

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