targate-maths booklet (non dowloadable)

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ENGINEERING

MATHEMATICS

Objective Paper –“Topic & Level-wise”

GATEFor “Electrical”, “Mechanical”, “CS/IT” & “Electronics & Comm.”

Engg.

Product of,

TARGATE EDUCATION

a team of



Copyright © TARGATE EDUCATION, Bilaspur-2013

All rights reserved

No part of this publication may be reproduced, stored in retrieval system, or transmitted in any form or byany means, electronics, mechanical, photocopying, digital, recording or otherwise without the prior

permission of the TARGATE EDUCATION.

Authors:

Subject Experts @TRGATE EDUCATION, BILASPUR

TARGATE EDUCATION

Ground Floor, Below Old Arpa Bridge,Jabdapara,

SARKANDA RD. Bilaspur (Chhattisgarh) 495001

Phone No: 07752406380,093004-32128 (01:30 PM - 07:30 PM, Wed-Off)Web Address: www.targate.org, E-Contact: [email protected]



SYLLABUS: ENGG. MATHEMATICS

GATE – 2013

EE /ECEC

Linear Algebra: Matrix Algebra, Systems of linear equations, Eigen values and eigen vectors.

Calculus: Mean value theorems, Theorems of integral calculus, Evaluation of definite and improper integrals, Partial

Derivatives, Maxima and minima, Multiple integrals, Fourier series. Vector identities, Directional derivatives, Line,

Surface and Volume integrals, Stokes, Gauss and Green’s theorems.

Differential equations: First order equation (linear and nonlinear), Higher order linear differential equations with

constant coefficients, Method of variation of parameters, Cauchy’s and Euler’s equations, Initial and boundary value

problems, Partial Differential Equations and variable separable method.

Complex variables: Analytic functions, Cauchy’s integral theorem and integral formula, Taylor’s and Laurent’ series,

Residue theorem, solution integrals.

Probability and Statistics: Sampling theorems, Conditional probability, Mean, median, mode and standard deviation,

Random variables, Discrete and continuous distributions, Poisson,Normal and Binomial distribution, Correlation and

regression analysis.

Numerical Methods: Solutions of non-linear algebraic equations, single and multi-step methods for differential

equations.

Transform Theory: Fourier transform,Laplace transform, Z-transform.

Mechanical Engineering (ME)

Linear Algebra: Matrix algebra, Systems of linear equations, Eigen values and eigen vectors.

Calculus: Functions of single variable, Limit, continuity and differentiability, Mean value theorems, Evaluation of

definite and improper integrals, Partial derivatives, Total derivative, Maxima and minima, Gradient, Divergence and

Curl, Vector identities, Directional derivatives, Line, Surface and Volume integrals, Stokes, Gauss and Green’s theorems.

Differential equations: First order equations (linear and nonlinear), Higher order linear differential equations with

constant coefficients, Cauchy’s and Euler’s equations, Initial and boundary value problems, Laplace transforms,

Solutions of one dimensional heat and wave equations and Laplace equation.

Complex variables: Analytic functions, Cauchy’s integral theorem, Taylor and Laurent series.

Probability and Statistics: Definitions of probability and sampling theorems, Conditional probability, Mean, median,

mode and standard deviation, Random variables, Poisson,Normal and Binomial distributions.

Numerical Methods: Numerical solutions of linear and non-linear algebraic equations Integration by trapezoidal and Simpson’s rule, single and multi-step methods for differential equations.



Computer Science and Information Technology (CS)

Mathematical Logic: Propositional Logic; First Order Logic.

Probability: Conditional Probability; Mean, Median, Mode and Standard Deviation; Random Variables; Distributions;

uniform, normal, exponential, Poisson, Binomial.

Set Theory & Algebra: Sets; Relations; Functions; Groups; Partial Orders; Lattice; Boolean Algebra.

Combinatorics: Permutations; Combinations; Counting; Summation; generating functions; recurrence relations;

asymptotics.

Graph Theory: Connectivity; spanning trees; Cut vertices & edges; covering; matching; independent sets; Colouring;

Planarity; Isomorphism.

Linear Algebra: Algebra of matrices, determinants, systems of linear equations, Eigen values and Eigen vectors.

Numerical Methods: LU decomposition for systems of linear equations; numerical solutions of non-linear algebraic

equations by Secant, Bisection and Newton-Raphson Methods; Numerical integration by trapezoidal and Simpson’s

rules.

Calculus: Limit, Continuity & differentiability, Mean value Theorems, Theorems of integral calculus, evaluation of

definite & improper integrals, Partial derivatives, Total derivatives, maxima & minima.

Expert CommentComparing to the ME syllabus EE/EC has an extra topic “Transform Theory”. ME students need not to read this topics.

CS students have to refer topics from this booklet which is listed in there syllabus. Remaining topic for CS will be

covered in separate booklet.


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Table of Contents

LINEAR ALGEBRA 7

1.1 PROPERTY BASED PROBLEM 7

1.2 DETERMINANTE 10

1.3 ADJOINT - INVERSE 11

1.4 EIGEN VALUES & EIGEN VECTORS 13

1.5 RANK 19

1.6 SOLUTION OF LINEAR EQUATION 21

1.7 MISCELLANEOUS 26

1.8 CALY- HAMILTON 31

CALCULUS 32

2.1 MEAN VALUE THEOREM 32

2.2 MAXIMA AND MINIMA 32

2.3 DIFFERENTIAL CALCULUS 34

2.4 INTEGRAL CALCULUS 36

2.5 LIMIT AND CONTINUITY 39

2.6 SERIES 43

2.7 VECTOR CALCULUS 44

2.8 AREA / VOLUME 51


DIFFERENTIAL EQUATIONS 55

3.1 DEGREE AND ORDER OF DE 55

3.2 HIGHER ORDER DE 56

3.3 LEIBNITZ LINEAR EQUATION 61


COMPLEX VARIABLE 66

4.1CAUCHY’S THEOREM 66


PROBABILITY AND STATISTICS 74

5.2 COMBINATION 74

5.3 PROBABILITY RELATED PROBLEMS 75

5.4 BAYS THEOREMS 80

5.5 PROBABILITY DISTRIBUTION 80

5.6 RANDOM VARIABLE 82

5.7 EXPECTION 85



5.8 SET THEORY 86

NUMERICAL METHODS 87

6.1 CLUBBED PROBLEM 87

6.2 NEWTON-RAP SON 89

6.3 D

IFFERENTIAL93

6.4 INTEGRATION 93

TRANSFORM THEORY 95



01Linear A lgebr a

Complete subtopic in this chapter, is in the scope of “GATE- CS/ME/EC/EE SYLLABUS”

.

1.1 Property Based Problem

Question Level – 0 (Basic Problems)

eE1 / T1 / K1 / L0 / V1 / R11 / AD [GATE – CS – 1994]

(01) If A and B are real symmetric matrices of order n

then which of the following is true.

(A) A AT = I (B) A = A-1

(C) AB = BA (D) (AB)T = BTAT

eE1 / T1 / K1 / L0 / V1 / R11 / AA [GATE – PI – 1994]

(02) If for a matrix, rank equals both the number of

rows and number of columns, then the matrix is

called

(A) Non-singular (B) singular

(C) Transpose (D) Minor

eE1 / T1 / K1 / L0 / V1 / R11 / A [GATE – CE – 2000]

(03) If A, B, C are square matrices of the same order

then 1( ) ABC is equal be

(A) 1 1 1C A B

(B) 1 1 1C B A

(C) 1 1 1 A B C

(D) 1 1 1 A C B

eE1 / T1 / K1 / L0 / V1 / R11 / AB [GATE – CE – 2008]

(04) The product of matrices 1( )PQ P is

(A) 1P (B) 1Q

(C) 1 1P Q P (D) 1P Q P

-----00000-----

Question Level – 01


(01) If A is a real square matrix then AAT is

(A) Un symmetric

(B) Always symmetric

(C) Skew – symmetric

(D) Sometimes symmetric

eE1 / T1 / K1 / L1 / V1 / R11 / AC [GATE – CE – 1998] (02) In matrix algebra AS = AT (A, S, T, are matrices

of appropriate order) implies S = T only if

(A) A is symmetric

(B) A is singular

(C) A is non-singular

(D) A is skew=symmetric



ENGINEERING MATHEMATICS

TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)

eE1 / T1 / K1 / L1 / V1 / R11 / AA [GATE – IN – 2001]

(03) The necessary condition to diagonalizable a

matrix is that

(A) Its all Eigen values should be distinct

(B) Its Eigen values should be independent

(C) Its Eigen values should be real

(D) The matrix is non-singular

eE1 / T1 / K1 / L1 / V1 / R11 / AD [GATE – EC – 2005]

(04) Given an orthogonal matrix A =

1 1 1 1

1 1 1 1

1 1 0 0

0 0 1 1

1( )T AA Is ____

(A) 4

1

4 I (B)

4

1

2 I

(C) I (D) 4

1

3 I

eE1 / T1 / K1 / L1 / V1 / R11 / AA [GATE – ME – 2007]

(05) If a square matrix A is real and symmetric then

the Eigen values

(A) Are always real

(B) Are always real and positive

(C) Are always real and non-negative

(D) Occur in complex conjugate pairs

eE1 / T1 / K1 / L1 / V1 / R11 / AC [GATE – EC – 2010]

(06) The Eigen values of a skew-symmetric matrix are

(A) Always zero

(B) Always pure imaginary

(C) Either zero (or) pure imaginary

(D) Always real

eE1 / T1 / K1 / L1 / V1 / R11 / AC [GATE – ME – 2011]

(07) Eigen values of a real symmetric matrix are

always

(A) Positive (B) Negative

(C) Real (D) 162. [A] is square

-----00000-----


eE1 / T1 / K1 / L2 / V2 / R11 / AA [GATE – CS – 2001]

(01) Consider the following statements

S1: The sum of two singular matrices may be

singular.

S2: The sum of two non-singulars may be non-

singular.

This of the following statements is true.

(A) S1 & S2 are both true

(B) S1 & S2 are both false

(C) S1 is true and S2 is false

(D) S1 is false and S2 is true



TOPIC. 01 – LINEAR ALGEBRA

www.targate.org

eE1 / T1 / K1 / L2 / V2 / R11 / AD [GATE – EE – 2008]

(02) A is m x n full rank matrix with m > n and I is an

identity matrix. Let matrix 1( ) .T T A A A A

then which one of the following statements is

false?

(A) AA+A = A (B) (AA+)2 = AA+

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


(03) A square matrix B is symmetric if -------------

(A)B

T

= B(B)

B

T

= B

(C) B 1 = B (D) B 1 = BT

-----00000-----


eE1 / T1 / K1 / L3 / V2 / R11 / AD [GATE – CE – 1998]

(01) The real symmetric matrix C corresponding to

the quadratic form Q = 1 2 1 24 5 x x x x is

(A) 1 2

2 5

(B) 2 0

0 5

(C) 1 1

1 2

(D) 0 2

2 5


(02) Consider the following two statements.

(I) The maximum number of linearly

independent column vectors of a matrix A is

called the rank of A.

(II) If A is n n square matrix then it will be

non-singular is rank of A = n

(A) Both the statements are false

(B) Both the statements are true

(C) (I) is true but (II) is false

(D) (I) is false but (II) is true

eE1 / T1 / K1 / L3 / V2 / R11 / AA [GATE – CE – 2004]

(03) Real matrices 3 1, 3 3 3 5

, , , A B C D

5

, E 1

F

are given. Matrices [B] and [E]

are symmetric. Following statements are made

with respect to their matrices.

(I) Matrix product [F]T[C]T[B] [C] [F] is a scalar.

Matrix product [D]T[F] [D] is always

symmetric. With reference to above

statements which of the following applies?

(A) Statement (I) is true but (II) is false

(B) Statement (I) is false but (II) is true

(C) Both the statements are true

(D) Both the statements are false

eE1 / T1 / K1 / L3 / V2 / R11 / AB [GATE – EE – 2008]

(04) Let P be 2x2 real orthogonal matrix and x is a

real vector 1 2

T x x with length || || x =

2 2 1/21 2( ) x x Then which one of the following

statement is correct?

(A) || || || || px x where at least one vector

satisfies || || || || px x





(B) || || || || px x for all vectors x

(C) || || || || px x when atleast one vector satisfies

|| || x and || || px

(D) No relationship can be established between

|| || x and || || px

eE1 / T1 / K6 / L3 / V2 / R11 / A [GATE – CS – 2008]

(05) The following system of equations

1 2 32 1, x x x 1 1 32 3 x x x ,

1 1 34 4 x x αx has a unique solution solution.

The only possible value(s) for α is/are

(A) 0 (B) either 0 (or) 1

(C) one of 0, 1 (or) – 1 (D) any real number


(06) [A] is a square matrix which is neither symmetric

nor skew-symmetric and [A]T is its transpose.

The sum and differences of these matrices are

defined as [S] = [A] + [A]T and [D] = [A] – [A]T

respectively. Which of the following statements

is true?

(A) Both [S] and [D] are symmetric

(B) Both [S] and [D] are skew-symmetric

(C) [S] is skew-symmetric and [D] is symmetric

(D) [S] is symmetric and [D] is skew-symmetric

-----00000-----

1.2 Determinante

Question Level – 00 (Basic Problem)

eE1 / T1 / K2 / L0 / V1 / R11 / AD [GATE – PI – 1994]

(01) The value of the following determinant

1 4 9

4 9 16

9 16 25

is

(A) 8 (B) 12

(C) – 12 (D) – 8


eE1 / T1 / K2 / L1 / V2 / R11 / AB [GATE – CS – 1997]

(01) The determinant of the matrix

6 8 1 1

0 2 4 6

0 0 4 8

0 0 0 1

(A) 11 (B) – 48

(C) 0 (D) – 24


(02) If the determinant of the matrix

1 3 2

0 5 6

2 7 8

is

26 then the determinant of the matrix

2 7 8

0 5 6

1 3 2

is

(A) – 26 (B) 26

(C) 0 (D) 52




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(01) If =

1

1

1

a bc

b ca

c ab

then which of the following is

a factor of .

(A) a + b (B) a - b

(C) abc (D) a + b + c


(02) The determinant

1 1

1 1

1 2 1

b b

b b

b

equals to

(A) 0 (B) 2b(b – 1)

(C) 2(1 – b)(1 + 2b) (D) 3b(1 + b)

eE1 / T1 / K2 / L2 / V2 / R11 / AB [GATE – PI – 2009]

(03) The value of the determinant

1 3 2

4 1 1

2 1 3

is

(A) – 28 (B) – 24

(C) 32 (D) 36

-----00000-----

1.3 Adjoint - Inverse



(01) The inverse of 2 2 matrix1 2

5 7

is

(A) 7 21

5 13

(B) 7 21

5 13

(C)

7 21

5 13

(D)

7 21

5 13



(01) The matrix1 4

1 5

is an inverse of the matrix

5 4

1 1

(A) True (B) False



(01) The inverse of the matrix S =

1 1 0

1 1 1

0 0 1

is

(A)

1 0 1

0 0 0

0 1 1

(B)

0 1 1

1 1 1

1 0 1





(C)

2 2 2

2 2 2

0 2 2

(D) 1/ 2 1/ 2 1/ 2

1 / 2 1/ 2 1/ 2

0 0 1


(02) Inverse of matrix0 1 00 0 1

1 0 0

is

(A)

0 0 1

1 0 0

0 1 0

(B)

1 0 0

0 0 1

0 1 0

(C)

1 0 0

0 1 0

0 0 1

(D)

0 0 1

0 1 0

1 0 0

eE1 / T1 / K3 / L2 / V2 / R11 / AA [GATE – EE – 1998]

(03) If A =

5 0 2

0 3 0

2 0 1

then 1 A

=

(A)

1 0 2

0 1/ 3 0

2 0 5

(B)

5 0 2

0 1/ 3 0

2 0 1

(C)

1/ 5 0 1/ 2

0 1/ 3 0

1/ 2 0 1

(D) 1/ 5 0 1/ 2

0 1 / 3 0

1 / 2 0 1


(04) If A =

1 2 1

2 3 1

0 5 2

and ad (A) =

11 9 1

4 2 3

10 7k

Then k =

(A) – 5 (B) 3

(C) – 3 (D) 5


(05) The inverse of matrix0 1 01 0 0

0 0 1

is

(A)

0 1 0

1 0 0

0 0 1

(B)

0 1 0

1 0 0

0 0 1

(C)

0 1 0

0 0 1

1 0 0

(D)

0 1 0

0 0 1

1 0 0


(06) For a matrix [M] =3 / 4 4 / 5

3 / 5 x

. The transpose

of the matrix is equal to the inverse of the matrix,

1[ ] [ ] .T M M The value of x is given by

(A) 4

5 (B)

3

5

(C) 35

(D) 45


(07) The inverse of the matrix3 2

3 2

i i

i i

is

(A) 3 21

3 22

i i

i i

(B) 3 21

3 212

i i

i i




www.targate.org

(C) 3 21

3 214

i i

i i

(D) 3 21

3 214

i i

i i

-----00000-----

1.4 Eigen Values & Eigen Vectors



(01) The Eigen values of the matrix1

1

a

a

are

(A) ( 1),0a (B) ,0a

(C) ( 1),0a (D) 0,0


(02) A =

2 0 0 1

0 1 0 0

0 0 3 0

1 0 0 4

the sum of the Eigen

Values of the matrix A is

(A) 10 (B) – 10

(C) 24 (D) 22

eE1 / T1 / K4 / L0 / V1 / R11 / AB [GATE – ME – 2004]

(03) The sum of the eigen values of the matrix given

below is

1 1 3

1 5 1

3 1 1

(A) 5 (B) 7

(C) 9 (D) 18

eE1 / T1 / K4 / L0 / V1 / R11 / AC [GATE – CE – 2004]

(04) The eigen values of the matrix4 2

2 1

are

(A) 1, 4 (B) – 1, 2

(C) 0, 5 (D) cannot be determined


(05) What are the Eigen values of the following 2 x 2

matrix?2 1

4 5

(A) – 1, 1 (B) 1, 6

(C) 2, 5 (D) 4, -1

eE1 / T1 / K4 / L0 / V1 / R11 / AC [GATE – PI – 2005]

(06) The Eigen values of the matrix M given below

are 15, 3 and 0. M =

8 6 2

6 7 4

2 4 3

, the value of

the determinant of a matrix is

(A) 20 (B) 10

(C) 0 (D) – 10

eE1 / T1 / K4 / L0 / V1 / R11 / A [GATE – CS – 2008]

(07) How many of the following matrices have an

Eigen value 1?

1 0 0 1 1 1 1 0, , &

0 0 0 0 1 1 0 1





(A) One (B) Two

(C) Three (D) Four

eE1 / T1 / K4 / L0 / V1 / R11 / AC [GATE – EE – 2009]

(08) The trace and determinant of a 2x2 matrix are

shown to be -2 and -35 respectively. Its eigen

values are

(A) -30, -5 (B) -37, -1

(C) -7, 5 (D) 17.5, -2

-----00000-----


eE1 / T1 / K4 / L1 / V1 / R11 / AA [GATE – –1993]

(01) The eigen vector (s) of the matrix

0 0

0 0 0 , 0

0 0 0

α

α

Is (are);

(A) 0,0,α (B) ,0,0α

(C) 0,0,1 (D) 0, ,0α


(02) The eigen values of

1 1 1

1 1 1

1 1 1

are

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

(C) 0, 0,3 (D) 1, 1, 1


(03) The eigen values of the matrix A =0 1

1 0

are

(A) 1, 1 (B) -1, -1

(C) , j j (D) 1, 1

eE1 / T1 / K4 / L1 / V1 / R11 / AD [GATE – CE – 2001

]

(04) The eigen values of the matrix5 3

2 9

are

(A) (5.13,9.42) (B) (3.85,2.93)

(C) (9.00,5.00) (D) (10.16,3.84)


(05) Eigen values of the following matrix are

1 4

4 1

(A) 3, -5 (B) -3, 5

(C) -3, -5 (D) 3, 5

eE1 / T1 / K4 / L1 / V1 / R11 / A [GATE – IN – 2005]

(06) Identify which one of the following is an eigen

vector of the matrix A =1 0

1 2

(A) 1 1T

(B) 3 1T

(C) 1 1T

(D) 2 1T




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(07) The minimum and maximum Eigen values of

Matrix

1 1 3

1 5 1

3 1 1

are -2 and 6 respectively.

What is the other Eigen value?

(A) 5 (B) 3

(C) 1 (D) -1


(08) All the four entries of 2 x 2 matrix P =

11 12

21 22

p p

p p

are non-zero and one of the Eigen

values is zero. Which of the following statement

is true?

(A) 11 22 12 21 1P P P P (B) 11 22 12 21 1P P P P

(C) 11 22 21 12 0P P P P (D) 11 22 12 21 0P P P P


(09) The eigen values of the matrix [P] =4 5

2 5

are

(A) – 7 and 8 (B) – 6 and 5

(C) 3 and 4 (D) 1 and2


(10) One of the eigen vector of the matrix A =

2 2

1 3

is

(A) 2

1

(B) 2

1

(C) 4

1

(D) 1

1


(11) Consider the following matrix A =2 3

. x y

If

the eigen values of A are 4 and 8 then

(A) x = 4, y = 10 (B) x = 5, y = 8

(C) x = -3, y = 9 (D) x = -4, y = 10


(12) Obtain the eigen values of the matrix A =

1 2 34 49

0 2 43 940 0 2 104

0 0 0 1

(A) 1,2,-2,-1 (B) -1,-2,-1,-2

(C) 1,2,2,1 (D) None

-----00000-----



(01) The vector

1

2

1

is an eigen vector of A =




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(A) 0 (B) 1/2

(C) 1 (D) 2


(08) The eigen vector pair of the matrix3 4

4 3

is

(A) 2 1

1 2

(B) 2 1

1 2

(C) 2 11 2

(D) 2 1

1 2


(09) For the matrix4 2

.2 4

The eigen value

corresponding to the eigen vector 101101

is

(A) 2 (B) 4

(C) 6 (D) 8


(10) For a given matrix A =

2 2 3

2 1 6

1 2 0

, one of the

eigen value is 3. The other two eigen values are

(A) 2, -5 (B) 3, -5

(C) 2, 5 (D) 3, 5


(11) An eigen vector of p =

1 1 0

0 2 2

0 0 3

is

(A) 1 1 1T

(B) 1 2 1T

(C) 1 1 2T

(D) 2 1 1T


(12) The Eigen values of the following matrix

10 4

18 12

are

(A) 4, 9 (B) 6, - 8

(C) 4, 8 (D) – 6, 8


(13) The eigen values of the following matrix

1 3 5

3 1 6

0 0 3

are

(A) 3, 3 5 ,6 j j (B) 6 5 ,3 ,3 j j j

(C) 3 ,3 ,5 j j j (D) 3, 1 3 , 1 3 j j

eE1 / T1 / K4 / L2 / V2 / R11 / AB [GATE – IN – 2011]

(14) The matrix M =

2 2 3

2 1 6

1 2 0

has eigen values

-3, -3, 5. An eigen vector corresponding to the

eigen value 5 is 1 2 1 .T One of the eigen

vector of the matrix M3 is





(A) 1 8 1T

(B) 1 2 1T

(C) 31 2 1T

(D) 1 1 1T


(15) Consider the matrix as given below

1 2 30 4 7

0 0 3

.

Which one of the following options provides the

correct values of the eigen values of the matrix?

(A) 1, 4, 3 (B) 3, 7, 3

(C) 7, 3, 2 (D) 1, 2, 3


(16) If {1,0, 1}T is an eigen vector of the following

matrix

1 1 0

1 2 1

0 1 1

then the corresponding

eigen value is

(A) 1 (B) 2

(C) 3 (D) 5

eE1 / T1 / K4 / L2 / V3 / R11 / AC [GATE – IN – 2009] (17) The eigen values of a 2 2 matrix X are -2 and -

3. The eigen values of matrix 1( ) ( 5 ) X I X I

are

(A) – 3, - 4 (B) -1, -2

(C) -1, -3 (D) -2, -4

-----00000-----



(01) If A is square symmetric real valued matrix of

dimension 2n, then the eigen values of A are

(A) 2n distinct real values

(B) 2n real values not necessarily distinct

(C) n distinct pairs of complex conjugate

numbers

(D) n pairs of complex conjugate numbers, not

necessarily distinct

eE1 / T1 / K4 / L3 / V2 / R11 / AA [GATE – EC – 2006]

(02) The eigen values and the corresponding eigen

vectors of a 2x2 matrix are given by

Eigen Value Eigen Vector

1 8 λ 1

1

1V

2 4 λ 2

1

1V

The matrix is

(A) 6 2

2 6

(B) 4 6

6 4

(C) 2 4

4 2

(D) 4 8

8 4


(03) Which one of the following is an eigen vector of




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the matrix

5 0 0 0

0 5 0 0

0 0 2 1

0 0 3 1

is

(A) 1 2 0 0T

(B) 0 0 1 0T

(C) 1 0 0 2T

(D) 1 1 2 1T


(04) A real nxn matrix A = ija is defined as follows

,

0,

ija i i j

otherwise

The sum of all n eigen values of A is

(A) ( 1)

2

n n (B)

( 1)

2

n n

(C) ( 1)(2 1)

2

n n n (D) 2

n

eE1 / T1 / K4 / L3 / V2 / R11 / A [GATE – EE – 2011]

(05) The two vectors 1 1 1 and 21 a a

where1 3

2 2a j and 1 j are

(A) Orthonormal (B) Orthogonal

(C) Parallel (D) Collinear


(06) 1 2 3, , ,........ mq q q q are n-dimensional vectors with

m < n. This set of vectors is linearly dependent.

Q is the matrix with 1 2 3, , ,....... mq q q q as the

columns. The rank of Q is(A) Less than m (B) m

(C) Between m and n (D) n


(07) X = 1 2 ...........T

n x x x is an n – tuple non zero

vector. The n x n matrix V = XXT

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

(C) is orthogonal (D) has rank n

-----00000-----

1.5 Rank



(01) The rank of (m x n) matrix (m < n) cannot be

more than

(A) m (B) n

(C) mn (D) None

eE1 / T1 / K5 / L0 / V1 / R11 / AC [GATE – CS – 2002]

(02) The rank of the matrix1 1

0 0

is

(A) 4 (B) 2

(C) 1 (D) 0


(03) A 5x7 matrix has all its entries equal to -1. Then

the rank of a matrix is

(A) 7 (B) 5

(C) 1 (D) Zero







(01) The number of Linearly independent solutions of

the system of equations

1 0 2

1 1 0

2 2 0

1

2

3

x

x

x

=0 is

equal to

(A) 1 (B) 2

(C) 3 (D) 0


(02) The rank of matrix

0 0 3

9 3 5

3 1 1

is

(A) 0 (B) 1

(C) 2 (D) 3

eE1 / T1 / K5 / L1 / V1 / R11 / AB [GATE – EC –]

(03) Rank of the matrix

0 2 2

7 4 8

7 0 4

is 3

(A) True (B) False

eE1 / T1 / K5 / L1 / V1 / R11 / AC [GATE – IN – 2000]

(04) The rank of matrix A =

1 2 3

3 4 5

4 6 8

is

(A) 0 (B) 1

(C) 2 (D) 3


(05) The rank of the following (n+1) x (n+1) matrix,

where ‘a’ is a real number is

2

2

2

1 . . .

1 . . .

.

.

1 . . .

n

n

n

a a a

a a a

a a a

(A) 1 (B) 2

(C) n (D) depends on the value of a

----00000-----



(01) The rank of the matrix

1 4 8 7

0 0 3 0

4 2 3 1

3 12 24 2

is

(A) 3 (B) 1

(C) 2 (D) 4


(02) The rank of the matrix

1 1 1

1 1 0

1 1 1

is

(A) 0 (B) 1

(C) 2 (D) 3




www.targate.org



(01) Given matrix [A] =

4 2 1 3

6 3 4 7

2 1 0 1

, the rank of

the matrix is

(A) 4 (B) 3

(C) 2 (D) 1


(02) Let A = [ ],1 ,ija i j n with 3n and .ija i j .

Then the rank of A is

(A) 0 (B) 1

(C) n – 1 (D) n

eE1 / T1 / K5 / L3 / V2 / R11 / AA [GATE – EE – 2008] (03) If the rank of a 5x6 matrix Q is 4 then which one

of the following statements is correct?

(A) Q will have four linearly independent rows

and four linearly independent columns

(B) Q will have four linearly independent rows

and five linearly independent columns

(C) QQT will be invertible.

(D) QT Q will be invertible.

-----00000-----

1.6 Solution of Linear Equation


eE1 / T1 / K6 / L1 / V1 / R11 / AB [GATE – EC – 1994]

(01) Solve the following system

1 2 3 3 x x x

1 3 0 x x

1 2 3 1 x x x

(A) Unique solution

(B) No solution

(C) Infinite number of solutions

(D) Only one solution


(02) In the Gauss – elimination for a solving system of linear algebraic equations, triangularization leads

to

(A) diagonal matrix

(B) lower triangular matrix

(C) upper triangular matrix

(D) singular matrix


(03) Let A be 3 3 matrix with rank 2. Then AX = O

has

(A) Only the trivial solution X = 0

(B) One independent solution





(C) Two independent solutions

(D) Three independent solutions

eE1 / T1 / K6 / L1 / V1 / R11 / A [GATE – CS – 2004]

(04) How many solutions does the following system

of linear equations have

5 1 x y

2 x y

3 3 x y

(A) Infinitely many

(B) Two distinct solutions

(C) Unique

(D) None

eE1 / T1 / K6 / L1 / V1 / R11 / AC [GATE – EC –]

(05) The value of q for which the following set of

linear equations 2x + 3y = 0, 6x + qy = 0 can

have non-trival solution is

(A) 2 (B) 7

(C) 9 (D) 11

-----00000-----



(01) A system of equations represented by AX = 0

where X is a column vector of unknown and A is

a matrix containing coefficient has a non-trivial

solution when A is.

(A) non-singular (B) singular

(C) symmetric (D) Hermitian


(02) Consider the following set of equations

2 5, x y 4 8 12, x y 3 6 3 15. x y z This

set

(A) has unique solution

(B) has no solution

(C) has infinite number of solutions

(D) has 3 solutions


(03) Consider the following system of equations in

three real variable 1 2 3, : x x and x

1 2 32 3 1 x x x

1 2 33 2 5 2 x x x

1 2 34 3 x x x

This system of equations has

(A) No solution

(B) A unique solution




www.targate.org

(C) More than one but a finite number of

solutions.

(D) An infinite number of solutions.


(04) Consider a non-homogeneous system of linear equations represents mathematically an over

determined system. Such a system will be

(A) Consistent having a unique solution

(B) Consistent having many solutions.

(C) Inconsistent having a unique solution.

(D) Inconsistent having no solution.


(05) In the matrix equation PX = Q which of the

following is a necessary condition for the

existence of at least one solution one solution for

the unknown vector X.

(A) Augmented matrix [P|Q] 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


(06) A is a 3 4 matrix and AX = B is an inconsistent

system of equations. The highest possible rank of

A is

(A) 1 (B) 2

(C) 3 (D) 4

eE1 / T1 / K6 / L2 / V2 / R11 / AD [GATE – IN – 2006]

(07) A system of linear simultaneous equations is

given as AX = b

Where A =

1 0 1 0

0 1 0 1

1 1 0 0

0 0 0 1

& b =

0

0

0

1

Then the rank of matrix A is

(A) 1 (B) 2

(C) 3 (D) 4


(08) A system of linear simultaneous equations is

given as Ax b

Where A =

1 0 1 0

0 1 0 1

1 1 0 0

0 0 0 1

& b =

0

0

0

1

Which of the following statement is true?

(A) x is a null vector

(B) x is unique

(C) x does not exist

(D) x has infinitely many values


(09) Solution for the system defined by the set of

equations 4 3 8,2 2 y z x z & 3 2 5 x y

is

(A) 0, 1, 4 / 5 x y z

(B) 0, 1/ 2, 2 x y z





(C) 1, 1/ 2, 2 x y z

(D) Non existent


(10) For what values of α and β the following

simultaneous equations have an infinite number

of solutions 5, x y z 3 3 9, x y z

2 x y αz = β

(A) 2, 7 (B) 3, 8

(C) 8, 3 (D) 7, 2

eE1 / T1 / K6 / L2 / V2 / R11 / A [GATE – CE – 2008]

(11) The following system of equations 3, x y z

2 3 4, x y z 4 6 x y k will not have a

unique solution for k equal to

(A) 0 (B) 5

(C) 6 (D) 7


(12) For the set of equations 1 2 3 42 4 2, x x x x

1 2 3 43 6 3 12 6. x x x x The following

statement is true

(A) Only the trivial solution 1 2 3 4 0 x x x x

exist

(B) There are no solutions

(C) A unique non-trivial solution exist

(D) Multiple non-trivial solution exist

eE1 / T1 / K6 / L2 / V2 / R11 / A [GATE – IN – 2010]

(13) X and Y are non-zero square matrices of size

nxn. If XY = Onxn then

(A) | | 0 X and | | 0Y

(B) | | 0 X and | | 0Y

(C) | | 0 X and | | 0Y

(D) | | 0 X and | | 0Y


(14) Consider the following system of equations

1 2 3 2 32 0, 0 x x x x x and 1 2 0 x x .

This system has

(A) A unique solution

(B) No solution

(C) Infinite number of solution

(D) Five solutions


(15) The system of linear equations4 2 7

2 6

x y

x y

has

(A) A unique solution

(B) No solution

(C) An infinite no. of solution

(D) Exactly two distinct solution.




www.targate.org


(16) The value of x3 obtained by solving the following

system of linear equations is

1 2 32 2 4 x x x

1 2 32 2 x x x

1 2 3 2 x x x

(A) – 12 (B) - 2

(C) 0 (D) 12


(17) The system of equations 6 x y z ,

4 6 20, x y z and 4 x y λz μ has no

solution for values of λ and given by

(A) 6, 20 λ μ (B) 6, 20 λ μ

(C) 6, 20 λ μ = (D) 6, 20 λ μ

eE1 / T1 / K6 / L2 / V2 / R11 / AC [GATE – EC –]

(18) For the following set of simultaneous equations

1.5 0.5 2 x y z

4 2 3 0 x y z

7 5 10 x y z

(A) the solution is unique

(B) infinitely many solutions exist

(C) the equations are incompatible

(D) finite many solutions exist


(19) In the solution of the following set of linear

equations by Gauss-elimination using partial

pivoting 5 2 34, x y z 4 3 12 y z and

10 2 4. x y z The pivots for elimination of

x and y are

(A) 10 and 4 (B) 10 and 2

(C) 5 and 4 (D) 5 and – 4



(01) Let AX = B be a system of linear equationswhere A is an m n matrix B is an 1m column

matrix which of the following is false?

(A) The system has a solution, if ( ) ( / ) ρ A ρ A B

(B) If m = n and B is a non – zero vector then the

system has a unique solution

(C) If m < n and B is a zero vector then the

system has infinitely many solutions.

(D) The system will have a trivial solution when

m = n , B is the zero vector and rank of A is

n.


(02) A set of linear equations is represented by the

matrix equations Ax = b. The necessary condition

for the existence of a solution for this system is

(A) must be invertible

(B) b must be linearly dependent on the columns

of A

(C) b must be linearly independent on the

columns of A





(D) None


(03) Let A be an n x n real matrix such that A2 = I and

Y be an n-dimensional vector. Then the linear

system of equations Ax = Y has

(A) No solution

(B) unique solution

(C) More than one but infinitely many dependent

solutions.

(D) Infinitely many dependent solutions


(04) Let x and y be two vectors in a 3 – dimensional

space and , x y denote their dot product. Then

the determinant det, ,

, ,

x x x y

y x y y

=_____

(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


(05) For what values of ‘a’ if any will the following

system of equations in x, y are z have a solution?

2 3 4, 4, 2 x y x y z x y z a

(A) Any real number

(B) 0

(C) 1

(D) There is no such value

1.7 Miscellaneous



(01) Let A, B,C, D be n n matrices, each with non-

zero determinant. ABCD = I then B 1 =

(A) 1 1 1 D C A

(B) CDA

(C) ABC (D) Does not exist


(02) If A and B are two matrices and if AB exist then

BA exists.

(A) Only if A has as many rows as B has

columns

(B) Only if both A and B are square matrices

(C) Only if A and B are skew matrices

(D) Only if both A and B are symmetric

-----00000-----



(01) Let Anxn be matrix of order n and I12 be the matrix

obtained by interchanging the first.

(A) Row is the same as its second row

(B) row is the same as second row of A




www.targate.org

(C) column is the same as the second column of

(D) Row is a zero row.


(02) If A is any n n matrix and k is a scalar then

| | | |kA α A where α is

(A) kn (B) k n

(C) nk (D)

k

n


(03) The number of terms in the expansion of general

determinant of order n is

(A) 2n (B) !n

(C) n (D) 2( 1)n


(04) The determinant of the following matrix

5 3 2

1 2 6

3 5 10

(A) – 76 (B) – 28

(C) 28 (D) 72


(05) The product [P] [Q]T of the following two

matrices [P] and [Q] is where [P] =2 3

,4 5

4 8[ ]

9 2Q

(A) 32 24

56 46

(B) 46 56

24 32

(C) 35 22

61 42

(D) 32 56

24 46

eE1 / T1 / K7 / L1 / V1 / R11 / AC [GATE – CS – 2004] (06) The number of different n n symmetric

matrices with each elements being either 0 or 1 is

(A) 2n (B) 2

2n

(C)

2

22n n

(D)

2

22n n


(07) Given the matrix4 2

,4 3

the eigen vector is

(A) 3

2

(B) 4

3

(C) 2

1

(D) 2

1


eE1 / T1 / K7 / L2 / V1 / R11 / A [GATE – PI – 1994]

(01) For the following matrix 1 12 3

the number of

positive roots is

(A) One (B) Two

(C) Four (D) Cannot be found





www.targate.org

(A) 7

20 (B)

3

20

(C) 19

60 (D)

11

20


(10) For a given 2x2 matrix A, it is observed that

1 11

1 1 A

and 1

2 A

and

1 12

2 2 A

then the matrix A is

(A) 2 1 1 0 1 1

1 1 0 2 1 2 A

(B) 1 1 1 0 2 1

1 2 1 2 1 1 A

(C)

1 1 1 0 2 1

1 2 0 2 1 1 A

(D) 0 2

1 3 A

eE1 / T1 / K7 / L2 / V2 / R11 / AD [GATE – ME – 2011]

(11) If a matrix A =2 4

1 3

and matrix B =4 6

5 9

the transpose of product of these two matrices

i.e., ( )T AB is equal to

(A) 28 19

34 47

(B) 19 34

47 28

(C)

48 33

28 19

(D)

28 19

48 33


(12) The matrix [A] =2 1

4 1

is decomposed into a

product of lower triangular matrix [L] and an

upper triangular [U]. The property decomposed

[L] and [U] matrices respectively are

(A) 1 0

4 1

and 1 1

0 2

(B) 1 0

2 1

and 2 1

0 3

(C) 1 0

4 1

and

2 1

0 1

(D) 2 0

4 3

and 1 0.5

0 1

-----00000-----



(01) The matricescos sin

sin cos

θ θ

θ θ

and 0

0

a

b

commute under multiplication.

(A) If a = b (or) ,θ nπ n is an integer

(B) Always

(C) never

(D) If a cos sinθ b θ






(02) The equation2

2 1 1

1 1 1 0

y x x

represents a

parabola passing through the points.

(A) (0,1), (0,2),(0,-1) (B) (0,0), (-1,1),(1,2)

(C) (1,1), (0,0), (2,2) (D) (1,2), (2,1), (0,0)

eE1 / T1 / K7 / L3 / V2 / R11 / AC [GATE – EC –2004]

(03) What values of x, y, z satisfy the following

system of linear equations

1 2 3 61 3 4 8

2 2 3 12

x y

z

(A) x = 6, y = 3, z = 2

(B) x = 12, y = 3, z = -4

(C) x = 6, y= 6, z = -4

(D) x = 12, y = -3, z = 4

eE1 / T1 / K7 / L3 / V2 / R11 / AB [GATE – EC –2004]

(04) If matrix X =2

1

1 1

a

a a a

and

2 0. X X I Then the inverse of X is

(A) 2

1 1a

a a

(B) 2

1 1

1

a

a a a

(C) 2

1

1 1

a

a a a

(D) 2 1

1 1

a a a

a


(05) Consider the system of equations,

1 1n n n n A X λX where λ is a scalar. Let

,i i λ X be an eigen value and its corresponding

eigen vector for real matrix A. Let Inxn be unit

matrix. Which one of the following statement is

not correct?

(A) For a homogeneous nxn system of linear

equations (A- λ I) is less than n.

(B) For matrix Am, m being a positive integer, (

,mi λ m

i X ) will be eigen pair for all i.

(C) If 1T A A then | | 1i λ for all i.

(D) If T A A then i λ are real for all i.


(06) Multiplication of matrices E and F is G. Matrices

E and G are E =

cos sin 0

sin cos 0

0 0 1

θ θ

θ θ

and G =

1 0 0

0 1 0

0 0 1

. What is the matrix F?

(A)

cos sin 0

sin cos 0

0 0 1

θ θ

θ θ

(B)

cos cos 0

cos sin 0

0 0 1

θ θ

θ θ

(C)

cos sin 0

sin cos 0

0 0 1

θ θ

θ θ

(D)

sin cos 0

cos sin 0

0 0 1

θ θ

θ θ




www.targate.org


(07) An n n array V is defined as follows V[i,j] =

i j for all i, j, 1 ,i j n then the sum of the

elements of the array V is

(A) 0 (B) n – 1

(C) 2 3 2n n (D) ( 1)n n

1.8 CALY- HAMILTON



(01) If A =3 2

1 0

then A satisfies the relation

(A) A + 3I + 21

A

= O (B) 2 2 2 A A I O

(C) ( )( 2 ) A I A I O (D) Ae O


(02) If A =3 2

1 0

then9

A equals

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

(C) 154 A + 155 I (D) 9 A

e



(01) The characteristic equation of a 3x3 matrix P is

defined as

3 2( ) | | 2 1 0.α λ λI P λ λ λ

If I denotes identity matrix then the inverse of P

will be

(A) 2 2P P I (B)

2P P I

(C) 2( )P P I (D) 2( 2 )P P I

-----00000-----


http://slidepdf.com/reader/full/targate-maths-booklet-non-dowloadable 32/102 TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)

02Calculus

Complete subtopic in this chapter, is in the scope of “GATE- CS/ME/EC/EE SYLLABUS”, Note: Subtopic “2.7 Vector Calculus” is excluded in GATE- CS SYLLABUS.

2.1 Mean Value theorem


eE1 / T2 / K1 / L1 / V1 / R11 / AC [GATE – – 1994]

(01) The value of ε in the mean value theorem of

f(B) – f(A) = (b – a) f’( )ε for

2( ) f x Ax Bx C in (a, b) is

(A) b a (B) b a

(C) 2

b a (D)

2

b a

-----00000-----


eE1 / T2 / K1 / L3 / V2 / R11 / AB [GATE – – 1995]

(01) If f(0) = 2 and f’(x) =2

1,

5 xthen the lower

and upper bounds of f(1) estimated by the mean

value theorem are ____________

(A) 1.9, 2.2 (B) 2.2, 2.25

(C) 2.25, 2.5 (D) None of the above

-----00000-----

2.2 Maxima and Minima


eE1 / T2 / K2 / L0 / V1 / R11 / AB [GATE – – ]

(01) A point on the 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

4 3 23 16 24 37 x x x is ___________

(A) 0 (B) 1

(C) 2 (D) 3

-----00000-----


eE1 / T2 / K2 / L2 / V2 / R11 / AB [GATE – – 1994]

(01) The function 2 250 y x

x at x = 5 attains

(A) Maximum (B) Minimum

(C) Neither (D) 1

eE1 / T2 / K2 / L2 / V2 / R11 / AA [GATE – – 1995]

(02) The function f(x) = 3 26 9 25 x x x has



TOPIC. 02 – CALCULUS

www.targate.org

(A) A maxima at x = 1 and minima at x = 3

(B) A maxima at x = 3 and a minima at x = 1

(C) No maxima, but a minima at x = 3

(D) A maxima at x = 1, but no minima


(03) What is the maximum value of the function

2( ) 2 2 6 f x x x in the interval [0, 2]?

(A) 6 (B) 10

(C) 12 (D) 5.5


(04) The function f(x) = 3 22 3 36 2 x x x has its

maxima at

(A) x = - 2 only

(B) x = 0 only

(C) x = 3 only

(D) both x = - 2 and x = 3


(05) For the function f(x) = 2 , x x e the maximum

occurs when x is equal to

(A) – 2 (B) 1

(C) 0 (D) – 1


(06) Consider the function f(x) = 2 2. x x the

maximum value of f(x) in the closed interval [-

4, 4] is

(A) 18 (B) 10

(C) – 2.25 (D) indeterminate


(07) Consider the function 2 6 9. y x x The

maximum value of y obtained when x varies

over the internal 2 to 5 is

(A) 1 (B) 3

(C) 4 (D) 9


(08) For real values of x, the minimum value of

function f(x) = x xe e

is

(A) 2 (B) 1

(C) 0.5 (D) 0


(09) If 1/ y xe x then y has a

(A) Maximum at x = e

(B) Minimum at x = e

(C)Maximum at x =

1

e

(D) Minimum at x = 1e






eE1 / T2 / K2 / L3 / V2 / R11 / A [GATE – ME – 1993]

(01) The function 2( , ) 3 2 f x y x y xy y x has

(A) No local extremism

(B) One local maximum but no local minimum

(C) One local minimum but no local maximum

(D)One local minimum and one local maximum

eE1 / T2 / K2 / L3 / V2 / R11 / A [GATE – CS – 1998] (02) Find the points of local maxima and minima if

any of the following function defined in

0 6, x 3 26 9 15. x x x

eE1 / T2 / K2 / L3 / V2 / R11 / AB [GATE – – 2002]

(03) The function f(x, y) = 2 32 2 x xy y has

(A) Only one stationary point at (0, 0)

(B) Two stationary points at (0, 0) and 1 1

,6 3

(C) Two stationary points at (0, 0) and (1, -1)

(D) No stationary point.


(04) For real x, the maximum value of sin

cos

x

x

e

eis

(A) 1 (B) e

(C) 2e (D)


(05) For the function f(x, y) = 2 2 x y defined on R 2,

the point (0, 0) is

(A) A local minimum

(B) Neither a local minimum (nor) a local

maximum.

(C) A local maximum

(D) Both a local minimum and a local maximum


(06) Consider the function 22( ) 4 f x x where x

is a real number. Then the function has

(A) Only one minimum (B) Only two minima

(C) Three minima (D) Three maxima

-----00000-----

2.3 Differential Calculus


eE1 / T2 / K3 / L0 / V1 / R11 / AA [GATE – – 1996]

(01) If a function is continuous at a point its first

derivative

(A) May or may not exist

(B) Exists always

(C) Will not exist

(D) Has a unique value




www.targate.org



(01) Given y = 2 2 10 x x the value of 1 X

dy

dx

is

equal to

(A) 0 (B) 4

(C) 12 (D) 13


(02) The total derivative of the function ‘xy’ is

(A) xdy ydx (B) xdx ydy

(C) dx dy (D) dx dy


eE1 / T2 / K3 / L2 / V2 / R11 / AA [GATE – – 1997]

(01) If 2

0( )

x x t dt then __________ d

dx

(A) 22 x (B) x

(C) 0 (D) 1

eE1 / T2 / K3 / L2 / V2 / R11 / AA [GATE – – 2000]

(02) If f(x, y, z) =

2 2 22 2 2 1/2

2 2 2( ) ,

f f f x y z

x y z

is equal to

_______

(A) 0 (B) 1

(C) 2 (D) 2 2 2 5/23( ) x y z


eE1 / T2 / K3 / L3 / V2 / R11 / AC [GATE – – 2004]

(01) If x = ( sin )a θ θ and (1 cos ) y a θ then

______ dy

dx

(A) sin2

θ (B) cos

2

θ

(C) tan2

θ (D) cot

2

θ

eE1 / T2 / K3 / L3 / V2 / R11 / AC [GATE – – 2005]

(02) By a change of variables x(u, v) = uv,

( , ) / y u v v u in a double integral, the integral

( , ) f x y changes to , .u f uvv

Then

( , )u v is _______

(A) 2v

u

(B) 2 u v

(C) 2V (D) 1


(03) If (x) = sin | | x then the value of df

dxat

4

π x

is

(A) 0 (B) 1

2

(C) 1

2 (D) 1


(04) Given a function

2 2( , ) 4 6 8 4 8, f x y x y x y the optimal

values of f(x, y) is





(A) a minimum equal to10

3

(B) a maximum equal to10

3

(C) a minimum equal to8

3

(D) a maximum equal to8

3


(05) The function f(x) = 22 3 x x has

(A) A maxima at x = 1 and a minima at x = 5

(B) A maxima at x = 1 and a minima at x = - 5

(C) Only a maximum at x = 1

(D) Only a minima at x = 0

-----00000-----

2.4 Integral Calculus



(01) IF S = 3

1

α

X dx then S has the value

(A) 1

3

(B)

1

4

(C) 1

2 (D) 1


(02) The value of the integral1

21

1dx

x is

(A) 2 (B) does not exists

(C) - 2 (D)

-----00000-----


eE1 / T2 / K4 / L1 / V1 / R11 / A [GATE – PI – 1995]

(01) Given2

1cos , x y t dt then ________ dy

dx


(02) If at every point of a certain curve, the slope of

the tangent equals2 x

y

, the curve is _________

(A) A straight line (B) A parabola

(C) A circle (D) An Ellipse


(03) The value of the integral/2

/2( cos )

π

π x x dx

is

(A) 0 (B) 2π

(C) π (D) 2π



α

α

dx

x

(A) π (B) 2π

(C) 2

π (D) π




www.targate.org


eE1 / T2 / K4 / L2 / V2 / R11 / AA [GATE – – 1994]

(01) The integration of log xdx has the value

(A) ( log 1) x x (B) log x x

(C) (log 1) x x (D) None of the above

eE1 / T2 / K4 / L2 / V2 / R11 / AC [GATE – – 1995]

(02) By reversing the order of integration

2

2 2

0( , )

x

x f x y dydx may be represented as _____

(A) 2

2 2

0( , )

x

x f x y dydx

(B) 2

0( , )

y

y f x y dxdy

(C) 4

0 /2( , )

y

y f x y dxdy

(D) 2

2 2

0( , )

x

x f x y dydx

eE1 / T2 / K4 / L2 / V2 / R11 / AD [GATE – – 2000]

(03) /2 /2

0 0sin( )

π π

x y dxdy

(A) 0 (B) π

(C) 2π (D) 2

eE1 / T2 / K4 / L2 / V2 / R11 / AC [GATE – – 2002]

(04) The value of the following definite integral in

/2

/2sin2 _______ 1 cos

π

π x dx

(A) - 2 log 2 (B) 2

(C) 0 (D) None

eE1 / T2 / K4 / L2 / V2 / R11 / AC [GATE – – 2002]

(05) The value of the following improper integral is

1

0 log x x dx

= ________

(A) 1

4 (B) 0

(C) 1

4 (D) 1

eE1 / T2 / K4 / L2 / V2 / R11 / AA [GATE – – 2005] (06) Changing the order of integration in the double

integral I =8 2

0 /4( , )

x f x y dy dx leads to

I = ( , ) .s q

r p f x y dy dx What is q?

(A) 4y (B) 16 y2

(C) x (D) 8


(07) 6 7sin sina

a x x dx

is equal to

(A) 60

2 sina

xdx

(B) 7

02 sin

a

xdx

(C) 6 7

02 sin sin

a

x x dx

(D) zero





eE1 / T2 / K4 / L2 / V2 / R11 / AA [GATE – – ]

(08) The value of the integral I =2 /8

0

1

2

xe

π

dx is

____

(A) 1 (B) π

(C) 2 (D) 2π


(09) The value of 3

0 0(6 )

x

x y is _____

(A) 13.5 (B) 27.0

(C) 40.5 (D) 54.0


(10) The following plot shows a function y which

varies linearly with x. The value of the integral I

=2

1

ydx

(A) 1 (B) 2.5

(C) 4 (D) 5


(11) The integral 6sin( )6

α

α

π

t t dt

evaluates to

(A) 6 (B) 3

(C) 1.5 (D) 0

-----00000-----


eE1 / T2 / K4 / L3 / V2 / R11 / A [GATE – – 1994]

(01) The value of 3 1/2

0. y

e y dy

is ________


(02) The value of 2 2

0 0

α α x y

e e dx dy is

(A) 2

π (B) π

(C) π (D)

4

π


(03) The integral2

0

1sin( )cos

2t τ τdτ

equals

(A) Sin cost (B) 0

(C) 1 cos2

t (D) 1 sin2

t


(04) The value of the integral of the function

3 4( , ) 4 10g x y x y along the straight line

segment from the point (0, 0) to the point (1, 2)

in the xy-plane is

(A) 33 (B) 35

(C) 40 (D) 56


(05) Which of the following integrals is unbounded?

(A) 0

tanπ / 4

dx (B) 20

1

1

α

dx x




www.targate.org

(C) 0

.α

x x e dx

(D) 1

0

1

1dx

x


(06) Given 1,i what will be the evaluation of

the definite integral 20

cos sincos sin

π

x i x dx x i x ?

(A) 0 (B) 2

(C) – i (D) i

-----00000-----

2.5 Limit and Continuity


eE1 / T2 / K5 / L0 / V1 / R11 / AB [GATE – – 1995]

(01) 0

1lim sin ______ x

x x

(A) (B) 0

(C) 1 (D) Does not exist

eE1 / T2 / K5 / L0 / V1 / R11 / AD [GATE – – ]

(02) Limit of the following series as x approaches

2

is

3 5 7

( )3! 5! 7!

x x x f x x

(A) 2

3

π (B)

2

π

(C) 3

π (D) 1

eE1 / T2 / K5 / L0 / V1 / R11 / AD [GATE – – 2000]

(03) Limit of the function4

4

1( )

a f x

x

as x

is given

(A) 1 (B) 4

ae

(C) (D) 0

eE1 / T2 / K5 / L0 / V1 / R11 / AA [GATE – – 2003]

(04) 2

0

sinlim ____ x

x

x

(A) 0 (B)

(C) (D) – 1


(05) Consider the function f(x) = 3| | , x where x is

real. Then the function f(x) at x = 0 is

(A) Continuous but not differentiable

(B) Once differentiable but not twice.

(C) Twice differentiable but not thrice.

(D) Thrice differentiable


(06) The minimum value of function 2 y x in the

interval [1, 5] is

(A) 0 (B) 1

(C) 25 (D) Undefined






(07)

2

30

12

lim

x

x

xe x

x

(A) 0 (B) 1

6

(C) 1

3 (D) 1

eE1 / T2 / K5 / L0 / V1 / R11 / AA [GATE – – ]

(08) sin

lim ______ cos x

x x

x x

(A) 1 (B) - 1

(C) (D)

eE1 / T2 / K5 / L0 / V1 / R11 / AC [GATE – – ]

(09) 0

sinlim x

x

xis

(A) Indeterminate (B) 0

(C) 1 (D)


(10) The value of 1/3

8

2lim

8 x

x

x

is

(A) 1

16 (B)

1

12

(C) 1

8 (D)

1

4


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

t has

(A) A minimum (B) A discontinuity

(C) A point of inflection (D) A Maximum


(12) What is0

sinlimθ

θ

θ equal to?

(A) θ (B) sin θ

(C) 0 (D) 1

-----00000-----


eE1 / T2 / K5 / L1 / V1 / R11 / AB [GATE – – 1995]

(01) The function f(x) = | 1| x on the interval

[ 2,0] is _________

(A) Continuous and differentiable

(B) Continuous on the interval but not

differentiable at all points

(C) Neither continuous nor differentiable

(D) Differentiable but not continuous

eE1 / T2 / K5 / L1 / V1 / R11 / AA [GATE – – 1997]

(02) 0

sinlim ,θ

mθ

θ where m is an integer, is one of the

following:

(A) m (B) m π

(C) mθ (D) 1




www.targate.org

eE1 / T2 / K5 / L1 / V1 / R11 / AC [GATE – – 1997]

(03) If y=|x| for x < 0 and y = x for 0 x then

(A) dy

dxis discontinuous at x = 0

(B) y is discontinuous at x = 0

(C) y is not defined at x = 0

(D) Both y and dy

dxare discontinuous at x = 0


(04) 0

sin( / 2)limθ

θ

θ

(A) 0.5 (B) 1

(C) 2 (D) Not defined

eE1 / T2 / K5 / L1 / V1 / R11 / AB [GATE – – 2004]

(05) The value of the function.3 2

3 20( ) lim

2 7 x

x x f x

x x

is _____

(A) 0 (B) 1

7

(C) 1

7 (D)


(06) The value of the expression0

sin( )lim

x x

x

e x

is

(A) 0 (B) 1

2

(C) 1 (D) 1

1 e



(01) 5

0

1 1lim _____

10 1

j x

jx x

e

e

(A) 0 (B) 1.1

(C) 0.5 (D) 1

eE1 / T2 / K5 / L2 / V2 / R11 / AD [GATE – – 1999]\

(02) Limit of the function,2

limn

n

n n is _____

(A) 12

(B) 0

(C) (D) 1

eE1 / T2 / K5 / L2 / V2 / R11 / AA [GATE – – 2001]

(03) The value of the integral is I = 2

0cos

π / 4

x dx

(A) 5

2

(B) 5

(C) 5

2 (D)

5

2


(04) Limit of the following sequence as n is

___________ 1/n x n

(A) 0 (B) 1

(C) (D) -






(05) What is the value of /4

cos sinlim

/ 4 x π

x x

x π

(A) 2

(B) 0

(C) 2

(D) Limit does not exist


(06) The function | 2 3 | y x

(A) is continuous x R and differential

x R

(B) is continuous x R and differential

x R except at x =3

2

(C) is continuous x R and differential

x R except at x =2

3

(D) Is continuous x R and except at x = 3

and differential x R

-----00000-----


eE1 / T2 / K5 / L3 / V2 / R11 / A [GATE – ME – 1993]

(01) 0

( 1) 2(cos 1)lim ________

(1 cos )

x

x

x e x

x x


(02) Value of the function limx a

x a x a

is

________

(A) 1 (B) 0

(C) (D) a

eE1 / T2 / K5 / L3 / V2 / R11 / AD [GATE – – 2002]

(03) Which of the following functions is not

differentiable in the domain [-1, 1]?

(A) f(x) = x2

(B) f(x) = x – 1

(C) f(x) = 2

(D) f(x) = maximum (x – x)


(04) What should be the value of λ such that the

function defined below is continuous at ?2

π x =

cos

2( ) 2

1

2

λ x π if x

π x

f x

π if x

(A) 0 (B) 2π

(C) 1 (D) 2

π

-----00000-----




www.targate.org

2.6 Series



(01) The infinite sires 1 112 3

(A) Converges (B) Diverges

(C) Oscillates (D) Unstable


(02) A series expansion for the function sin θ is ______

(A) 2 4

1 ........2! 4!

θ θ

(B) 3 6

........3! 5!

θ θ θ

(C)2 3

1 ........2! 3!

θ θ θ

(D) 3 5

......3! 5!

θ θ θ

-----00000-----


eE1 / T2 / K6 / L1 / V1 / R11 / AB [GATE – – 1995]

(01) The third term in the taylor’s series expansion of

xe about ‘a’ would be _______

(A) ( )ae x a (B) 2( )

2

ae x a

(C) 2

ae (D) 3( )

6

ae x a


(02) Which of the following function would have

only odd powers of x in its Taylor series

expansion about the point x = 0?

(A)

3sin x (B)

2sin x

(C) 3cos x (D) 2cos x

-----00000-----


eE1 / T2 / K6 / L2 / V2 / R11 / AB [GATE – – ]

(01) Consider the following integral

4

1lim ___

a

x x dx

(A) Diverges (B) converges to 1/3

(C) Converges to 31a (D) converges to 0


(02) If ........ y x x x x α then y(2) =

_____

(A) 4 (or) 1 (B) 4 only

(C) 1 only (D) Undefined


(03) The series 2

0

1( 1)

4

αm

mm

x

converges for

(A) 2 2 x (B) 1 3 x

(C) 3 1 x (D) 3 x






(04) The infinite series

3 5 7

( )3! 5! 7!

x x x f x x Converges to

(A) cos( ) x (B) sin( ) x

(C) sinh( ) x (D) xe



(01) In the Taylor series expansion of sin xe x

about the point x = π , the coefficient of

2

x π is

(A) π e (B) 0.5 π

e

(C) 1π

e (D) 1π

e

eE1 / T2 / K6 / L3 / V2 / R11 / AC [GATE – – ]

(02) In the Taylor series expansion of ex about x = 2,

the coefficient of (x – 2)4 is

(A) 1

4! (B)

42

4!

(C) 2

4!

e (D)

4

4!

e


(03) The Taylor series expansion of sin x about

6 x

is given by

(A) 2 3

1 3 1 3

2 2 6 4 6 12 6 x x x

(B) 3 5 7

3! 5! 7!

x x x x

(C)

3 5 7

6 6 661! 3! 5! 7!

x x x x

(D) 1

2


(04)The Taylor series expansion of sin

x π at x π is

given by

(A)

2( )

1 3!

x π

(B) 2( )

13!

x π

(C) 2( )

13!

x π

(D) 2( )

13!

x π

-----00000-----

2.7 Vector Calculus


eE1 / T2 / K7 / L0 / V1 / R11 / AD[GATE – ME – 1996]

(01) The expression curl (grad ) f where f is a

scalar function is

(A) Equal to 2 f

(B) Equal to div (grad f )

(C) A scalar of zero magnitude

(D) A vector of zero magnitude




www.targate.org

eE1 / T2 / K7 / L0 / V1 / R11 / AA [GATE – –]

(02) Stokes theorem connects

(A) A line integral and a surface integral

(B) A surface integral and a volume integral

(C) A line integral and a volume integral

(D) Gradient of a function and its surface

integral.

-----00000-----


eE1 / T2 / K7 / L1 / V1 / R11 / AA [GATE – – ]

(01) Given a vector field F

, the divergence theorem

states that

(A) . .S V

F ds F dv

(B) .S V

F ds F dv

(C) S V

F ds F dv

(D) S V

F ds F dv

eE1 / T2 / K7 / L1 / V1 / R11 / AA [GATE – – ]

(02) If a vector ( ) R t has a constant magnitude than

(A) . 0dR

Rdt

(B) . 0d R

Rdt

(C) .d R

R Rdt

(D) d R

R Rdt


(03) Which one of the following is Not associated

with vector calculus?

(A) Stoke’s theorem

(B) Gauss Divergence theorem

(C) Green’s theorem

(D) Kennedy’s theorem


(04) P where P is a vector is equal to

(A) 2P P P (B) ( )P P

(C) 2 )P P (D) 2( )P P


(05) The area of a triangle formed by the tips of

vectors ,a b and c is

(A) 1

( ) ( )2

a b a c

(B) 1

| ( ) ( ) |2

a b a c

(C) 1

| |2

a b c

(D) 1

( )2

a b c

eE1 / T2 / K7 / L1 / V1 / R11 / AD [GATE – – ]

(06) The angle (in degrees) between two planar

vectors3 1

2 2a i j and

3 1

2 2b i j

is





(A) 30 (B) 60

(C) 90 (D) 120


(07) The divergence of the vector field

( ) ( ) ( ) x y i y x j x y z k is

(A) 0 (B) 1

(C) 2 (D) 3

eE1 / T2 / K7 / L1 / V1 / R11 / AD [GATE – – ]

(08) If r is the position vector of any point on a

closed surface S that encloses the volume V

then ( . )S

r d s is equal to

(A) 1

2V (B) V

(C) 2V (D) 3V

eE1 / T2 / K7 / L1 / V1 / R11 / AB [GATE – – ]

(09) If a vector field V is related to another field A

through V = A , which of the following is

true?

Note: C and SC refer to any closed contour and

any surface whose boundary is C.

(A) . .Sc

C

V dl A ds

(B) . .Sc

C

A dl V ds

(C) . .Sc

C

V dl A ds

(D) . .Sc

C

A dl V ds

eE1 / T2 / K7 / L1 / V1 / R11 / AA [GATE – – 1993]

(10) A sphere of unit radius is centred at the origin.

The unit normal at a point (x, y, z) on the

surface of the sphere is the vector.

(A) (x, y, z) (B)

1 1 1

, ,3 3 3

(C) , ,3 3 3

x y z

(D) , ,2 2 2

x y z

-----00000-----


eE1 / T2 / K7 / L2 / V2 / R11 / AB [GATE – – 1995]

(01) The directional derivative of the function f(x, y,

z) = x + y at the point P(1, 1, 0) along the

direction i j

is

(A) 1/ 2 (B) 2

(C) - 2 (D) 2

eE1 / T2 / K7 / L2 / V2 / R11 / AD [GATE – – 1999]

(02) For the function 2 3ax y y to represent the

velocity potential of an ideal fluid, 2 should

be equal to zero. In that case, the value of ‘a’has to be

(A) -1 (B) 1

(C) – 3 (D) 3

eE1 / T2 / K7 / L2 / V2 / R11 / AB [GATE – – 2002]

(03) The directional derivative of the following

function at (1, 2) in the direction of (4i + 3j) is:

F(x, y) = 2 2 x y




www.targate.org

(A) 4/5 (B) 4

(C) 2/5 (D) 1

eE1 / T2 / K7 / L2 / V2 / R11 / AC [GATE – – 2003]

(04) The vector field F = xi yj (where i and j are

unit vectors) is

(A) Divergence free, but not irrotational

(B) Irrotational, but divergence free

(C) Divergence free and irrotational

(D) Neither divergence free nor irrotational

eE1 / T2 / K7 / L2 / V2 / R11 / AC [GATE – – ]

(05) For the scalar field u =2 3

,2 3

x y the magnitude

of the gradient at the point (1, 3) is

(A) 13

9 (B)

9

2

(C) 5 (D) 9

2

eE1 / T2 / K7 / L2 / V2 / R11 / AC [GATE – – ]

(06) The directional derivative of

2 2 2( , , ) 2 3 f x y z x y z at the point p(2, 1, 3)

in the direction of the vector 2a i k is

_____.

(A) – 2.785 (B) – 2.145

(C) – 1.789 (D) 1.000


(07) The expression V =2

2 1 H

o

hπR dh

H

for the

volume of a cone is equal to _______.

(A) 2

2 1 R

o

hπR dr

H

(B) 2

2 1 R

o

hπR dh

H

(C) 1 R

o

r 2πrH dh

R

(D) 2

1 R

o

r 2πrH dr

R


(08) The velocity vector is given as

2 25 2 3 .v xyi y j yz k The divergence of this

velocity vector at (1, 1, 1) is

(A) 9 (B) 10

(C) 14 (D) 15

eE1 / T2 / K7 / L2 / V2 / R11 / AA [GATE – – ]

(09) Divergence of the vector field ( , , )v x y z

( cos ) ( cos ) x xy y i y xy j 2 2 2[(sin ) ] z x y k is

(A) 22 cos z z (B) 2sin 2 cos xy z z

(C) sin cos x xy z (D) None of these





eE1 / T2 / K7 / L2 / V2 / R11 / AB [GATE – – ]

(10) The directional derivative of the scalar function

2 2( , , ) 2 f x y z x y z at the point P = (1, 1,

2) in the direction of the vector 3 4a i j is

(A) – 4 (B) - 2

(C) – 1 (D) 1

eE1 / T2 / K7 / L2 / V2 / R11 / AB [GATE – –]

(11) For a scalar function f(x, y, z) = 2 2 23 2 , x y z

the gradient at the point P (1, 2, -1) is

(A) 2 6 4i j k (B) 2 12 4i j k

(C) 2 12 4i j k (D) 56

eE1 / T2 / K7 / L2 / V2 / R11 / AB [GATE – –]

(12) For a scalar function f(x, y, z) = 2 2 23 2 , x y z

the directional derivative at the point P (1, 2, -1)

in the direction of a vector 2i j k is

(A) - 18 (B) 3 6

(C) 3 6 (D) 18

eE1 / T2 / K7 / L2 / V2 / R11 / AC [GATE – –]

(13) The divergence of the vector field

2 ˆˆ ˆ3 2 xzi xyj yz k at a point (1, 1, 1) is equal to

(A) 7 (B) 4

(C) 3 (D) 0

eE1 / T2 / K7 / L2 / V2 / R11 / AD [GATE – –]

(14) F(x, y) = 2 2ˆ ˆ( ) ( ) . x y x xy a y xy a its line

integral over the straight line from ( , ) (0,2) x y

to (x, y) = (2, 0) evaluates to

(A) - 8 (B) 4

(C) 8 (D) 0

eE1 / T2 /K7 / L2 / V2 / R11 / AC [GATE – –]

(15) The line integral of the vector function

2ˆ ˆ2F xi x j along the x – axis from x = 1 to x

= 2 is

(A) 0 (B) 2.33

(C) 3 (D) 5.33

eE1 / T2 /K7 / L2 / V2 / R11 / AA [GATE – –]

(16) Divergence of the 3 – dimensional radial vector

fields r is

(A) 3 (B) 1

r

(C) ˆˆ ˆi j k (D) ˆˆ ˆ3 i j k

eE1 / T2 /K7 / L2 / V2 / R11 / AA [GATE – – ]

(17) If a and b are two arbitrary vectors with

magnitudes a and b respectively, 2| |a b will

be equal to

(A) 2 2 2( . )a b a b (B) .ab ab

(C) 2 2 2( . )a b a b (D) .ab a b




www.targate.org

eE1 / T2 /K7 / L2 / V2 / R11 / AA [GATE – PI – 2011]

(18) If A (0, 4, 3), B (0, 0, 0) and C (3, 0, 4) are there

points defined in x, y, z coordinate system, then

which one of the following vectors is

perpendicular to both the vectors B A and BC

(A) 16 9 12i j K (B) 16 9 12i j K

(C) 16 9 12i j K (D) 16 9 12i j K

eE1 / T2 /K7 / L2 / V2 / R11 / AD [GATE – – ]

(19) Consider a closed surface ‘S’ surrounding a

volume V. If r is the position vector of a point

inside S with n the unit normal on ‘S’, the

value of the integral ˆ5 .r n ds is

(A) 3V (B) 5V

(C) 10V (D) 15V

eE1 / T2 /K7 / L2 / V2 / R11 / AB [GATE – – ]

(20) The two vectors [1, 1, 1] and [1, a, a2] where a =

1 3

2 2 j

are

(A) Orthonormal (B) Orthogonal

(C) Parallel (D) Collinear

-----00000-----


eE1 / T2 / K7 / L3 / V2 / R11 / AB [GATE – – 1994]

(01) The directional derivative of f(x, y) =

2 2 22 3 x y z at point P(2, 1, 3) in the

direction of the vector 2a i k

is

(A) 4 / 5 (B) 4 / 5

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

(02) The derivative of f(x, y) at point (1, 2) in the

direction of vector i + j is 2 2 and in the

direction of the vector -2j is -3. Then the

derivative of f(x, y) in direction –i-2j is

(A) 2 2 3 / 2 (B) 7 / 5

(C) 2 2 3 / 2 (D) 1 / 5

eE1 / T2 / K7 / L3 / V2 / R11 / AC [GATE – – 2005]

(03) Value of the integral 2 ,c

xydy y dx where, c is

the square cut from the first quadrant by th line

x= 1 and y = 1 will be (Use Green’s theorem to

change the line integral into double integral)

(A) 1/2 (B) 1

(C) 3/2 (D) 5/3

eE1 / T2 / K7 / L3 / V2 / R11 / AA [GATE – – 2005]

(04) The line integral .V dr of the vector function

V(r) = 2 22 xyzi x zj x yk from the origin to

the point P (1, 1, 1)

(A) is 1

(B) is Zero

(C) is – 1





(D) Cannot be determined without specifying

the path

eE1 / T2 / K7 / L3 / V2 / R11 / AA [GATE – –]

(05) A scalar field is given by f = 2/3 2/3 , x y where

x and y are the Cartesian coordinates. The

derivative of ‘f’ along the line y = x directed

away from the origin at the point (8, 8) is

(A) 2

3 (B)

3

2

(C) 2

3 (D)

3

2

eE1 / T2 / K7 / L3 / V2 / R11 / AB [GATE – –]

(06) Consider points P and Q in xy – plane with P =

(1, 0) and Q = (0, 1). The line integral

2 ( )Q

P xdx ydy along the semicircle with the

line segment PQ as its diameter

(A) is – 1

(B) is 0

(C) 1

(D) Depends on the direction (clockwise (or)

anti-clockwise) of the semi circle

eE1 / T2 /K7 / L3 / V2 / R11 / AC [GATE – – ]

(07) If 2ˆ ˆ x y A xy a x a

then . A dl

over the path

shown in the figure is

(A) 0 (B) 2

3

(C) 1 (D) 2 3

eE1 / T2 /K7 / L3 / V2 / R11 / AA [GATE – – ]

(08) The line integral2

1

( )P

P ydx xdy from 1 1 1( , )P x y

to 2 2 2( , )P x y along the semi-circle P1P2 shown

in the figure is

(A) 2 2 1 1 x y x y

(B) 2 2 2 22 1 2 1( ) ( ) y y x x

(C) 2 1 2 1( )( ) x x y y

(D) 2 22 1 2 1( ) ( ) y y x x

eE1 / T2 /K7 / L3 / V2 / R11 / AA [GATE – PI – 2011]

(09) If T(x, y, z) = 2 2 22 x y z defines the

temperature at any location (x, y, z) then the




www.targate.org

magnitude of the temperature gradient at point

P(1, 1, 1) is -----

(A) 2 6 (B) 4

(C) 24 (D) 6

-----00000-----

2.8 AREA / VOLUME


E1 / T2 / K8 / L3 / V2 / R11 / AD [GATE – – 1994]

(01) The volume generated by revolving he area

bounded by the parabola 2 8 y x and the line

2 x about y-axis is

(A) 128

5

π (B)

5

128π

(C) 127

5π (D) None of the above


(02) The area bounded by the parabola 22 y x and

the lines 4 x y is equal to _________

(A) 6 (B) 18

(C) (D) None of the above

eE1 / T2 / K8 / L3 / V2 / R11 / AB [GATE – – 1997]

(03) Area bounded by the curve y = x2 and the lines

x = 4 and y = 0 is given by

(A) 64 (B) 64

3

(C) 128

3 (D)

128

4

eE1 / T2 / K8 / L3 / V2 / R11 / AB [GATE – – 2004]

(04) The area enclosed between the parabola y = x2

ad the straight line y = x is _____

(A) (B)

(C) (D)

eE1 / T2 / K8 / L3 / V2 / R11 / AA [GATE – – 2004]

(05) The volume of an object expressed in spherical

co-ordinates is given by

2 /3 12

0 0 0

sinπ π

V r drd d θ

The value of the integral

(A) 3

π (B)

6

π

(C) 2

3

π (D)

4

π


(06) Consider the shaded triangular region P shown

in the figure. What is ?P

xy dx dy

(A) 1

6 (B)

2

9

(C) 7

16 (D) 1






(07) If (x, y) is continuous function defined over (x,

y) [0,1] [0,1] Given two constraints, 2 x y

and 2 , y x the volume under f(x, y) is

(A) 2

1

0( , ) y x y

y x y f x y dxdy

(B) 2 2

1 1( , )

y x

y x x y f x y dxdy

(C) 1 1

0 0( , )

y x

y x f x y dxdy

(D) 0 0

( , ) y x x y

x x f x y dxdy


(08) The area enclosed between the curves 2 4 y x

and 2 4 x y is

(A) 16

3 (B) 8

(C) 32

3 (D) 16


(09) The parabolic arcy = , 1 2 x x is revolved

around the x-axis. The volume of the solid of

revolution is

(A) 4

π (B)

2

π

(C) 3

4

π (D)

3

2

π

-----00000-----

2.9 Miscellaneous


eE1 / T2 / K9 / L0 / V1 / R11 / AC [GATE – – 1999]

(01) The function f(x) = ex is ________

(A) Even (B) Odd

(C) Neither even nor odd (D) None

eE1 / T2 / K9 / L0 / V1 / R11 / AB [GATE – – 1998]

(02) The continuous function f(x, y) is said to have

saddle point at (a, b) if

(A) ( , ) ( , ) 0 x y f a b f a b

2

0 xy xx yy f f f at (a, b)

(B) ( , ) 0, ( , ) 0, x y f a b f a b 2 0 xy xx yy f f f at (a,b)

(C) ( , ) 0, ( , ) 0, x y f a b f a b 2 0 xy xx yy f f f at (a, b)

(D) ( , ) 0, ( , ) 0, x y f a b f a b 2 0 xy xx yy f f f at (a,b)


(03) The expression ln xe

for x > 0 is equal to

(A) – x (B) x

(C) 1 x (D) 1

x

eE1 / T2 / K9 / L0 / V1 / R11 / AD [GATE – – 1998]

(04) A discontinuous real function can be expressed

as

(A) Taylor’s series and Fourier’s series

(B) Taylor’s series and not by Fourier’s series

(C) Neither Taylor’s series nor Fourier’s series

(D) Not by Taylor’s series, but by Fourier’s

series




www.targate.org


eE1 / T2 / K9 / L1 / V1 / R11 / AD [GATE – – 1998]

(01) The taylor’s series expansion of sin x

is_________

(A) 2 4

1 ..........2! 4!

x x (B)

2 4

1 ......2! 4!

x x

(C) 3 5

....2! 4!

x x x (D)

3 5

....3! 5!

x x x



(01) The infinite series2

1

( !)

(2 )!n

n

n

(A) Converges (B) Diverges

(C) Is unstable (D) Oscillate


(02) What is the value of 2

1lim 1 ?

n

n α n

(A) 0 (B) 2e

(C) 1/2e

(D) 1


(03) Roots of the algebraic equation

3 2 1 0 x x x are

(A) (1, j, -j) (B) (1, -1, 1)

(C) (0, 0, 0) (D) (-1, j, -j)

-----00000-----


eE1 / T2 / K9 / L3 / V2 / R11 / AC [GATE – – 1997]

(01) The curve given by the equation 2 2 3 x y axy

is

(A) Symmetrical about x –axis

(B) Symmetrical about y – axis

(C) Symmetrical about the line y = x

(D) Tangential to x = y = a/3


(02) For the function , xe the linear approximation

around x = 2 is

(A) (3 – x ) 2e

(B) 1 x

(C) 23 2 2 1 2 x e

(D) 2e


(03) For | | 1,coth( ) x x can be approximated as

(A) x (B) x2

(C) 1

x (D)

2

1

x


(04) The length of the curvey y = 3/ 22

3 x between x =

0 & x = 1 is

(A) 0.27 (B) 0.67

(C) 1 (D) 1.22






(05) A parabolic cable is held between two supports

at the same level. The horizontal span between

the supports is L.

The sag at the mid-span is h. The equation of

the parabola is y =

2

24 ,

x

h L where x is the

horizontal coordinate and y is the vertical

coordinate with the origin at the centre of the

cable. The expression for the total length of the

cable is

(A) 2 2

401 64

L h xdx

L

(B) 2 2/2

402 1 64

L h xdx

L

(C) 2 2/2

401 64

L h xdx

L

(D) 2 2/2

402 1 64

L h xdx

L


(06) The distance between the origin and the point

nearest to it on the surface Z2 = 1 + xy is

(A) 1 (B) 3

2

(C) 3 (D) 2

-----00000-----



03L

D iffer ent ia l Equat ions Complete subtopic in this chapter, is in the scope of “GATE- ME/EC/EE SYLLABUS”

3.1 Degree and order of DE



(01) The partial differential equation

2 2

2 20

y y x y

has

(A) degree 1 and order 2

(B) degree 1 and order 1

(C) degree 2 and order 1

(D) degree 2 and order 1

eE1 / T3 / K1 / L0 / V1/ R11 / AC [GATE – PI – 2005]

(02) The differential equation

32

1 dydx

=

222

2

d yC

dx

is of

(A) 2nd order and 3rd degree

(B) 3rd order and 2nd degree

(C) 2nd order and 2nd degree

(D) 3rd order and 3rd degree

eE1 / T3 / K1 / L0 / V1/ R11 / AB [GATE – EC – 2009]

(03) The order of differential equation

32 4

2

t d y dy y edxdx

is

(A) 1 (B) 2

(C) 3 (D) 4


(04) The following differential equation has

322

23 4 2

d y dy y x

dt dt

(A) degree = 2, order = 1

(B) degree = 1, order = 2

(C) degree = 4, order = 3

(D) degree = 2, order = 3

eE1 / T3 / K1 / L0 / V3 / R11 / A [GATE – CE – 2010]

(05) The order and degree of a differential equation

332

34 0

d y dy y

dxdx

are respectively

(A) 3 and 2 (B) 2 and 3

(C) 3 and 3 (D) 3 and 1





eE1 / T3 / K1 / L1 / V1/ R11 / AB [GATE – CE – 2007]

(06) The degree of the differential equation

23

22 0

d x x

dt

is

(A) 0 (B) 1

(C) 2 (D) 3

eE1 / T3 / K1 / L1 / V1/ R11 / AB [GATE – ME – 2007]

(07) The differential equation4 2

4 20

d y d yP ky

dx dx

is

(A) Linear of Fourth order

(B) Non – Linear of fourth order

(C) Non – Homogeneous

(D) Linear and Fourth degree


(08) The equation2

2 8

2( 4 ) 8

d y dy x x y x

dxdx is a

(A) partial differential equation

(B) non-linear differential equation

(C) non-homogeneous differential equation

(D) ordinary differential equation

eE1 / T3 / K1 / L1 / V2 / R11 / AC [GATE – – 1995]

(09) The differential equation

11 3 5 1 3( sin ) cos y S x y y x is

(A) homogeneous

(B) non – linear

(C) 2nd order linear

(D) non – homogeneous with constant

coefficients

-----00000-----

3.2 Higher Order DE


eE1 / T3 / K2 / L1 / V1/ R11 / AC [GATE – PI – 2011]

(01) The solution of the differential equation2

26 9 9 6

d y dy y x

dxdx with C1 and C2 as

constants is

(A) 31 2( ) x y C x C e

(B) 3 31 2

x x y C e C e

(C) 31 2( ) x y C x C e x

(D) 31 2( ) x y C x C e x

eE1 / T3 / K2 / L1 / V1/ R11 / AD [GATE – CE – 1998]

(02) The general solution of the differential equation

22

20

d y dy x x y

dxdx is

(A) Ax + Bx2 (A, B are constants)

(B) Ax + B logx (A, B are constants)

(C) Ax + Bx2logx (A, B are constants)

(D) Ax + Bxlog (A, B are constants)’’



TOPIC. 03 – DIFFERENTIAL EQUATIONS

www.targate.org



(01) 2 x y e is a solution of the differential equation

11 1 2 0 y y y

(A) True (B) False


(02) The general solution of the differential equation

2( 4 4) 0 D D y is of the form (given D =

d

dxan C1, C2 are constants)

(A) 21

xC e (B) 2 21 2

x xC e C e

(C) 2 21 2

x xC e C e (D) 2 21 2

x xC e C x e


(03) For the differential equation2

2

20,

d yk y

dx

the

boundary conditions are

(i) 0 y for 0 x and

(ii) 0 y for x a

The form of non-zero solution of y (where m

varies over all integers) are

(A) sinm

m

mπx y Aa

(B) cosm

m

mπx y A

a

(C) mπ

am

m

y A x

(D) mπx

am

m

y A e

eE1 / T3 / K2 / L2 / V1/ R11 / AA [GATE – EC – 1994]

(04) Match each of the items A, B, C with an appropriate

item from 1, 2, 3, 4 and 5

(A)

2

1 2 3 42

d y dya a y a y a

dxdx

(B)

3

1 2 33

d ya a y a

dx

(C)

22

1 2 320

d y dya a x a x y

dxdx

(1) Non – linear differential equation

(2) Linear differential equation with constant

coefficients

(3) Linear homogeneous differential equation

(4) Non – linear homogeneous differential equation

(5) Non – linear first order differential equation

(A) A – 1, B – 2, C – 3 (B) A – 3, B – 4, C - 2

(C) A – 2, B – 5, C – 3 (D) A – 3, B – 1, C – 2

eE1 / T3 / K2 / L2 / V1/ R11 / AD [GATE – EC – 2007]

(05) The solution of the differential equation

22

22

d yk y y

dx

under the boundary conditions

(i) 1 y yat 0 x and

(ii) 2 y yat x where k, 1 y

and 2 yare

constant is

(A) 2

1 2 2( ) x

k y y y e y

(B) 2 1 1( ) x

k y y y e y

(C) 1 2 1( ) sin

x y y y h y

k





(D) 1 2 2( ) x

k y y y e y

eE1 / T3 / K2 / L2 / V1/ R11 / AA [GATE – PI – 2008]

(06) The solutions of the differential equation

2

22 2 0

d y dy y

dxdx

are

(A) (1 ) (1 ),i x i xe e

(B) (1 ) (1 ),i x i xe e

(C) (1 ) (1 ),i x i xe e

(D) (1 ) , (1 )i x i x

e e

eE1 / T3 / K2 / L2 / V1/ R11 / AB [GATE – EE – 2010]

(07) For the differential equation

2

2 6 8 0

d x dx

xdt dt

with initial conditions x(0) = 1 and 0

0t

dx

dt

the

solution

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

2 4( ) 2 t t x t e e

(C) 6 4( ) 2t t

x t e e

(D) 2 4( ) 2t t

x t e e


(08) A function n(x) satisfies the differential equation

2

2 2

( ) ( )0

d n x n x

dx L

where L is a constant. The

boundary conditions are n(0) = k and n( )

= 0. The

solution to this equation is 06. A function n(x)

satisfies the differential equation.

This equation is

(A) ( ) exp /n x k x L

(B) ( ) exp /n x k x L

(C) 2

( ) exp / x n k x L

(D) 2( ) exp /n x k x L


(09) For

22

24 3 3 , xd y dy

y edxdx

the particular integral

is

(A)

21

15

xe (B)

21

5

xe

(C) 23 x

e (D) 3

1 2 x xc e c e


(10) The homogeneous part of the differential equation

2

22

d y dy p q r

dxdx

(p, q, r are constants) has real

distinct roots if

(A) 2 4 0 p q (B)

2 4 0 p q

(C) 2 4 0 p q (D)

2 4 p q r


(11) A solution of the differential equation

2

25 6 0

d y dy y

dxdx

is given by

(A) 2 3 x x y e e (B)

2 3 x x y e e

(C) 2 3 x x y e e

(D) None of these.


(12) It is given that" 2 ' 0, y y y

(0) 0 y

(1) 0 y

what is(0.5)? y

(A) 0 (B) 0.37

(C) 0.62 (D) 1.13




www.targate.org

eE1 / T3 / K2 / L2 / V1/ R11 / AC [GATE – ME – 2007]

(13) The solution of 2dy y

dx with initial value y(0) =

1 is bounded in the internal is

(A) x (B) 1 x

(C) 1, 1 x x (D) 2 2 x


(14) For initial value problem 2 101 10.4 , x y y y e

y(0)=1.1 and y(0) = - 0.9. Various solutions are

written in the following groups. Match the type

of solution with the correct expression.

Group – I Group – II

P. General solution

of Homogeneous

equations

(1)0.1

xe

Q. Particular integral (2) xe

[A

cos10 sin10 x B x ]

R. Total solutionsatisfying

boundary

conditions

(3) cos10 0.1 x x

e x e

Codes:

(A) P – 2, Q – 1, R -3 (B) P -1, Q -3, R – 2

(C) P – 1, Q – 2, R – 3 (D) P -3 , Q – 2, R – 1

eE1 / T3 / K2 / L2 / V2 / R11 / AC [GATE – – ]


2

20

d y

dx

with boundary conditions

(i)1

dy

dx

at x = 0

(ii)1

dy

dx

at x = 1 is

(A) y = 1

(B) y = x

(C) y = x + c where c is an arbitrary constants are

arbitrary constants

(D) y = 1 2C x C where C1, C2 are arbitrary

constants


(16) The solution to the differential equation

11 1( ) 4 ( ) 4 ( ) 0 f x f x f x

(A) 21( ) x f x e

(B) 2 21 2( ) , ( ) x x f x e f x e

(C) 2 21 2( ) , ( ) x x f x e f x xe

(D) 21 2( ) , ( ) x x f x e f x e

eE1 / T3 / K2 / L2 / V2 / R11 / AC [GATE – – 1995]

(17) The solution of a differential equation

11 13 2 0 y y y is of the form

(A) 21 2

x xc e c e (B) 31 2

x xc e c e

(C) 21 2

x xc e c e (D) 21 2 2 x xc e c


(18) The particular solution for the differential

equation2

23 2

d y dy y

dxdt sx is

(A) 0.5 cos 1.5sin x x (B) 1.5cos 0.5sin x x

(C) 1.5sin x (D) 0.5cos x






(19) Solve for y if 2

22 0

d y dy y

dt dt with y(0) = 1

and 1(0) 2 y

(A) (1 ) t t e (B) (1 ) t t e

(C) (1 ) t t e (D) (1 ) t t e


(20) Which of the following is a solution of the

differential equation

2

( 1) 0?d y dy

P qdx dx

Where p = 4, q = 3

(A) 3 xe

(B) x xe

(C) 2 x x e (D) 2 2 x x e


(21) For the equation ( ) 3 ( ) 2 ( ) 5, x t x t x t the

solution x(t) approaches the following values as

t

(A) 0 (B) 5/2

(C) 5 (D) 10

eE1 / T3 / K2 / L2 / V2 / R11 / A [GATE – EE – 2005]

(22) The solution to the ordinary differential equation

2

26 0d y dy y

dxdx

is

(A) 3 2

1 2 x x y C e C e

(B) 3 2

2 2 x x y C e C e

(C)

3 21 2

x x y C e C e

(D) 3 2

1 2 x x y C e C e


(23) The solution for the following differential

equation with boundary conditions y(0) = 2 and

1(1) 3 y is where2

23 2

d y x

dx

(A) 3 2

3 23 2 x x y x

(B) 233 5 22

x y x x

(C) 3 2 5 23 2

x x y x

(D) 2

3 352 2 x y x x


(24) The solution2

22 17 0;

d y dy y

dxdx

4

(0) 1, 0π

x

dy y

dx

in the range 0

4π x is

given by

(A) 1

[cos4 sin 4 ]4

xe x x

(B) 1

[cos4 sin 4 ]4

xe x x

(C) 4 1[cos4 sin ]

4 xe x x

(D) 4 1[cos4 sin4 ]

4

xe x x


(25) The complete solution of the ordinary differential

equation2

20

d y dyP qy

dxdx is

31 2

x x y C e C e then P and q are

(A) P = 3, q = 3 (B) P = 3, q = 4

(C) P = 4, q = 3 (D) P = 4, q = 4




www.targate.org

3.3 Leibnitz linear equation



(01) Which of the following is a solution to thedifferential equation

( ) 3 ( ) 0, (0) 2?d

x t x t xdt

(A) ( ) 3 t x t e (B) 3( ) 2 t x t e

(C) 23( )

2

x t t

(D) 2( ) 3 x t t


(02) For the differential equation 5 0dy

ydt

with

(0) 1, y the general solution is:

(A) 5t e (B) 5t

e

(C) 55 t e (D) 5t e

eE1 / T3 / K3 / L2 / V2 / R112 / A [GATE – EE – 1994]


dy y x

dx x with the condition that 1 y at x = 1

is:

(A) 2

2

33

x y

x (B)

1

2 2

x y

x

(C) 2

3 3

x y (D)

22

3 3

x y

x



2

2 xdy xy e

dx

with (0) 1 y is

(A) 2

(1 ) x x e (B) 2

(1 ) x x e

(C) 2

(1 ) x x e (D) 2

(1 ) x x e

eE1 / T3 / K3 / L2 / V1/ R11 / AD [GATE – ME – 2005]

(05) If 2 2ln2dy x x xydx x

and y(1) = 0 then what

is y(e)?

(A) e (B) 1

(C) 1

e (D)

2

1

e


(06) The differential equation,

dy py Q

dx

is a linear

equation of first order only if,

(A) P is a constant but Q is a function of y

(B) P and Q are functions of y (or) constants

(C) P is a functions of y but Q is a constant

(D) P and Q are functions of x (or) constants


(07) The solution of 4dy x y x

dx with condition

6(1)5

y

(A) 4 1

5

x y

x (B)

44 4

5 5

x y

x

(C) 4

15

x y (D)

5

15

x y


(08) Transformation to linear form by substituting v =

1 n y of the equation ( ) ( ) , 0ndyP t y q t y n

dt

will be





(A) (1 ) (1 )dv

n pv n qdt

(B) (1 ) (1 )dv

n pv n qdt

(C) (1 ) (1 )dv

n pv n qdt

(D) (1 ) (1 )dv

n pv n qdt


(09) Consider the differential equation xdy y e

dx

with (0) 1. y Then the value of (1) y is

(A) 1e e

(B) 11

2e e

(C) 11

2e e (D) 12 e e

eE1 / T3 / K3 / L2 / V2 / R11 / AD [GATE – PI – 2010] (10) The solution of the differential equation

2 1dy

ydx

satisfying the condition y(0) = 1 is

(A) 2

x y e (B) y x

(C) cot

4

π y x (D) tan

4

π y x

-----00000-----

3.4 Miscellaneous


eE1 / T3 / K4 / L1 / V1/ R11 / AA [GATE – ME – 2003]


2 0dy

ydx

is

(A) 1

y x c

(B) 3

3

x y c

(C) xc e

(D) Unsolvable as equations is non – linear

eE1 / T4 / K4 / L1 / V1/ R11 / AB [GATE – CE – 1997]

(02) For the differential equation

( , ) ( , ) 0dy

f x y g x ydx

to be exact is

(A) f g y x (B) f g

x y

(C) f g (D) 2 2

2 2

f g

x y

eE1 / T4 / K4 / L1 / V1/ R11 / AC [GATE – CE – 1999]

(03) If C is a constant, then the solution of

21dy

ydx

is

(A) sin( ) y x c (B) cos( ) y x c

(C) tan( ) y x c (D) x y e c

-----00000-----




www.targate.org



(01) The solution for the differential equation

2dy x y

dx with the condition that y = 1 at x = 0 is

(A) 1

2 x y e (B) 3

ln( ) 43

x y

(C) 2

ln( )2

x y (D)

3

3 x

y e


(02) A body originally at 060 cools down to 40 in 15

minutes when kept in air at a temperature of

025 .c What will be the temperature of the body

at the and of 30 minutes?

(A) 035.2 C (B) 031.5 C

(C) 028.7 C (D) 015 C


(03) Consider the differential equation 21 .dy

ydx

Which one of the following can be particular

solution of this differential equation?

(A) tan( 3) y x (B) tan( 3) y x

(C) tan( 3) x y (D) tan( 3) x y

-----00000-----



(01) Solution of the differential equation

3 2 0dy

y xdx

represents a family of

(A) ellipses (B) circles

(C) parabolas (D) hyperbolas


(02) Which one of the following differential equations

has a solution given by the function

5sin 35

π y x

(A) 5

cos(3 ) 03

dy x

dx (B)

5(cos3 ) 0

3

dy x

dx

(C) 2

29 0

d y y

dx (D)

2

29 0

d y y

dx

eE1 / T3 / K4 / L3 / V2 / R11 / AC [GATE – – ]

(03) Let . x f y What is x

x y

at x = 2, y = 1?

(A) 0 (B) ln 2

(C) 1 (D) 2

1

ln


(04) Match each differential equation in Group I to its

family of solution curves from Group II.

Group I Group II

P: dy y

dx x

(1) Circles

Q: dy y

dx x

(2) Straight lines





R: dy x

dx y

(3) Hyperbolas

S: dy x

dx y

(A) P-2, Q-3, R-3, S-1 (B) P-1, Q-3, R-2, S-1

(C) P-2,Q-1,R-3, S-3 (D) P-3, Q-2, R-1, S-2


(05) With K as constant, the possible solution for the

first order differential equation 3 xdye

dx

is

(A) 31

3 xe K

(B) 31( 1)

3 xe K

(C) 33 xe K (D) kx y Ce


(06) The differential2

2sin 0

d y dy y

dxdx is

(A) linear (B) non – linear

(C) homogeneous (D) of degree two


(07) The necessary & sufficient for the differential

equation of the form M(x, y)dx + N(x, y) dy = 0to be exact is

(A) M = N (B) M N

x y

(C) M N

y x

(D)

2 2

2 2

M N

x y


(08) The solution of differential equation

, (0)dy

Ky y C dx

is

(A) Ky x Ce (B) cy x Ke

(C) kx y e C (D) kx y Ce


(09) Consider the differential equation 2 0 y y y

with boundary conditions (0) 1 y (0) 0 y .The

value of (2) y is

(A) – 1 (B) - e 1

(C) 2e

(D) 2e


(10) Consider the differential equation 2(1 ) .dy

y x

dx

The general solution with constant “C” is

(A) 2

tan2

x y C

(B) 2tan

2

x y C

(C) 2tan2

x y C

(D) 2

tan2

x y C


(11) The one dimensional heat conduction partial

differential equation2

T T

t x

is

(A) parabolic (B) hyperbolic

(C) elliptic (D) mixed




www.targate.org


(12) The number of boundary conditions required to

solve the differential equation2 2

2 20

x y

is

(A) 2 (B) 0

(C) 4 (D) 1


(13) Biotransformation of an organic compound

having concentration (x) can be modelled using

an ordinary differentia equation 2 0,dx

kxdt

where k is the reaction rate constant. If x = a at t

= 0 then solution of the equation is

(A) kt x a e (B) 1 1 kt

a x

(C) (1 )kt x a e (D) x a kt

-----00000-----



(1) 1 1

0 1 1n n n n

n n f a x a n a y a y

where

ia (i = 0 to n) are constants then v f f

x y x y

is

(A) f

n (B)

n

f

(C) n f (D) n f

-----00000-----




04L

Complex V ar iable Complete subtopic in this chapter, is in the scope of “GATE- ME/EC/EE SYLLABUS”

4.1Cauchy’s Theorem


eE1 / T4 / K1 / L2 / V2 / R11 / AD [GATE – – ]

(01) The value of the contour integral2| | 2

1

4 z jdz

z

in the positive sense is

(A) 2

jπ (B) 2π

(C) 2

jπ (D)

2

π


(02) Using Cauchy’s integral theorem, the value of the

integral (integration being taken in contour clock

wise direction)

3 6

3C

zdz

z i

is where C is |z| = 1

(A) 2

481

π πi (B) 6

8

π πi

(C) 4

681

π πi (D) 1


(03) For the function3

sin z

zof a complex variable z,

the point z = 0 is

(A) a pole of order 3 (B) a pole of order 2

(C) a pole of order 1 (D) not a singularity


(04) The value of 2

1

(1 )C

dz z where C is the contour

| / 2| 1 z i

(A) 2 π i (B) π

(C) 1tan ( ) z (D) 1tanπ i z


(05) If the semi – circulator controur D of radius 2 is

as shown in the figure. Then the value of the

integral2

1

1 D

s ds is



TOPIC. 04 – COMPLEX VARIABLE

www.targate.org

(A) i π (B) i π

(C) π (D) π


(06) The value of the integralcos(2 )

(2 1)( 3)C

πz

z z dz

where C is a closed curve given by 1 1 1 is

(A) π i (B) 5

π i

(C) 2

5

π i (D) π i


(07) If f(z) = 10 1C C z then

1( ( )

unit

f zdz

z is given

(A) 12 π C (B) 02 1π C

(C) 12π j C (D) 02 (1 )π j C


(08) The contour integral1 z

C

e dz with C as the

counter clock – wise unit circle in the z – plane is

equal to

(A) 0 (B) 2 π

(C) 2 1π (D)


(09) For an analytic function f(x + iy) = u(x, y) + iv(x,

y), u is given by u = 2 23 3 . x y The expression

for v. Considering k is to be constant is

(A)

2 2

3 3 y x k (B) 6 6 x y k

(C) 6 6 y x k (D) 6 xy k

-----00000-----



(01) Consider likely applicability of Cauchy’s Integral

theorem to evaluate the following integral

counter clock wise around the unit circle C I =

sec ,C

zdz z being a complex variable. The value

of I will be

(A) I = 0; Singularities set =

(B) I = 0; Singularities set =

(2 1)/ 0,1, 2,........

2

nπ n

(C) I = / 2π ; Singularities set =

; 0,1,2, ...........nπ n

(D) None of the above.


(02) The value of sin

,a

zdz

zwhere the contour of the

integration is a simple closed curve around the

origin is





(A) 0 (B) 2 π j

(C) (D) 1

2 π j

eE1 / T4 / K1 / L3 / V2 / R11 / AA [GATE – – ]


3 4,

4 5C

zdz

z z

when

C is the circle | | 1 z is given by

(A) 0 (B) 110

(C) 4

5 (D) 1


(04) The value of 2

4,

1C

zdz

z using Cauchy’s integral

around the circle | 1| 1 z where z x iy is

(A) 2 π i (B) 2πi

(C) 32

πi (D) 2π i

-----00000-----

4.2 Miscellaneous



(01) The real part of the complex number z x iy is

given by

(A) Re( ) * z z z (B) *

Re( )2

z z z

(C) *

Re( )2

z z z

(D) Re( ) * z z z


(02) coscan be represented as

(A) 2

i ie e (B)

2

i ie e

i

(C) i ie e

i

(D)

2

i ie e


(03) If Z = x + jy where x, y are real then the value of

| | jze is

(A) 1 (B) 2 2

x ye

(C) ye (D) y

e


(04) The product of complex numbers (3 – 21) & (3 +

i4) results in

(A) 1 + 6i (B) 9 – 8i

(C) 9 + 8i (D) 17 + i 6


(05) The analytical function has singularities at, where

f(z) =2

1

1

z

z

(A) 1 and -1 (B) 1 and i

(C) 1 and – i (D) i and – i

-----00000-----




www.targate.org



(01) ,ii where i = 1 is given by

(A) 0 (B) /2π e

(C) 2

π (D) 1


(02) ze is a periodic with a period of

(A) 2π (B) 2πi

(C) π (D) iπ


(03) The function

2 2 11 log( ) tan2

yw u iv x y i

x is not

analytic at the point.

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

(C) (1, 0) (D) (2, α )


(04) The value of the expression5 10

3 4

i

i

(A) 1 2i (B) 1 2i

(C) 2 i (D) 2 i

eE1 / T4 / K2 / L1 / V1 / R11 / AD [GATE – – ]

(05) The equation sin( ) 10 z has

(A) no real (or) complex solution

(B) exactly two distinct complex solutions

(C) a unique solution

(D) an infinite number of complex solutions


(06) The contour C in the adjoining figure is described

by 2 2 16. x y Then the value of

2 8

(0.5) (1.5)C

z

dz z j

(A) 2 π j (B) 2 π j

(C) 4 π j (D) 4 π j


(07) Let j = 1. Then one value of j j is

(A) 3 (B) 1

(C) 12 (D) 2

π

e


(08) If a complex number z =3 1

2 2i then 4 z is

(A) 2 2 2i (B) 1 3

2 2i

(C)

3 1

2 2i (D)

3 1

8 8i


(09) The product of two complex numbers 1 + i & 2 –

5 i is

(A) 7 – 3i (B) 3 – 4i

(C) – 3 – 4 i (D) 7 + 3i






(10) The residue of the function f(z) =

2 2

1

( 2) ( 2) z z at z = 2 is

(A) 1

32

(B) 1

16

(C) 116 (D) 1

32

-----00000-----



(01) The complex number z x jy which satisfy

the equation | 1| 1 z lie on

(A) a circle with (1, 0) as the centre and radius 1

(B) a circle with (-1, 0) as the centre and radius 1

(C) y – axis

(D) x – axis


(02) The bilinear transformation w =1

1

z

z

(A) Maps the inside of the unit circle in the z –

plane to the left half of the w - plane

(B) Maps the outside the unit circle in the z –

plane to the left half of the w – plane

(C) maps the inside of the unit circle in the z –

plane to right half of the w – plane

(D) maps the outside of the unit circle in the z –

plane to the right half of the w – plane


(03) Which one of the following is Not true for the

complex numbers z1 and z2?

(A) 1 1 22

2 2| |

z z z

z z

(B) 1 2 1 2| | | | | | z z z z

(C) 1 2 1 2| | | | | | z z z z

(D) 2 2 2 21 2 1 2 1 2| | | | 2 | | 2 | | z z z z z z


(04) Consider the circle | 5 5 | 2 z i in the complex

number plane (x, y) with z = x+iy. The minimum

distance from the origin to the circle is

(A) 5 2 2 (B) 54

(C) 34 (D) 5 2


(05) Let 3 , z z where z is a complex number not

equal to zero. Then z is a solution of

(A) 2 1 z (B)

3 1 z

(C) 4 1 z (D)

9 1 z


(06) For the function of a complex variable w = l nz

(where w = u jv and z x jy ) the u =

constant lines get mapped i the z – plane as




www.targate.org

(A) Set of radial straight lines

(B) Set of concentric circles

(C) Set of co focal hyperbolas

(D) Set of co focal ellipses


(07) Potential function is given 2 2. x y What

will be the stream function with the condition

0 at x = 0, y = 0?

(A) 2xy (B) 2 2 x y

(C) 2 2 x y (D) 2 22 x y


(08) The modulus of the complex number 3 41 2

ii

is

(A) 5 (B) 5

(C) 1

5 (D)

1

5


(09) If complex number satisfies the equation

3 1 then the value of 1

1

is _______

(A) 0 (B) 1

(C) 2 (D) 4

eE1 / T4 / K2 / L2 / V2 / R11 / AA [GATE – – ]

(10) The integral ( ) f z dz evaluated around the unit

circle on the complex plane for ( )Coz z

p z z

is

(A) 2 π i (B) 4 π i

(C) 2 π i (D) 0

-----00000-----



(01) If ( , ) x y and ( , ) x y are functions with

continuous 2nd derivatives then

( , ) ( , ) x y i x y can be expressed as an

analytic function of ( 1) x iy i when

(A) , x x y y

(B) , x x y y

(C) 2 2 2 2

2 2 2 21

x y x y

(D) 0 x y x y

eE1 / T4 / K2 / L3 / V2 / R11 / AB [GATE – – ]

(02) A complex variable z x j (0.1) has its real

part x varying in the range to . Which one

of the following is the locus (shown in thick

lines) of 1

zin the complex plane?






(03) One of the roots of equation 3 , x j where j is

the +ve square root of – 1 is

(A) j (B) 3

2 2

j

(C) 3

2 2

j (D)

3

2 2

j


(04) An analytic function of a complex variable z =

x iy is expressed as f(z) = ( , ) ( , )u x y i v x y

where 1i . If u = xy then the expression for

v should be

(A) 2( )

2

x yk

(B)

2

2

x yk

(C) 2 2

2

y xk

(D)

2( )

2

x yk


(05) If f(x + iy) = 3 23 ( , ) x xy i x y where 1i

and ( ) f x iy is an analytic function then

( / ) x y is

(A) 3 23 y x y (B) 2 33 x y y

(C) 4 34 x x y (D) 2 xy y


(06) A point z has been plotted in the complex plane

as shown in the figure below




www.targate.org

The plot of the complex number

-----00000-----





05Pr obabilit y and Stat ist ics

Complete subtopic in this chapter, is in the scope of “GATE- CS/ME/EC/EE SYLLABUS

5.2 Combination


eE1 / T5 / K2 / L1 / V1 / R11 / AD [GATE – – 2004]

(01) From a pack of regular playing cards, two cards

are drawn at random. What is the probability that

both cards will be kings, if the card is NOT

replaced?

(A) 1/26 (B) 1/52

(C) 1/169 (D) 1/221

-----00000-----


eE1 / T5 / K2 / L2 / V1 / R11 / AD [GATE – – 2003]

(01) A box contains 10 screws, 3 of which are

defective. Two screws are drawn at random with

replacement. The probability that none of the two

screws is defective will be

(A) 100% (B) 50%

(C) 49% (D) None of these

-----00000-----


eE1 / T5 / K2 / L3 / V2 / R11 / AB [GATE – IT – 2005]

(01) A bag contains 10 blue marbles, 20 black marblesand 30 red marbles. A marble is drawn from the

bag, its colour recorded and it is put back in the

bag. This process is repeated 3 times. The

probability that no two no two of the marbles

drawn have the same colour is

(A) 1

36

(B) 1

6

(C) 1

4 (D)

1

3


(02) A box contains 2 washers, 3 nuts and 4 bolts.

Items are drawn from the box at random one at a

time without replacement. The probability of drawing 2 washers first followed by 3 nuts and

subsequently the 4 bots is

(A) 2/315 (B) 1/630

(C) 1/1260 (D) 1/2520


(03) A box contains 4 while balls and 3 red balls. In

succession, two balls are randomly selected and

removed from the box. Given that first removed



TOPIC. 05 – PROBABILITY AND STATISTICS

www.targate.org

ball is white, the probability that the 2nd removed

ball is red is

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

(C) ¼ (D) 4/7


(04) Two white and two black balls, kept in two bins,

are arranged in four ways as shown below. In

each arrangement, a bin has to be chosen

randomly and only one ball needs to be picked

randomly from the chosen bin. Which one of the

following arrangements has the highest

probability for getting a white ball picked?


(05) There are two containers with one containing 4red and 3 green balls and the other containing 3

blue balls and 4 green balls. One ball is drawn at

random from each container. The probabilities

that one of the balls is red and the other is blue

will be ________

(A) 1

7 (B)

4

49

(C) 12

49 (D)

3

7

-----00000-----

5.3 Probability related problems



(01) If P and Q are two random events, then the

following is true

(A) Independence of P and Q implies that

probability 0P Q

(B) Probability P Q probability (P) +

probability (Q)

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

must be independent

(D) Probability P Q probability (P)


(02) A fair coin is tossed 3 times in succession. If the

first toss produces a head, then the probability of

getting exactly two heads in three tosses is

(A) 1

8 (B)

1

2

(C) 3

8 (D)

3

4






(03) A fair dice is rolled twice. The probability that an

odd number will follow an even number is

(A) 1

2 (B)

1

6

(C) 1

3 (D)

1

4

-----00000-----


eE1 / T5 / K3 / L1 / V1 / R11 / AD [GATE – – 1997]

(01) The probability that it will rain today is 0.5, the

probability that it will rain tomorrow is 0.6. The

probability that it will rain either today or

tomorrow is 0.7. What is the probability that it

will rain today and tomorrow?

(A) 0.3 (B) 0.25

(C) 0.35 (D) 0.4

eE1 / T5 / K3 / L1 / V1 / R11 / AD [GATE – – 2000]

(02) E1 and E2 are events in a probability space

satisfying the following constraints

1 2( ) ( );P E P E 1 2( ) 1P E Y E : 1 2& E E are

independent then 1( )P E

(A) 0 (B) 1

4

(C) 1

2 (D) 1

eE1 / T5 / K3 / L1 / V2 / R11 / AD [GATE – – 2003]

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

Given P(A) = 1, P(B) =1

2the values of P(A/B)

and P(B/A) respectively are

(A) 1 1

,4 2

(B) 1 1

,2 4

(C) 1

,12

(D) 1

1,2

eE1 / T5 / K3 / L1 / V1 / R11 / AC [GATE – – 2004]

(04) A hydraulic structure has four gates which

operate independently. The probability of failure

of each gate is 0.2. Given that gate 1 has failed,

the probability that both gates 2 and 3 will fail is

(A) 0.240 (B) 0.200

(C) 0.040 (D) 0.008

eE1 / T5 / K3 / L1 / V1 / R11 / AB [GATE – – 2001]

(05) Seven car accidents occurred in a week, what is

the probability that they all occurred on same

day?

(A) 7

1

7 (B)

6

1

7

(C) 7

1

2 (D)

7

7

2


(06) If a fair coin is tossed 4 times, what is the

probability that two heads and two tails will

result?




www.targate.org

(A) 3

8 (B)

1

2

(C) 5

8 (D)

3

4


(07) Two dice are thrown simultaneously. The

probability that the sum of numbers on both

exceeds 8 is

(A) 4

36 (B)

7

36

(C) 936

(D) 1036


(08) A coin is tossed 4 times. What is the probability

of getting heads exactly 3 times?

(A) 1/4 (B) 3/8

(C) 1/2 (D) ¾


(09) An examination consists of two papers, paper 1

and paper 2. The probability of failing in

probability of failing in paper 1 is 0.6. The

probability of a student failing in both the papers

is

(A) 0.5 (B) 0.18

(C) 0.12 (B) 0.06


(10) A fair coin is tossed independently four times.

The probability of the event “The number of

times heads show up is more than the number of

times tails show up” is

(A) 1/16 (B) 1/8

(C) ¼ (D) 5/16


(11) Two coins are simultaneously tossed. The

probability of two heads simultaneously

appearing is

(A) 1/8 (B) 1/6

(C) 1/4 (D) ½


(12) An unbiased coin is tossed five times. The

outcome of each loss is either a head or a tail.

Probability of getting at least one head is

________

(A) 1

32 (B)

13

32

(C) 16

32 (D)

31

32


(13) It two fair coins are flipped and at least one of the

outcomes is known to be a head, what is the

probability that both outcomes are heads?

(A) 1

3 (B)

1

4

(C) 1

2 (D)

2

3

-----00000-----






eE1 / T5 / K3 / L2 / V2 / R11 / AD [GATE – – 1995]

(01) The probability that a number selected at random

between 100 and 999 (both inclusive) will not

contain the digit 7 is

(A) 16

25 (B)

39

10

(C) 27

75 (D)

18

25

eE1 / T5 / K3 / L2 / V1 / R11 / AB [GATE – – 1998]

(02) A die is rolled three times. The probability that

exactly one odd number turns up among the three

outcomes is

(A) 1

6 (B)

3

8

(C)

1

8 (D)

1

2

eE1 / T5 / K3 / L2 / V1 / R11 / AB [GATE – – 1998]

(03) The probability that two friends share the same

birth-month is

(A) 1/6 (B) 1/12

(C) 1/144 (D) 1/24

eE1 / T5 / K3 / L2 / V2 / R11 / AB [GATE – IT – 2004]

(04) In a population of N families, 50% of the families

have three children, 30% of families have two

children and the remaining families have one

child. What is the probability that a randomly

picked child belongs to a family with two

children?

(A) 3

23 (B)

6

23

(C) 3

10 (D)

3

5


(05) The probability that there are 53 Sundays in a

randomly chosen leap year is

(A) 1

7 (B)

1

14

(C) 1

28 (D)

2

7


(06) A fair dice is tossed two times. The probability

that the 2nd toss results in a value that is higher

than the first toss is

(A) 2

36

(B) 2

6

(C) 5

12 (D)

1

2

eE1 / T5 / K3 / L2 / V1 / R11 / AC [GATE – – 1999]

(07) Consider two events E1 and E2 such that

1

1( ) ,

2

p E 2

1( )

3

p E and 1 2

1( ) .

5

E I E Which

of the following statement is true?

(A) 1 2

2( )

3 p E Y E

(B) 1 E and 2 E are independent

(C) 1 E and 2 E are not independent

(D) 1 2( / ) 4 / 5P E E




www.targate.org


eE1 / T5 / K3 / L3 / V2 / R11 / AC [GATE – – 2002]

(01) Four fair coins are tossed simultaneously. The

probability that at least one heads and at least one

tails turn up is

(A) 1

16 (B)

1

8

(C) 7

8 (D)

15

16


(02) Two cards are drawn at random in succession

with replacement from a deck of 52 well shuffled

cards Probability of getting both ‘Aces’ is

(A) 1

169 (B)

2

169

(C) 1

13 (D)

2

13


(03) In a game, two players X and Y toss a coin

alternately. Whosever gets a ‘heat’ first, wins the

game and the game is terminated. Assuming that

player X starts the game the probability of player

X winning the game is

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

(C) 2/3 (D) 3/4

eE1 / T5 / K3 / L3 / V2 / R11 / AC [GATE – –]

(04) A fair coin is tossed 10 time. What is the

probability that only the first two tosses will yield

heads?

(A) 2

1

2

(B) 2

2

110

2c

(C) 10

1

2

(D) 10

2

110

2c


(05)What is the probability that a divisor of 10

99

is a

multiple of 1096?

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

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


(06) The box 1 contains chips numbered 3, 6, 9, 12

and 15. The box 2 contains chips numbered 6, 11,

16, 21 and 26. Two chips, one from each box are

drawn at random.

The numbers written on these chips are

multiplied. The probability for the product to be

an even number is ___________ .

(A) 6

25 (B)

2

5

(C) 3

5 (D)

19

25






(07) It is estimated that the average number of events

during a year is three. What is the probability of

occurrence of not more than two events over a

two-year duration? Assume that the number of

events follow a poisson distribution.

(A) 0.052 (B) 0.062

(C) 0.072 (D) 0.082


(08) A single die is thrown two times. What is the

probability that the sum is neither 8 nor 9?

(A) 1

9 (B)

5

36

(C) 1

4 (D)

3

4


(09) Assume for simplicity that N people, all born in

April (a month of 30 days) are collected in a

room, consider the event of at least two people in

the room being born on the same date of the

month even if in different years e.g. 1980 and

1985. What is the smallest N so that the

probability of this exceeds 0.5 is?

(A) 20 (B) 7

(C) 15 (D) 16

-----00000-----

5.4 Bays theorems

No Question

5.5 Probability Distribution



(01) Assume that the duration in minutes of a

telephone conversation follows the exponential

distribution f(x) =

/51

, .5

x

e x o

The probability

that the conversation will exceed five minutes is

(A) 1

e (B)

11

e

(C) 2

1

e

(D) 2

11

e

eE1 / T5 / K5 / L0 / V1 / R11 / AB [GATE – – 2005]

(02) Lot has 10% defective items. Ten items are

chosen randomly from this lot. The probability

that exactly 2 of the chosen items are defective is

(A) 0.0036 (B) 0.1937

(C) 0.2234 (D) 0.3874

-----00000-----




www.targate.org


eE1 / T5 / K5 / L1 / V1 / R11 / AA [GATE – – 2000]

(01) In a manufacturing plant, the probability of

making a defective bolt is 0.1. The mean and

standard deviation of defective bolts in a total of

900 bolts are respectively

(A) 90 and 9 (B) 9 and 90

(C) 81 and 9 (D) 9 and 81


(02) A lot had 10% defective items. Ten items are

chosen randomly from this lot. The probability

that exactly 2 of the chosen items are defective is

(A) 0.0036 (B) 0.1937

(C) 0.2234 (D) 0.3874


(03) The life of a bulb (in hours) is a random variable

with an exponential distribution f(t) = ,αt α e

0 .t The probability that its value lies b/w

100 and 200 hours is

(A) 100 200α αe e (B) 100 200

e e

(C) 100 200α αe e (D) 200 100α αe e


(04) If the standard deviation of the spot speed of

vehicles in a highway is 8.8 kemps and the mean

speed of the vehicles is 33 kmph, the coefficient

of variation in speed is

(A) 0.1517 (B) 0.1867

(C) 0.2666 (D) 0.3646



(01) If X is a continuous random variable whose

probability density function is given by

2(5 2 ), 0 2( ) 0,

k x x x

f x otherwise

Then P(x >

1) is

(A) 3/14 (B) 4/5

(C) 14/17 (D) 17/28

-----00000-----


eE1 / T5 / K5 / L3 / V2 / R11 / AB [GATE – – 1999]

(01) Four arbitrary points 1 1( , ) x y , 2 2 3 3( , ),( , ) x y x y ,

4 4( , ) x y , are given in the xy – plane using the

method of least squares, if, regressing y upon x

gives the fitted line y = ax + b; and regressing x

upon y gives the fitted line x = cy + d, then

(A) The two fitted lines must coincide

(B) the two fitted lines need not coincide

(C) It is possible that ac = 0

(D) a must be 1/c

-----00000-----





5.6 Random Variable


eE1 / T5 / K6 / L0 / V1 / R11 / AB [GATE – – 2009]

(01) Using given data points tabulated below, astraight line passing through the origin is fitted

using least squares method. The slope of the line

x 1 2 3

y 1.5 2.2 2.7

(A) 0.9 (B) 1

(C) 1.1 (D) 1.5


(02) Let X and Y be two independent random

variables. Which one of the relations b/w

expectation (E), variance (Var ) and covariance

(Cov) given below is FALSE?

(A) E(XY) = E(X) E(Y)

(B) cov (X, Y) = 0

(C) Var (X + Y) = Var (X) + Var (Y)

(D) E(X2Y2) = (E(X))2(E(y))2

eE1 / T5 / K6 / L0 / V2 / R11 / A [GATE – – 2008]

(03) Three values of x and y are to be fitted in a

straight line in the form y a bx by the method

of least squares. Given 6, 21, x y

2 14, 46, x xy the values of a and b are

respectively

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

(C) 2, 1 (D) 3, 2


(04) The standard deviation of a uniformly distributed

random variable b/w 0 and 1 is

(A) 1

12 (B)

1

3

(C) 5

12 (D)

7

12

-----00000-----



(01) X is uniformly distributed random variable that

takes values between 0 and 1. The value of E(X3)

will be

(A) 0 (B) 1/8

(C) 1/4 (D) ½


(02) A random variable is uniformly distributed over

the interval 2 to 10. Its variance will be

(A) 16/3 (B) 6

(C) 256/9 (D) 36


(03) Consider a Gaussian distributed random variable

with zero mean and standard deviation . The

value of its cumulative distribution function at

the origin will be

(A) 0 (B) 0.5

(C) 1 (D) 10




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(04) Px(X) = Me(-2|x|) + Ne(-3|x|) is the probability

density function for the real random variable X,

over the entire x-axis, M and N are both positive

real numbers. The equation relating M and N is

(A) 2

13

M N (B) 1

2 13

M N

(C) 1 M N (D) 3 M N


(05) For a random variable ( ) x x following

normal distribution, the mean is 100 μ If the

probability is P = α for 110. x Then the

probability of x lying b/w 90 and 110 i.e.

(90 110)P x and equal to

(A) 1 2α (B) 1 α

(C) 1 / 2α (D) 2α


(06) If three coins are tossed simultaneously, the

probability of getting at least one head is

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

(C) ½ (D) 7/8


(07) Consider a company that assembles computers.

The probability of a faulty assembly of any

computer is p. The company therefore 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 + (1 – p) (1 – q) (B) (1 – q)p

(C) (1 – p)q (D) pq

-----00000-----



(01) If a random variable X satisfies the poission’s

distribution with a mean value of 2, then the

probability that X > 2 is

(A) 22e (B) 21 2e

(C) 23e (D) 21 3e

eE1 / T5 / K6 / L2 / V2 / R11 / AD [GATE – CS – 2011] (02) If the difference between the expectation of the

square of a random variable 2| E (X ) | and the

square of the expectation of the random variable

2E(X ) is denoted by R, then,

(A) R = 0 (B) R < 0

(C) R 0 (D) R > 0






(03) The random variable X taken on the values 1, 2

(or) 3 with probabilities2 5 1 3

,5 5

P P and

1.5 2

5

Prespectively the values of P and E(X)

are respectively

(A) 0.05, 1.87 (B) 1.90, 5.87

(C) 0.05, 1.10 (D) 0.25, 1.40

-----00000-----



(01) The standard normal probability function can be

approximated as

F(X N) = 0.012

1

1 exp 1.7255 | | N N X X where

X N = standard normal deviate. If mean and standard deviation of annual precipitation are 102

cm and 27 cm respectively, the probability that

the annual precipitation will be b/w 90 cm and

102 cm is

(A) 66.7% (B) 50.0%

(C) 33.3% (D) 16.7%


(02) Consider two independent random variable X and

Y with identical distributions. The variables X

and Y take values 0,1 and 2 with probability 1/2,

¼ and ¼ respectively. What is the conditional

probability P(X + Y = 2/X – Y = 0)?

(A) 0 (B) 1/16

(C) 1/6 (D) 1


(03) A discrete random variable X takes value from 1

to 5 with probabilities as shown in the table. A

student calculates the mean of X as 3.5 and her

teacher calculates the variance to X as 1.5. Which

of the following statements is true?

K 1 2 3 4 5

P(X = K) 0.1 0.2 0.4 0.2 0.1

(A) Both the student and the teacher are right

(B) Both the student and the teacher are wrong

(C) The student is wrong but the teacher is right

(D) The student is right but the teacher is wrong


(04) A screening test is carried out to detect a certain

disease. It is found that 12% of the positive

reports and 15% of the negative reports are

incorrect. Assuming that the probability of a

person getting positive report is 0.01, the

probability that a person tested gets an incorrect

report is

(A) 0.0027 (B) 0.0173

(C) 0.1497 (D) 0.2100


(05) Consider a finite sequence of random values X =

1 2 3 n{x ,x ,x ,...........x }. Let x μ be the mean and

x be the standard deviation of X. Let another




www.targate.org

finite sequence Y of equal length be derived from

this .i i y a x b , where a and b are positive

constants. Let y μ be the mean and y be the

standard deviation of this sequence. Which one

of the following statements is incorrect?

(A) Index position of mode of X in X is the same

as the index position of mode of Y in Y.

(B) Index position of median of X i X is the same

as the index position of median of Y in Y.

(C) y x μ aμ b

(D) y xa b

-----00000-----

5.7 EXPECTION



(01) If E denotes expectation, the variance of a

random variable X is given by

(A) 2 2( ) ( ) E X E X (B) 2 2( ) ( ) E X E X

(C) 2( ) E X (D) 2 ( ) E X

-----00000-----


eE1 / T5 / K2 / L2 / V2 / R11 / AC [GATE – – 1999]

(01) Suppose that the expectation of a random

variable X is 5, width of the following statement

is true?

(A) There is a sample point at which X has the

value = 5

(B) There is a sample point at which X has the

value > 5

(C) There is a sample point at which X has a

value 5

(D) None of the above

-----00000-----



(01) An exam paper has 150 multiple choice questions

of 1 mark each, with each question having four

choices. Each incorrect answer fetches – 0.25

marks. Suppose 1000 students choose all their

answers randomly with uniform probability. The

sum total of the expected marks obtained by all

the students is

(A) 0 (B) 2550

(C) 7525 (D) 9375

-----00000-----





5.8 SET THEORY


eE1 / T5 / K8 / L3 / V2 / R11 / AC [GATE – IT – 2004]

(01) In a class of 200 students, 125 students havetaken programming language course, 85 students

have taken data structures course, 65 students

have taken computer organization course, 50

students have taken both programming languages

and data structures, 35 students Have taken both

programming languages and computer

organization, 30 students have taken both data

structures and computer organization, 15 students

have taken all the three courses. How many

students have not taken any of the three courses?

(A) 15 (B) 20

(C) 25 (D) 35

-----00000-----




06N umer ical M ethods

Complete subtopic in this chapter, is in the scope of “GATE-CS/ ME/EC/EE SYLLABUS”

6.1 Clubbed problem


eE1 / T6 / K / L1 / V1 / R11 / A [GATE – –]

(01) In the interval [0, ]π the equation cos x x has

(A) No solution

(B) Exactly one solution

(C) Exactly 2 solutions

(D) An infinite number of solutions

-----00000-----


eE1 / T6 / K / L2 / V2 / R11 / AB [GATE – – ] (01) For solving algebraic and transcendental equation

which one of the following is used?

(A) Coulomb’s theorem

(B) Newton-Raphson method

(C) Euler’s method

(D) Stoke’s theorem

eE1 / T6 / K / L2 / V2 / R11 / AC [GATE – –]

(02) The polynomial 5( ) 2 p x x x has

(A) all real roots

(B) 3 real and 2 complex roots

(C) 1 real and 4 complex roots

(D) all complex roots

eE1 / T6 / K / L2 / V2 / R11 / AC [GATE – –]

(03) It is known that two roots of the non-linear

equation3 26 11 6 0 x x x are 1 and 3. The

third root will be

(A) j (B) j

(C) 2 (D) 4

-----00000-----






eE1 / T6 / K / L3 / V2 / R11 / AC [GATE – – ]

(01) Match the following and choose the correct

combination

E. Newton –

Raphso

n

method

(1) Solving non-linear

equations

F. Runge-Kutta

method

(2) Solving linear

simultaneous

equations

G. Simpson’s Rule (3) Solving ordinary

differential

equations

H. Gauss

elimina

tion

(4) Numerical

intergration

method

(5) Interpolation

(A) E – 6, F – 1, G

– 5, H –

3

(B) E – 1, F – 6, G – 4, H

– 3

(C) E – 1, F – 3, G

– 4, H –

2

(D) E – 5, F – 3, G – 4, H

– 1

eE1 / T6 / K / L3 / V2 / R11 / AA [GATE – – ]

(02) Given that one root of the equation

3 210 31 30 0 x x x is 5 then other roots

arc

(A) 2 and 3 (B) 2 and 4

(C) 3 and 4 (D) 2 and 3

eE1 / T6 / K / L3 / V2 / R11 / AC [GATE – – ]

(03) Matching exercise choose the correct one out of

the alternatives A, B, C, D

Group – I Group – II

P. 2n order

differe

ntial

equatio

ns

(1) Runge –

Kutta

method

Q. Non-linear

algebra

ic

equatio

ns

(2) Newton –

Raphso

n

method

R. Linear

algebra

ic

equatio

ns

(3) Gauss

Elimin

ation

S. Numerical

integration

(4) Simpson’s

Rule

(A) P-3, Q-2, R-4, S-1 (B) P-2, Q-4, R-3, S-1

(C) P-1, Q-2, R-3, S-4 (D) P-1, Q-3, R-2, S-4

eE1 / T6 / K2 / L1 / V1 / R11 / AB [GATE – – ]

(04) Back ward Euler method for solving the

differential equation ( , )dy

f x ydx

is specified

by

(A) 1 ,( )n n n n

y y h f x y

(B) 1 1 1( , )n n n n y y h f x y

(C) 1 1 2 ( , )n n n n y y h f x y

(D) 1 (1 )n y h 1 1( , )n n f x y



TOPIC. 06 – NUMERICAL METHODS

www.targate.org

eE1 / T6 / K3 / L2 / V2 / R11 / AA [GATE – – ]

(05) The following equation needs to be numerically

solved using the Newton – Raphson method

3 4 9 0. x x The iterative equation for this

purpose is ( k indicates the iteration level)

(A) 3

1 2

2 9

3 4k

k

k

x x

x

(B)

3

1 2

3 9

2 9k

k

k

x x

x

(C) 21 3 4

k k k x x (D)

2

1 2

4 3

9 2k

k

k

x x

x

-----00000-----

6. 2 Newton-Rap son


eE1 / T6 / K4 / L0 / V1 / R11 / AD [GATE – – ]

(01) The Newton-Raphson method is to be used to

find the root of the equation and '( ) f x is the

derivative of . f the method converges

(A) Always

(B) Only is f is a polynomial

(C) Only if 0( ) 0 f x

(D) None of the above

-----00000-----


eE1 / T6 / K4 / L1 / V1 / R11 / AB [GATE – – ]

(01) The iteration formula to find the square root of a

positive real number by using the Newton-

Raphson method is

(A) 1

3( )

2k

k

k

x b x

x

(B)

22

12

k

k

x b x

x

(C) 11 2

2k k

k

k

x x x

x b

(D) None

eE1 / T6 / K4 / L1 / V1 / R11 / AC [GATE – – ] (02) Given a > 0, we wish to calculate it reciprocal

value1

aby using Newton – Raphson method for

( ) 0. f x The Newton-Raphson algorithm for

the function will be

(A) 1

1

2k k k

a

x x x

(B)

2

1 2k k k

a

x x x

(C) 21 2

k k k x x ax (D) 2

12

k k k

a x x x

eE1 / T6 / K4 / L1 / V1 / R11 / AA [GATE – – ]

(03) Identify the Newton – Raphson iteration scheme

for the finding the square root of 2

(A) 1

1 2

2n n

n

x x x

(B) 1

1 2

2n n

n

x x x

(C) 1

2n

n

x

x

(D) 1 2n n x x





eE1 / T6 / K4 / L1 / V1 / R11 / A [GATE – – ]

(04) The Newton – Raphson iteration

1

1

2n n

n

R x x

x

can be used to compute the

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

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

eE1 / T6 / K4 / L1 / V1 / R11 / A [GATE – – ]

(05) Let2 117 0. x The iterative steps for the

solution using Newton – Raphson’s method

given by

(A) 1

1 117

2k k

k

x x x

(B) 1

117k k

k

x x x

(C) 1117

k k k

x x x

(D) 1

1 117

2k k k

k

x x x x

eE1 / T6 / K4 / L1 / V1 / R11 / AA [GATE – – ]

(06) Newton-Raphson formula to find the roots of an

equation ( ) 0 f x is given by

(A) 1 1

( )

( )n

n n

n

f x x x

f x

(B) 1 1

( )

( )n

n n

n

f x x x

f x

(C) 1 1

( )

( )n

nn n

f x

x x f x

(D) none of the above

eE1 / T6 / K4 / L1 / V1 / R11 / AC [GATE – – ]

(07) The recursion relation to solve x

x e using

Newton – Raphson method is

(A) 1n x

n x e

(B) 1n x

n n x x e

(C) 1

(1 )

(1 )

n

n

x

n

n x

x e x

e

(D) 2

1

(1 ) 1n

n

x

n nn x

n

x e x x

x e

eE1 / T6 / K4 / L1 / V1 / R11 / A [GATE – – ]

(08) The integral3

1

1dx

x when evaluated by using

simpson’s 1/ 3rd rule on two equal sub intervals

each of length 1, equal to

(A) 1.000 (B) 1.008

(C) 1.1111 (D) 1.120

-----00000-----


eE1 / T6 / K4 / L2 / V2 / R11 / AD [GATE – – ]

(01) The formula used to compute an approximation

for the second derivative of a function f at a

point 0 x is

(A) 0 0( ) ( )

2

f x h f x h

(B) 0 0( ) ( )

2

f x h f x h

h




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(C) 0 0 02

( ) 2 ( ) ( ) f x h f x f x h

h

(D) 0 0 02

( ) 2 ( ) ( ) f x h f x f x h

h

eE1 / T6 / K4 / L2 / V2 / R11 / AC [GATE – – ]

(02) The Newton-Raphson iteration formula for

finding 3 ,c where c > 0 is ,

(A) 3 3

1 2

2

3n

n

n

x c x

x

(B)

3 3

1 2

2

3n

n

n

x c x

x

(C) 3

1 2

2

3n

x

n

x c x

x

(D) 3

1 2

2

3n

n

n

x c x

x

eE1 / T6 / K4 / L2 / V1 / R11 / AC [GATE – – ]

(03) Starting from 0 1 x , one step of Newton –

Raphson method in solving the equation

3 3 7 0 x x gives the next value 1 x as

(A) 1 0.5 x (B) 1 1.406 x

(C) 1 1.5 x (D) 1 2 x

eE1 / T6 / K4 / L2 / V2 / R11 / AB [GATE – – ]

(04) The real root of the equation 2 x

xe is

evaluated using Newton – Raphson’s method. If

the first approximation of the value of x is

0.8679, the 2 nd approximation of the value of x

correct to three decimal places is

(A) 0.865 (B) 0.853

(C) 0.849 (D) 0.838

eE1 / T6 / K4 / L2 / V2 / R11 / AB [GATE – – ]

(05) The equation3 2 4 4 0 x x x is to be solved

using the Newton – Raphson method using 2 x

taken as the initial approximation of the solution

then the next approximation using this method,

will be

(A) 2/3 (B) 4/3

(C) 1 (D) 3/2

eE1 / T6 / K4 / L2 / V2 / R11 / AA [GATE – – ]

(06) Equation 1 0 xe is required to be solved

using Newton’s method with an initial guess

0 1. x Then after one step of Newton’s

method estimate 1 x of the solution will be given

by

(A) 0.71828 (B) 0.36784

(C) 0.20587 (D) 0.0000

eE1 / T6 / K4 / L2 / V2 / R11 / A [GATE – – ]

(07) Newton – Raphson method is used to compute a

root of the equation2 13 0 x 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





eE1 / T6 / K4 / L2 / V2 / R11 / A [GATE – – ]

(08) The square root of a number N is to be obtained

by applying the Newton – Raphson iteration to

the equation2 0. x N If i denotes the

iteration index, the correct iterative scheme will

be

(A) 1

1

2i i

i

N x x

x

(B) 21 2

1

2i i

i

N x x

x

(C) 2

1

1

2i i

i

N x x

x

(D) 1 1

( )

( )n n

n

n

x f x x

f x

-----00000-----


eE1 / T6 / K4 / L3 / V2 / R11 / A [GATE – – ]

(01) A numerical solution of the equation

( ) 3 0 f x x x can be obtained using

Newton – Raphson method. If the starting value

is x = 2 for the iteration then the value of x that is

to be used in the next step is

(A) 0.306 (B) 0.739

(C) 1.694 (D) 2.306

eE1 / T6 / K4 / L3 / V2 / R11 / A [GATE – – ]

(02) Solution, the variable 1 2 x and x for the following

equations is to be obtained by employing the

Newton – Raphson iteration method

Equation (i) 2 110 sin 0.8 0 x x

22 2 110 10 cos 0.6 0 x x x

Assuming the initial values 1 0.0 x and

2 1.0 x the Jacobian matrix is

(A) 10 0.8

0 0.6

(B) 10 0

0 10

(C) 0 0.8

10 0.6

(D) 10 0

10 10

eE1 / T6 / K4 / L3 / V2 / R11 / AB [GATE – – ]

(03) Give a > 0, we wish to calculate its reciprocal

value1

aby using Newton – Raphson method for

( ) f x = 0. For 7a and starting with 0 0.2 x

the first two iteration will be

(A) 0.11, 0.1299 (B) 0.12, 0.1392

(C) 0.12, 0.1416 (D) 0.13, 0.1428

-----00000-----




www.targate.org

6.3 Differential


eE1 / T6 / K5 / L0 / V1 / R11 / A [GATE – – ]

(01) During the numerical solution of a first order

differential equation using the Euler (also known

as Euler Cauchy) method with step size h, the

local truncation error is of the order of

(A) 2h (B)

3h

(C) 4h (D)

5h

-----00000-----


eE1 / T6 / K5 / L2 / V1 / R11 / A [GATE – – ]

(01) Consider a differential equation

( )( )

dy x y x x

dx with initial condition

(0) 0. y Using Euler’s first order method with

a step size of 0.1 then the value of y(0.3) is

(A) 0.01 (B) 0.031

(C) 0.0631 (D) 0.1

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6.4 Integration


eE1 / T6 / K6 / L0 / V1 / R11 / AC [GATE – – ]

(01) The trapezoidal rule for integration give exact

result when the integrand is a polynomial of

degree

(A) but not 1 (B) 1 but not 0

(C) 0 (or) 1 (D)2

-----00000-----


eE1 / T6 / K6 / L1 / V1 / R11 / AC [GATE – – ]

(01) The Newton – Raphson method is used to find

the root of the equation2 2. x if the iterations

are started from 1, then the iteration will

(A) Converge to – 1 (B) Converge to 2

(C) Converge to 2 (D) not converge

-----00000-----


eE1 / T6 / K6 / L3 / V2 / R11 / A [GATE – – ]

(01) The following algorithm computes the integral J

= ( )b

a f x dx from the given values ( ) j j f f x

at equidistant points 0 1 0, , x a x x h

2 0 2 , x x h 2 0......... 2m x x mh b

Compute 0 0 2mS f f





1 1 3 2 1......mS f f f

2 2 4 2 2.........mS f f f

J = 0 1 24( ) 2( )3

hS S S

The rule of numerical integration, which uses theabove algorithm is

(A) Rectangle rule (B) Trapezoidal rule

(C) Four – point rule (D) Simpson’s rule

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07

Transform Theory

Complete subtopic in this chapter, is in the scope of “GATE- EC/EE SYLLABUS”



(01) The Laplace transform of f(t) is F(s). Given F(s)

=2 2

,s

the final value of f(t) is ________.

(A) Initially (B) Zero

(C) One (D) None

eE1 / T7 / K2 / L0 / V1 / R11 / AC [GATE – – ]

(02) Let Y(s) be the laplace transform of function y(t),

then the final value of the function is

(A) 0

( )s

LimY s

(B) ( )s

LimY s

(C) 0

( )s

LimsY s

(D) ( )s

LimsY s

eE1 / T7 / K2 / L0 / V1 / R11 / AB [GATE – – ]

(03) If L denotes the Laplace transform of a function.

L{sin at} will be equal to

(A) 2 2

a

s a (B)

2 2

a

s a

(C) 2 2

s

s a (D)

2 2

s

s a

eE1 / T7 / K2 / L0 / V1 / R11 / AB [GATE – – 2004]

(04) A delayed unit step function is defined as

( )u t a =

Its Laplace transform is __________ .

(A) aas

e

(B) /ase s

(C) /ase s (D) /as

e a


(05) Consider the function f(t) having Laplace

transform F(s) = 02 2

0

,s

Re(s) > 0. The final

value of f(t) would be _________

(A) 0 (B) 1

(C) 1 ( ) 1 f (D)





eE1 / T7 / K2 / L0 / V1 / R11 / AA [GATE – – 2007]

(06) If F(s) is the Laplace transform of the function

f(t) than Laplace transform of 0

( )t

f t dx is

(A) 1 ( )F ss

(B) 1 ( ) (0)F s f s

(C) ( ) (0)sF s f (D) ( )F s ds

eE1 / T7 / K2 / L0 / V1 / R11 / AD [GATE – – 2008]

(07) Laplace transform of 38t is

(A) 4

8

s (B)

4

16

s

(C) 4

24

s (D)

4

48

s

eE1 / T7 / K2 / L0 / V1 / R11 / AA [GATE – – 2008]

(08) Laplace transform of sin ht is

(A) 2

1

1s (B)

2

1

1 s

(C) 2 1

s

s (D)

21

s

s


(09) If L ( ) f t =2 2

w

s wthen the value of

( )t

Lim f t

________.

(A) can not be determined (B) Zero

(C) unity (D) Infinite

eE1 / T7 / K2 / L0 / V1 / R11 / AC [GATE – – 2010]

(10) u(t) represents the unit function. The Laplace

transform of ( )u t τ is

(A) 1

sτ (B)

1

s τ

(C) sτ e

s

(D) sτ

e

-----00000-----


eE1 / T7 / K2 / L1 / V1 / R11 / A [GATE – IN – 1995]

(01) Find L { cosat e t } when L{ cos t } =

2 2

s

s

eE1 / T7 / K2 / L1 / V1 / R11 / AD [GATE – – ]

(02) 2( 1)s is the Laplace transform of

(A) 2t (B)

3t

(C) 2t

e

(D) t

te

\

eE1 / T7 / K2 / L1 / V1 / R11 / AB [GATE – – ]

(03) If L{f(t)} =2

2,

1

s

s

2 1

{ ( )}

( 3)( 2)

s L g t

s s

,

h(t)=0

( ) ( )t

f T g t T dT

Then { ( )} L h t is ___________

(A) 2 1

3

s

s

(B) 1

3s

(C) 2

2

1 2

( 3)( 2) 1

s s

s s s

(D) None



TOPIC. 07 – TRANSFORM THEORY

www.targate.org


(04) The Dirac delta Function ( )t is defined as

(A) 1, 0

( )0,

t t

other wise

(B) , 0( )0,

t t other wise

(C) 1, 0

( ) ( ) 10,

t t and t dt

other wise

(D) , 0

( ) ( ) 10,

t t and t dt

other wise


(05) The inverse Laplace transform of the function

5

( 1)( 3)

s

s s

is _______

(A) 32 t t

e e (B)

32 t t e e

(C) 32t t

e e (D)

32t t e e


(06) If { ( )} ( ) L f t F s then { ( )} L f t T is equal to

(A) ( )sT e F s (B) ( )sT

e F s

(C) ( )

1 sT

F s

e (D)

( )

1 sT

F s

e


(07) The Laplace transform of cosαt e αt is equal to

________

(A) 2 2( )

s α

s α α

(B) 2 2( )

s α

s α α

(C) 2

1

( )s α (D) None

eE1 / T7 / K2 / L1 / V1 / R11 / AC [GATE – – 2009]

(08) The inverse Laplace transform of 2

1

( )s sis

(A) 1 t e (B) 1 t

e

(C) 1 t e

(D) 1 t e

-----00000-----


eE1 / T7 / K2 / L2 / V2 / R11 / A [GATE – – 1994] (01) If f(t) is a finite and continuous Function for

0t the Laplace transformation is given by

F =0

( ),st e f t then for ( ) cos , f t h mt the

Laplace Transformation is _____________

eE1 / T7 / K2 / L2 / V2 / R11 / AD [GATE – – 1999]

(02) The Laplace transform of the function

( ) , 0 . f t k t c

(A) ( / ) sck s e

(B) ( / ) sck s e

(C) sck e

(D) ( / )(1 )sck s e

eE1 / T7 / K2 / L2 / V2 / R11 / AC [GATE – – 1999]

(03) Laplace transform of 2( )a bt where ‘a’ and ‘b’

are constants is given by:

(A) 2( )a bs

(B) 21/ ( )a bs

(C) 2 2 2 3( / ) (2 / ) (2 / )a s ab s b s

(D) 2 2 2 3( / ) (2 / ) ( / )a s ab s b s





eE1 / T7 / K2 / L2 / V2 / R11 / AD [GATE – – 2001]

(04) The inverse Laplace transforms of 21/ ( 2 )s s is

(A) 2(1 )t e

(B) 2(1 ) / 2t e

(C) 2(1 ) / 2t e (D) 2(1 ) / 2t e

eE1 / T7 / K2 / L2 / V2 / R11 / AC [GATE – – 2002]

(05) The Laplace transform of the following function

is

sin 0( )

0

t for t π f t

for t π

(A) 21 (1 )s for all x > 0

(B) 21/ (1 )s for all s < π

(C) 2(1 ) / (1 )πse s for all s > 0

(D) 2/ (1 )πse s

for all s > 0

eE1 / T7 / K2 / L2 / V2 / R11 / A [GATE – EE – 2002]

(06) Using Laplace transforms, solve

2 2( / ) 4 12d y dt y t


(07) The Laplace transform of i(t) is given by I(s) =

2

(1 )s sAs ,t the value of i(t) tends to

____ .

(A) 0 (B) 1

(C) 2 (D)

eE1 / T7 / K2 / L2 / V2 / R11 / AA [GATE – – 2005]

(08) The Laplace transform of a function f(t) is F(s) =

2

2

5 23 6.

( 2 2)

s s

s s s

As t , f(t) approaches

(A) 3 (B) 5

(C) 17/2 (D)

eE1 / T7 / K2 / L2 / V2 / R11 / AD [GATE – – 2010]

(09) Given f(t) = 1

3 2

3 1.

4 ( 3)

s L

s s k s

If

( )t Lt f t

= 1 then value of k is

(A) 1 (B) 2

(C) 3 (D) 4

eE1 / T7 / K2 / L2 / V2 / R11 / AB [GATE – – 2009]

(10) Laplace transform of f(x) = cos h(ax) is

(A) 2 2

a

s a (B)

2 2

s

s a

(C) 2 2

a

s a (D)

2 2

s

s a

eE1 / T7 / K2 / L2 / V2 / R11 / AB [GATE – – 2009]

(11) Given that F(s) is the one-sided Laplace

transform of f(t), the Laplace transform of

0( )

t

f τ dτ is

(A) ( ) (0)sF s f (B) 1

( )F ss

(C) 0

( )s

f τ dτ (D) 1

[ ( ) (0)]F s f s



TOPIC. 07 – TRANSFORM THEORY

www.targate.org


(12) In what range should Re(s) remain so that the

laplace transform of the function ( 2) 5a t e

exists?

(A) Re(s) > a + 2 (B) Re (s) > a + 7

(C) Re (s) < 2 (D) Re (s) > a + 5

eE1 / T7 / K2 / L2 / V2 / R11 / AB [GATE – – 2011]

(13) If F(s) = L{f(t)} =2

2( 1)

4 7

s

s s

then the initial

and final values of f(t) are respectively

(A) 0,2 (B) 2, 0

(C) 0,2

7 (D)

2,0

7

eE1 / T7 / K2 / L2 / V2 / R11 / AA [GATE – – 1998]

(14) The Laplace Transform of a unit step function

( ),au t defined as

( ) 0au t for t < a is

= 1 for t > a,

(A) /ase s

(B)

asse

(C) (0)s u (D) 1asse

-----00000-----


eE1 / T7 / K2 / L3 / V2 / R11 / A [GATE – – 1993]

(01) The Laplace transform of the periodic function

f(t) described by the curve below i.e. (Gate

– 1993)

sin , (2 1) 2 ( 1, 2,3,..)( )

0

t if n π t nπ n f t

other wise


(02) The inverse Laplace transform of

2( 9) / ( 6 13)s s s is

(A) cos 2 9 sin 2t t

(B) 3 3cos2 3 sin 2t t

e t e t

(C) 3 3sin2 3 cos2t t

e t e t

(D) 3 3cos2 3 sin 2t t

e t e t


(03) If L{f(t)} =2

2( 1)

2

s

s s s

then f(0 ) and f( ) are

given by _______

(A) 0, 2 respectively (B) 2, 0 respectively

(C) 0, 1 respectively (D) 2

5, 0 respectively

eE1 / T7 / K2 / L3 / V2 / R11 / A [GATE – – 1996]

(04) Using Laplace Transform, solve the initial value

problem 11 19 6 0 y y y (0) 3 y and

1(0) 1, y where prime denotes derivative with

respect to t.

eE1 / T7 / K2 / L3 / V2 / R11 / A [GATE – ME – 1997]

(05) Solve the initial value problem

2

24 3 0

d y dy y

dx dx

with y = 3 and 7dy

dt

at

0 x






(06) The laplace transform of 2( 2 ) ( 1)t t u t is

________ .

(A) 3 2

2 2s se e

s s

(B) 2

3 2

2 2s se e

s s

(C) 3

2 2s se e

s s

(D) None

eE1 / T7 / K2 / L3 / V2 / R11 / AD [GATE – – ]

(07) Let F(s) = £[f(t)] denote the Laplace transform of

the function f(t). Which of the following

statements is correct?

(A) £[ / ] 1/ ( );df dt s F s 0£ ( (

t

f τ dτ

= sF(s) f(0)

(B) £[ / ]df dt = sF(s) – F(0). 0£ ( )

t

f τ dτ

(C) £[ / ]df dt = s F(s) – F(0);

0£ ( ) ( )

t

f τ dτ F s a

(D) £[ / ]df dt = s F(s) – F(0);

0

£ ( ) 1/ ( )t

f τ dτ s F s

eE1 / T7 / K2 / L3 / V2 / R11 / AA [GATE – – 2005]

(08) Laplace transform of f(t) = cos( ) pt q is

(A) 2 2

cos sins q p q

s p

(B) 2 2

cos sins q p q

s p

(C) 2 2

sin coss q p q

s p

(D) 2 2

sin coss q p q

s p

eE1 / T7 / K2 / L3 / V2 / R11 / AA [GATE – – 2010]

(09) The Laplace transform of f(t) is 2

1.

( 1)s s The

function

(A) 1 t t e

(B) 1 t t e

(C) 1 t e

(D) 2 t t e

eE1 / T7 / K2 / L3 / V2 / R11 / AA [GATE – – 2011]

(10) Given two continuous time signals x(t) =t

e

and

y(t) =2t

e which exists for t > 0 then the

convolution z(t) = f(t) * y(t) is ___________ .

(A) 2t t

e e (B)

2t e

(C) t

e

(D) 3t t

e e

------THE END ------



FINALLY HISTORY HAS BEEN CHANGED IN BILASPUR

FIRST TIME IN BILASPUR

62% of students are qualified in GATE (28 students out of 45) .

Min 5 students will be securing seats in IIT out of 45.

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Highest rank of 400.

GATE - 2013 RESULT (@ TARGATE EDU)EC/EE/CS:

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(EC - 99.54 %ile)

AJAY TIWARI

(EC - 99.38 %ile)

ANKUR GUPTA

(EC - 99.00 %ile)

AMIT JAISWAL

(EC - 98.83 %ile)

VARUN DAS

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NARENDRA PATEL

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SURYAKANT

(CS - 99.81 %ile)

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(CS - 98.43 %ile)

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(CS - 97.71 %ile)

Many more…………..(28 students qualified out of 45)

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