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7/28/2019 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
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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]
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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.
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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
2.9 MISCELLANEOUS 52
DIFFERENTIAL EQUATIONS 55
3.1 DEGREE AND ORDER OF DE 55
3.2 HIGHER ORDER DE 56
3.3 LEIBNITZ LINEAR EQUATION 61
3.4 MISCELLANEOUS 62
COMPLEX VARIABLE 66
4.1CAUCHY’S THEOREM 66
4.2 MISCELLANEOUS 68
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
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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
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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
eE1 / T1 / K1 / L1 / V1 / R11 / AB [GATE – CE – 1998]
(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
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ENGINEERING MATHEMATICS
Page 8 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-----
Question Level – 02
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
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TOPIC. 01 – LINEAR ALGEBRA
www.targate.org Page 9
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+
eE1 / T1 / K1 / L2 / V1 / R11 / AB [GATE – CE – 2009]
(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-----
Question Level – 03
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
eE1 / T1 / K1 / L3 / V2 / R11 / AB [GATE – CE – 2000]
(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
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ENGINEERING MATHEMATICS
Page 10 TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
(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
eE1 / T1 / K1 / L3 / V2 / R11 / AD [GATE – CS – 2011]
(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
Question Level – 01
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
eE1 / T1 / K2 / L1 / V1 / R11 / AA [GATE – CE – 1997]
(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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TOPIC. 01 – LINEAR ALGEBRA
www.targate.org Page 11
Question Level – 02
eE1 / T1 / K2 / L2 / V2 / R11 / AB [GATE – CS – 1998]
(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
eE1 / T1 / K2 / L2 / V2 / R11 / AA [GATE – PI – 2007]
(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
Question Level – 00 (Basic Problem)
eE1 / T1 / K3 / L0 / V1 / R11 / AA [GATE – CE – 2007]
(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
Question Level – 01
eE1 / T1 / K3 / L1 / V1 / R11 / AA [GATE – PI – 1994]
(01) The matrix1 4
1 5
is an inverse of the matrix
5 4
1 1
(A) True (B) False
Question Level – 02
eE1 / T1 / K3 / L2 / V2 / R11 / AD [GATE – EE – 1995]
(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
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ENGINEERING MATHEMATICS
Page 12 TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
(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
eE1 / T1 / K3 / L2 / V1 / R11 / AA [GATE – CE – 1997]
(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
eE1 / T1 / K3 / L2 / V2 / R11 / AA [GATE – EE – 1999]
(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
eE1 / T1 / K3 / L2 / V2 / R11 / AA [GATE – PI – 2008]
(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
eE1 / T1 / K3 / L2 / V2 / R11 / AA [GATE – ME – 2009]
(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
eE1 / T1 / K3 / L2 / V2 / R11 / AB [GATE – CE – 2010]
(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
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TOPIC. 01 – LINEAR ALGEBRA
www.targate.org Page 13
(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
Question Level – 00 (Basic Problem)
eE1 / T1 / K4 / L0 / V1 / R11 / AA [GATE – EE – 1994]
(01) The Eigen values of the matrix1
1
a
a
are
(A) ( 1),0a (B) ,0a
(C) ( 1),0a (D) 0,0
eE1 / T1 / K4 / L0 / V1 / R11 / AA [GATE – EE – 1998]
(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
eE1 / T1 / K4 / L0 / V1 / R11 / AB [GATE – CS – 2005]
(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
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ENGINEERING MATHEMATICS
Page 14 TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
(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-----
Question Level – 01
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
eE1 / T1 / K4 / L1 / V1 / R11 / AC [GATE – ME – 1996]
(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
eE1 / T1 / K4 / L1 / V1 / R11 / AD [GATE – EC – 1998]
(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)
eE1 / T1 / K4 / L1 / V1 / R11 / AA [GATE – CE – 2002]
(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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TOPIC. 01 – LINEAR ALGEBRA
www.targate.org Page 15
eE1 / T1 / K4 / L1 / V1 / R11 / AB [GATE – CE – 2007]
(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
eE1 / T1 / K4 / L1 / V1 / R11 / AD [GATE – EC – 2008]
(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
eE1 / T1 / K4 / L1 / V1 / R11 / AB [GATE – CE – 2008]
(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
eE1 / T1 / K4 / L1 / V1 / R11 / AA [GATE – ME – 2010]
(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
eE1 / T1 / K4 / L1 / V1 / R11 / AD [GATE – CS – 2010]
(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
eE1 / T1 / K4 / L1 / V2 / R11 / AA [GATE – CS – 2002]
(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-----
Question Level – 02
eE1 / T1 / K4 / L2 / V2 / R11 / AC [GATE – EE – 1998]
(01) The vector
1
2
1
is an eigen vector of A =
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TOPIC. 01 – LINEAR ALGEBRA
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(A) 0 (B) 1/2
(C) 1 (D) 2
eE1 / T1 / K4 / L2 / V2 / R11 / AA [GATE – PI – 2008]
(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
eE1 / T1 / K4 / L2 / V2 / R11 / AC [GATE – EC – 2006]
(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
eE1 / T1 / K4 / L2 / V2 / R11 / AB [GATE – CE – 2006]
(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
eE1 / T1 / K4 / L2 / V2 / R11 / AB [GATE – EE – 2010]
(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
eE1 / T1 / K4 / L2 / V2 / R11 / AB [GATE – PI – 2011]
(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
eE1 / T1 / K4 / L2 / V2 / R11 / AD [GATE – EC – 2009]
(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
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ENGINEERING MATHEMATICS
Page 18 TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
(A) 1 8 1T
(B) 1 2 1T
(C) 31 2 1T
(D) 1 1 1T
eE1 / T1 / K4 / L2 / V2 / R11 / AA [GATE – CS – 2011]
(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
eE1 / T1 / K4 / L2 / V2 / R11 / AA [GATE – PI – 2010]
(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-----
Question Level – 03
eE1 / T1 / K4 / L3 / V2 / R11 / AB [GATE – PI – 2007]
(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
eE1 / T1 / K4 / L3 / V2 / R11 / AA [GATE – ME – 2005]
(03) Which one of the following is an eigen vector of
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TOPIC. 01 – LINEAR ALGEBRA
www.targate.org Page 19
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
eE1 / T1 / K4 / L3 / V2 / R11 / AA [GATE – IN – 2010]
(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
eE1 / T1 / K4 / L3 / V3 / R11 / AA [GATE – EE – 2007]
(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
eE1 / T1 / K4 / L3 / V3 / R11 / AB [GATE – CE – 2007]
(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
Question Level – 00 (Basic Problem)
eE1 / T1 / K5 / L0 / V1 / R11 / AA [GATE – EC – 1994]
(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
eE1 / T1 / K5 / L0 / V1 / R11 / AC [GATE – EE – 1994]
(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
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ENGINEERING MATHEMATICS
Page 20 TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
Question Level – 01
eE1 / T1 / K5 / L1 / V1 / R11 / AA [GATE – EE – 1994]
(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
eE1 / T1 / K5 / L1 / V1 / R11 / AC [GATE – CS – 1994]
(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
eE1 / T1 / K5 / L1 / V1 / R11 / AA [GATE – EE – 1995]
(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-----
Question Level – 02
eE1 / T1 / K5 / L2 / V2 / R11 / AD [GATE – CS – 1998]
(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
eE1 / T1 / K5 / L2 / V2 / R11 / AC [GATE – EC – 2006]
(02) The rank of the matrix
1 1 1
1 1 0
1 1 1
is
(A) 0 (B) 1
(C) 2 (D) 3
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TOPIC. 01 – LINEAR ALGEBRA
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Question Level – 03
eE1 / T1 / K5 / L3 / V2 / R11 / AC [GATE – CE – 2003]
(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
eE1 / T1 / K5 / L3 / V2 / R11 / AB [GATE – IN – 2007]
(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
Question Level – 01
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
eE1 / T1 / K6 / L1 / V1 / R11 / AC [GATE – ME – 1996]
(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
eE1 / T1 / K6 / L1 / V1 / R11 / AB [GATE – IN – 2005]
(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
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ENGINEERING MATHEMATICS
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(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-----
Question Level – 02
eE1 / T1 / K6 / L2 / V2 / R11 / AB [GATE – CS – 2003]
(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
eE1 / T1 / K6 / L2 / V2 / R11 / AB [GATE – CS – 1998]
(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
eE1 / T1 / K6 / L2 / V2 / R11 / AB [GATE – CE – 2005]
(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
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TOPIC. 01 – LINEAR ALGEBRA
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(C) More than one but a finite number of
solutions.
(D) An infinite number of solutions.
eE1 / T1 / K6 / L2 / V2 / R11 / AD [GATE – CE – 2005]
(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.
eE1 / T1 / K6 / L2 / V2 / R11 / AA [GATE – EE – 2005]
(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
eE1 / T1 / K6 / L2 / V1 / R11 / AB [GATE – ME – 2005]
(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
eE1 / T1 / K6 / L2 / V2 / R11 / AB [GATE – EC –]
(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
eE1 / T1 / K6 / L2 / V2 / R11 / AD [GATE – CE – 2006]
(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
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ENGINEERING MATHEMATICS
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(C) 1, 1/ 2, 2 x y z
(D) Non existent
eE1 / T1 / K6 / L2 / V2 / R11 / AA [GATE – CE – 2007]
(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
eE1 / T1 / K6 / L2 / V2 / R11 / AD [GATE – EE – 2010]
(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
eE1 / T1 / K6 / L2 / V2 / R11 / AC [GATE – ME – 2011]
(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
eE1 / T1 / K6 / L2 / V2 / R11 / AB [GATE – EC – 2008]
(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.
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eE1 / T1 / K6 / L2 / V2 / R11 / AB [GATE – EC –]
(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
eE1 / T1 / K6 / L2 / V2 / R11 / AB [GATE – EC – 2011]
(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
eE1 / T1 / K6 / L2 / V3 / R11 / AA [GATE – CE – 2009]
(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
Question Level – 03
eE1 / T1 / K6 / L3 / V2 / R11 / AB [GATE – CS – 1996]
(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.
eE1 / T1 / K6 / L3 / V2 / R11 / AB [GATE – EE – 1998]
(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
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ENGINEERING MATHEMATICS
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(D) None
eE1 / T1 / K6 / L3 / V2 / R11 / AB [GATE – IN – 2007]
(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
eE1 / T1 / K6 / L3 / V2 / R11 / AB [GATE – EE – 2007]
(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
eE1 / T1 / K6 / L3 / V2 / R11 / AB [GATE – ME – 2008]
(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
Question Level – 00 (Basic Problem)
eE1 / T1 / K7 / L0 / V1 / R11 / AB [GATE – CS – 2004]
(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
eE1 / T1 / K7 / L0 / V1 / R11 / AA [GATE – CE – 1997]
(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-----
Question Level – 01
eE1 / T1 / K7 / L1 / V1 / R11 / AC [GATE – CS – 1997]
(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
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TOPIC. 01 – LINEAR ALGEBRA
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(C) column is the same as the second column of
(D) Row is a zero row.
eE1 / T1 / K7 / L1 / V1 / R11 / AC [GATE – CE – 1999]
(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
eE1 / T1 / K7 / L1 / V1 / R11 / AB [GATE – CE – 1999]
(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
eE1 / T1 / K7 / L1 / V1 / R11 / AB [GATE – CE – 2001]
(04) The determinant of the following matrix
5 3 2
1 2 6
3 5 10
(A) – 76 (B) – 28
(C) 28 (D) 72
eE1 / T1 / K7 / L1 / V1 / R11 / AA [GATE – CE – 2001]
(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
eE1 / T1 / K7 / L1 / V1 / R11 / AD [GATE – EC – 2005]
(07) Given the matrix4 2
,4 3
the eigen vector is
(A) 3
2
(B) 4
3
(C) 2
1
(D) 2
1
Question Level – 02
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
eE1 / T1 / K7 / L2 / V2 / R11 / AB [GATE – PI – 1995]
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TOPIC. 01 – LINEAR ALGEBRA
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(A) 7
20 (B)
3
20
(C) 19
60 (D)
11
20
eE1 / T1 / K7 / L2 / V2 / R11 / AC [GATE – IN – 2006]
(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
eE1 / T1 / K7 / L2 / V2 / R11 / AB [GATE – EE – 2011]
(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-----
Question Level – 03
eE1 / T1 / K7 / L3 / V2 / R11 / AA [GATE – CS – 1996]
(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
eE1 / T1 / K7 / L3 / V2 / R11 / AB [GATE – CE – 1999]
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ENGINEERING MATHEMATICS
Page 30 TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
(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
eE1 / T1 / K7 / L3 / V2 / R11 / AB [GATE – CE – 2005]
(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.
eE1 / T1 / K7 / L3 / V2 / R11 / AC [GATE – ME – 2006]
(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
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TOPIC. 01 – LINEAR ALGEBRA
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eE1 / T1 / K7 / L3 / V2 / R11 / AA [GATE – CS – 2000]
(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
Question Level – 01
eE1 / T1 / K8 / L1 / V1 / R11 / AC [GATE – CE – 2007]
(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
eE1 / T1 / K8 / L1 / V1 / R11 / AA [GATE – EE – 2007]
(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
Question Level – 02
eE1 / T1 / K8 / L2 / V2 / R11 / AD [GATE – EE – 2008]
(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-----
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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
Question Level – 01
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-----
Question Level – 03
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
Question Level – 00 (Basic Problem)
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-----
Question Level – 02
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
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TOPIC. 02 – CALCULUS
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(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
eE1 / T2 / K2 / L2 / V2 / R11 / AC [GATE – CS – 1997]
(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
eE1 / T2 / K2 / L2 / V2 / R11 / AA [GATE – ME – 2005]
(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
eE1 / T2 / K2 / L2 / V2 / R11 / AA [GATE – EE – 2005]
(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
eE1 / T2 / K2 / L2 / V2 / R11 / AA [GATE – EC – 2007]
(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
eE1 / T2 / K2 / L2 / V2 / R11 / AC [GATE – IN – 2008]
(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
eE1 / T2 / K2 / L2 / V2 / R11 / AA [GATE – EC – 2008]
(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
eE1 / T2 / K2 / L2 / V2 / R11 / AA [GATE – EC – 2010]
(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
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ENGINEERING MATHEMATICS
Page 34 TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
Question Level – 03
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.
eE1 / T2 / K2 / L3 / V2 / R11 / AC [GATE – IN – 2007]
(04) For real x, the maximum value of sin
cos
x
x
e
eis
(A) 1 (B) e
(C) 2e (D)
eE1 / T2 / K2 / L3 / V2 / R11 / AB [GATE – PI – 2007]
(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
eE1 / T2 / K2 / L3 / V2 / R11 / AB [GATE – EE – 2007]
(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
Question Level – 00 (Basic Problem)
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
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TOPIC. 02 – CALCULUS
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Question Level – 01
eE1 / T2 / K3 / L1 / V1 / R11 / AB [GATE – IN – 2008]
(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
eE1 / T2 / K3 / L1 / V1 / R11 / AA [GATE – PI – 2009]
(02) The total derivative of the function ‘xy’ is
(A) xdy ydx (B) xdx ydy
(C) dx dy (D) dx dy
Question Level – 02
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
Question Level – 03
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
eE1 / T2 / K3 / L3 / V2 / R11 / AC [GATE – PI – 2010]
(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
eE1 / T2 / K3 / L3 / V2 / R11 / AA [GATE – CE – 2010]
(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
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ENGINEERING MATHEMATICS
Page 36 TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
(A) a minimum equal to10
3
(B) a maximum equal to10
3
(C) a minimum equal to8
3
(D) a maximum equal to8
3
eE1 / T2 / K3 / L3 / V2 / R11 / AC [GATE – EE – 2011]
(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
Question Level – 00 (Basic Problem)
eE1 / T2 / K4 / L0 / V1 / R11 / AC [GATE – EE – 2005]
(01) IF S = 3
1
α
X dx then S has the value
(A) 1
3
(B)
1
4
(C) 1
2 (D) 1
eE1 / T2 / K4 / L0 / V2 / R11 / AD [GATE – EC – 2005]
(02) The value of the integral1
21
1dx
x is
(A) 2 (B) does not exists
(C) - 2 (D)
-----00000-----
Question Level – 01
eE1 / T2 / K4 / L1 / V1 / R11 / A [GATE – PI – 1995]
(01) Given2
1cos , x y t dt then ________ dy
dx
eE1 / T2 / K4 / L1 / V1 / R11 / AA [GATE – CS – 1995]
(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
eE1 / T2 / K4 / L1 / V1 / R11 / AA [GATE – PI – 2008]
(03) The value of the integral/2
/2( cos )
π
π x x dx
is
(A) 0 (B) 2π
(C) π (D) 2π
eE1 / T2 / K4 / L1 / V1 / R11 / AD [GATE – ME – 2010]
(04) The value of the integral21
α
α
dx
x
(A) π (B) 2π
(C) 2
π (D) π
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TOPIC. 02 – CALCULUS
www.targate.org Page 37
Question Level – 02
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
eE1 / T2 / K4 / L2 / V2 / R11 / AA [GATE – ME – 2005]
(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
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ENGINEERING MATHEMATICS
Page 38 TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
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
eE1 / T2 / K4 / L2 / V2 / R11 / AA [GATE – CS – 2008]
(09) The value of 3
0 0(6 )
x
x y is _____
(A) 13.5 (B) 27.0
(C) 40.5 (D) 54.0
eE1 / T2 / K4 / L2 / V2 / R11 / AB [GATE – EC – 2007]
(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
eE1 / T2 / K4 / L2 / V2 / R11 / AB [GATE – IN – 2010]
(11) The integral 6sin( )6
α
α
π
t t dt
evaluates to
(A) 6 (B) 3
(C) 1.5 (D) 0
-----00000-----
Question Level – 03
eE1 / T2 / K4 / L3 / V2 / R11 / A [GATE – – 1994]
(01) The value of 3 1/2
0. y
e y dy
is ________
eE1 / T2 / K4 / L3 / V2 / R11 / AD [GATE – IN – 2007]
(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
eE1 / T2 / K4 / L3 / V2 / R11 / AB [GATE – EE – 2007]
(03) The integral2
0
1sin( )cos
2t τ τdτ
equals
(A) Sin cost (B) 0
(C) 1 cos2
t (D) 1 sin2
t
eE1 / T2 / K4 / L3 / V2 / R11 / AA [GATE – EC – 2008]
(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
eE1 / T2 / K4 / L3 / V2 / R11 / AD [GATE – ME – 2008]
(05) Which of the following integrals is unbounded?
(A) 0
tanπ / 4
dx (B) 20
1
1
α
dx x
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TOPIC. 02 – CALCULUS
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(C) 0
.α
x x e dx
(D) 1
0
1
1dx
x
eE1 / T2 / K4 / L3 / V2 / R11 / AD [GATE – CS – 2011]
(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
Question Level – 00 (Basic Problem)
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
eE1 / T2 / K5 / L0 / V1 / R11 / AC [GATE – IN – 2007]
(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
eE1 / T2 / K5 / L0 / V1 / R11 / AB [GATE – ME – 2007]
(06) The minimum value of function 2 y x in the
interval [1, 5] is
(A) 0 (B) 1
(C) 25 (D) Undefined
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ENGINEERING MATHEMATICS
Page 40 TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
eE1 / T2 / K5 / L0 / V1 / R11 / AB [GATE – ME – 2007]
(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)
eE1 / T2 / K5 / L0 / V1 / R11 / AB [GATE – ME – 2008]
(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
eE1 / T2 / K5 / L0 / V1 / R11 / AD [GATE – EE – 2010]
(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
eE1 / T2 / K5 / L0 / V1 / R11 / AD [GATE – ME – 2011]
(12) What is0
sinlimθ
θ
θ equal to?
(A) θ (B) sin θ
(C) 0 (D) 1
-----00000-----
Question Level – 01
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
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TOPIC. 02 – CALCULUS
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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
eE1 / T2 / K5 / L1 / V1 / R11 / AA [GATE – EC – 2007]
(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)
eE1 / T2 / K5 / L1 / V1 / R11 / AC [GATE – PI – 2008]
(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
Question Level – 02
eE1 / T2 / K5 / L2 / V2 / R11 / AC [GATE – IN – 1999]
(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
eE1 / T2 / K5 / L2 / V2 / R11 / AB [GATE – CE – 2002]
(04) Limit of the following sequence as n is
___________ 1/n x n
(A) 0 (B) 1
(C) (D) -
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ENGINEERING MATHEMATICS
Page 42 TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
eE1 / T2 / K5 / L2 / V2 / R11 / AC [GATE – PI – 2007]
(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
eE1 / T2 / K5 / L2 / V2 / R11 / AB [GATE – ME – 2010]
(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-----
Question Level – 03
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
eE1 / T2 / K5 / L3 / V2 / R11 / AA [GATE – CE – 2000]
(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)
eE1 / T2 / K5 / L3 / V2 / R11 / AC [GATE – CE – 2011]
(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-----
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TOPIC. 02 – CALCULUS
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2.6 Series
Question Level – 00 (Basic Problem)
eE1 / T2 / K6 / L0 / V1 / R11 / AB [GATE – CE – 1998]
(01) The infinite sires 1 112 3
(A) Converges (B) Diverges
(C) Oscillates (D) Unstable
eE1 / T2 / K6 / L0 / V1 / R11 / AB [GATE – ME – 2011]
(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-----
Question Level – 01
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
eE1 / T2 / K6 / L1 / V1 / R11 / AA [GATE – EC – 2008]
(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-----
Question Level – 02
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
eE1 / T2 / K6 / L2 / V2 / R11 / AB [GATE – ME – 2007]
(02) If ........ y x x x x α then y(2) =
_____
(A) 4 (or) 1 (B) 4 only
(C) 1 only (D) Undefined
eE1 / T2 / K6 / L2 / V2 / R11 / AB [GATE – IN – 2011]
(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
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ENGINEERING MATHEMATICS
Page 44 TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
eE1 / T2 / K6 / L2 / V2 / R11 / AB [GATE – ME – 2010]
(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
Question Level – 03
eE1 / T2 / K6 / L3 / V2 / R11 / AB [GATE – EC – 2008]
(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
eE1 / T2 / K6 / L3 / V2 / R11 / AA [GATE – CE – 2000]
(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
eE1 / T2 / K6 / L3 / V2 / R11 / AD [GATE – EC – 2009]
(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
Question Level – 00 (Basic Problem)
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
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TOPIC. 02 – CALCULUS
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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-----
Question Level – 01
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
eE1 / T2 / K7 / L1 / V1 / R11 / AD [GATE – PI – 2005]
(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
eE1 / T2 / K7 / L1 / V1 / R11 / AD [GATE – EC – 2005]
(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
eE1 / T2 / K7 / L1 / V1 / R11 / AB [GATE – CE – 2007]
(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
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ENGINEERING MATHEMATICS
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(A) 30 (B) 60
(C) 90 (D) 120
eE1 / T2 / K7 / L1 / V1 / R11 / AD [GATE – ME – 2008]
(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-----
Question Level – 02
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
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TOPIC. 02 – CALCULUS
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(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
eE1 / T2 / K7 / L2 / V2 / R11 / AD [GATE – EE – 2006]
(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
eE1 / T2 / K7 / L2 / V2 / R11 / AD [GATE – CE – 2007]
(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
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ENGINEERING MATHEMATICS
Page 48 TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
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
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TOPIC. 02 – CALCULUS
www.targate.org Page 49
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-----
Question Level – 03
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
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ENGINEERING MATHEMATICS
Page 50 TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
(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
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TOPIC. 02 – CALCULUS
www.targate.org Page 51
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
Question Level – 03
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
eE1 / T2 / K8 / L3 / V2 / R11 / AB [GATE – ME – 1995]
(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
eE1 / T2 / K8 / L3 / V2 / R11 / AA [GATE – ME – 2008]
(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
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ENGINEERING MATHEMATICS
Page 52 TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
eE1 / T2 / K8 / L3 / V2 / R11 / AA [GATE – EE – 2009]
(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
eE1 / T2 / K8 / L3 / V2 / R11 / AA [GATE – ME – 2009]
(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
eE1 / T2 / K8 / L3 / V2 / R11 / AD [GATE – ME – 2010]
(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
Question Level – 00 (Basic Problem)
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)
eE1 / T2 / K9 / L0 / V1 / R11 / AC [GATE – IN – 2008]
(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
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TOPIC. 02 – CALCULUS
www.targate.org Page 53
Question Level – 01
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
Question Level – 02
eE1 / T2 / K9 / L2 / V2 / R11 / AA [GATE – CE – 1999]
(01) The infinite series2
1
( !)
(2 )!n
n
n
(A) Converges (B) Diverges
(C) Is unstable (D) Oscillate
eE1 / T2 / K9 / L2 / V2 / R11 / AB [GATE – CS – 2010]
(02) What is the value of 2
1lim 1 ?
n
n α n
(A) 0 (B) 2e
(C) 1/2e
(D) 1
eE1 / T2 / K9 / L2 / V2 / R11 / AA [GATE – EE – 2011]
(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-----
Question Level – 03
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
eE1 / T2 / K9 / L3 / V2 / R11 / AA [GATE – EC – 2007]
(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
eE1 / T2 / K9 / L3 / V2 / R11 / AC [GATE – EC – 2007]
(03) For | | 1,coth( ) x x can be approximated as
(A) x (B) x2
(C) 1
x (D)
2
1
x
eE1 / T2 / K9 / L3 / V2 / R11 / AD [GATE – ME – 2008]
(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
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ENGINEERING MATHEMATICS
Page 54 TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
eE1 / T2 / K9 / L3 / V2 / R11 / AD [GATE – CE – 2010]
(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
eE1 / T2 / K9 / L3 / V2 / R11 / AA [GATE – ME – 2009]
(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-----
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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
Question Level – 00 (Basic Problem)
eE1 / T3 / K1 / L0 / V2 / R11 / AB [GATE – ME – 2007]
(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
eE1 / T3 / K1 / L0 / V2 / R11 / AB [GATE – EC – 2005]
(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
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ENGINEERING MATHEMATICS
Page 56 TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
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
eE1 / T3 / K1 / L1 / V2 / R11 / AD [GATE – ME – 1999]
(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
Question Level – 01
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)’’
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TOPIC. 03 – DIFFERENTIAL EQUATIONS
www.targate.org Page 57
Question Level – 02
eE1 / T3 / K2 / L2 / V2 / R11 / AA [GATE – EC – 1994]
(01) 2 x y e is a solution of the differential equation
11 1 2 0 y y y
(A) True (B) False
eE1 / T3 / K2 / L2 / V2 / R11 / AD [GATE – IN – 2005]
(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
eE1 / T3 / K2 / L2 / V2 / R11 / AA [GATE – EC – 2006]
(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
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ENGINEERING MATHEMATICS
Page 58 TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
(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
eE1 / T3 / K2 / L2 / V2 / R11 / AA [GATE – EC – 2010]
(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
eE1 / T3 / K2 / L2 / V2 / R11 / AB [GATE – ME – 2006]
(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
eE1 / T3 / K2 / L2 / V2 / R11 / AA [GATE – PI – 2009]
(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
eE1 / T3 / K2 / L2 / V2 / R11 / AB [GATE – EC – 2005]
(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.
eE1 / T3 / K2 / L2 / V2 / R11 / AA [GATE – ME – 2008]
(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
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TOPIC. 03 – DIFFERENTIAL EQUATIONS
www.targate.org Page 59
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
eE1 / T3 / K2 / L2 / V2 / R11 / AA [GATE – IN – 2006]
(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 – – ]
(15) The solution of the differential equation
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
eE1 / T3 / K2 / L2 / V2 / R11 / AC [GATE – ME – 1995]
(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
eE1 / T3 / K2 / L2 / V2 / R11 / AA [GATE – ME – 1996]
(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
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ENGINEERING MATHEMATICS
Page 60 TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
eE1 / T3 / K2 / L2 / V2 / R11 / AA [GATE – ME – 1994]
(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
eE1 / T3 / K2 / L2 / V2 / R11 / AC [GATE – EC – 2005]
(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
eE1 / T3 / K2 / L2 / V2 / R11 / AB [GATE – EE – 2005]
(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
eE1 / T3 / K2 / L2 / V2 / R11 / AC [GATE – CE – 2001]
(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
eE1 / T3 / K2 / L2 / V2 / R11 / AA [GATE – CE – 2005]
(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
eE1 / T3 / K2 / L2 / V2 / R11 / AC [GATE – ME – 2005]
(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
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TOPIC. 03 – DIFFERENTIAL EQUATIONS
www.targate.org Page 61
3.3 Leibnitz linear equation
Question Level – 02
eE1 / T3 / K3 / L2 / V2 / R11 / AB [GATE – EC – 2008]
(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
eE1 / T3 / K3 / L2 / V2 / R112 / AB [GATE – ME – 1994]
(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]
(03) The solution of the differential equation
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
eE1 / T3 / K3 / L2 / V2 / R11 / AB [GATE – ME – 2006]
(04) The solution of the differential equation
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
eE1 / T3 / K3 / L2 / V2 / R11 / AD [GATE – CE – 1997]
(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
eE1 / T3 / K3 / L2 / V2 / R11 / AA [GATE – ME – 2009]
(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
eE1 / T3 / K3 / L2 / V2 / R11 / AA [GATE – CE – 2005]
(08) Transformation to linear form by substituting v =
1 n y of the equation ( ) ( ) , 0ndyP t y q t y n
dt
will be
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ENGINEERING MATHEMATICS
Page 62 TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
(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
eE1 / T3 / K3 / L2 / V2 / R11 / AC [GATE – IN – 2010]
(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
Question Level – 01
eE1 / T3 / K4 / L1 / V1/ R11 / AA [GATE – ME – 2003]
(01) The solution of the differential equation
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-----
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TOPIC. 03 – DIFFERENTIAL EQUATIONS
www.targate.org Page 63
Question Level – 02
eE1 / T3 / K4 / L2 / V2 / R11 / AD [GATE – CE – 2007]
(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
eE1 / T3 / K4 / L2 / V2 / R11 / AB [GATE – CE – 2007]
(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
eE1 / T3 / K4 / L2 / V2 / R11 / AA [GATE – IN – 2008]
(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-----
Question Level – 03
eE1 / T3 / K4 / L3 / V2 / R11 / AA [GATE – CE – 2009]
(01) Solution of the differential equation
3 2 0dy
y xdx
represents a family of
(A) ellipses (B) circles
(C) parabolas (D) hyperbolas
eE1 / T3 / K4 / L3 / V2 / R11 / AC [GATE – PI – 2010]
(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
eE1 / T3 / K4 / L3 / V2 / R11 / AA [GATE – EC – 2009]
(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
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ENGINEERING MATHEMATICS
Page 64 TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
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
eE1 / T3 / K4 / L3 / V2 / R11 / AA [GATE – EE – 2011]
(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
eE1 / T3 / K4 / L3 / V2 / R11 / AB [GATE – ME – 1993]
(06) The differential2
2sin 0
d y dy y
dxdx is
(A) linear (B) non – linear
(C) homogeneous (D) of degree two
eE1 / T3 / K4 / L3 / V2 / R11 / AC [GATE – ME – 1994]
(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
eE1 / T3 / K4 / L3 / V2 / R11 / AC [GATE – EC – 2011]
(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
eE1 / T3 / K4 / L3 / V2 / R11 / AC [GATE – IN – 2011]
(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
eE1 / T3 / K4 / L3 / V2 / R11 / AD [GATE – ME – 2011]
(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
eE1 / T3 / K4 / L3 / V2 / R11 / AC [GATE – ME – 1996]
(11) The one dimensional heat conduction partial
differential equation2
T T
t x
is
(A) parabolic (B) hyperbolic
(C) elliptic (D) mixed
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TOPIC. 03 – DIFFERENTIAL EQUATIONS
www.targate.org Page 65
eE1 / T3 / K4 / L3 / V2 / R11 / AA [GATE – CE – 2001]
(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
eE1 / T3 / K4 / L3 / V2 / R11 / AB [GATE – CE – 2004]
(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-----
Question Level – 03
eE1 / T3 / K9 / L3 / V2 / R11 / AC [GATE – IN – 2005]
(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-----
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Page 66 TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
04L
Complex V ar iable Complete subtopic in this chapter, is in the scope of “GATE- ME/EC/EE SYLLABUS”
4.1Cauchy’s Theorem
Question Level – 02
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
eE1 / T4 / K1 / L2 / V2 / R11 / AA [GATE – EC – 2006]
(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
eE1 / T4 / K1 / L2 / V2 / R11 / AB [GATE – IN – 2007]
(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
eE1 / T4 / K1 / L2 / V2 / R11 / AB [GATE – EC – 2007]
(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
eE1 / T4 / K1 / L2 / V2 / R11 / AA [GATE – EC – 2007]
(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
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TOPIC. 04 – COMPLEX VARIABLE
www.targate.org Page 67
(A) i π (B) i π
(C) π (D) π
eE1 / T4 / K1 / L2 / V2 / R11 / AC [GATE – CE – 2009]
(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
eE1 / T4 / K1 / L2 / V2 / R11 / AD [GATE – EC – 2009]
(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
eE1 / T4 / K1 / L2 / V2 / R11 / AC [GATE – IN – 2011]
(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)
eE1 / T4 / K1 / L2 / V2 / R11 / AD [GATE – CE – 2011]
(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-----
Question Level – 03
eE1 / T4 / K1 / L3 / V2 / R11 / AA [GATE – CE – 2005]
(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.
eE1 / T4 / K1 / L3 / V2 / R11 / AA [GATE – IN – 2009]
(02) The value of sin
,a
zdz
zwhere the contour of the
integration is a simple closed curve around the
origin is
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ENGINEERING MATHEMATICS
Page 68 TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
(A) 0 (B) 2 π j
(C) (D) 1
2 π j
eE1 / T4 / K1 / L3 / V2 / R11 / AA [GATE – – ]
(03) The value of the integral2
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
eE1 / T4 / K1 / L3 / V2 / R11 / AB [GATE – PI – 2011]
(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
Question Level – 00 (Basic Problem)
eE1 / T4 / K2 / L0 / V1 / R11 / AC [GATE – IN – 1994]
(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
eE1 / T4 / K2 / L0 / V1 / R11 / AD [GATE – IN – 1994]
(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
eE1 / T4 / K2 / L0 / V1 / R11 / AD [GATE – IN – 2009]
(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
eE1 / T4 / K2 / L0 / V1 / R11 / AD [GATE – PI – 2009]
(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
eE1 / T4 / K2 / L0 / V1 / R11 / AD [GATE – CE – 2009]
(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-----
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TOPIC. 04 – COMPLEX VARIABLE
www.targate.org Page 69
Question Level – 01
eE1 / T4 / K2 / L1 / V1 / R11 / AB [GATE – ME – 1996]
(01) ,ii where i = 1 is given by
(A) 0 (B) /2π e
(C) 2
π (D) 1
eE1 / T4 / K2 / L1 / V1 / R11 / AB [GATE – CE – 1997]
(02) ze is a periodic with a period of
(A) 2π (B) 2πi
(C) π (D) iπ
eE1 / T4 / K2 / L1 / V1 / R11 / AA [GATE – IN – 2005]
(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 )
eE1 / T4 / K2 / L1 / V1 / R11 / AB [GATE – PI – 2008]
(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
eE1 / T4 / K2 / L1 / V1 / R11 / AD [GATE – EC – 2010]
(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
eE1 / T4 / K2 / L1 / V1 / R11 / AD [GATE – IN – 2007]
(07) Let j = 1. Then one value of j j is
(A) 3 (B) 1
(C) 12 (D) 2
π
e
eE1 / T4 / K2 / L1 / V1 / R11 / AB [GATE – PI – 2007]
(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
eE1 / T4 / K2 / L1 / V1 / R11 / AA [GATE – ME – 2011]
(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
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ENGINEERING MATHEMATICS
Page 70 TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
eE1 / T4 / K2 / L1 / V1 / R11 / AA [GATE – EC – 2008]
(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-----
Question Level – 02
eE1 / T4 / K2 / L2 / V2 / R11 / AB [GATE – IN – 1997]
(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
eE1 / T4 / K2 / L2 / V2 / R11 / AA [GATE – IN – 2002]
(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
eE1 / T4 / K2 / L2 / V2 / R11 / AC [GATE – CE – 2005]
(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
eE1 / T4 / K2 / L2 / V2 / R11 / AA [GATE – IN – 2005]
(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
eE1 / T4 / K2 / L2 / V2 / R11 / AC [GATE – IN – 2005]
(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
eE1 / T4 / K2 / L2 / V2 / R11 / AB [GATE – EC – 2006]
(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
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TOPIC. 04 – COMPLEX VARIABLE
www.targate.org Page 71
(A) Set of radial straight lines
(B) Set of concentric circles
(C) Set of co focal hyperbolas
(D) Set of co focal ellipses
eE1 / T4 / K2 / L2 / V2 / R11 / AA [GATE – CE – 2007]
(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
eE1 / T4 / K2 / L2 / V2 / R11 / AB [GATE – CE – 2010]
(08) The modulus of the complex number 3 41 2
ii
is
(A) 5 (B) 5
(C) 1
5 (D)
1
5
eE1 / T4 / K2 / L2 / V2 / R11 / AA [GATE – PI – 2010]
(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-----
Question Level – 03
eE1 / T4 / K2 / L3 / V2 / R11 / AB [GATE – ME – 2007]
(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?
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ENGINEERING MATHEMATICS
Page 72 TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
eE1 / T4 / K2 / L3 / V2 / R11 / AB [GATE – IN – 2009]
(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
eE1 / T4 / K2 / L3 / V2 / R11 / AC [GATE – ME – 2009]
(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
eE1 / T4 / K2 / L3 / V2 / R11 / AB [GATE – PI – 2010]
(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
eE1 / T4 / K2 / L3 / V2 / R11 / AD [GATE – EE – 2011]
(06) A point z has been plotted in the complex plane
as shown in the figure below
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TOPIC. 04 – COMPLEX VARIABLE
www.targate.org Page 73
The plot of the complex number
-----00000-----
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ENGINEERING MATHEMATICS
Page 74 TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
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
Question Level – 01
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-----
Question Level – 02
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-----
Question Level – 03
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
eE1 / T5 / K2 / L3 / V2 / R11 / AC [GATE – ME – 2010]
(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
eE1 / T5 / K2 / L3 / V1 / R11 / AC [GATE – EE – 2010]
(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
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TOPIC. 05 – PROBABILITY AND STATISTICS
www.targate.org Page 75
ball is white, the probability that the 2nd removed
ball is red is
(A) 1/3 (B) 3/7
(C) ¼ (D) 4/7
eE1 / T5 / K2 / L3 / V2 / R11 / AC [GATE – PI – 2010]
(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?
eE1 / T5 / K2 / L3 / V2 / R11 / AC [GATE – CE – 2011]
(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
Question Level – 00 (Basic Problem)
eE1 / T5 / K3 / L0 / V1 / R11 / AD [GATE – EE – 2005]
(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)
eE1 / T5 / K3 / L0 / V1 / R11 / AB [GATE – EE – 2005]
(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
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ENGINEERING MATHEMATICS
Page 76 TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
eE1 / T5 / K3 / L0 / V1 / R11 / AD [GATE – EC – 2005]
(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-----
Question Level – 01
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
eE1 / T5 / K3 / L1 / V2 / R11 / AA [GATE – CS – 2004]
(06) If a fair coin is tossed 4 times, what is the
probability that two heads and two tails will
result?
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TOPIC. 05 – PROBABILITY AND STATISTICS
www.targate.org Page 77
(A) 3
8 (B)
1
2
(C) 5
8 (D)
3
4
eE1 / T5 / K3 / L1 / V2 / R11 / AD [GATE – PI – 2005]
(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
eE1 / T5 / K3 / L1 / V1 / R11 / AA [GATE – ME – 2008]
(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
eE1 / T5 / K3 / L1 / V1 / R11 / AC [GATE – EC – 2007]
(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
eE1 / T5 / K3 / L1 / V1 / R11 / AD [GATE – EC – 2010]
(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
eE1 / T5 / K3 / L1 / V1 / R11 / AC [GATE – CE – 2010]
(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
eE1 / T5 / K3 / L1 / V1 / R11 / AD [GATE – ME – 2011]
(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
eE1 / T5 / K3 / L1 / V1 / R11 / AA [GATE – CS – 2011]
(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-----
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ENGINEERING MATHEMATICS
Page 78 TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
Question Level – 02
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
eE1 / T5 / K3 / L2 / V2 / R11 / AD [GATE – ME – 2005]
(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
eE1 / T5 / K3 / L2 / V2 / R11 / AC [GATE – EC – 2011]
(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
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TOPIC. 05 – PROBABILITY AND STATISTICS
www.targate.org Page 79
Question Level – 03
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
eE1 / T5 / K3 / L3 / V2 / R11 / AA [GATE – PI – 2007]
(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
eE1 / T5 / K3 / L3 / V2 / R11 / AC [GATE – PI – 2008]
(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
eE1 / T5 / K3 / L3 / V2 / R11 / AA [GATE – CS – 2010]
(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
eE1 / T5 / K3 / L3 / V2 / R11 / AD [GATE – IN – 2011]
(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
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ENGINEERING MATHEMATICS
Page 80 TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
eE1 / T5 / K3 / L3 / V2 / R11 / AC [GATE – PI – 2011]
(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
eE1 / T5 / K3 / L3 / V1 / R11 / AD [GATE – ME – 2005]
(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
eE1 / T5 / K3 / L3 / V1 / R11 / AB [GATE – EE – 2009]
(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
Question Level – 00 (Basic Problem)
eE1 / T5 / K5 / L0 / V1 / R11 / AA [GATE – IN – 2007]
(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-----
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TOPIC. 05 – PROBABILITY AND STATISTICS
www.targate.org Page 81
Question Level – 01
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
eE1 / T5 / K5 / L1 / V1 / R11 / AB [GATE – ME – 2005]
(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
eE1 / T5 / K5 / L1 / V1 / R11 / AA [GATE – PI – 2005]
(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
eE1 / T5 / K5 / L1 / V1 / R11 / AC [GATE – CE – 2007]
(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
Question Level – 02
eE1 / T5 / K5 / L2 / V1 / R11 / AD [GATE – PI – 2007]
(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-----
Question Level – 03
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-----
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ENGINEERING MATHEMATICS
Page 82 TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
5.6 Random Variable
Question Level – 00 (Basic Problem)
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
eE1 / T5 / K6 / L0 / V1 / R11 / AD [GATE – ME – 2007]
(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
eE1 / T5 / K6 / L0 / V1 / R11 / AA [GATE – ME – 2009]
(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-----
Question Level – 01
eE1 / T5 / K6 / L1 / V1 / R11 / AC [GATE – EC – 2008]
(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
eE1 / T5 / K6 / L1 / V1 / R11 / AA [GATE – IN – 2008]
(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
eE1 / T5 / K6 / L1 / V1 / R11 / AB [GATE – IN – 2008]
(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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TOPIC. 05 – PROBABILITY AND STATISTICS
www.targate.org Page 83
eE1 / T5 / K6 / L1 / V1 / R11 / AA [GATE – IN – 2008]
(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
eE1 / T5 / K6 / L1 / V1 / R11 / AA [GATE – PI – 2008]
(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
eE1 / T5 / K6 / L1 / V1 / R11 / AD [GATE – IN – 2009]
(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
eE1 / T5 / K6 / L1 / V1 / R11 / AA [GATE – CS – 2010]
(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-----
Question Level – 02
eE1 / T5 / K6 / L2 / V2 / R11 / AD [GATE – PI – 2010]
(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
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ENGINEERING MATHEMATICS
Page 84 TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
eE1 / T5 / K6 / L2 / V2 / R11 / AA [GATE – PI – 2007]
(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-----
Question Level – 03
eE1 / T5 / K6 / L3 / V2 / R11 / AD [GATE – CE – 2009]
(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%
eE1 / T5 / K6 / L3 / V2 / R11 / AC [GATE – EC – 2009]
(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
eE1 / T5 / K6 / L3 / V2 / R11 / AB [GATE – EC – 2009]
(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
eE1 / T5 / K6 / L3 / V2 / R11 / AC [GATE – IN – 2009]
(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
eE1 / T5 / K6 / L3 / V2 / R11 / AD [GATE – CS – 2011]
(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
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TOPIC. 05 – PROBABILITY AND STATISTICS
www.targate.org Page 85
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
Question Level – 01
eE1 / T5 / K2 / L1 / V1 / R11 / AA [GATE – EC – 2007]
(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-----
Question Level – 02
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-----
Question Level – 03
eE1 / T5 / K2 / L3 / V2 / R11 / AD [GATE – CS – 2004]
(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-----
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ENGINEERING MATHEMATICS
Page 86 TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
5.8 SET THEORY
Question Level – 03
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-----
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Page 87 TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
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
Question Level – 01
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-----
Question Level – 02
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-----
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ENGINEERING MATHEMATICS
Page 88 TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
Question Level – 03
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
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TOPIC. 06 – NUMERICAL METHODS
www.targate.org Page 89
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
Question Level – 00 (Basic Problem)
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-----
Question Level – 01
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
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ENGINEERING MATHEMATICS
Page 90 TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
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-----
Question Level – 02
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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TOPIC. 06 – NUMERICAL METHODS
www.targate.org Page 91
(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
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ENGINEERING MATHEMATICS
Page 92 TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
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-----
Question Level – 03
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-----
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TOPIC. 06 – NUMERICAL METHODS
www.targate.org Page 93
6.3 Differential
Question Level – 00 (Basic Problem)
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-----
Question Level – 02
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
-----00000-----
6.4 Integration
Question Level – 00 (Basic Problem)
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-----
Question Level – 01
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-----
Question Level – 03
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
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ENGINEERING MATHEMATICS
Page 94 TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
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
-----00000-----
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Page 95 TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
07
Transform Theory
Complete subtopic in this chapter, is in the scope of “GATE- EC/EE SYLLABUS”
Question Level – 00 (Basic Problem)
eE1 / T7 / K2 / L0 / V1 / R11 / AB [GATE – EE – 1995]
(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
eE1 / T7 / K2 / L0 / V1 / R11 / AA [GATE – EC – 2006]
(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)
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ENGINEERING MATHEMATICS
Page 96 TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
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
eE1 / T7 / K2 / L0 / V1 / R11 / AB [GATE – EC – 1998]
(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-----
Question Level – 01
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
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TOPIC. 07 – TRANSFORM THEORY
www.targate.org Page 97
eE1 / T7 / K2 / L1 / V1 / R11 / AD [GATE – EC – 2005]
(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
eE1 / T7 / K2 / L1 / V1 / R11 / AA [GATE – EC – 1997]
(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
eE1 / T7 / K2 / L1 / V1 / R11 / AB [GATE – EC – 1999]
(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
eE1 / T7 / K2 / L1 / V1 / R11 / AA [GATE – EC – 1997]
(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-----
Question Level – 02
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
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ENGINEERING MATHEMATICS
Page 98 TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
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
eE1 / T7 / K2 / L2 / V2 / R11 / AC [GATE – EC – 2003]
(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
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TOPIC. 07 – TRANSFORM THEORY
www.targate.org Page 99
eE1 / T7 / K2 / L2 / V2 / R11 / AA [GATE – EC – 2005]
(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-----
Question Level – 03
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
eE1 / T7 / K2 / L3 / V2 / R11 / AD [GATE – EE – 1995]
(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
eE1 / T7 / K2 / L3 / V2 / R11 / AB [GATE – EC – 1995]
(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
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ENGINEERING MATHEMATICS
Page 100 TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
eE1 / T7 / K2 / L3 / V2 / R11 / AC [GATE – EC – 1998]
(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 ------
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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.
8 students scored above 99 percentile.
Highest rank of 400.
GATE - 2013 RESULT (@ TARGATE EDU)EC/EE/CS:
PARAS JAIN
(EC - 99.54 %ile)
AJAY TIWARI
(EC - 99.38 %ile)
ANKUR GUPTA
(EC - 99.00 %ile)
AMIT JAISWAL
(EC - 98.83 %ile)
VARUN DAS
(EC - 98.34 %ile)
NARENDRA PATEL
(EC - 98.28 %ile)
SURYAKANT
(CS - 99.81 %ile)
SAURABH SINGH
(CS - 98.43 %ile)
SUMIT CHAURASIA
(CS - 98.43 %ile)
SHAHRUKH KHAN
(CS - 97.71 %ile)
Many more…………..(28 students qualified out of 45)
NAVEEN YADAV
(CS - 96.86 %ile)
PRAKASH PURI
(EE - 93.95 %ile)
LAVLEEN DHALLA
(EE - 89.25 %ile)
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