ENGINEERING MATHEMATICS Objective Paper –“Topic & Level-wise”
GATE For “Electrical”, “Mechanical”, “CS/IT” & “Electronics & Comm.” Engg.
Product of,
TARGATE EDUCATION a team of
Copyright © TARGATE EDUCATION, Bilaspur-2013
All rights reserved No part of this publication may be reproduced, stored in retrieval system, or transmitted in any form or by any means, electronics, mechanical, photocopying, digital, recording or otherwise without the prior permission of the TARGATE EDUCATION.
Authors: Subject Experts @TRGATE EDUCATION, BILASPUR
TARGATE EDUCATION Ground Floor, Below Old Arpa Bridge,Jabdapara, SARKANDA RD. Bilaspur (Chhattisgarh) 495001 Phone No: 07752406380,093004-32128 (01:30 PM - 07:30 PM, Wed-Off) Web Address: www.targate.org, E-Contact:
[email protected]
SYLLABUS: ENGG. MATHEMATICS GATE – 2013 EE /ECEC Linear Algebra: Matrix Algebra, Systems of linear equations, Eigen values and eigen vectors. Calculus: Mean value theorems, Theorems of integral calculus, Evaluation of definite and improper integrals, Partial Derivatives, Maxima and minima, Multiple integrals, Fourier series. Vector identities, Directional derivatives, Line, Surface and Volume integrals, Stokes, Gauss and Green’s theorems. Differential equations: First order equation (linear and nonlinear), Higher order linear differential equations with constant coefficients, Method of variation of parameters, Cauchy’s and Euler’s equations, Initial and boundary value problems, Partial Differential Equations and variable separable method. Complex variables: Analytic functions, Cauchy’s integral theorem and integral formula, Taylor’s and Laurent’ series, Residue theorem, solution integrals. Probability and Statistics: Sampling theorems, Conditional probability, Mean, median, mode and standard deviation, Random variables, Discrete and continuous distributions, Poisson,Normal and Binomial distribution, Correlation and regression analysis. Numerical Methods: Solutions of non-linear algebraic equations, single and multi-step methods for differential equations. Transform Theory: Fourier transform,Laplace transform, Z-transform.
Mechanical Engineering (ME) Linear Algebra: Matrix algebra, Systems of linear equations, Eigen values and eigen vectors. Calculus: Functions of single variable, Limit, continuity and differentiability, Mean value theorems, Evaluation of definite and improper integrals, Partial derivatives, Total derivative, Maxima and minima, Gradient, Divergence and Curl, Vector identities, Directional derivatives, Line, Surface and Volume integrals, Stokes, Gauss and Green’s theorems. Differential equations: First order equations (linear and nonlinear), Higher order linear differential equations with constant coefficients, Cauchy’s and Euler’s equations, Initial and boundary value problems, Laplace transforms, Solutions of one dimensional heat and wave equations and Laplace equation. Complex variables: Analytic functions, Cauchy’s integral theorem, Taylor and Laurent series. Probability and Statistics: Definitions of probability and sampling theorems, Conditional probability, Mean, median, mode and standard deviation, Random variables, Poisson,Normal and Binomial distributions. Numerical Methods: Numerical solutions of linear and non-linear algebraic equations Integration by trapezoidal and Simpson’s rule, single and multi-step methods for differential equations.
Computer Science and Information Technology (CS) Mathematical Logic: Propositional Logic; First Order Logic. Probability: Conditional Probability; Mean, Median, Mode and Standard Deviation; Random Variables; Distributions; uniform, normal, exponential, Poisson, Binomial. Set Theory & Algebra: Sets; Relations; Functions; Groups; Partial Orders; Lattice; Boolean Algebra. Combinatorics: Permutations; Combinations; Counting; Summation; generating functions; recurrence relations; asymptotics. Graph Theory: Connectivity; spanning trees; Cut vertices & edges; covering; matching; independent sets; Colouring; Planarity; Isomorphism. Linear Algebra: Algebra of matrices, determinants, systems of linear equations, Eigen values and Eigen vectors. Numerical Methods: LU decomposition for systems of linear equations; numerical solutions of non-linear algebraic equations by Secant, Bisection and Newton-Raphson Methods; Numerical integration by trapezoidal and Simpson’s rules. Calculus: Limit, Continuity & differentiability, Mean value Theorems, Theorems of integral calculus, evaluation of definite & improper integrals, Partial derivatives, Total derivatives, maxima & minima.
Expert Comment Comparing 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.
Table of Contents LINEAR ALGEBRA
7
1.1 PROPERTY BASED PROBLEM 1.2 DETERMINANTE 1.3 ADJOINT - INVERSE 1.4 EIGEN VALUES & EIGEN VECTORS 1.5 RANK 1.6 SOLUTION OF LINEAR EQUATION 1.7 MISCELLANEOUS 1.8 CALY- HAMILTON
7 10 11 13 19 21 26 31
CALCULUS
32
2.1 MEAN VALUE THEOREM 2.2 MAXIMA AND MINIMA 2.3 DIFFERENTIAL CALCULUS 2.4 INTEGRAL CALCULUS 2.5 LIMIT AND CONTINUITY 2.6 SERIES 2.7 VECTOR CALCULUS 2.8 AREA / VOLUME 2.9 MISCELLANEOUS
32 32 34 36 39 43 44 51 52
DIFFERENTIAL EQUATIONS
55
3.1 DEGREE AND ORDER OF DE 3.2 HIGHER ORDER DE 3.3 LEIBNITZ LINEAR EQUATION 3.4 MISCELLANEOUS
55 56 61 62
COMPLEX VARIABLE
66
4.1CAUCHY’S THEOREM 4.2 MISCELLANEOUS
66 68
PROBABILITY AND STATISTICS
74
5.2 COMBINATION 5.3 PROBABILITY RELATED PROBLEMS 5.4 BAYS THEOREMS 5.5 PROBABILITY DISTRIBUTION 5.6 RANDOM VARIABLE 5.7 EXPECTION
74 75 80 80 82 85
Page 5
www.targate.org
5.8 SET THEORY
86
NUMERICAL METHODS
87
6.1 CLUBBED PROBLEM
87
6.2 NEWTON-RAP SON
89
6.3 DIFFERENTIAL 6.4 INTEGRATION
93 93
TRANSFORM THEORY
95
01 Linear Algebra Complete subtopic in this chapter, is in the scope of “GATE- CS/ME/EC/EE SYLLABUS”
.
1.1 Property Based Problem
1 (A) P
(B) Q 1
Question Level – 0 (Basic Problems)
(C) P 1Q 1 P
(D) P Q P 1
eE1 / T1 / K1 / L0 / V1 / R11 / AD [GATE – CS – 1994]
-----00000-----
(01) If A and B are real symmetric matrices of order n then which of the following is true.
Question Level – 01 T
-1
(A) A A = I
(B) A = A
eE1 / T1 / K1 / L1 / V1 / R11 / AB [GATE – CE – 1998] T
(C) AB = BA
T
T
(D) (AB) = B A
eE1 / T1 / K1 / L0 / V1 / R11 / AA [GATE – PI – 1994]
(01) If A is a real square matrix then AAT is
(A) Un symmetric
(02) If for a matrix, rank equals both the number of rows and number of columns, then the matrix is
(B) Always symmetric
called (A) Non-singular
(B) singular
(C) Transpose
(D) Minor
(C) Skew – symmetric
(D) Sometimes symmetric eE1 / T1 / K1 / L1 / V1 / R11 / AC [GATE – CE – 1998]
eE1 / T1 / K1 / L0 / V1 / R11 / A [GATE – CE – 2000]
(03) If A, B, C are square matrices of the same order then ( ABC )
1
(02) In matrix algebra AS = AT (A, S, T, are matrices of appropriate order) implies S = T only if
is equal be (A) A is symmetric
1
1
(A) C A B
1
1
1
1
(B) C B A
(B) A is singular (C) A1 B1C 1
(D) A1C 1 B1 (C) A is non-singular
eE1 / T1 / K1 / L0 / V1 / R11 / AB [GATE – CE – 2008]
(04) The product of matrices ( PQ) 1 P is
Page 7
(D) A is skew=symmetric
www.targate.org
ENGINEERING MATHEMATICS eE1 / T1 / K1 / L1 / V1 / R11 / AA [GATE – IN – 2001]
(A) Always zero
(03) The necessary condition to diagonalizable a matrix is that
(B) Always pure imaginary
(A) Its all Eigen values should be distinct
(C) Either zero (or) pure imaginary
(B) Its Eigen values should be independent
(D) Always real
(C) Its Eigen values should be real
eE1 / T1 / K1 / L1 / V1 / R11 / AC [GATE – ME – 2011]
(07) Eigen values of a real symmetric matrix are (D) The matrix is non-singular
always
eE1 / T1 / K1 / L1 / V1 / R11 / AD [GATE – EC – 2005]
(A) Positive
(B) Negative
(C) Real
(D) 162. [A] is square
(04) Given an orthogonal matrix A =
1 1 1 1 1 1 1 1 T 1 1 1 0 0 ( AA ) Is ____ 0 0 1 1
-----00000-----
Question Level – 02 1 (A) I 4 4
1 (B) I 4 2
(C) I
(D)
1 I4 3
eE1 / T1 / K1 / L2 / V2 / R11 / AA [GATE – CS – 2001]
(01) Consider the following statements
S1: The sum of two singular matrices may be eE1 / T1 / K1 / L1 / V1 / R11 / AA [GATE – ME – 2007]
singular.
(05) If a square matrix A is real and symmetric then the Eigen values
S2: The sum of two non-singulars may be nonsingular.
(A) Are always real This of the following statements is true. (B) Are always real and positive (A) S1 & S2 are both true (C) Are always real and non-negative (B) S1 & S2 are both false (D) Occur in complex conjugate pairs (C) S1 is true and S2 is false eE1 / T1 / K1 / L1 / V1 / R11 / AC [GATE – EC – 2010]
(06) The Eigen values of a skew-symmetric matrix are
Page 8
(D) S1 is false and S2 is true
TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
TOPIC. 01 – LINEAR ALGEBRA 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 A ( AT A) 1 AT .
(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
then which one of the following statements is (B) Both the statements are true
false? (A) AA+A = A
(B) (AA+)2 = AA+
(C) (I) is true but (II) is false
(C) A+A = I
(D) AA+A = A+
(D) (I) is false but (II) is true
eE1 / T1 / K1 / L2 / V1 / R11 / AB [GATE – CE – 2009]
(03) A square matrix B is symmetric if ------------(A) BT = B
eE1 / T1 / K1 / L3 / V2 / R11 / AA [GATE – CE – 2004]
(03) Real
E 5 , F 1
(B) BT = B
A31, B 33 , C 35 , D ,
matrices
are given. Matrices [B] and [E]
are symmetric. Following statements are made with respect to their matrices.
(D) B 1 = BT
(C) B 1 = B
(I) Matrix product [F]T[C]T[B] [C] [F] is a scalar. Matrix product [D]T[F] [D] is always -----00000-----
symmetric. With reference to above statements which of the following applies?
Question Level – 03 (A) Statement (I) is true but (II) is false eE1 / T1 / K1 / L3 / V2 / R11 / AD [GATE – CE – 1998]
(01) The real symmetric matrix C corresponding to
(B) Statement (I) is false but (II) is true
the quadratic form Q = 4x1 x2 5x1 x2 is (C) Both the statements are true
1 2 (A) 2 5
2 0 (B) 0 5
1 1 (C) 1 2
0 2 (D) 2 5
(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
eE1 / T1 / K1 / L3 / V2 / R11 / AB [GATE – CE – 2000]
(02) Consider the following two statements.
x1
T
x2
with length || x || =
( x12 x22 )1/2 Then which one of the following
statement is correct? (I)
The
maximum
number
of
linearly
independent column vectors of a matrix A is called the rank of A.
(A) || px |||| x || where at least one vector satisfies || px |||| x ||
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Page 9
ENGINEERING MATHEMATICS (B) || px |||| x || for all vectors x
1.2 Determinante
(C) || px |||| x || when atleast one vector satisfies
Question Level – 00 (Basic Problem)
|| x || and || px || eE1 / T1 / K2 / L0 / V1 / R11 / AD [GATE – PI – 1994]
(01) The (D) No relationship can be established between
1
|| x || and || px ||
value
4
of
the
following
determinant
9
4 9 16 is 9 16 25
eE1 / T1 / K6 / L3 / V2 / R11 / A [GATE – CS – 2008]
(05) The
following
system
x1 x2 2 x3 1,
of
equations
x1 2x1 3x3
(A) 8
(B) 12
(C) – 12
(D) – 8
,
x1 4 x1 αx3 4 has a unique solution solution. The only possible value(s) for α is/are
Question Level – 01 (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.
eE1 / T1 / K2 / L1 / V2 / R11 / AB [GATE – CS – 1997]
6 8 0 2 (01) The determinant of the matrix 0 0 0 0
1 4
1 6 4 8 0 1
The sum and differences of these matrices are defined as [S] = [A] + [A]T and [D] = [A] – [A]T
(A) 11
(B) – 48
(C) 0
(D) – 24
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
eE1 / T1 / K2 / L1 / V1 / R11 / AA [GATE – CE – 1997]
1 3 2 (02) If the determinant of the matrix 0 5 6 is 2 7 8 26
(D) [S] is symmetric and [D] is skew-symmetric
-----00000-----
Page 10
then
the
determinant
of
2 7 8 0 5 6 is 1 3 2
(A) – 26
(B) 26
(C) 0
(D) 52
TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
the
matrix
TOPIC. 01 – LINEAR ALGEBRA Question Level – 02
1.3 Adjoint - Inverse
eE1 / T1 / K2 / L2 / V2 / R11 / AB [GATE – CS – 1998]
Question Level – 00 (Basic Problem) 1 a bc (01) If = 1 b ca then which of the following is 1 c ab
1 2 (01) The inverse of 2 2 matrix is 5 7
a factor of .
(A) a + b
(B) a - b
(C) abc
1 b (02) The determinant
b 1
b
1
1 7 2 3 5 1
(B)
1 7 2 3 5 1
(C)
1 7 2 3 5 1
(D)
1 7 2 3 5 1
Question Level – 01
1 b 1 equals to 2b 1
(A) 0
(B) 2b(b – 1)
(C) 2(1 – b)(1 + 2b)
(D) 3b(1 + b)
eE1 / T1 / K3 / L1 / V1 / R11 / AA [GATE – PI – 1994]
1 4 (01) The matrix is an inverse of the matrix 1 5 5 4 1 1
eE1 / T1 / K2 / L2 / V2 / R11 / AB [GATE – PI – 2009]
1 3 2
(A) True
(B) False
Question Level – 02
(03) The value of the determinant 4 1 1 is 2 1 3
(C) 32
(A) (D) a + b + c
eE1 / T1 / K2 / L2 / V2 / R11 / AA [GATE – PI – 2007]
(A) – 28
eE1 / T1 / K3 / L0 / V1 / R11 / AA [GATE – CE – 2007]
eE1 / T1 / K3 / L2 / V2 / R11 / AD [GATE – EE – 1995]
(B) – 24
(D) 36
-----00000-----
1 1 0 (01) The inverse of the matrix S = 1 1 1 is 0 0 1
1 0 1 (A) 0 0 0 0 1 1
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0 1 1 (B) 1 1 1 1 0 1
Page 11
ENGINEERING MATHEMATICS 2 2 2 (C) 2 2 2 0 2 2
1 / 2 1 / 2 1 / 2 (D) 1 / 2 1 / 2 1 / 2 0 0 1
eE1 / T1 / K3 / L2 / V1 / R11 / AA [GATE – CE – 1997]
0 1 0 (02) Inverse of matrix 0 0 1 is 1 0 0 0 0 1 (A) 1 0 0 0 1 0
1 0 0 (B) 0 0 1 0 1 0
1 0 0 (C) 0 1 0 0 0 1
0 0 1 (D) 0 1 0 1 0 0
(A) – 5
(B) 3
(C) – 3
(D) 5
eE1 / T1 / K3 / L2 / V2 / R11 / AA [GATE – PI – 2008]
0 1 0 (05) The inverse of matrix 1 0 0 is 0 0 1
0 1 0 (A) 1 0 0 0 0 1
0 1 0 (B) 1 0 0 0 0 1
0 1 0 (C) 0 0 1 1 0 0
0 1 0 (D) 0 0 1 1 0 0
eE1 / T1 / K3 / L2 / V2 / R11 / AA [GATE – EE – 1998] eE1 / T1 / K3 / L2 / V2 / R11 / AA [GATE – ME – 2009]
5 0 2 1 (03) If A = 0 3 0 then A = 2 0 1 0 2 1 0 1/ 3 0 (A) 2 0 5
0 2 5 0 1/ 3 0 (B) 2 0 1
3 / 4 4 / 5 (06) For a matrix [M] = . The transpose x 3 / 5 of the matrix is equal to the inverse of the matrix, [ M ]T [ M ]1. The value of x is given by
(A)
1/ 5 0 1/ 2 (C) 0 1/ 3 0 1/ 2 0 1
0 1 / 2 1/ 5 (D) 1/ 3 0 0 1 / 2 0 1
(C)
3 5
4 5
(B)
(D)
3 5
4 5
eE1 / T1 / K3 / L2 / V2 / R11 / AA [GATE – EE – 1999] eE1 / T1 / K3 / L2 / V2 / R11 / AB [GATE – CE – 2010]
1 2 1 (04) If A = 2 3 1 and ad(A) = 0 5 2
11 9 1 4 2 3 Then k = 10 k 7
Page 12
i 3 2i (07) The inverse of the matrix is 3 2i i
(A)
i 1 3 2i 3 2i 2 i
(B)
i 1 3 2i 3 2i 12 i
TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
TOPIC. 01 – LINEAR ALGEBRA (C)
i 1 3 2i 3 2i 14 i
(D)
i 1 3 2i 14 i 3 2i
(A) 5
(B) 7
(C) 9
(D) 18
eE1 / T1 / K4 / L0 / V1 / R11 / AC [GATE – CE – 2004]
-----00000-----
4 2 (04) The eigen values of the matrix are 2 1
1.4 Eigen Values & Eigen Vectors (A) 1, 4
(B) – 1, 2
(C) 0, 5
(D) cannot be determined
Question Level – 00 (Basic Problem) eE1 / T1 / K4 / L0 / V1 / R11 / AA [GATE – EE – 1994] eE1 / T1 / K4 / L0 / V1 / R11 / AB [GATE – CS – 2005]
a 1 (01) The Eigen values of the matrix are a 1
(A) (a 1),0
(B) a, 0
(C) (a 1),0
(D) 0, 0
2 1 matrix? 4 5
eE1 / T1 / K4 / L0 / V1 / R11 / AA [GATE – EE – 1998]
2 0 (02) A = 0 1
(05) What are the Eigen values of the following 2 x 2
0 0 1 1 0 0 the sum of the Eigen 0 3 0 0 0 4
Values of the matrix A is
(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
8 6 2 are 15, 3 and 0. M = 6 7 4 , the value of 2 4 3 the determinant of a matrix is
(A) 10
(B) – 10
(C) 24
(D) 22
eE1 / T1 / K4 / L0 / V1 / R11 / AB [GATE – ME – 2004]
(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 (03) The sum of the eigen values of the matrix given
1 1 3 below is 1 5 1 3 1 1
Eigen value 1?
1 0 0 1 1 1 1 0 0 0 , 0 0 , 1 1 & 0 1
www.targate.org
Page 13
ENGINEERING MATHEMATICS (A) One
(B) Two
(C) Three
(D) Four
eE1 / T1 / K4 / L1 / V1 / R11 / AD [GATE – EC – 1998]
0 1 (03) The eigen values of the matrix A = are 1 0
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
(A) 1, 1
(B) -1, -1
(C) j , j
(D) 1, 1
values are eE1 / T1 / K4 / L1 / V1 / R11 / AD [GATE – CE – 2001
(A) -30, -5
(B) -37, -1
(C) -7, 5
(D) 17.5, -2
-----00000-----
]
5 3 (04) The eigen values of the matrix are 2 9
(A) (5.13,9.42)
(B) (3.85,2.93)
(C) (9.00,5.00)
(D) (10.16,3.84)
Question Level – 01 eE1 / T1 / K4 / L1 / V1 / R11 / AA [GATE – CE – 2002] eE1 / T1 / K4 / L1 / V1 / R11 / AA [GATE – –1993]
(01) The eigen vector (s) of the matrix
0 0 α 0 0 0 , α 0 Is (are); 0 0 0
(A) 0,0, α
(B) α ,0,0
(C) 0,0,1
(D) 0, α ,0
(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] eE1 / T1 / K4 / L1 / V1 / R11 / AC [GATE – ME – 1996]
1 1 1 (02) The eigen values of 1 1 1 are 1 1 1
(A) 0, 0, 0
(B) 0, 0, 1
(C) 0, 0,3
(D) 1, 1, 1
Page 14
(06) Identify which one of the following is an eigen
1 0 vector of the matrix A = 1 2
T
(B) 3 1
T
(D) 2 1
(A) 1 1
(C) 1 1
T
TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
T
TOPIC. 01 – LINEAR ALGEBRA eE1 / T1 / K4 / L1 / V1 / R11 / AB [GATE – CE – 2007]
2 (A) 1
2 (B) 1
4 (C) 1
1 (D) 1
(07) The minimum and maximum Eigen values of
1 1 3 Matrix 1 5 1 are -2 and 6 respectively. 3 1 1
eE1 / T1 / K4 / L1 / V1 / R11 / AD [GATE – CS – 2010]
What is the other Eigen value?
(A) 5
(B) 3
(C) 1
(D) -1
2 (11) Consider the following matrix A = x
3 . If y
the eigen values of A are 4 and 8 then
eE1 / T1 / K4 / L1 / V1 / R11 / AD [GATE – EC – 2008]
(A) x = 4, y = 10
(B) x = 5, y = 8
(C) x = -3, y = 9
(D) x = -4, y = 10
(08) All the four entries of 2 x 2 matrix P =
p11 p 21
p12 are non-zero and one of the Eigen p22
values is zero. Which of the following statement
eE1 / T1 / K4 / L1 / V2 / R11 / AA [GATE – CS – 2002]
(12) Obtain the eigen values of the matrix A =
is true?
(A) P11P22 P12 P21 1
(B) P11P22 P12 P21 1
(C) P11P22 P21P12 0
(D) P11P22 P12 P21 0
eE1 / T1 / K4 / L1 / V1 / R11 / AB [GATE – CE – 2008]
4 5 (09) The eigen values of the matrix [P] = are 2 5
(A) – 7 and 8
(B) – 6 and 5
(C) 3 and 4
(D) 1 and2
1 0 0 0
2 34 2 43
49 94 0 2 104 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 / L1 / V1 / R11 / AA [GATE – ME – 2010]
(10) One of the eigen vector of the matrix A =
2 2 1 3 is
eE1 / T1 / K4 / L2 / V2 / R11 / AC [GATE – EE – 1998]
1 (01) The vector 2 is an eigen vector of A = 1
www.targate.org
Page 15
ENGINEERING MATHEMATICS eE1 / T1 / K4 / L2 / V2 / R11 / AA [GATE – ME – 2006]
2 2 3 2 1 6 one of the eigen value of A is 1 2 0
3 2 (04) Eigen values of a matrix S = are 5 and 1. 2 3
(A) 1
(B) 2
What are the Eigen values of the matrix S2 = SS?
(C) 5
(D) -1
(A) 1 and 25
(B) 6, 4
(C) 5, 1
(D) 2, 10
eE1 / T1 / K4 / L2 / V2 / R11 / AB [GATE – EC – 2000]
2 1 0 0 3 0 (02) The eigen values of the matrix 0 0 2 0 0 1
0 0 0 4
are
(A) 2, -2, 1, -1
(B) 2, 3, -2, 4
(C) 2, 3, 1, 4
(D) None
eE1 / T1 / K4 / L2 / V1 / R11 / AB [GATE – ME – 2007]
(05) The number of linearly independent eigen vectors
2 1 of is 0 2
(A) 0
(B) 1
(C) 2
(D) Infinite
eE1 / T1 / K4 / L2 / V2 / R11 / AD [GATE – EE – 2005] eE1 / T1 / K4 / L2 / V2 / R11 / AC [GATE – ME – 2008]
3 2 2 (03) For the matrix P = 0 2 1 , one of the Eigen 0 0 1
1 2 4 (06) The matrix 3 0 6 has one eigen value to 3. 1 1 p
A value is – 2. Which of the following is an The sum of the other two eigen values is
Eigen vector?
3 (A) 2 1
1 (C) 2 3
Page 16
3 (B) 2 1
2 (D) 5 0
(A) p
(B) p – 1
(C) p – 2
(D) p – 3
eE1 / T1 / K4 / L2 / V2 / R11 / AB [GATE – ME – 2008]
1 2 (07) The eigen vectors of the matrix are 0 2 1 1 written in the form & . What is a+ b? a b
TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
TOPIC. 01 – LINEAR ALGEBRA (A) 0
(B) 1/2
(C) 1
(D) 2
eE1 / T1 / K4 / L2 / V2 / R11 / AB [GATE – EE – 2010]
eE1 / T1 / K4 / L2 / V2 / R11 / AA [GATE – PI – 2008]
3 4 (08) The eigen vector pair of the matrix is 4 3
1 1 0 (11) An eigen vector of p = 0 2 2 is 0 0 3
T
(B) 1 2 1
T
(D) 2 1 1
(A) 1 1 1
(C) 1 1 2
2 1 (A) 1 2
2 1 (B) 1 2
T
T
eE1 / T1 / K4 / L2 / V2 / R11 / AB [GATE – PI – 2011]
(12) The Eigen values of the following matrix
2 1 (C) 1 2
10 4 18 12 are
2 1 (D) 1 2
(A) 4, 9
(B) 6, - 8
(C) 4, 8
(D) – 6, 8
eE1 / T1 / K4 / L2 / V2 / R11 / AC [GATE – EC – 2006]
4 2 (09) For the matrix . The eigen value 2 4 101 corresponding to the eigen vector is 101
(A) 2
(B) 4
(C) 6
(D) 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 are 0 0 3
(A) 3, 3 5 j,6 j
(B) 6 5 j,3 j,3 j
(C) 3 j,3 j,5 j
(D) 3, 1 3 j, 1 3 j
eE1 / T1 / K4 / L2 / V2 / R11 / AB [GATE – CE – 2006]
2 2 3 (10) For a given matrix A = 2 1 6 , one of the 1 2 0 eigen value is 3. The other two eigen values are
(A) 2, -5
(B) 3, -5
eE1 / T1 / K4 / L2 / V2 / R11 / AB [GATE – IN – 2011]
2 2 3 (14) The matrix M = 2 1 6 has eigen values 1 2 0 -3, -3, 5. An eigen vector corresponding to the T
eigen value 5 is 1 2 1 . One of the eigen (C) 2, 5
(D) 3, 5
vector of the matrix M3 is
www.targate.org
Page 17
ENGINEERING MATHEMATICS T
T
(A) 1 8 1
(C) 1
3
1
2
(B) 1 2 1 T
T
(D) 1 1 1
Question Level – 03 eE1 / T1 / K4 / L3 / V2 / R11 / AB [GATE – PI – 2007]
(01) If A is square symmetric real valued matrix of
eE1 / T1 / K4 / L2 / V2 / R11 / AA [GATE – CS – 2011]
1 2 3 (15) Consider the matrix as given below 0 4 7 . 0 0 3 Which one of the following options provides the
dimension 2n, then the eigen values of A are (A) 2n distinct real values
(B) 2n real values not necessarily distinct
correct values of the eigen values of the matrix? (C) n distinct pairs of complex conjugate (A) 1, 4, 3
(B) 3, 7, 3
(C) 7, 3, 2
(D) 1, 2, 3
numbers
(D) n pairs of complex conjugate numbers, not necessarily distinct
eE1 / T1 / K4 / L2 / V2 / R11 / AA [GATE – PI – 2010]
(16) If {1,0, 1}T is an eigen vector of the following
eE1 / T1 / K4 / L3 / V2 / R11 / AA [GATE – EC – 2006]
(02) The eigen values and the corresponding eigen
1 1 0 matrix 1 2 1 then the corresponding 0 1 1
eigen value is
(A) 1
(C) 3
vectors of a 2x2 matrix are given by
Eigen Value
Eigen Vector
λ1 8
1 V1 1
λ2 4
1 V2 1
(B) 2
(D) 5
eE1 / T1 / K4 / L2 / V3 / R11 / AC [GATE – IN – 2009]
The matrix is
(17) The eigen values of a 2 2 matrix X are -2 and 3. The eigen values of matrix ( X I ) 1 ( X 5 I ) are
(A) – 3, - 4
(C) -1, -3
6 2 (A) 2 6
4 6 (B) 6 4
2 4 (C) 4 2
4 8 (D) 8 4
(B) -1, -2
(D) -2, -4
eE1 / T1 / K4 / L3 / V2 / R11 / AA [GATE – ME – 2005]
-----00000-----
Page 18
(03) Which one of the following is an eigen vector of
TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
TOPIC. 01 – LINEAR ALGEBRA 5 0 the matrix 0 0
eE1 / T1 / K4 / L3 / V3 / R11 / AB [GATE – CE – 2007]
0 0 0 5 0 0 is 0 2 1 0 3 1 T
T
(07) X = x1 x2 ........... xn is an n – tuple non zero vector. The n x n matrix V = XXT T
(A) 1 2 0 0
(B) 0 0 1 0
T
(A) has rank zero
(B) has rank 1
(C) is orthogonal
(D) has rank n
T
(C) 1 0 0 2
(D) 1 1 2 1
eE1 / T1 / K4 / L3 / V2 / R11 / AA [GATE – IN – 2010]
-----00000-----
(04) A real nxn matrix A = aij is defined as follows
aij i, i j 0, otherwise
1.5 Rank
The sum of all n eigen values of A is
(A)
n(n 1) 2
(B)
Question Level – 00 (Basic Problem)
n(n 1) 2
eE1 / T1 / K5 / L0 / V1 / R11 / AA [GATE – EC – 1994]
(01) The rank of (m x n) matrix (m < n) cannot be more than
n( n 1)(2n 1) (C) 2
2
(D) n
(A) m
(B) n
(C) mn
(D) None
eE1 / T1 / K4 / L3 / V2 / R11 / A [GATE – EE – 2011]
(05) The two vectors
1
1 1 and 1 a a 2
1 3 where a j and j 1 are 2 2
eE1 / T1 / K5 / L0 / V1 / R11 / AC [GATE – CS – 2002]
(A) Orthonormal
(B) Orthogonal
(C) Parallel
(D) Collinear
1 1 (02) The rank of the matrix is 0 0
(A) 4
(B) 2
(C) 1
(D) 0
eE1 / T1 / K4 / L3 / V3 / R11 / AA [GATE – EE – 2007]
(06)
q1 , q2 , q3 ,........qm are n-dimensional vectors with
eE1 / T1 / K5 / L0 / V1 / R11 / AC [GATE – EE – 1994]
m < n. This set of vectors is linearly dependent.
(03) A 5x7 matrix has all its entries equal to -1. Then
Q is the matrix with q1 , q2 , q3 ,.......qm as the
the rank of a matrix is
columns. The rank of Q is (A) Less than m
(B) m
(A) 7
(B) 5
(C) Between m and n
(D) n
(C) 1
(D) Zero
www.targate.org
Page 19
ENGINEERING MATHEMATICS eE1 / T1 / K5 / L1 / V1 / R11 / AA [GATE – EE – 1995]
Question Level – 01
(05) The rank of the following (n+1) x (n+1) matrix, eE1 / T1 / K5 / L1 / V1 / R11 / AA [GATE – EE – 1994]
where ‘a’ is a real number is
(01) The number of Linearly independent solutions of
1 0 2 x1 the system of equations 1 1 0 x2 =0 is 2 2 0 x3 equal to
1 a a 2 2 1 a a . . 2 1 a a
.
.
.
.
.
.
.
.
.
an an an
(A) 1
(B) 2
(A) 1
(B) 2
(C) 3
(D) 0
(C) n
(D) depends on the value of a
eE1 / T1 / K5 / L1 / V1 / R11 / AC [GATE – CS – 1994]
----00000-----
0 0 3 (02) The rank of matrix 9 3 5 is 3 1 1
Question Level – 02 eE1 / T1 / K5 / L2 / V2 / R11 / AD [GATE – CS – 1998]
(A) 0
(C) 2
(B) 1
(D) 3
eE1 / T1 / K5 / L1 / V1 / R11 / AB [GATE – EC –]
0 2 2 (03) Rank of the matrix 7 4 8 is 3 7 0 4
(A) True
1 4 8 0 0 3 (01) The rank of the matrix 4 2 3 3 12 24
(A) 3
(B) 1
(C) 2
(D) 4
7 0 is 1 2
(B) False eE1 / T1 / K5 / L2 / V2 / R11 / AC [GATE – EC – 2006]
eE1 / T1 / K5 / L1 / V1 / R11 / AC [GATE – IN – 2000]
1 2 3 (04) The rank of matrix A = 3 4 5 is 4 6 8
(A) 0
(C) 2
Page 20
1 1 1 (02) The rank of the matrix 1 1 0 is 1 1 1
(A) 0
(B) 1
(C) 2
(D) 3
(B) 1
(D) 3
TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
TOPIC. 01 – LINEAR ALGEBRA Question Level – 03
1.6 Solution of Linear Equation
eE1 / T1 / K5 / L3 / V2 / R11 / AC [GATE – CE – 2003]
4 2 1 3 (01) Given matrix [A] = 6 3 4 7 , the rank of 2 1 0 1 the matrix is
Question Level – 01 eE1 / T1 / K6 / L1 / V1 / R11 / AB [GATE – EC – 1994]
(01) Solve the following system
x1 x2 x3 3
(A) 4
x1 x3 0
(B) 3
x1 x2 x3 1 (C) 2
(D) 1 (A) Unique solution
eE1 / T1 / K5 / L3 / V2 / R11 / AB [GATE – IN – 2007]
(02) Let A = [ aij ],1 i, j n with n 3 and aij i. j .
(B) No solution
Then the rank of A is (C) Infinite number of solutions (A) 0
(B) 1 (D) Only one solution
(C) n – 1
(D) n eE1 / T1 / K6 / L1 / V1 / R11 / AC [GATE – ME – 1996]
eE1 / T1 / K5 / L3 / V2 / R11 / AA [GATE – EE – 2008]
(02) In the Gauss – elimination for a solving system of
(03) If the rank of a 5x6 matrix Q is 4 then which one
linear algebraic equations, triangularization leads
of the following statements is correct?
to
(A) Q will have four linearly independent rows
(A) diagonal matrix
and four linearly independent columns (B) lower triangular matrix (B) Q will have four linearly independent rows and five linearly independent columns (C) QQT will be invertible. (D) QT Q will be invertible.
(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 -----00000----(A) Only the trivial solution X = 0
(B) One independent solution
www.targate.org
Page 21
ENGINEERING MATHEMATICS Question Level – 02 (C) Two independent solutions eE1 / T1 / K6 / L2 / V2 / R11 / AB [GATE – CS – 2003]
(D) Three independent solutions
(01) A system of equations represented by AX = 0 where X is a column vector of unknown and A is
eE1 / T1 / K6 / L1 / V1 / R11 / A [GATE – CS – 2004]
(04) How many solutions does the following system of linear equations have
x 5 y 1 x y 2
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
x 3y 3
the
following
set
of
equations
x 2 y 5, 4x 8 y 12, 3x 6 y 3z 15. This set (A) Infinitely many (A) has unique solution (B) Two distinct solutions (B) has no solution (C) Unique (C) has infinite number of solutions (D) None (D) has 3 solutions 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
eE1 / T1 / K6 / L2 / V2 / R11 / AB [GATE – CE – 2005]
(03) Consider the following system of equations in three real variable x1 , x2 and x3 :
have non-trival solution is
2 x1 x2 3x3 1 (A) 2
(B) 7
3x1 2x2 5x3 2 x1 4 x2 x3 3
(C) 9
(D) 11 This system of equations has
(A) No solution -----00000----(B) A unique solution
Page 22
TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
TOPIC. 01 – LINEAR ALGEBRA (C) More than one but a finite number of
eE1 / T1 / K6 / L2 / V2 / R11 / AD [GATE – IN – 2006]
(07) A system of linear simultaneous equations is
solutions.
given as AX = b (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
1 0 Where A = 1 0
0 1 0 1 0 1 &b= 1 0 0 0 0 1
0 0 0 1
Then the rank of matrix A is
determined system. Such a system will be (A) Consistent having a unique solution
(A) 1
(B) 2
(B) Consistent having many solutions.
(C) 3
(D) 4
(C) Inconsistent having a unique solution.
eE1 / T1 / K6 / L2 / V2 / R11 / AB [GATE – EC –]
(08) A system of linear simultaneous equations is
(D) Inconsistent having no solution.
given as Ax b 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
1 0 Where A = 1 0
0 1 0 1 0 1 &b= 1 0 0 0 0 1
0 0 0 1
the unknown vector X. Which of the following statement is true? (A) Augmented matrix [P|Q] must have the same rank as matrix P.
(A) x is a null vector
(B) Vector Q must have only non-zero elements.
(B) x is unique
(C) Matrix P must be singular
(C) x does not exist
(D) x has infinitely many values
(D) Matrix p must be square
eE1 / T1 / K6 / L2 / V2 / R11 / AD [GATE – CE – 2006] 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
(09) Solution for the system defined by the set of equations 4 y 3z 8,2x z 2 & 3x 2 y 5 is
A is
(A) 1
(B) 2
(C) 3
(D) 4
(A) x 0, y 1, z 4 / 5
(B) x 0, y 1/ 2, z 2
www.targate.org
Page 23
ENGINEERING MATHEMATICS (C) x 1, y 1/ 2, z 2 eE1 / T1 / K6 / L2 / V2 / R11 / A [GATE – IN – 2010]
(13) X and Y are non-zero square matrices of size
(D) Non existent
nxn. If XY = Onxn then eE1 / T1 / K6 / L2 / V2 / R11 / AA [GATE – CE – 2007]
(10) For what values of α and β the following
(A) | X | 0 and | Y | 0
simultaneous equations have an infinite number of
solutions
x y z 5,
x 3y 3z 9,
x 2 y αz = β
(B) | X | 0 and | Y | 0
(C) | X | 0 and | Y | 0
(A) 2, 7
(B) 3, 8
(C) 8, 3
(D) 7, 2
(D) | X | 0 and | Y | 0
eE1 / T1 / K6 / L2 / V2 / R11 / AC [GATE – ME – 2011]
(14) Consider the following system of equations eE1 / T1 / K6 / L2 / V2 / R11 / A [GATE – CE – 2008]
(11) The following system of equations x y z 3,
x 2 y 3z 4, x 4 y k 6 will not have a unique solution for k equal to
2 x1 x2 x3 0, x2 x3 0 and x1 x2 0 . This system has
(A) A unique solution
(A) 0
(B) 5
(B) No solution
(C) 6
(D) 7
(C) Infinite number of solution
eE1 / T1 / K6 / L2 / V2 / R11 / AD [GATE – EE – 2010]
(D) Five solutions
(12) For the set of equations x1 2 x2 x3 4 x4 2,
3x1 6 x2 3x3 12 x4 6.
The
following
eE1 / T1 / K6 / L2 / V2 / R11 / AB [GATE – EC – 2008]
statement is true (15) The system of linear equations (A) Only the trivial solution x1 x2 x3 x4 0
4x 2 y 7 has 2x y 6
exist (A) A unique solution (B) There are no solutions (B) No solution (C) A unique non-trivial solution exist (C) An infinite no. of solution (D) Multiple non-trivial solution exist (D) Exactly two distinct solution.
Page 24
TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
TOPIC. 01 – LINEAR ALGEBRA (19) In the solution of the following set of linear eE1 / T1 / K6 / L2 / V2 / R11 / AB [GATE – EC –]
(16) The value of x3 obtained by solving the following
equations by Gauss-elimination using partial pivoting
5x y 2z 34,
4 y 3z 12
and
system of linear equations is
10x 2 y z 4. The pivots for elimination of
x1 2x2 2 x3 4
x and y are
2 x1 x2 x3 2
(A) 10 and 4
(B) 10 and 2
(C) 5 and 4
(D) 5 and – 4
x1 x2 x3 2 (A) – 12
(B) - 2
(C) 0
(D) 12
Question Level – 03 eE1 / T1 / K6 / L3 / V2 / R11 / AB [GATE – CS – 1996] eE1 / T1 / K6 / L2 / V2 / R11 / AB [GATE – EC – 2011]
(01) Let AX = B be a system of linear equations
x yz 6 ,
where A is an m n matrix B is an m 1 column
(17) The
system
of
equations
x 4 y 6z 20, and x 4 y λz μ has no solution for values of λ and μ given by
(A) λ 6, μ 20
(B) λ 6, μ 20
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) λ 6, μ= 20
(D) λ 6, μ 20 (C) If m < n and B is a zero vector then the
eE1 / T1 / K6 / L2 / V2 / R11 / AC [GATE – EC –]
system has infinitely many solutions.
(18) For the following set of simultaneous equations (D) The system will have a trivial solution when
1.5x 0.5 y z 2
m = n , B is the zero vector and rank of A is
4 x 2 y 3z 0
n.
7x y 5z 10 eE1 / T1 / K6 / L3 / V2 / R11 / AB [GATE – EE – 1998]
(A) the solution is unique
(02) A set of linear equations is represented by the matrix equations Ax = b. The necessary condition
(B) infinitely many solutions exist
for the existence of a solution for this system is
(C) the equations are incompatible
(A) must be invertible
(D) finite many solutions exist
(B) b must be linearly dependent on the columns of A
eE1 / T1 / K6 / L2 / V3 / R11 / AA [GATE – CE – 2009]
(C) b must be linearly independent on the columns of A
www.targate.org
Page 25
ENGINEERING MATHEMATICS (C) 1 (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
(D) There is no such value
1.7 Miscellaneous
Y be an n-dimensional vector. Then the linear system of equations Ax = Y has (A) No solution (B) unique solution
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 nonzero determinant. ABCD = I then B 1 =
(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
(A) D1C 1 A1
(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
space and x, y denote their dot product. Then
BA exists.
x, x x , y the determinant det =_____ y, x y , y
(A) Only if A has as many rows as B has columns
(A) Is zero when x and y are linearly independent (B) Only if both A and B are square matrices (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]
(C) Only if A and B are skew matrices
(D) Only if both A and B are symmetric
-----00000-----
Question Level – 01
(05) For what values of ‘a’ if any will the following system of equations in x, y are z have a solution?
2x 3 y 4, x y z 4, x 2 y z a
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) Any real number (A) Row is the same as its second row (B) 0 (B) row is the same as second row of A
Page 26
TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
TOPIC. 01 – LINEAR ALGEBRA (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]
35 22 (C) 61 42
32 56 (D) 24 46
(02) If A is any n n matrix and k is a scalar then
| kA | α | A | where α is
eE1 / T1 / K7 / L1 / V1 / R11 / AC [GATE – CS – 2004]
(06) The (A) kn
(B) nk
(C) k n
(D)
number
of
different
n n symmetric
matrices with each elements being either 0 or 1 is
k n
n (A) 2
eE1 / T1 / K7 / L1 / V1 / R11 / AB [GATE – CE – 1999]
(03) The number of terms in the expansion of general
(C)
n2 n 2 2
2
n (B) 2
(D)
n2 n 2 2
determinant of order n is eE1 / T1 / K7 / L1 / V1 / R11 / AD [GATE – EC – 2005]
(A) n2
(B) n!
(C) n
(D) (n 1) 2
4 2 (07) Given the matrix , the eigen vector is 4 3
eE1 / T1 / K7 / L1 / V1 / R11 / AB [GATE – CE – 2001]
(04) The
determinant
of
the following
(B) – 28
(C) 28
(D) 72
4 (B) 3
2 (C) 1
2 (D) 1
matrix
5 3 2 1 2 6 3 5 10
(A) – 76
3 (A) 2
Question Level – 02 eE1 / T1 / K7 / L2 / V1 / R11 / A [GATE – PI – 1994]
1 1 (01) For the following matrix the number of 2 3 eE1 / T1 / K7 / L1 / V1 / R11 / AA [GATE – CE – 2001]
(05) The product [P] [Q]T of the following two
2 3 matrices [P] and [Q] is where [P] = , 4 5 4 8 [Q] 9 2
positive roots is
(A) One
(B) Two
(C) Four
(D) Cannot be found
eE1 / T1 / K7 / L2 / V2 / R11 / AB [GATE – PI – 1995]
32 24 (A) 56 46
46 56 (B) 24 32
www.targate.org
Page 27
ENGINEERING MATHEMATICS 2 1 3 2 (02) Given matrix L = 3 2 and M = then 0 1 4 5 L x M is
8 1 (A) 13 2 22 5
6 5 (B) 9 8 12 13
1 8 (C) 2 13 5 22
6 2 (D) 9 4 0 5
(A) 4
(B) 0
(C) 15
(D 20
eE1 / T1 / K7 / L2 / V2 / R11 / AA [GATE – CE – 2005]
(06) Consider the matrices X4x3, Y4x3, and P2x3. The T
order of P ( X T Y ) 1 P T will be
eE1 / T1 / K7 / L2 / V2 / R11 / AC [GATE – ME – 1995]
(03) Among the following, the pair of the vector orthogonal to each other is
(A) 2x2
(B) 3x3
(C) 4x3
(D) 3x4
eE1 / T1 / K7 / L2 / V2 / R11 / AA [GATE – EC –2005]
(07) The determinant of the matrix given below is 0 1 1 1 0 0 1 2
(A) 3, 4, 7 , 3, 4, 7
0 2 1 3 0 1 0 1
(B) 1, 0, 0 , 1, 1, 0 (C) 1, 0, 2 , 0, 5, 0 (D) 1, 1, 1 , 1, 1, 1
eE1 / T1 / K7 / L2 / V2 / R11 / AC [GATE – EE – 2002]
(04) The
determinant
0 0 1 100 1 0 100 200 1 100 200 300
of
the
matrix
0 0 is 0 1
(A) 100
(B 200
(C) 1
(D) 300
Page 28
(B) 0
(C) 1
(D) 2
eE1 / T1 / K7 / L2 / V2 / R11 / AB [GATE – EE – 2005]
1 0 1 (08) If R = 2 1 1 then the top row of R 1 is 2 3 2 (A) 5 6 4
(B) 5 3 1
(C) 2 0 1
(D) 2 1 0
eE1 / T1 / K7 / L2 / V2 / R11 / AA [GATE – EE – 2005]
eE1 / T1 / K7 / L2 / V2 / R11 / AA [GATE – CS – 2000]
2 8 (05) The determinant of the matrix 2 9
(A) -1
0 0 0 1 7 2 is 0 2 0 0 6 1
2 0.1 (09) If A = and 3 0
1 / 2 a A1 0 b
a b __________
TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
then
TOPIC. 01 – LINEAR ALGEBRA (A)
7 20
(B)
3 20
2 1 (12) The matrix [A] = is decomposed into a 4 1 product of lower triangular matrix [L] and an
(C)
19 60
(D)
11 20
upper triangular [U]. The property decomposed [L] and [U] matrices respectively are
eE1 / T1 / K7 / L2 / V2 / R11 / AC [GATE – IN – 2006]
1 0 1 1 (A) and 4 1 0 2
(10) For a given 2x2 matrix A, it is observed that
1 1 A 1 1 1
and
1 A 2
and
1 0 2 1 (B) and 2 1 0 3
1 1 A 2 then the matrix A is 2 2 1 0 2 1 (C) and 4 1 0 1
2 1 1 0 1 1 (A) A 1 1 0 2 1 2
2 0 (D) and 4 3
1 0.5 0 1
1 1 1 0 2 1 (B) A 1 2 1 2 1 1 -----00000-----
1 1 1 0 2 1 (C) A 1 2 0 2 1 1
Question Level – 03 eE1 / T1 / K7 / L3 / V2 / R11 / AA [GATE – CS – 1996]
0 2 (D) A 1 3
(01) The
eE1 / T1 / K7 / L2 / V2 / R11 / AD [GATE – ME – 2011]
2 4 4 6 (11) If a matrix A = and matrix B = 1 3 5 9
matrices
cos θ sin θ sin θ cos θ
and
a 0 0 b
commute under multiplication.
(A) If a = b (or) θ nπ , n is an integer
the transpose of product of these two matrices (B) Always
i.e., ( AB)T is equal to
(C) never
28 19 (A) 34 47
19 34 (B) 47 28
48 33 (C) 28 19
28 19 (D) 48 33
(D) If a cos θ b sin θ
eE1 / T1 / K7 / L3 / V2 / R11 / AB [GATE – CE – 1999]
eE1 / T1 / K7 / L2 / V2 / R11 / AB [GATE – EE – 2011]
www.targate.org
Page 29
ENGINEERING MATHEMATICS
(02) The
equation
2
1
1
1
1
1 0
y
2
x
(05) Consider represents
a
x
system
of
equations,
Ann X n1 λX n1 where λ is a scalar. Let
λi , X i
parabola passing through the points.
the
be an eigen value and its corresponding
eigen vector for real matrix A. Let Inxn be unit (A) (0,1), (0,2),(0,-1)
(B) (0,0), (-1,1),(1,2) matrix. Which one of the following statement is
(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
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, ( λim , X im ) will be eigen pair for all i.
1 2 3 x 6 1 3 4 y 8 2 2 3 z 12
T 1 (C) If A A then | λi | 1 for all i. T (D) If A A then λi are real for all i.
(A) x = 6, y = 3, z = 2
eE1 / T1 / K7 / L3 / V2 / R11 / AC [GATE – ME – 2006]
(06) Multiplication of matrices E and F is G. Matrices (B) x = 12, y = 3, z = -4
cos θ sin θ 0 E and G are E = sin θ cos θ 0 and G = 0 0 1
(C) x = 6, y= 6, z = -4
1 0 0 0 1 0 . What is the matrix F? 0 0 1
(D) x = 12, y = -3, z = 4 eE1 / T1 / K7 / L3 / V2 / R11 / AB [GATE – EC –2004]
(04) If
matrix
X
=
a 1 2 a a 1 1 a
and
cos θ sin θ 0 (A) sin θ cos θ 0 0 0 1
X 2 X I 0. Then the inverse of X is
1 a 1 (A) 2 a a
1 1 a (B) 2 a a 1 a
a 1 a2 a 1 a (C) 2 (D) 1 1 a a a 1 a 1 eE1 / T1 / K7 / L3 / V2 / R11 / AB [GATE – CE – 2005]
Page 30
cos θ cos θ 0 (B) cos θ sin θ 0 0 0 1 cos θ sin θ 0 (C) sin θ cos θ 0 0 0 1
sin θ cos θ 0 (D) cos θ sin θ 0 0 0 1
TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
TOPIC. 01 – LINEAR ALGEBRA eE1 / T1 / K7 / L3 / V2 / R11 / AA [GATE – CS – 2000]
2 (A) P P 2I
2 (B) P P I
(C) ( P 2 P I )
(D) ( P 2 P 2 I )
(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) n2 3n 2
(D) n(n 1)
-----00000-----
1.8 CALY- HAMILTON Question Level – 01 eE1 / T1 / K8 / L1 / V1 / R11 / AC [GATE – CE – 2007]
3 2 (01) If A = then A satisfies the relation 1 0
1
(A) A + 3I + 2 A = O
(B) A2 2 A 2I O
(C) ( A I )( A 2I ) O (D) e A O
eE1 / T1 / K8 / L1 / V1 / R11 / AA [GATE – EE – 2007]
3 2 9 (02) If A = then A equals 1 0
(A) 511 A + 510 I
(B) 309 A + 104 I
(C) 154 A + 155 I
(D) e9 A
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 α ( λ) | λI P | λ3 2 λ λ 2 1 0.
If I denotes identity matrix then the inverse of P will be
www.targate.org
Page 31
02 Calculus 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
2.2 Maxima and Minima
Question Level – 01
Question Level – 00 (Basic Problem)
eE1 / T2 / K1 / L1 / V1 / R11 / AC [GATE – – 1994]
eE1 / T2 / K2 / L0 / V1 / R11 / AB [GATE – – ]
(01) The value of ε in the mean value theorem of
(01) A point on the curve is said to be an extremum
f(B)
–
f(A)
=
(b
–
a)
f’( ε)
for
f ( x) Ax 2 Bx C in (a, b) is
if it is a local minimum (or) a local maximum. The number of distinct extreme for the curve
3x4 16 x3 24 x2 37 is ___________ (A) b a
(C)
(B) b a
ba 2
(D)
ba 2
(A) 0
(B) 1
(C) 2
(D) 3
-----00000-----
-----00000-----
Question Level – 02
Question Level – 03 eE1 / T2 / K1 / L3 / V2 / R11 / AB [GATE – – 1995]
eE1 / T2 / K2 / L2 / V2 / R11 / AB [GATE – – 1994]
(01) The function y x 2
250 at x = 5 attains x
value theorem are ____________
(A) Maximum
(B) Minimum
(A) 1.9, 2.2
(B) 2.2, 2.25
(C) Neither
(D) 1
(C) 2.25, 2.5
(D) None of the above
eE1 / T2 / K2 / L2 / V2 / R11 / AA [GATE – – 1995]
(01) If f(0) = 2 and f’(x) =
1 , then the lower 5 x2
and upper bounds of f(1) estimated by the mean
(02) The function f(x) = x3 6 x2 9 x 25 has -----00000-----
Page 32
TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
TOPIC. 02 – CALCULUS (A) A maxima at x = 1 and minima at x = 3
eE1 / T2 / K2 / L2 / V2 / R11 / AA [GATE – EC – 2007]
(06) Consider the function f(x) = x2 x 2. the (B) A maxima at x = 3 and a minima at x = 1
maximum value of f(x) in the closed interval [4, 4] is
(C) No maxima, but a minima at x = 3 (A) 18
(B) 10
(C) – 2.25
(D) indeterminate
(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 eE1 / T2 / K2 / L2 / V2 / R11 / AC [GATE – IN – 2008]
f ( x) 2 x 2 2 x 6 in the interval [0, 2]?
(07) Consider the function
y x 2 6 x 9. The
maximum value of y obtained when x varies (A) 6
(B) 10
over the internal 2 to 5 is
(C) 12
(D) 5.5
(A) 1
(B) 3
(C) 4
(D) 9
eE1 / T2 / K2 / L2 / V2 / R11 / AA [GATE – ME – 2005] 3
2
(04) The function f(x) = 2 x 3x 36 x 2 has its maxima at
eE1 / T2 / K2 / L2 / V2 / R11 / AA [GATE – EC – 2008]
(08) For real values of x, the minimum value of (A) x = - 2 only
function f(x) = ex e x is
(B) x = 0 only
(A) 2
(B) 1
(C) x = 3 only
(C) 0.5
(D) 0
(D) both x = - 2 and x = 3
eE1 / T2 / K2 / L2 / V2 / R11 / AA [GATE – EC – 2010]
(09) If e y x1/ x then y has a eE1 / T2 / K2 / L2 / V2 / R11 / AA [GATE – EE – 2005]
(05) For the function f(x) = x 2 e x , the maximum
(A) Maximum at x = e
occurs when x is equal to (B) Minimum at x = e (A) – 2
(B) 1
(C) 0
(D) – 1
(C) Maximum at x = e 1
(D) Minimum at x = e 1
www.targate.org
Page 33
ENGINEERING MATHEMATICS eE1 / T2 / K2 / L3 / V2 / R11 / AB [GATE – PI – 2007]
Question Level – 03
(05) For the function f(x, y) = x 2 y 2 defined on R2, eE1 / T2 / K2 / L3 / V2 / R11 / A [GATE – ME – 1993]
the point (0, 0) is
(01) The function f ( x, y ) x 2 y 3xy 2 y x has (A) A local minimum (A) No local extremism (B) Neither a local minimum (nor) a local (B) One local maximum but no local minimum
maximum.
(C) One local minimum but no local maximum
(C) A local maximum
(D)One local minimum and one local maximum
(D) Both a local minimum and a local maximum
eE1 / T2 / K2 / L3 / V2 / R11 / A [GATE – CS – 1998]
eE1 / T2 / K2 / L3 / V2 / R11 / AB [GATE – EE – 2007]
(02) Find the points of local maxima and minima if
(06) Consider the function f ( x ) x 2 4
any of the following function defined in 3
2
where x
is a real number. Then the function has
2
0 x 6, x 6x 9 x 15. eE1 / T2 / K2 / L3 / V2 / R11 / AB [GATE – – 2002]
(03) The function f(x, y) = 2 x 2 2 xy y 3 has
(A) Only one minimum
(B) Only two minima
(C) Three minima
(D) Three maxima
(A) Only one stationary point at (0, 0)
-----00000-----
1 1 (B) Two stationary points at (0, 0) and , 6 3
2.3 Differential Calculus
(C) Two stationary points at (0, 0) and (1, -1)
Question Level – 00 (Basic Problem) eE1 / T2 / K3 / L0 / V1 / R11 / AA [GATE – – 1996]
(D) No stationary point.
(01) If a function is continuous at a point its first derivative eE1 / T2 / K2 / L3 / V2 / R11 / AC [GATE – IN – 2007]
(04) For real x, the maximum value of
esin x is ecos x
(A) May or may not exist
(B) Exists always (A) 1
(B) e (C) Will not exist
(C) e
2
(D) (D) Has a unique value
Page 34
TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
TOPIC. 02 – CALCULUS Question Level – 01
Question Level – 03
eE1 / T2 / K3 / L1 / V1 / R11 / AB [GATE – IN – 2008]
(01) Given y = x2 2x 10 the value of
dy dx
is X 1
(B) 4
(C) 12
(D) 13
(01) If x = a(θ sin θ) and y a(1 cos θ ) then dy ______ dx
equal to
(A) 0
eE1 / T2 / K3 / L3 / V2 / R11 / AC [GATE – – 2004]
(A) sin
θ 2
(B) cos
θ 2
(C) tan
θ 2
(D) cot
θ 2
eE1 / T2 / K3 / L1 / V1 / R11 / AA [GATE – PI – 2009]
(02) The total derivative of the function ‘xy’ is eE1 / T2 / K3 / L3 / V2 / R11 / AC [GATE – – 2005]
(A) xdy ydx
(B) xdx ydy
(02) By a change of variables x(u, v) = uv,
y(u, v) v / u in a double integral, the integral (C) dx dy
(D) dx dy
f ( x, y)
eE1 / T2 / K3 / L2 / V2 / R11 / AA [GATE – – 1997]
(01) If ( x)
x
0
to
(A)
d t dt then __________ dx
2v u
v
.
Then
(B) 2 u v
(C) V 2 (A) 2x
f uv, u
(u, v) is _______
Question Level – 02
2
changes
2
(B)
(C) 0
x
(D) 1
eE1 / T2 / K3 / L3 / V2 / R11 / AC [GATE – PI – 2010]
(03) If (x) = sin | x | then the value of
(D) 1
df π at x dx 4
is eE1 / T2 / K3 / L2 / V2 / R11 / AA [GATE – – 2000]
(02) If
f(x,
y,
z)
= (A) 0
2 f 2 f 2 f ( x 2 y 2 z 2 ) 1/2 , 2 2 2 is equal to x y z
(C)
(B)
1
1 2
(D) 1
2
_______ eE1 / T2 / K3 / L3 / V2 / R11 / AA [GATE – CE – 2010]
(04) Given (A) 0
a
function
(B) 1 f ( x, y ) 4 x 2 6 y 2 8 x 4 y 8, the optimal
(C) 2
(D) 3( x 2 y 2 z 2 )5/2
values of f(x, y) is
www.targate.org
Page 35
ENGINEERING MATHEMATICS (A) a minimum equal to
(B) a maximum equal to
eE1 / T2 / K4 / L0 / V2 / R11 / AD [GATE – EC – 2005]
10 3
(02) The value of the integral
10 3
(C) a minimum equal to
8 3
(D) a maximum equal to
8 3
1
1
1 dx is x2
(A) 2
(B) does not exists
(C) - 2
(D)
-----00000-----
Question Level – 01 eE1 / T2 / K3 / L3 / V2 / R11 / AC [GATE – EE – 2011]
(05) The function f(x) = 2 x x2 3 has
eE1 / T2 / K4 / L1 / V1 / R11 / A [GATE – PI – 1995]
(01) Given y
x2
1
cos t dt , then
(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-----
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 equals
2x , the curve is _________ y
(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
2.4 Integral Calculus Question Level – 00 (Basic Problem) eE1 / T2 / K4 / L0 / V1 / R11 / AC [GATE – EE – 2005]
(01) IF S =
α
1
X 3 dx then S has the value
1 3
(B)
1 2
Page 36
π / 2
( x cos x) dx is
(B) π 2
(C) π
(D) π 2
eE1 / T2 / K4 / L1 / V1 / R11 / AD [GATE – ME – 2010]
1 4
(A) π (C)
π/2
(A) 0
(04) The value of the integral (A)
dx α 1 x 2
α
(B)
(D) 1 (C)
π 2
π 2
(D) π
TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
TOPIC. 02 – CALCULUS Question Level – 02 (C) 0
(D) None
eE1 / T2 / K4 / L2 / V2 / R11 / AA [GATE – – 1994]
(01) The integration of
eE1 / T2 / K4 / L2 / V2 / R11 / AC [GATE – – 2002]
log xdx has the value
(05) The value of the following improper integral is 1
x log x dx = ________
(A) ( x log x 1)
(B) log x x
(C) x(log x 1)
(D) None of the above
(A)
0
1 4
(B) 0
eE1 / T2 / K4 / L2 / V2 / R11 / AC [GATE – – 1995]
(02) By
reversing
2
2x
0
x2
the
order
of
(C)
integration
1 4
(D) 1
f ( x, y ) dydx may be represented as _____ eE1 / T2 / K4 / L2 / V2 / R11 / AA [GATE – – 2005]
(A)
(B)
2
2x
0
x2
2
0
integral I = y
y
4
y
0
y /2
2x
2
(D)
x
f ( x, y )dxdy
I=
(C)
2
(06) Changing the order of integration in the double f ( x, y ) dydx
0
(03)
π/2
0
0
r
p
0
x/4
f ( x, y )dy dx leads to
f ( x, y)dy dx. What is q?
(A) 4y
(B) 16 y2
f ( x, y ) dydx
(C) x
(D) 8
a
sin 6 x sin 7 x dx is equal to
a
a
(A) 2 sin 6 xdx
(B) π
2
eE1 / T2 / K4 / L2 / V2 / R11 / AA [GATE – ME – 2005]
(07)
sin( x y )dxdy
(A) 0
(C) π
q
2
f ( x, y )dxdy
eE1 / T2 / K4 / L2 / V2 / R11 / AD [GATE – – 2000] π/2
s
8
0
(D) 2
a
(B) 2 sin 7 xdx
0
eE1 / T2 / K4 / L2 / V2 / R11 / AC [GATE – – 2002]
(04) The value of the following definite integral in
π /2
sin 2 x dx _______ cos
π /2 1
(A) - 2 log 2
(C) 2
a
sin 0
6
x sin 7 x dx
(D) zero (B) 2
www.targate.org
Page 37
ENGINEERING MATHEMATICS eE1 / T2 / K4 / L2 / V2 / R11 / AA [GATE – – ]
(08) The value of the integral I =
1
2π
0
e x
2
/8
Question Level – 03
dx is
____
eE1 / T2 / K4 / L3 / V2 / R11 / A [GATE – – 1994]
(01) The value of
0
3
e y . y1/ 2 dy is ________
(B) π
(A) 1
eE1 / T2 / K4 / L3 / V2 / R11 / AD [GATE – IN – 2007]
(C) 2
(D) 2π
eE1 / T2 / K4 / L2 / V2 / R11 / AA [GATE – CS – 2008]
(09) The value of
3
x
0
0
(02) The value of
(B) 27.0
(C) 40.5
(D) 54.0
α
0
0
2
2
e x e y dx dy is
π 2
(A)
(6 x y ) is _____
(A) 13.5
α
π
(B)
(C) π
(D)
π 4
eE1 / T2 / K4 / L3 / V2 / R11 / AB [GATE – EE – 2007]
(03) The integral eE1 / T2 / K4 / L2 / V2 / R11 / AB [GATE – EC – 2007]
1 2
2
0
sin(t τ ) cos τdτ equals
(10) The following plot shows a function y which varies linearly with x. The value of the integral I =
2
1
(A) Sin cost
ydx
(C)
(A) 1
1 cos t 2
(B) 0
(D)
1 sin t 2
(B) 2.5 eE1 / T2 / K4 / L3 / V2 / R11 / AA [GATE – EC – 2008]
(04) The value of the integral of the function (C) 4
(D) 5 g ( x, y ) 4 x3 10 y 4 along the straight line
eE1 / T2 / K4 / L2 / V2 / R11 / AB [GATE – IN – 2010]
(11) The integral
π t 6sin(t ) dt evaluates to α 6
segment from the point (0, 0) to the point (1, 2)
α
in the xy-plane is
(A) 6
(B) 3
(A) 33
(B) 35
(C) 1.5
(D) 0
(C) 40
(D) 56
eE1 / T2 / K4 / L3 / V2 / R11 / AD [GATE – ME – 2008]
-----00000-----
(05) Which of the following integrals is unbounded?
(A)
Page 38
π/4
0
tan dx
(B)
TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
α
0
1 dx 1 x2
TOPIC. 02 – CALCULUS (C)
α
0
x.e x dx
(D)
1
eE1 / T2 / K5 / L0 / V1 / R11 / AD [GATE – – 2000]
1
1 x dx 0
(03) Limit of the function f ( x ) eE1 / T2 / K4 / L3 / V2 / R11 / AD [GATE – CS – 2011]
is given
(06) Given i 1, what will be the evaluation of the definite integral
π 2 0
1 a4 as x x4
cos x i sin x dx ? cos x i sin x
(A) 1
(B) e a
(D) 0
4
(A) 0
(B) 2
(C)
(C) – i
(D) i
eE1 / T2 / K5 / L0 / V1 / R11 / AA [GATE – – 2003]
(04) -----00000-----
2.5 Limit and Continuity
sin 2 x ____ x 0 x
lim
(A) 0
(B)
(C)
(D) – 1
Question Level – 00 (Basic Problem) eE1 / T2 / K5 / L0 / V1 / R11 / AC [GATE – IN – 2007]
eE1 / T2 / K5 / L0 / V1 / R11 / AB [GATE – – 1995]
(01)
(05) Consider the function f(x) = | x |3 , where x is real. Then the function f(x) at x = 0 is
lim x sin 1 ______ x x 0
(A) Continuous but not differentiable
(A)
(B) 0
(C) 1
(D) Does not exist
(B) Once differentiable but not twice.
eE1 / T2 / K5 / L0 / V1 / R11 / AD [GATE – – ]
(C) Twice differentiable but not thrice.
(02) Limit of the following series as x approaches (D) Thrice differentiable
is 2 x3 x5 x7 f ( x) x 3! 5! 7!
eE1 / T2 / K5 / L0 / V1 / R11 / AB [GATE – ME – 2007]
(06) The minimum value of function y x 2 in the interval [1, 5] is
(A)
(C)
2π 3
π 3
(B)
π 2
(D) 1
(A) 0
(B) 1
(C) 25
(D) Undefined
www.targate.org
Page 39
ENGINEERING MATHEMATICS eE1 / T2 / K5 / L0 / V1 / R11 / AB [GATE – ME – 2007]
eE1 / T2 / K5 / L0 / V1 / R11 / AD [GATE – EE – 2010]
(11) At t = 0, the function f(t) =
x2 e x 1 x 2 (07) lim x 0 x3
(A) A minimum
(B) A discontinuity
(C) A point of inflection (D) A Maximum
1 (B) 6
(A) 0
sin t has t
eE1 / T2 / K5 / L0 / V1 / R11 / AD [GATE – ME – 2011]
(C)
1 3
(D) 1
(12) What is lim
θ 0
eE1 / T2 / K5 / L0 / V1 / R11 / AA [GATE – – ]
(08)
x sin x ______ x x cos x lim
(A) 1
(B) - 1
(C)
(D)
sin θ equal to? θ
(A) θ
(B) sin θ
(C) 0
(D) 1
-----00000-----
Question Level – 01 eE1 / T2 / K5 / L1 / V1 / R11 / AB [GATE – – 1995]
eE1 / T2 / K5 / L0 / V1 / R11 / AC [GATE – – ]
(01) The function f(x) = | x 1| on the interval
[2,0] is _________ (09)
lim
x 0
sin x is x
(A) Continuous and differentiable (B) Continuous on the interval but not
(A) Indeterminate
(B) 0
(C) 1
(D)
differentiable at all points
(C) Neither continuous nor differentiable
eE1 / T2 / K5 / L0 / V1 / R11 / AB [GATE – ME – 2008]
(D) Differentiable but not continuous x1/3 2 is x 8 x 8
(10) The value of lim
eE1 / T2 / K5 / L1 / V1 / R11 / AA [GATE – – 1997]
(02) (A)
1 16
1 (C) 8
Page 40
(B)
1 12
1 (D) 4
sin mθ , where m is an integer, is one of the θ 0 θ lim
following:
(A) m
(B) m π
(C) mθ
(D) 1
TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
TOPIC. 02 – CALCULUS eE1 / T2 / K5 / L1 / V1 / R11 / AC [GATE – – 1997]
Question Level – 02
(03) If y=| x| for x < 0 and y = x for x 0 then eE1 / T2 / K5 / L2 / V2 / R11 / AC [GATE – IN – 1999]
dy (A) is discontinuous at x = 0 dx
(01)
1 1 e j 5 x _____ x 0 10 1 e jx
lim
(B) y is discontinuous at x = 0 (A) 0
(B) 1.1
(C) 0.5
(D) 1
(C) y is not defined at x = 0
(D) Both y and
dy are discontinuous at x = 0 dx
eE1 / T2 / K5 / L1 / V1 / R11 / AA [GATE – EC – 2007]
eE1 / T2 / K5 / L2 / V2 / R11 / AD [GATE – – 1999]\
n
(02) Limit of the function, lim
n
(04)
lim
θ 0
sin(θ / 2) θ
(A) 1 (A) 0.5
(B) 1
(C) 2
(D) Not defined
2
(C)
is _____
2
n n
(B) 0
(D) 1
eE1 / T2 / K5 / L2 / V2 / R11 / AA [GATE – – 2001] eE1 / T2 / K5 / L1 / V1 / R11 / AB [GATE – – 2004]
3
2
x x (05) The value of the function. f ( x ) lim 3 x 0 2 x 7 x 2
(03) The value of the integral is I =
is _____
(A) 0
(C)
1 7
(B)
(A)
5 2
(C)
5 2
1 7
(B)
π/4
0
cos 2 x dx
5
(D)
5 2
(D) eE1 / T2 / K5 / L2 / V2 / R11 / AB [GATE – CE – 2002]
eE1 / T2 / K5 / L1 / V1 / R11 / AC [GATE – PI – 2008]
sin( x ) (06) The value of the expression lim x is x0 e x
(04) Limit of the following sequence as n is ___________ x n1/n
(A) 0
(B)
1 2
(A) 0
(B) 1
(C) 1
(D)
1 1 e
(C)
(D) -
www.targate.org
Page 41
ENGINEERING MATHEMATICS eE1 / T2 / K5 / L2 / V2 / R11 / AC [GATE – PI – 2007]
cos x sin x x π /4 xπ / 4
(05) What is the value of lim
(02) Value
of
the
function
lim x a
xa
x a
is
________
2
(A)
eE1 / T2 / K5 / L3 / V2 / R11 / AA [GATE – CE – 2000]
(B) 0
(C) 2
(A) 1
(B) 0
(C)
(D) a
eE1 / T2 / K5 / L3 / V2 / R11 / AD [GATE – – 2002]
(03) Which of the following functions is not (D) Limit does not exist
differentiable in the domain [-1, 1]?
eE1 / T2 / K5 / L2 / V2 / R11 / AB [GATE – ME – 2010]
(A) f(x) = x2
(06) The function y | 2 3x | (B) f(x) = x – 1 (A) is continuous x R and differential
xR
(C) f(x) = 2
(B) is continuous x R and differential
x R except at x =
3 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 (C) is continuous x R and differential function defined below is continuous at x =
x R except at x =
2 3
(D) Is continuous x R and except at x = 3 and differential x R
-----00000-----
π λ cos x if x π 2 x f ( x) 2 π if x 1 2
(A) 0
(B) 2π
(C) 1
(D)
Question Level – 03 eE1 / T2 / K5 / L3 / V2 / R11 / A [GATE – ME – 1993]
-----00000----(01)
x (e x 1) 2(cos x 1) ________ x 0 x (1 cos x )
lim
Page 42
TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
π 2
π ? 2
TOPIC. 02 – CALCULUS eE1 / T2 / K6 / L1 / V1 / R11 / AA [GATE – EC – 2008]
2.6 Series
(02) Which of the following function would have only odd powers of x in its Taylor series
Question Level – 00 (Basic Problem)
expansion about the point x = 0?
eE1 / T2 / K6 / L0 / V1 / R11 / AB [GATE – CE – 1998]
(B) sin x2
(D) cos x 2
(A) sin x3
1 1 (01) The infinite sires 1 2 3
(C) cos x3 (A) Converges
(B) Diverges
(C) Oscillates
(D) Unstable -----00000-----
eE1 / T2 / K6 / L0 / V1 / R11 / AB [GATE – ME – 2011]
Question Level – 02
(02) A series expansion for the function sin θ is ______
eE1 / T2 / K6 / L2 / V2 / R11 / AB [GATE – – ]
(01) Consider 2
(A) 1
(B) θ
4
θ θ ........ 2! 4!
lim
a
x 1
the
(B) converges to 1/3
(C) Converges to 1 (C) 1 θ
integral
x4 dx ___
(A) Diverges
θ3 θ6 ........ 3! 5!
2
following
3
θ θ ........ 2! 3!
a3
(D) converges to 0
eE1 / T2 / K6 / L2 / V2 / R11 / AB [GATE – ME – 2007]
(D) θ
θ3 θ 5 ...... 3! 5!
(02) If y x x x x ........α
then y(2) =
_____ -----00000----(A) 4 (or) 1
(B) 4 only
(C) 1 only
(D) Undefined
Question Level – 01 eE1 / T2 / K6 / L1 / V1 / R11 / AB [GATE – – 1995]
(01) The third term in the taylor’s series expansion of eE1 / T2 / K6 / L2 / V2 / R11 / AB [GATE – IN – 2011]
ex about ‘a’ would be _______
α
(03) The series (B)
ea (C) 2
ea (D) ( x a )3 6
m
( x 1) 2m converges for
m 0
ea ( x a )2 2
(A) e a ( x a)
1
4
(A) 2 x 2
(B) 1 x 3
(C) 3 x 1
(D) x 3
www.targate.org
Page 43
ENGINEERING MATHEMATICS eE1 / T2 / K6 / L2 / V2 / R11 / AB [GATE – ME – 2010]
(B) x
x3 x5 x7 3! 5! 7!
(04) The infinite series 3
f ( x) x
5
3
7
x x x Converges to 3! 5! 7!
(A) cos(x)
(B) sin( x)
(C) sinh( x)
(D) ex
5
(C)
(D)
1 2
eE1 / T2 / K6 / L3 / V2 / R11 / AD [GATE – EC – 2009]
(04)The Taylor series expansion of
eE1 / T2 / K6 / L3 / V2 / R11 / AB [GATE – EC – 2008]
(A) 1
(01) In the Taylor series expansion of ex sin x about the point x = π , the coefficient of 2
( x π )2 3!
(B) 1
is (C) 1
(A) eπ
(B) 0.5 eπ
( x π )2 3!
( x π)2 3!
(D) 1 (C) eπ 1
sin at x π is xπ
given by
Question Level – 03
x π
7
x x x 6 6 6 6 1! 3! 5! 7!
x
( x π )2 3!
(D) eπ 1 -----00000-----
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)
2.7 Vector Calculus Question Level – 00 (Basic Problem)
24 4!
eE1 / T2 / K7 / L0 / V1 / R11 / AD [GATE – ME – 1996]
e2 (C) 4!
(01) The expression curl (grad f ) where f is a
e4 (D) 4!
scalar function is
eE1 / T2 / K6 / L3 / V2 / R11 / AA [GATE – CE – 2000]
(A) Equal to 2 f
(03) The Taylor series expansion of sin x about x
is given by 6
(A)
1 3 1 3 x x x 2 2 6 4 6 12 6
(B) Equal to div (grad f )
2
3
(C) A scalar of zero magnitude
(D) A vector of zero magnitude
Page 44
TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
TOPIC. 02 – CALCULUS eE1 / T2 / K7 / L0 / V1 / R11 / AA [GATE – –]
(02) Stokes theorem connects
eE1 / T2 / K7 / L1 / V1 / R11 / AD [GATE – PI – 2005]
(03) Which one of the following is Not associated with vector calculus?
(A) A line integral and a surface integral (A) Stoke’s theorem (B) A surface integral and a volume integral (B) Gauss Divergence theorem (C) A line integral and a volume integral (C) Green’s theorem (D) Gradient of a function and its surface integral.
(D) Kennedy’s theorem
-----00000-----
eE1 / T2 / K7 / L1 / V1 / R11 / AD [GATE – EC – 2005]
(04)
P where P is a vector is equal to
Question Level – 01 (A) P P 2 P
(B) P (P)
(C) 2 P P)
(D) ( P) 2 P
eE1 / T2 / K7 / L1 / V1 / R11 / AA [GATE – – ]
(01) Given a vector field F , the divergence theorem states that
(A)
S
F . ds
V
eE1 / T2 / K7 / L1 / V1 / R11 / AB [GATE – CE – 2007]
. F dv
(05) The area of a triangle formed by the tips of vectors a, b and c is
(B)
(C)
(D)
S
F . ds
F ds
F ds
S
S
V
F dv
F dv
F dv
V
V
(A)
1 ( a b ) (a c) 2
(B)
1 | ( a b) ( a c ) | 2
(C)
1 | abc | 2
(D)
1 ( a b) c 2
eE1 / T2 / K7 / L1 / V1 / R11 / AA [GATE – – ]
(02) If a vector R (t ) has a constant magnitude than
(A) R.
dR 0 dt
(B) R.
dR 0 dt
eE1 / T2 / K7 / L1 / V1 / R11 / AD [GATE – – ]
(06) The angle (in degrees) between two planar (C) R.R
dR dt
(D) R R
dR dt
vectors a
www.targate.org
3 1 3 1 i j and b i j is 2 2 2 2
Page 45
ENGINEERING MATHEMATICS (A) 30
eE1 / T2 / K7 / L1 / V1 / R11 / AA [GATE – – 1993]
(B) 60
(10) A sphere of unit radius is centred at the origin. (C) 90
The unit normal at a point (x, y, z) on the
(D) 120
surface of the sphere is the vector. eE1 / T2 / K7 / L1 / V1 / R11 / AD [GATE – ME – 2008]
(07) The
divergence
of
the
vector
field
(A) (x, y, z)
1 1 1 , , (B) 3 3 3
x y z , , (C) 3 3 3
x y z , , (D) 2 2 2
( 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 – – ]
-----00000-----
(08) If r is the position vector of any point on a closed surface S that encloses the volume V
Question Level – 02
(r . d s) is equal to
then
S
eE1 / T2 / K7 / L2 / V2 / R11 / AB [GATE – – 1995]
(01) The directional derivative of the function f(x, y, 1 (A) V 2
(B) V
(C) 2V
(D) 3V
z) = x + y at the point P(1, 1, 0) along the direction i j is
(A) 1/ 2
(B)
2
eE1 / T2 / K7 / L1 / V1 / R11 / AB [GATE – – ]
(09) If a vector field V is related to another field A
(C) -
2
(D) 2
through V = A , which of the following is eE1 / T2 / K7 / L2 / V2 / R11 / AD [GATE – – 1999]
true? Note: C and SC refer to any closed contour and any surface whose boundary is C.
(A)
V .dl
(02) For the function ax 2 y y 3 to represent the velocity potential of an ideal fluid, 2 should
A.ds
be equal to zero. In that case, the value of ‘a’
V .ds
(A) -1
(B) 1
(C) – 3
(D) 3
Sc
has to be
C
(B)
A.dl
Sc
C
(C)
V .dl
Sc
A.ds
eE1 / T2 / K7 / L2 / V2 / R11 / AB [GATE – – 2002]
C
(03) The directional derivative of the following (D)
C
Page 46
A.dl
Sc
V .ds
function at (1, 2) in the direction of (4i + 3j) is: F(x, y) = x 2 y 2
TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
TOPIC. 02 – CALCULUS (A) 4/5
(B) 4
(C) 2/5
(D) 1
eE1 / T2 / K7 / L2 / V2 / R11 / AD [GATE – EE – 2006]
(07) The expression V =
H
o
2
h πR 2 1 dh for the H
eE1 / T2 / K7 / L2 / V2 / R11 / AC [GATE – – 2003]
(04) The vector field F = xi yj (where i and j are
volume of a cone is equal to _______.
unit vectors) is (A) (A) Divergence free, but not irrotational
(B) Irrotational, but divergence free
(B)
R
o
R
o
2
h πR 2 1 dr H
2
h πR 2 1 dh H
(C) Divergence free and irrotational (C) (D) Neither divergence free nor irrotational eE1 / T2 / K7 / L2 / V2 / R11 / AC [GATE – – ]
x 2 y3 (05) For the scalar field u = , the magnitude 2 3
(D)
R
o
R
o
(08) The
(A)
(C)
(B)
5
(D)
directional
vector
is
given
as
velocity vector at (1, 1, 1) is
9 2
derivative
velocity
v 5 xyi 2 y 2 j 3 yz 2 k . The divergence of this
9 2
eE1 / T2 / K7 / L2 / V2 / R11 / AC [GATE – – ]
(06) The
2
r 2πrH 1 dr R
eE1 / T2 / K7 / L2 / V2 / R11 / AD [GATE – CE – 2007]
of the gradient at the point (1, 3) is
13 9
r 2πrH 1 dh R
(A) 9
(B) 10
(C) 14
(D) 15
of eE1 / T2 / K7 / L2 / V2 / R11 / AA [GATE – – ]
f ( x, y , z ) 2 x 2 3 y 2 z 2 at the point p(2, 1, 3)
in the direction of the vector a i 2k is _____.
(09) Divergence of the vector field v( x, y, z)
( x cos xy y)i ( y cos xy) j [(sin z 2 ) x 2 y 2 ]k
(A) – 2.785
(B) – 2.145
(C) – 1.789
(D) 1.000
is
(A) 2 z cos z 2
(B) sin xy 2 z cos z 2
(C) x sin xy cos z
(D) None of these
www.targate.org
Page 47
ENGINEERING MATHEMATICS eE1 / T2 / K7 / L2 / V2 / R11 / AB [GATE – – ]
eE1 / T2 / K7 / L2 / V2 / R11 / AD [GATE – –]
(10) The directional derivative of the scalar function
(14) F(x, y) = ( x 2 xy )aˆ x ( y 2 xy ) aˆ y . its line
f ( x, y, z ) x 2 2 y 2 z at the point P = (1, 1,
integral over the straight line from ( x, y) (0,2)
2) in the direction of the vector a 3i 4 j is
(A) – 4
to (x, y) = (2, 0) evaluates to
(A) - 8
(B) 4
(C) 8
(D) 0
(B) - 2
(C) – 1
(D) 1 eE1 / T2 /K7 / L2 / V2 / R11 / AC [GATE – –]
(15) The line integral of the vector function eE1 / T2 / K7 / L2 / V2 / R11 / AB [GATE – –]
(11) For a scalar function f(x, y, z) = x 2 3 y 2 2 z 2 ,
F 2 xiˆ x 2 ˆj along the x – axis from x = 1 to x
= 2 is
the gradient at the point P (1, 2, -1) is
(A) 2 i 6 j 4k
(B) 2 i 12 j 4k
(C) 2 i 12 j 4k
(D)
(A) 0
(B) 2.33
(C) 3
(D) 5.33
eE1 / T2 /K7 / L2 / V2 / R11 / AA [GATE – –]
56
(16) Divergence of the 3 – dimensional radial vector eE1 / T2 / K7 / L2 / V2 / R11 / AB [GATE – –]
fields r is
(12) For a scalar function f(x, y, z) = x 2 3 y 2 2 z 2 , the directional derivative at the point P (1, 2, -1)
1 r
(A) 3
(B)
(C) iˆ ˆj kˆ
(D) 3 iˆ ˆj kˆ
in the direction of a vector i j 2k is
(B) 3 6
(A) - 18
eE1 / T2 /K7 / L2 / V2 / R11 / AA [GATE – – ]
(C) 3 6
(17) If a and b are two arbitrary vectors with
(D) 18
magnitudes a and b respectively, | a b |2 will eE1 / T2 / K7 / L2 / V2 / R11 / AC [GATE – –]
(13) The
divergence
of
the
vector
field
be equal to
3 xziˆ 2 xyjˆ yz 2 kˆ at a point (1, 1, 1) is equal to
(A) 7
(B) 4
(C) 3
(D) 0
Page 48
(A) a 2 b 2 ( a.b) 2
(B) ab a.b
(C) a 2 b 2 ( a.b) 2
(D) ab a.b
TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
TOPIC. 02 – CALCULUS 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
(01) The directional
derivative
of
f(x,
direction of the vector a i 2k is
which
(A) 4 / 5
(B) 4 / 5
(C)
(D) 5 / 4
of
the
following vectors
is
=
2 x 2 3 y 2 z 2 at point P(2, 1, 3) in the
points defined in x, y, z coordinate system, then one
y)
perpendicular to both the vectors B A and BC
(A) 16i 9 j 12 K
(B) 16i 9 j 12 K
(C) 16i 9 j 12 K
(D) 16i 9 j 12 K
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
eE1 / T2 /K7 / L2 / V2 / R11 / AD [GATE – – ]
(19) Consider a closed surface ‘S’ surrounding a
(A) 2 2 3 / 2
(B) 7 / 5
(C) 2 2 3 / 2
(D) 1 / 5
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
5 r .nˆ ds is eE1 / T2 / K7 / L3 / V2 / R11 / AC [GATE – – 2005]
(A) 3V
(B) 5V
(C) 10V
(D) 15V
(03) Value of the integral
xydy y dx, where, c is 2
c
the square cut from the first quadrant by th line x= 1 and y = 1 will be (Use Green’s theorem to
eE1 / T2 /K7 / L2 / V2 / R11 / AB [GATE – – ]
change the line integral into double integral)
(20) The two vectors [1, 1, 1] and [1, a, a2] where a = 1 3 j are 2 2
(A) Orthonormal
(B) Orthogonal
(C) Parallel
(D) Collinear
(A) 1/2
(B) 1
(C) 3/2
(D) 5/3
eE1 / T2 / K7 / L3 / V2 / R11 / AA [GATE – – 2005]
(04) The line integral -----00000-----
V . dr of the vector function
V(r) = 2xyzi x 2 zj x 2 yk from the origin to the point P (1, 1, 1)
Question Level – 03 (A) is 1 eE1 / T2 / K7 / L3 / V2 / R11 / AB [GATE – – 1994]
(B) is Zero
(C) is – 1
www.targate.org
Page 49
ENGINEERING MATHEMATICS (D) Cannot be determined without specifying the path eE1 / T2 / K7 / L3 / V2 / R11 / AA [GATE – –]
(05) A scalar field is given by f = x 2/3 y 2/3 , 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
2 (A) 3
3 (B) 2
2
(C)
(D)
3
3
2
(A) 0
(B)
(C) 1
(D) 2 3
3
2 eE1 / T2 /K7 / L3 / V2 / R11 / AA [GATE – – ]
eE1 / T2 / K7 / L3 / V2 / R11 / AB [GATE – –]
(08) The line integral
(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
P2
P1
( ydx xdy ) from P1 ( x1 , y1 )
to P2 ( x2 , y2 ) along the semi-circle P1P 2 shown in the figure is
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 A xy aˆ x x 2 aˆ y then
shown in the figure is
A . dl over the path
(A) x2 y2 x1 y1
(B) ( y22 y12 ) ( x22 x12 )
(C) ( x2 x1 )( y2 y1 )
(D) ( y2 y1 )2 ( x2 x1 ) 2
eE1 / T2 /K7 / L3 / V2 / R11 / AA [GATE – PI – 2011]
(09) If T(x, y, z) = x 2 y 2 2 z 2 defines the temperature at any location (x, y, z) then the
Page 50
TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
TOPIC. 02 – CALCULUS magnitude of the temperature gradient at point eE1 / T2 / K8 / L3 / V2 / R11 / AB [GATE – – 2004]
P(1, 1, 1) is -----
(04) The area enclosed between the parabola y = x2 (A) 2 6
(B) 4
(C) 24
(D)
ad the straight line y = x is _____
6
(A)
(B)
(C)
(D)
-----00000----eE1 / T2 / K8 / L3 / V2 / R11 / AA [GATE – – 2004]
(05) The volume of an object expressed in spherical
2.8 AREA / VOLUME co-ordinates is given by
Question Level – 03 V E1 / T2 / K8 / L3 / V2 / R11 / AD [GATE – – 1994]
2π
π /3 1
0
0
r 0
2
sin drd dθ
The value of the integral
(01) The volume generated by revolving he area bounded by the parabola y 2 8 x and the line (A)
π 3
(B)
π 6
(C)
2π 3
(D)
π 4
x 2 about y-axis is
(A)
128π 5
(B)
(C)
127 5π
(D) None of the above
5 128π
eE1 / T2 / K8 / L3 / V2 / R11 / AA [GATE – ME – 2008]
(06) Consider the shaded triangular region P shown
eE1 / T2 / K8 / L3 / V2 / R11 / AB [GATE – ME – 1995]
in the figure. What is
xy dx dy ? P
(02) The area bounded by the parabola 2 y x 2 and the lines x y 4 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)
1 6
(B)
(C)
7 16
(D) 1
64 3
(A) 64
(B)
128 (C) 3
128 (D) 4
www.targate.org
2 9
Page 51
ENGINEERING MATHEMATICS
2.9 Miscellaneous eE1 / T2 / K8 / L3 / V2 / R11 / AA [GATE – EE – 2009]
(07) If (x, y) is continuous function defined over (x,
Question Level – 00 (Basic Problem)
y) [0,1] [0,1] Given two constraints, x y 2 and y x 2 , the volume under f(x, y) is (A)
y 1
x 1
(01) The function f(x) = ex is ________
f ( x , y ) dxdy
y0 x y2
y 1
(B)
x y
eE1 / T2 / K9 / L0 / V1 / R11 / AC [GATE – – 1999]
(A) Even
(B) Odd
(C) Neither even nor odd
(D) None
f ( x, y )dxdy
y x2 x y 2
eE1 / T2 / K9 / L0 / V1 / R11 / AB [GATE – – 1998]
(02) The continuous function f(x, y) is said to have y 1
(C)
x 1
y 0 x 0
saddle point at (a, b) if
f ( x, y)dxdy
(A) f x ( a, b) f y ( a, b) 0 (D)
y x
x 0
x y
x 0
f ( x, y)dxdy
f xy2 f xx f yy 0 at (a, b)
(B) f x (a, b) 0, f y (a, b) 0, f xy2 f xx f yy 0 at (a,b)
eE1 / T2 / K8 / L3 / V2 / R11 / AA [GATE – ME – 2009]
(C) f x (a, b) 0, f y (a, b) 0, f xy2 f xx f yy 0 at (a, b)
(08) The area enclosed between the curves y 2 4 x (D) f x (a, b) 0, f y (a, b) 0, f xy2 f xx f yy 0 at (a,b)
and x 2 4 y is
eE1 / T2 / K9 / L0 / V1 / R11 / AC [GATE – IN – 2008]
(A)
16 3
(B) 8
(C)
32 3
(D) 16
eE1 / T2 / K8 / L3 / V2 / R11 / AD [GATE – ME – 2010]
(09) The parabolic arcy =
(03) The expression e ln x for x > 0 is equal to
(A) – x
(B) x
(C) x1
(D) x1
x , 1 x 2 is revolved
around the x-axis. The volume of the solid of
eE1 / T2 / K9 / L0 / V1 / R11 / AD [GATE – – 1998]
(04) A discontinuous real function can be expressed
revolution is
as (A)
π 4
(B)
π 2
(C)
3π 4
(D)
3π 2
(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 -----00000-----
Page 52
series
TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
TOPIC. 02 – CALCULUS Question Level – 01
Question Level – 03
eE1 / T2 / K9 / L1 / V1 / R11 / AD [GATE – – 1998]
(01) The
taylor’s
series
expansion
of
sin x
is_________
(A) 1
eE1 / T2 / K9 / L3 / V2 / R11 / AC [GATE – – 1997]
(01) The curve given by the equation x 2 y 2 3axy is
x 2 x4 x2 x4 .......... (B) 1 ...... 2! 4! 2! 4!
(A) Symmetrical about x –axis (B) Symmetrical about y – axis
(C) x
x 3 x5 .... 2! 4!
(D) x
x3 x 5 .... 3! 5!
(C) Symmetrical about the line y = x (D) Tangential to x = y = a/3
Question Level – 02 eE1 / T2 / K9 / L3 / V2 / R11 / AA [GATE – EC – 2007] eE1 / T2 / K9 / L2 / V2 / R11 / AA [GATE – CE – 1999]
(01) The infinite series
( n!)
(02) For the function e x , the linear approximation
2
(2n)!
around x = 2 is
n 1
(A) Converges
(B) Diverges
(A) (3 – x ) e 2
(C) Is unstable
(D) Oscillate
(B) 1 x
eE1 / T2 / K9 / L2 / V2 / R11 / AB [GATE – CS – 2010] 2n
1 (02) What is the value of lim 1 ? n α n
(C) 3 2 2 1 2 x e 2 (D) e 2
(B) e 2
(A) 0
eE1 / T2 / K9 / L3 / V2 / R11 / AC [GATE – EC – 2007]
(C) e
1/2
(D) 1
(03) For | x | 1,coth( x) can be approximated as
eE1 / T2 / K9 / L2 / V2 / R11 / AA [GATE – EE – 2011]
(03) Roots
of
the
algebraic
equation
(A) x
x3 x2 x 1 0 are (C) (A) (1, j, -j)
(B) (1, -1, 1)
1 x
(B) x2 (D)
1 x2
eE1 / T2 / K9 / L3 / V2 / R11 / AD [GATE – ME – 2008]
(C) (0, 0, 0)
(D) (-1, j, -j)
(04) The length of the curvey y =
2 3/2 x between x = 3
0 & x = 1 is -----00000-----
(A) 0.27
(B) 0.67
(C) 1
(D) 1.22
www.targate.org
Page 53
ENGINEERING MATHEMATICS 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 = 4h
x2 , where x is the L2
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)
1 64
0
(B) 2
(C)
L
L /2
0
L /2
0
(D) 2
L /2
0
h2 x 2 dx L4
h2 x 2 1 64 4 dx L
1 64
h2 x 2 dx L4
1 64
h2 x 2 dx L4
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
(C)
(B)
3
3 2
(D) 2
-----00000-----
Page 54
TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
03 L
Differential Equations Complete subtopic in this chapter, is in the scope of “GATE- ME/EC/EE SYLLABUS”
eE1 / T3 / K1 / L0 / V1/ R11 / AB [GATE – EC – 2009]
3.1 Degree and order of DE
(03) The
differential
differential
equation
3
eE1 / T3 / K1 / L0 / V2 / R11 / AB [GATE – ME – 2007]
partial
of
d 2 y dy y 4 e t is dx 2 dx
Question Level – 00 (Basic Problem)
(01) The
order
equation
2 2 0 has x 2 y 2 y y
(A) 1
(B) 2
(C) 3
(D) 4
eE1 / T3 / K1 / L0 / V2 / R11 / AB [GATE – EC – 2005]
(A) degree 1 and order 2
(04) The
following
differential
equation
has
3
(B) degree 1 and order 1
d2y dy 3 2 4 y2 2 x dt dt
(C) degree 2 and order 1
(A) degree = 2, order = 1
(D) degree 2 and order 1
(B) degree = 1, order = 2
eE1 / T3 / K1 / L0 / V1/ R11 / AC [GATE – PI – 2005]
(02) The
differential
equation
2
d2y C 2 2 is of dx
(A) 2nd order and 3rd degree
dy 2 1 dx
3
(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 3
(B) 3rd order and 2nd degree
d3y dy 4 y2 0 3 dx dx are respectively
(C) 2nd order and 2nd degree
(A) 3 and 2
(B) 2 and 3
(C) 3 and 3
(D) 3 and 1
(D) 3rd order and 3rd degree
www.targate.org
Page 55
ENGINEERING MATHEMATICS eE1 / T3 / K1 / L1 / V1/ R11 / AB [GATE – CE – 2007]
(C) 2nd order linear
(06) The degree of the differential equation d2x 2x3 0 2 dt is
(D)
non
–
homogeneous
with
constant
coefficients
(A) 0
(B) 1
(C) 2
(D) 3
-----00000-----
eE1 / T3 / K1 / L1 / V1/ R11 / AB [GATE – ME – 2007]
(07) The differential equation
d4y d2y P ky 0 dx 4 dx 2
3.2 Higher Order DE Question Level – 01
is eE1 / T3 / K2 / L1 / V1/ R11 / AC [GATE – PI – 2011]
(A) Linear of Fourth order
(B) Non – Linear of fourth order
(01) The
solution
of
the
differential
equation
d2y dy 6 9 y 9 x 6 with C1 and C2 as 2 dx dx
constants is (C) Non – Homogeneous (A) y (C1 x C2 )e 3 x (D) Linear and Fourth degree (B) y C1e3 x C2e 3 x eE1 / T3 / K1 / L1 / V2 / R11 / AD [GATE – ME – 1999]
(08) The equation
d2y dy ( x 2 4 x ) y x8 8 is a 2 dx dx
(A) partial differential equation
(B) non-linear differential equation
(C) y (C1 x C2 )e 3 x x
(D) y (C1 x C2 )e3 x x
eE1 / T3 / K2 / L1 / V1/ R11 / AD [GATE – CE – 1998]
(02) The general solution of the differential equation (C) non-homogeneous differential equation
x2
d2y dy x y 0 is 2 dx dx
(D) ordinary differential equation eE1 / T3 / K1 / L1 / V2 / R11 / AC [GATE – – 1995]
(A) Ax + Bx2 (A, B are constants)
(09) The differential equation y11 ( S 3 sin x)5 y1 y cos x 3 is
(B) Ax + B logx (A, B are constants)
(A) homogeneous
(C) Ax + Bx2logx (A, B are constants)
(B) non – linear
(D) Ax + Bxlog (A, B are constants)’’
Page 56
TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
TOPIC. 03 – DIFFERENTIAL EQUATIONS eE1 / T3 / K2 / L2 / V1/ R11 / AA [GATE – EC – 1994]
Question Level – 02 (04)
item from 1, 2, 3, 4 and 5
eE1 / T3 / K2 / L2 / V2 / R11 / AA [GATE – EC – 1994]
(01)
Match each of the items A, B, C with an appropriate
y e2 x is a solution of the differential equation 11
1
y y 2y 0
a1
d2y dy a2 y a3 y a4 2 dx dx
a1
d3y a2 y a3 dx 3
a1
d2y dy a2 x a3 x 2 y 0 2 dx dx
(A)
(A) True
(B) False (B)
eE1 / T3 / K2 / L2 / V2 / R11 / AD [GATE – IN – 2005]
(02) The general solution of the differential equation ( D 2 4 D 4) y 0 is of the form (given D =
(C)
d an C1, C2 are constants) dx
(1) Non – linear differential equation
(2) Linear differential equation with constant
(B) C1 e 2 x C2 e 2 x
(A) C1 e 2 x
coefficients
(C) C1e 2 x C2e 2 x
(D) C1 e 2 x C2 x e 2 x
(3) Linear homogeneous differential equation (4) Non – linear homogeneous differential equation (5) Non – linear first order differential equation
eE1 / T3 / K2 / L2 / V2 / R11 / AA [GATE – EC – 2006]
(03) For the differential equation
d2y k 2 y 0, the 2 dx
(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
boundary conditions are
(i) y 0 for x 0 and
eE1 / T3 / K2 / L2 / V1/ R11 / AD [GATE – EC – 2007]
(ii) y 0 for x a
(05)
The form of non-zero solution of y (where m
The solution of the differential equation
k2
d2y y y2 dx 2 under the boundary conditions
(i)
y y1 at x 0 and
(ii)
y y2 at x where k, y1 and y2 are
varies over all integers) are (A) y
m
mπx Am sin a
constant is
(B) y
A
m
m
(C) y
mπx cos a mπ a
Am x
Am e
(B) y ( y2 y1 )e
m
(D) y
m
x
(A) y ( y1 y2 )e
mπx a
x k
k2
y2
y1
x y ( y1 y2 ) sin h y1 k (C)
www.targate.org
Page 57
ENGINEERING MATHEMATICS (D)
y ( y1 y2 )e
x
k
eE1 / T3 / K2 / L2 / V2 / R11 / AB [GATE – ME – 2006]
y2
eE1 / T3 / K2 / L2 / V1/ R11 / AA [GATE – PI – 2008] (06)
(09)
d2y dy 4 3 y 3e 2 x , 2 dx For dx the particular integral is
The solutions of the differential equation
d2y dy 2 2y 0 2 dx dx are
1 2x e (A) 15
(1 i ) x , e (1i ) x (A) e
(1 i ) x (1 i ) x ,e (B) e
(1 i ) x (1 i ) x ,e (C) e
(D) e
(C) 3e
1 2x e (B) 5
x 3 x (D) c1e c2 e
2x
(1i ) x , (1 i ) x
e
eE1 / T3 / K2 / L2 / V2 / R11 / AA [GATE – PI – 2009] (10)
The homogeneous part of the differential equation
eE1 / T3 / K2 / L2 / V1/ R11 / AB [GATE – EE – 2010]
(07)
d2y dy p 2q r 2 dx dx (p, q, r are constants) has real
d 2x dx 6 8x 0 2 dt For the differential equation dt
distinct roots if
dx 0 dt with initial conditions x(0) = 1 and t 0 the solution
6t 2 t (A) x(t ) 2e e
2t 4 t (B) x(t ) 2e e
6t 4t (C) x(t ) e 2e
2t 4t (D) x(t ) e 2e
2 (B) p 4q 0
2 (C) p 4q 0
2 (D) p 4q r
eE1 / T3 / K2 / L2 / V2 / R11 / AB [GATE – EC – 2005] (11)
eE1 / T3 / K2 / L2 / V2 / R11 / AA [GATE – EC – 2010] (08)
2 (A) p 4q 0
A solution of the differential equation
d2y dy 5 6y 0 2 dx dx is given by
A function n(x) satisfies the differential equation
d 2 n( x ) n( x ) 2 0 dx 2 L where L is a constant. The
() = 0. The boundary conditions are n(0) = k and n
2x 3 x (A) y e e
2x 3x (B) y e e
2 x 3x (C) y e e
(D) None of these.
solution to this equation is 06. A function n(x) satisfies the differential equation. This equation is eE1 / T3 / K2 / L2 / V2 / R11 / AA [GATE – ME – 2008] (A)
n( x ) k exp x / L
(B)
n( x ) k exp x / L
what is
x( n) k 2 exp x / L
(A) 0
(B) 0.37
(C)
(C) 0.62
(D) 1.13
(D)
(12)
n( x) k exp x / L2
Page 58
It is given that
y " 2 y ' y 0, y(0) 0 y(1) 0
y(0.5)?
TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
TOPIC. 03 – DIFFERENTIAL EQUATIONS eE1 / T3 / K2 / L2 / V1/ R11 / AC [GATE – ME – 2007]
(A) y = 1
dy y 2 with initial value y(0) = dx
(B) y = x
(13) The solution of
1 is bounded in the internal is (C) y = x + c where c is an arbitrary constants are (A) x
(B) x 1
(C) x 1, x 1
(D) 2 x 2
arbitrary constants
(D) y = C1 x C2 where C1, C2 are arbitrary constants
eE1 / T3 / K2 / L2 / V2 / R11 / AA [GATE – IN – 2006]
(14) For initial value problem y 2 y 101 y 10.4e x , y(0)=1.1 and y(0) = - 0.9. Various solutions are written in the following groups. Match the type
eE1 / T3 / K2 / L2 / V2 / R11 / AC [GATE – ME – 1995]
(16) The
solution
to
the
differential
equation
f 11 ( x) 4 f 1 ( x) 4 f ( x ) 0
of solution with the correct expression.
Group – I P.
General solution
(A) f1 ( x ) e 2 x
Group – II (1) 0.1 e x
(B) f1 ( x ) e 2 x , f 2 ( x ) e 2 x
of Homogeneous equations Q. Particular integral
(2) e x [A
(C) f1 ( x ) e2 x , f 2 ( x) xe 2 x
cos10 x B sin10 x ] R. Total solution
(3) e x cos10x 0.1e x
(D) f1 ( x ) e 2 x , f 2 ( x ) e x
satisfying boundary
eE1 / T3 / K2 / L2 / V2 / R11 / AC [GATE – – 1995]
conditions
(17) The
solution
of
a
differential
equation
y11 3 y1 2 y 0 is of the form
Codes: (A) P – 2, Q – 1, R -3
(C) P – 1, Q – 2, R – 3
(B) P -1, Q -3, R – 2
d2y 0 2 (15) The solution of the differential equation dx
dy 1 (i) dx at x = 0
dy 1 (ii) dx at x = 1 is
(B) c1e x c2 e3 x
(C) c1e x c2 e 2 x
(D) c1e 2 x c2 2 x
(D) P -3 , Q – 2, R – 1
eE1 / T3 / K2 / L2 / V2 / R11 / AC [GATE – – ]
with boundary conditions
(A) c1e x c2 e2 x
eE1 / T3 / K2 / L2 / V2 / R11 / AA [GATE – ME – 1996]
(18) The particular solution for the differential equation
d2y dy 3 2y 2 dx dt
sx is
(A) 0.5cos x 1.5sin x
(B) 1.5cos x 0.5sin x
(C) 1.5sin x
(D) 0.5cos x
www.targate.org
Page 59
ENGINEERING MATHEMATICS eE1 / T3 / K2 / L2 / V2 / R11 / AA [GATE – ME – 1994]
eE1 / T3 / K2 / L2 / V2 / R11 / AC [GATE – CE – 2001]
d2y dy 2 y 0 with y(0) = 1 2 dt dt
(23) The solution for the following differential
(19) Solve for y if and y1 (0) 2 (A) (1 t )e t
equation with boundary conditions y(0) = 2 and y1 (1) 3 is where
(B) (1 t ) et (A) y x
(C) (1 t ) e
d2y 3x 2 dx 2
t
(D) (1 t )e
3
3
t
eE1 / T3 / K2 / L2 / V2 / R11 / AC [GATE – EC – 2005]
x
(B) y 3x 3 x
2
2
2
(20) Which of the following is a solution of the
2
3x 2
5x 2
differential equation d2y dy P ( q 1) 0? Where p = 4, q = 3 dx dx
(C) y x
3
3
x2 5 x 2 2
(D) y x3 x 3x
(A) e
(B) xe
2
x
2
5x 3
2
eE1 / T3 / K2 / L2 / V2 / R11 / AA [GATE – CE – 2005]
(C) x e
2 x
2
(D) x e
2 x
(24) The eE1 / T3 / K2 / L2 / V2 / R11 / AB [GATE – EE – 2005]
x(t ) 3x(t) 2x(t) 5, the (21) For the equation
solution
d2y dy 2 17 y 0; 2 dx dx
dy y (0) 1, 0 in the range 0 x π 4 is dx x π 4
solution x(t) approaches the following values as t
given by
(A) 0
(B) 5/2
1 (A) e x [cos 4 x sin 4 x ] 4
(C) 5
(D) 10
1 (B) e x [cos 4 x sin 4 x ] 4
eE1 / T3 / K2 / L2 / V2 / R11 / A [GATE – EE – 2005]
(22) The solution to the ordinary differential equation 2
d y dy 6y 0 dx 2 dx is
1 (C) e 4 x [cos 4 x sin x ] 4 1 (D) e4 x [cos 4 x sin 4 x] 4 eE1 / T3 / K2 / L2 / V2 / R11 / AC [GATE – ME – 2005]
3x 2 x (A) y C1e C2e
(25) The complete solution of the ordinary differential equation
3x 2x (B) y C2 e C2e
d2y dy P qy 0 is 2 dx dx
y C1e x C2 e3 x then P and q are
3 x 2x (C) y C1e C2 e
(A) P = 3, q = 3
(B) P = 3, q = 4
(C) P = 4, q = 3
(D) P = 4, q = 4
3 x 2 x (D) y C1e C2 e
Page 60
TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
TOPIC. 03 – DIFFERENTIAL EQUATIONS
3.3 Leibnitz linear equation
(A) (1 x)e x
2
(B) (1 x)e x
Question Level – 02
(C) (1 x)e x
2
(D) (1 x)e x
eE1 / T3 / K3 / L2 / V2 / R11 / AB [GATE – EC – 2008]
(01) Which of the following is a solution to the differential equation d x(t ) 3x (t ) 0, x(0) 2? dt
(A) x(t ) 3e
2ln x dy (05) If x 2 2 xy and y(1) = 0 then what x dx
is y(e)?
(B) x(t ) 2 e
3 2 t 2
(D) x(t ) 3t 2
(02) For the differential equation
dy 5 y 0 with dt
y(0) 1, the general solution is:
(B) e5t
(C) 5 e 5 t
(D) e
of
the
differential
equation
is:
equation of first order only if,
(D) P and Q are functions of x (or) constants eE1 / T3 / K3 / L2 / V2 / R11 / AA [GATE – ME – 2009]
(07) The solution of x
y(1) 6 2 x 2 3 3x
(B) y
2 x 3 3
(D) y
x 1 2 2x
2 x2 3x 3
eE1 / T3 / K3 / L2 / V2 / R11 / AB [GATE – ME – 2006]
(04) The
dy py Q, (06) The differential equation dx is a linear
(C) P is a functions of y but Q is a constant
5t
dy y x with the condition that y 1 at x = 1 dx x
(C) y
solution
1 e2
(B) P and Q are functions of y (or) constants
eE1 / T3 / K3 / L2 / V2 / R112 / A [GATE – EE – 1994]
(A) y
(D)
(A) P is a constant but Q is a function of y
(A) e5t
solution
1 e
eE1 / T3 / K3 / L2 / V2 / R11 / AD [GATE – CE – 1997]
eE1 / T3 / K3 / L2 / V2 / R112 / AB [GATE – ME – 1994]
(03) The
(B) 1
3t
(C) (C) x(t )
2
eE1 / T3 / K3 / L2 / V1/ R11 / AD [GATE – ME – 2005]
(A) e t
2
of
the
differential
2 dy 2 xy e x with y(0) 1 is dx
equation
dy y x 4 with condition dx
5
(A) y
x4 1 5 x
(B) y
4 x4 4 5 5x
(C) y
x4 1 5
(D) y
x5 1 5
eE1 / T3 / K3 / L2 / V2 / R11 / AA [GATE – CE – 2005]
(08) Transformation to linear form by substituting v = y1 n of the equation
dy P (t ) y q (t ) y n , n 0 dt
will be
www.targate.org
Page 61
ENGINEERING MATHEMATICS (A)
dv (1 n) pv (1 n)q dt
3.4 Miscellaneous Question Level – 01
dv (B) (1 n) pv (1 n)q dt
eE1 / T3 / K4 / L1 / V1/ R11 / AA [GATE – ME – 2003]
(01) The dv (C) (1 n) pv (1 n)q dt
(D)
dy y ex dx
(C)
solution
the
differential
x3 c 3
(C) c e x
the
f ( x, y )
(D) 2 e e 1
of
(B) y
(02) For
eE1 / T3 / K3 / L2 / V2 / R11 / AD [GATE – PI – 2010]
(10) The
equation
(A)
differential
f g y x
(C) f g 2
equation
dy g ( x, y ) 0 to be exact is dx
(B)
f g x y
(D)
2 f 2 g x 2 y 2
dy y 2 1 satisfying the condition y(0) = 1 is dx
(A) y e x
equation
eE1 / T4 / K4 / L1 / V1/ R11 / AB [GATE – CE – 1997]
1 (B) e e1 2
1 e e 1 2
differential
(D) Unsolvable as equations is non – linear
with y(0) 1. Then the value of y(1) is
(A) e e
the
1 xc
(A) y
eE1 / T3 / K3 / L2 / V2 / R11 / AC [GATE – IN – 2010]
1
of
dy y 2 0 is dx
dv (1 n) pv (1 n)q dt
(09) Consider the differential equation
solution
(B) y x eE1 / T4 / K4 / L1 / V1/ R11 / AC [GATE – CE – 1999]
(C) y cot x π
4
(D) y tan x π
4
(03) If C is a constant, then the solution of dy 1 y 2 is dx
(A) y sin(x c)
(B) y cos( x c)
(C) y tan( x c)
(D) y e x c
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TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
TOPIC. 03 – DIFFERENTIAL EQUATIONS Question Level – 02
Question Level – 03
eE1 / T3 / K4 / L2 / V2 / R11 / AD [GATE – CE – 2007]
(01) The solution for
the differential equation
dy x 2 y with the condition that y = 1 at x = 0 is dx
(A) y e
1
2x
(C) ln( y )
x3 4 3
(B) ln( y )
x2 2
(D) y e
x3
eE1 / T3 / K4 / L3 / V2 / R11 / AA [GATE – CE – 2009]
(01) Solution 3y
of
the
differential
equation
dy 2 x 0 represents a family of dx
(A) ellipses
(B) circles
(C) parabolas
(D) hyperbolas
3
eE1 / T3 / K4 / L3 / V2 / R11 / AC [GATE – PI – 2010]
(02) Which one of the following differential equations eE1 / T3 / K4 / L2 / V2 / R11 / AB [GATE – CE – 2007]
(02) A body originally at 600 cools down to 40 in 15 minutes when kept in air at a temperature of
has
a
solution
given
by
the
function
π y 5sin 3 x 5
250 c. What will be the temperature of the body at the and of 30 minutes?
(A) 35.20 C
(A)
dy 5 cos(3x ) 0 dx 3
(B)
dy 5 (cos 3x) 0 dx 3
(C)
d2y 9y 0 dx 2
(D)
d2y 9y 0 dx 2
(B) 31.50 C
(C) 28.70 C
(D) 150 C eE1 / T3 / K4 / L3 / V2 / R11 / AC [GATE – – ]
eE1 / T3 / K4 / L2 / V2 / R11 / AA [GATE – IN – 2008]
(03) Consider the differential equation
(03) Let f y x . What is
dy 1 y2. dx
Which one of the following can be particular
x at x = 2, y = 1? xy
(A) 0
(B) ln 2
(C) 1
(D)
solution of this differential equation?
(A) y tan( x 3)
(B) y tan( x 3)
1 ln 2
eE1 / T3 / K4 / L3 / V2 / R11 / AA [GATE – EC – 2009]
(C) x tan( y 3)
(D) x tan( y 3)
(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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www.targate.org
Page 63
ENGINEERING MATHEMATICS R:
(3)
dy x dx y
S:
Hyperbolas
eE1 / T3 / K4 / L3 / V2 / R11 / AC [GATE – EC – 2011]
(08) The
solution
of
differential
equation
dy Ky , y (0) C is dx
dy x dx y
(A) P-2, Q-3, R-3, S-1
(B) P-1, Q-3, R-2, S-1
(A) x CeKy
(B) x Kecy
(C) P-2,Q-1,R-3, S-3
(D) P-3, Q-2, R-1, S-2
(C) y e kx C
(D) y Ce kx
eE1 / T3 / K4 / L3 / V2 / R11 / AA [GATE – EE – 2011]
(05) With K as constant, the possible solution for the dy first order differential equation e3x is dx
eE1 / T3 / K4 / L3 / V2 / R11 / AC [GATE – IN – 2011]
(09) Consider the differential equation y 2 y y 0 with boundary conditions y(0) 1 y(0) 0 .The value of y(2) is
(A)
1 3 x e K 3
(B)
(C) 3e3x K
1 ( 1)e 3 x K 3
(D) y Ce kx
eE1 / T3 / K4 / L3 / V2 / R11 / AB [GATE – ME – 1993]
(A) – 1
(B) - e
(C) e2
(D) e2
1
eE1 / T3 / K4 / L3 / V2 / R11 / AD [GATE – ME – 2011]
2
(06) The differential
d y dy sin y 0 is dx 2 dx
(10) Consider the differential equation
dy (1 y 2 ) x. dx
The general solution with constant “C” is (A) linear
(B) non – linear
(C) homogeneous
(D) of degree two
x2 (A) y tan C 2
x (B) y tan 2 C 2
x (C) y tan 2 C 2
x2 (D) y tan C 2
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 = 0
eE1 / T3 / K4 / L3 / V2 / R11 / AC [GATE – ME – 1996]
to be exact is
(11) The one dimensional heat conduction partial M N x y
(A) M = N
(B)
M N (C) y x
2M 2 N 2 (D) x 2 y
Page 64
differential equation
T T is t x 2
(A) parabolic
(B) hyperbolic
(C) elliptic
(D) mixed
TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
TOPIC. 03 – DIFFERENTIAL EQUATIONS eE1 / T3 / K4 / L3 / V2 / R11 / AA [GATE – CE – 2001]
(12) The number of boundary conditions required to solve the differential equation
2 2 0 is x 2 y 2
(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
dx kx 2 0, dt
where k is the reaction rate constant. If x = a at t = 0 then solution of the equation is
1 1 kt a x
(A) x a e kt
(B)
(C) x a(1 e kt )
(D) x a kt
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Question Level – 03 eE1 / T3 / K9 / L3 / V2 / R11 / AC [GATE – IN – 2005]
(1)
f a0 x n a1n n 1 an 1 y n 1 an y n
ai (i = 0 to n) are constants then v x
(A)
f n
(C) n f
(B)
where
f f y is x y
n f
(D) n f
-----00000-----
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Page 65
04 L
Complex Variable Complete subtopic in this chapter, is in the scope of “GATE- ME/EC/EE SYLLABUS”
4.1Cauchy’s Theorem
(C)
Question Level – 02
(01) The value of the contour integral
1 dz | z j| 2 z 4
(B)
(03) For the function
sin z of a complex variable z, z3
the point z = 0 is
2
in the positive sense is jπ 2
(D) 1
eE1 / T4 / K1 / L2 / V2 / R11 / AB [GATE – IN – 2007]
eE1 / T4 / K1 / L2 / V2 / R11 / AD [GATE – – ]
(A)
4π 6πi 81
π 2
(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]
(C)
jπ 2
(D)
π 2
(04) The value of
1
(1 z )dz where C is the contour 2
C
| z i / 2| 1 eE1 / T4 / K1 / L2 / V2 / R11 / AA [GATE – EC – 2006]
(02) Using Cauchy’s integral theorem, the value of the (A) 2 πi
(B) π
(C) tan 1 ( z )
(D) π i tan 1 z
integral (integration being taken in contour clock wise direction)
eE1 / T4 / K1 / L2 / V2 / R11 / AA [GATE – EC – 2007]
z3 6 dz is where C is |z| = 1 3z i C
(05) If the semi – circulator controur D of radius 2 is
(A)
2π 4πi 81
Page 66
(B)
π 6πi 8
as shown in the figure. Then the value of the
integral
s D
2
1 ds is 1
TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
TOPIC. 04 – COMPLEX VARIABLE 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 = 3 x 2 3 y 2 . The expression for v. Considering k is to be constant is
(A) i π
(B) i π
(C) π
(D) π
C
(C)
cos(2πz ) dz (2 z 1)( z 3)
(D) 6xy k
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 zdz, z being a complex variable. The value
πi (B) 5
2π i 5
(C) 6 y 6x k
Question Level – 03
where C is a closed curve given by 1 1 1 is
(A) π i
(B) 6x 6 y k
-----00000-----
eE1 / T4 / K1 / L2 / V2 / R11 / AC [GATE – CE – 2009]
(06) The value of the integral
(A) 3 y 2 3x 2 k
C
of I will be
(D) π i
eE1 / T4 / K1 / L2 / V2 / R11 / AD [GATE – EC – 2009]
(07) If f(z) = C0 C1 z 1 then
1( f ( z ) dz is given z unit
(A) 2 π C1
(B) 2π 1 C0
(C) 2π j C1
(D) 2 π j (1 C0 )
(A) I = 0; Singularities set =
(B) I = 0; Singularities set = (2n 1) π / n 0,1, 2,........ 2
(C) I = π / 2 ; Singularities set =
nπ ; n 0,1,2,........... eE1 / T4 / K1 / L2 / V2 / R11 / AC [GATE – IN – 2011]
(08) The contour integral
e
1
z
dz with C as the
(D) None of the above.
C
counter clock – wise unit circle in the z – plane is
eE1 / T4 / K1 / L3 / V2 / R11 / AA [GATE – IN – 2009]
equal to (02) The value of (A) 0
(B) 2 π
(C) 2 π 1
(D)
a
sin z dz, where the contour of the z
integration is a simple closed curve around the origin is
www.targate.org
Page 67
ENGINEERING MATHEMATICS (A) 0
eE1 / T4 / K2 / L0 / V1 / R11 / AD [GATE – IN – 1994]
(B) 2 π j (02)
(C)
(D)
cos can be represented as
1 2π j
(A)
ei e i 2
(B)
ei e i 2i
(C)
ei ei i
(D)
ei ei 2
eE1 / T4 / K1 / L3 / V2 / R11 / AA [GATE – – ]
(03) The value of the integral
C
3 z 4 dz , when 2 z 4z 5
C is the circle | z | 1 is given by eE1 / T4 / K2 / L0 / V1 / R11 / AD [GATE – IN – 2009]
(03) If Z = x + jy where x, y are real then the value of (B) 1 10
(A) 0
(C) 4
| e jz | is
(D) 1
5
eE1 / T4 / K1 / L3 / V2 / R11 / AB [GATE – PI – 2011]
(04) The value of
C
z2 dz , using Cauchy’s integral z4 1
around the circle | z 1| 1 where z x iy is
x2 y2
(A) 1
(B) e
(C) e y
(D) e y
eE1 / T4 / K2 / L0 / V1 / R11 / AD [GATE – PI – 2009]
(04) The product of complex numbers (3 – 21) & (3 + i4) results in
(B) πi
(A) 2 πi (C) 3πi
2
(A) 1 + 6i
(B) 9 – 8i
(C) 9 + 8i
(D) 17 + i 6
(D) π 2i
2
-----00000-----
eE1 / T4 / K2 / L0 / V1 / R11 / AD [GATE – CE – 2009]
(05) The analytical function has singularities at, where
4.2 Miscellaneous
f(z) =
Question Level – 00 (Basic Problem)
z 1 z2 1
(A) 1 and -1
(B) 1 and i
(C) 1 and – i
(D) i and – i
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 *
(C) Re( z )
Page 68
z z* 2
(B) Re( z )
z z* 2
-----00000-----
(D) Re( z) z z *
TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
TOPIC. 04 – COMPLEX VARIABLE eE1 / T4 / K2 / L1 / V1 / R11 / AD [GATE – EC – 2010]
Question Level – 01
(06) The contour C in the adjoining figure is described eE1 / T4 / K2 / L1 / V1 / R11 / AB [GATE – ME – 1996]
(01)
i i , where i =
1 is given by
(A) 0
(B) eπ /2
π (C) 2
(D) 1
by
x 2 y 2 16.
z2 8 dz (0.5) z (1.5) j
C
Then
the
value
of
eE1 / T4 / K2 / L1 / V1 / R11 / AB [GATE – CE – 1997]
(02)
ez is a periodic with a period of (A) 2π
(B) 2πi
(C) π
(D) iπ
(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]
1. Then one value of j j is
(07) Let j = eE1 / T4 / K2 / L1 / V1 / R11 / AA [GATE – IN – 2005]
(03) The function
w u iv 1 log( x 2 y 2 ) i tan 1 y 2 x is not analytic at the point.
(A)
(C) 1
(A) (0, 0)
(B) (0, 1)
(C) (1, 0)
(D) (2, α )
(B) 1
3
2
(D) e
π
2
eE1 / T4 / K2 / L1 / V1 / R11 / AB [GATE – PI – 2007]
(08) If a complex number z =
3 1 4 i then z is 2 2
eE1 / T4 / K2 / L1 / V1 / R11 / AB [GATE – PI – 2008]
(04) The value of the expression
(A) 1 2i
5 i10 3 4i
(A) 2 2 2i
(B) 1 2i (C)
(C) 2 i
(D) 2 i
eE1 / T4 / K2 / L1 / V1 / R11 / AD [GATE – – ]
3 1 i 2 2
1 3 (B) i 2 2
(D)
3 1 i 8 8
eE1 / T4 / K2 / L1 / V1 / R11 / AA [GATE – ME – 2011]
(09) The product of two complex numbers 1 + i & 2 –
(05) The equation sin(z) 10 has
5 i is (A) no real (or) complex solution (B) exactly two distinct complex solutions (C) a unique solution
(A) 7 – 3i
(B) 3 – 4i
(C) – 3 – 4 i
(D) 7 + 3i
(D) an infinite number of complex solutions
www.targate.org
Page 69
ENGINEERING MATHEMATICS eE1 / T4 / K2 / L1 / V1 / R11 / AA [GATE – EC – 2008]
(10) The
residue
of
the
function
f(z)
=
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?
1 at z = 2 is ( z 2) ( z 2) 2 2
(A) 1
(B) 1 16
32
(C) 1 16
(D) 1
(A)
z1 z1 z2 z2 | z2 |2
(B) | z1 z2 || z1 | | z2 |
32 (C) | z1 z2 | | z1 | | z2 |
-----00000-----
(D) | z1 z2 |2 | z1 z2 |2 2 | z1 |2 2 | z2 |2
Question Level – 02 eE1 / T4 / K2 / L2 / V2 / R11 / AB [GATE – IN – 1997]
eE1 / T4 / K2 / L2 / V2 / R11 / AA [GATE – IN – 2005]
(01) The complex number z x jy which satisfy
(04) Consider the circle | z 5 5i | 2 in the complex
the equation | z 1| 1 lie on
number plane (x, y) with z = x+iy. The minimum
(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
distance from the origin to the circle is
(A) 5 2 2
(B)
(C)
(D) 5 2
34
54
eE1 / T4 / K2 / L2 / V2 / R11 / AC [GATE – IN – 2005]
eE1 / T4 / K2 / L2 / V2 / R11 / AA [GATE – IN – 2002]
(02) The bilinear transformation w =
z 1 z 1
(A) Maps the inside of the unit circle in the z –
(05) Let z 3 z , where z is a complex number not equal to zero. Then z is a solution of
2 (A) z 1
3 (B) z 1
4 (C) z 1
9 (D) z 1
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
eE1 / T4 / K2 / L2 / V2 / R11 / AB [GATE – EC – 2006]
(06) For the function of a complex variable w = l nz (C) maps the inside of the unit circle in the z – plane to right half of the w – plane
(where w = u jv and z x jy ) the u = constant lines get mapped i the z – plane as
(D) maps the outside of the unit circle in the z – plane to the right half of the w – plane
Page 70
TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
TOPIC. 04 – COMPLEX VARIABLE eE1 / T4 / K2 / L2 / V2 / R11 / AA [GATE – – ]
(A) Set of radial straight lines
(10) The integral (B) Set of concentric circles
f ( z)dz evaluated around the unit
circle on the complex plane for p( z )
(C) Set of co focal hyperbolas (D) Set of co focal ellipses
(A) 2 π i
(B) 4 π i
(C) 2 π i
(D) 0
Coz z is z
eE1 / T4 / K2 / L2 / V2 / R11 / AA [GATE – CE – 2007]
(07) Potential function is given x 2 y 2 . What
-----00000-----
will be the stream function with the condition
0 at x = 0, y = 0? Question Level – 03 eE1 / T4 / K2 / L3 / V2 / R11 / AB [GATE – ME – 2007]
(B) x 2 y 2
(A) 2xy
(01) If ( x, y) and ( x, y) are functions with continuous 2
(C) x y
2
2
(D) 2x y
2
2nd
derivatives
then
( x, y) i ( x, y) can be expressed as an analytic function of x iy (i 1) when
eE1 / T4 / K2 / L2 / V2 / R11 / AB [GATE – CE – 2010]
(08) The modulus of the complex number
(A) 5
(C)
(B)
1 5
(D)
3 4i is 1 2i
5
(A)
, x x y y
(B)
, x x y y
(C)
2 2 2 2 1 x 2 y 2 x 2 y 2
(D)
0 x y x y
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 _______
eE1 / T4 / K2 / L3 / V2 / R11 / AB [GATE – – ]
(02) A complex variable z x j (0.1) has its real (A) 0
(B) 1
(C) 2
(D) 4
part x varying in the range to . Which one of the following is the locus (shown in thick lines) of
www.targate.org
1 in the complex plane? z
Page 71
ENGINEERING MATHEMATICS 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 i 1 . If u = xy then the expression for v should be
(A)
( x y )2 k 2
(B)
x y2 k 2
(C)
y 2 x2 k 2
(D)
( x y) 2 k 2
eE1 / T4 / K2 / L3 / V2 / R11 / AB [GATE – PI – 2010]
(05) If f(x + iy) = x 3 3 xy 2 i( x, y ) where i 1 and
f (x iy) is an analytic function then
(x / y) is
(A) y 3 3x 2 y
(B) 3x 2 y y 3
(C) x 4 4 x 3 y
(D) xy y 2
eE1 / T4 / K2 / L3 / V2 / R11 / AD [GATE – EE – 2011]
(06) A point z has been plotted in the complex plane eE1 / T4 / K2 / L3 / V2 / R11 / AB [GATE – IN – 2009]
as shown in the figure below
(03) One of the roots of equation x 3 j , where j is the +ve square root of – 1 is
(A) j
(C)
Page 72
(B)
3 j 2 2
3 j 2 2
(D)
3 j 2 2
TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
TOPIC. 04 – COMPLEX VARIABLE The plot of the complex number
-----00000-----
www.targate.org
Page 73
ENGINEERING MATHEMATICS
05 Probability and Statistics Complete subtopic in this chapter, is in the scope of “GATE- CS/ME/EC/EE SYLLABUS
Question Level – 03
5.2 Combination
eE1 / T5 / K2 / L3 / V2 / R11 / AB [GATE – IT – 2005]
Question Level – 01
(01) A bag contains 10 blue marbles, 20 black marbles
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
and 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
replaced?
(A) 1/26
(B) 1/52
(C) 1/169
(D) 1/221
(A)
1 36
(B)
1 6
(C)
1 4
(D)
1 3
-----00000----eE1 / T5 / K2 / L3 / V2 / R11 / AC [GATE – ME – 2010]
Question Level – 02
(02) A box contains 2 washers, 3 nuts and 4 bolts. Items are drawn from the box at random one at a
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
time without replacement. The probability of drawing 2 washers first followed by 3 nuts and subsequently the 4 bots is
replacement. The probability that none of the two screws is defective will be
(A) 2/315
(B) 1/630
(A) 100%
(B) 50%
(C) 1/1260
(D) 1/2520
(C) 49%
(D) None of these
eE1 / T5 / K2 / L3 / V1 / R11 / AC [GATE – EE – 2010]
(03) A box contains 4 while balls and 3 red balls. In -----00000-----
succession, two balls are randomly selected and removed from the box. Given that first removed
Page 74
TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
TOPIC. 05 – PROBABILITY AND STATISTICS ball is white, the probability that the 2nd removed
that one of the balls is red and the other is blue
ball is red is
will be ________
(A) 1/3
(B) 3/7
(C) ¼
(D) 4/7
(A)
1 7
(B)
4 49
(C)
12 49
(D)
3 7
eE1 / T5 / K2 / L3 / V2 / R11 / AC [GATE – PI – 2010]
(04) Two white and two black balls, kept in two bins, -----00000-----
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
5.3 Probability related problems
randomly from the chosen bin. Which one of the following
arrangements
has
the
highest
probability for getting a white ball picked?
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 P Q 0
(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 eE1 / T5 / K2 / L3 / V2 / R11 / AC [GATE – CE – 2011]
(05) There are two containers with one containing 4
(A)
1 8
(B)
1 2
(C)
3 8
(D)
3 4
red 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
www.targate.org
Page 75
ENGINEERING MATHEMATICS eE1 / T5 / K3 / L0 / V1 / R11 / AD [GATE – EC – 2005]
eE1 / T5 / K3 / L1 / V2 / R11 / AD [GATE – – 2003]
(03) A fair dice is rolled twice. The probability that an
(03) Let P(E) denote the probability of an event E.
odd number will follow an even number is Given P(A) = 1, P(B) =
1 the values of P(A/B) 2
(A)
1 2
(B)
1 6
and P(B/A) respectively are
(C)
1 3
(D)
1 4
(A)
1 1 , 4 2
(B)
(C)
1 ,1 2
(D) 1,
-----00000-----
1 1 , 2 4
1 2
Question Level – 01 eE1 / T5 / K3 / L1 / V1 / R11 / AC [GATE – – 2004] eE1 / T5 / K3 / L1 / V1 / R11 / AD [GATE – – 1997]
(04) A hydraulic structure has four gates which
(01) The probability that it will rain today is 0.5, the
operate independently. The probability of failure
probability that it will rain tomorrow is 0.6. The
of each gate is 0.2. Given that gate 1 has failed,
probability that it will rain either today or
the probability that both gates 2 and 3 will fail is
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
(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
eE1 / T5 / K3 / L1 / V1 / R11 / AD [GATE – – 2000]
the probability that they all occurred on same
(02) E1 and E2 are events in a probability space day? satisfying
the
following
constraints (A)
1 77
(B)
1 76
(C)
1 27
(D)
7 27
P( E1 ) P( E2 ); P( E1 Y E2 ) 1 : E1 & E2 are independent then P( E1 )
(A) 0
(B)
1 4
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
(C)
1 2
Page 76
result? (D) 1
TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
TOPIC. 05 – PROBABILITY AND STATISTICS (A)
(C)
3 8
5 8
(B)
(D)
1 2
3 4
(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 eE1 / T5 / K3 / L1 / V2 / R11 / AD [GATE – PI – 2005]
(07) Two dice are thrown simultaneously. The
probability
of
two
heads
simultaneously
appearing is
probability that the sum of numbers on both exceeds 8 is
(A)
4 36
(B)
7 36
(A) 1/8
(B) 1/6
(C) 1/4
(D) ½
eE1 / T5 / K3 / L1 / V1 / R11 / AD [GATE – ME – 2011]
9 (C) 36
10 (D) 36
(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
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
(A)
1 32
(B)
13 32
(C)
16 32
(D)
31 32
eE1 / T5 / K3 / L1 / V1 / R11 / AC [GATE – EC – 2007]
eE1 / T5 / K3 / L1 / V1 / R11 / AA [GATE – CS – 2011]
(09) An examination consists of two papers, paper 1
(13) It two fair coins are flipped and at least one of the
and paper 2. The probability of failing in
outcomes is known to be a head, what is the
probability of failing in paper 1 is 0.6. The
probability that both outcomes are heads?
probability of a student failing in both the papers is (A) 0.5
(B) 0.18
(C) 0.12
(B) 0.06
(A)
1 3
(B)
1 4
(C)
1 2
(D)
2 3
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
-----00000-----
times heads show up is more than the number of times tails show up” is
www.targate.org
Page 77
ENGINEERING MATHEMATICS Question Level – 02
(A)
3 23
(B)
6 23
(C)
3 10
(D)
3 5
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
eE1 / T5 / K3 / L2 / V2 / R11 / AD [GATE – ME – 2005]
16 (A) 25
9 (B) 10
27 (C) 75
18 (D) 25
3
eE1 / T5 / K3 / L2 / V1 / R11 / AB [GATE – – 1998]
(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
(02) A die is rolled three times. The probability that exactly one odd number turns up among the three
eE1 / T5 / K3 / L2 / V2 / R11 / AC [GATE – EC – 2011]
outcomes is
(06) A fair dice is tossed two times. The probability that the 2nd toss results in a value that is higher
(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
(C) 1/144
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]
(B) 1/12
(D) 1/24
(07) Consider two events E1 and E2 such that 1 1 1 p ( E1 ) , p ( E2 ) and ( E1 I E2 ) . Which 2 3 5
of the following statement is true? eE1 / T5 / K3 / L2 / V2 / R11 / AB [GATE – IT – 2004]
(04) In a population of N families, 50% of the families
(A) p ( E1 Y E2 )
2 3
have three children, 30% of families have two children and the remaining families have one
(B) E1 and E2 are independent
child. What is the probability that a randomly picked child belongs to a family with two
(C) E1 and E2 are not independent
children? (D) P( E1 / E2 ) 4 / 5
Page 78
TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
TOPIC. 05 – PROBABILITY AND STATISTICS eE1 / T5 / K3 / L3 / V2 / R11 / AC [GATE – –]
Question Level – 03
(04) A fair coin is tossed 10 time. What is the eE1 / T5 / K3 / L3 / V2 / R11 / AC [GATE – – 2002]
probability that only the first two tosses will yield
(01) Four fair coins are tossed simultaneously. The heads? probability that at least one heads and at least one tails turn up is
1 (A) 2 (A)
(C)
1 16
(B)
7 8
(D)
1 8
2
10
1 (C) 2
15 16
1 (B) 10c2 2
2
10
1 (D) 10c2 2
eE1 / T5 / K3 / L3 / V2 / R11 / AA [GATE – CS – 2010]
(05) What is the probability that a divisor of 1099 is a eE1 / T5 / K3 / L3 / V2 / R11 / AA [GATE – PI – 2007]
(02) Two cards are drawn at random in succession
multiple of 1096?
with replacement from a deck of 52 well shuffled cards Probability of getting both ‘Aces’ is
(A)
1 169
(B)
2 169
(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 (C)
1 13
(D)
2 13
and 15. The box 2 contains chips numbered 6, 11,
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
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 ___________ .
X winning the game is
(A) 1/3
(B) 1/3
(C) 2/3
(D) 3/4
(A)
6 25
(B)
2 5
(C)
3 5
(D)
19 25
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Page 79
ENGINEERING MATHEMATICS eE1 / T5 / K3 / L3 / V2 / R11 / AC [GATE – PI – 2011]
5.4 Bays theorems
(07) It is estimated that the average number of events during a year is three. What is the probability of
No Question
occurrence of not more than two events over a two-year duration? Assume that the number of
5.5 Probability Distribution
events follow a poisson distribution.
Question Level – 00 (Basic Problem) (A) 0.052
(B) 0.062 eE1 / T5 / K5 / L0 / V1 / R11 / AA [GATE – IN – 2007]
(C) 0.072
(D) 0.082
(01) Assume that the duration in minutes of a telephone conversation follows the exponential
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)
(C)
1 9
(B)
1 4
(D)
5 36
3 4
distribution f(x) =
1 x /5 e , x o. The probability 5
that the conversation will exceed five minutes is
(A)
1 e
(B) 1
1 e
(C)
1 e2
(D) 1
1 e2
eE1 / T5 / K3 / L3 / V1 / R11 / AB [GATE – EE – 2009]
(09) Assume for simplicity that N people, all born in
eE1 / T5 / K5 / L0 / V1 / R11 / AB [GATE – – 2005]
(02) Lot has 10% defective items. Ten items are April (a month of 30 days) are collected in a room, consider the event of at least two people in
chosen randomly from this lot. The probability that exactly 2 of the chosen items are defective is
the room being born on the same date of the (A) 0.0036
(B) 0.1937
(C) 0.2234
(D) 0.3874
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? -----00000----(A) 20
(B) 7
(C) 15
(D) 16
-----00000-----
Page 80
TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
TOPIC. 05 – PROBABILITY AND STATISTICS Question Level – 01
Question Level – 02
eE1 / T5 / K5 / L1 / V1 / R11 / AA [GATE – – 2000]
eE1 / T5 / K5 / L2 / V1 / R11 / AD [GATE – PI – 2007]
(01) In a manufacturing plant, the probability of
(01) If X is a continuous random variable whose
making a defective bolt is 0.1. The mean and standard deviation of defective bolts in a total of 900 bolts are respectively
probability density function is given by
k (5 x 2 x 2 ), f ( x) 0,
0 x2 otherwise
Then P(x >
1) is (A) 90 and 9
(B) 9 and 90
(C) 81 and 9
(D) 9 and 81
(A) 3/14
(B) 4/5
(C) 14/17
(D) 17/28
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
-----00000-----
that exactly 2 of the chosen items are defective is
(A) 0.0036
(B) 0.1937
(C) 0.2234
(D) 0.3874
Question Level – 03 eE1 / T5 / K5 / L3 / V2 / R11 / AB [GATE – – 1999]
(01) Four arbitrary points ( x1 , y1 ) , ( x2 , y2 ),( x3 , y3 ) , eE1 / T5 / K5 / L1 / V1 / R11 / AA [GATE – PI – 2005]
(03) The life of a bulb (in hours) is a random variable
( x4 , y4 ) , are given in the xy – plane using the
with an exponential distribution f(t) = α e αt ,
method of least squares, if, regressing y upon x
0 t . The probability that its value lies b/w
gives the fitted line y = ax + b; and regressing x
100 and 200 hours is upon y gives the fitted line x = cy + d, then (A) e100 α e200α
(B) e100 e200 (A) The two fitted lines must coincide
(C) e100 α e200α
(D) e200 α e100α (B) the two fitted lines need not coincide
eE1 / T5 / K5 / L1 / V1 / R11 / AC [GATE – CE – 2007]
(04) If the standard deviation of the spot speed of
(C) It is possible that ac = 0
vehicles in a highway is 8.8 kemps and the mean speed of the vehicles is 33 kmph, the coefficient
(D) a must be 1/c
of variation in speed is -----00000----(A) 0.1517
(B) 0.1867
(C) 0.2666
(D) 0.3646
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Page 81
ENGINEERING MATHEMATICS eE1 / T5 / K6 / L0 / V1 / R11 / AA [GATE – ME – 2009]
5.6 Random Variable
(04) The standard deviation of a uniformly distributed random variable b/w 0 and 1 is
Question Level – 00 (Basic Problem) eE1 / T5 / K6 / L0 / V1 / R11 / AB [GATE – – 2009]
(A)
1
(B)
12
1 3
(01) Using given data points tabulated below, a straight 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
(C)
5
(D)
12
7 12
-----00000-----
Question Level – 01
(A) 0.9
(B) 1
(C) 1.1
(D) 1.5
eE1 / T5 / K6 / L1 / V1 / R11 / AC [GATE – EC – 2008]
(01) X is uniformly distributed random variable that eE1 / T5 / K6 / L0 / V1 / R11 / AD [GATE – ME – 2007]
(02) Let X and Y be two independent random
takes values between 0 and 1. The value of E(X3) will be
variables. Which one of the relations b/w expectation (E), variance (Var) and covariance
(A) 0
(B) 1/8
(C) 1/4
(D) ½
(Cov) given below is FALSE? (A) E(XY) = E(X) E(Y) eE1 / T5 / K6 / L1 / V1 / R11 / AA [GATE – IN – 2008]
(02) A random variable is uniformly distributed over
(B) cov (X, Y) = 0
the interval 2 to 10. Its variance will be (C) Var(X + Y) = Var(X) + Var(Y) (A) 16/3
(B) 6
(C) 256/9
(D) 36
(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
eE1 / T5 / K6 / L1 / V1 / R11 / AB [GATE – IN – 2008]
(03) Consider a Gaussian distributed random variable with zero mean and standard deviation . The
of
least
squares.
Given
x 6, y 21,
2
x 14, xy 46, the values of a and b are
value of its cumulative distribution function at the origin will be
respectively
(A) 2, 3
(C) 2, 1
Page 82
(A) 0
(B) 0.5
(C) 1
(D) 10
(B) 1, 2
(D) 3, 2
TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
TOPIC. 05 – PROBABILITY AND STATISTICS eE1 / T5 / K6 / L1 / V1 / R11 / AA [GATE – IN – 2008] (-2|x|)
(04) Px(X) = Me
(-3|x|)
+ Ne
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) M
2 N 1 3
(C) M N 1
1 (B) 2M N 1 3
(D) M N 3
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
eE1 / T5 / K6 / L1 / V1 / R11 / AA [GATE – PI – 2008]
(05) For a random variable x( x ) following
-----00000-----
normal distribution, the mean is μ 100 If the probability is P = α for x 110. Then the probability of x lying b/w 90 and 110 i.e.
P(90 x 110) and equal to
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) 1 2α
(B) 1 α
(C) 1 α / 2
(D) 2α
eE1 / T5 / K6 / L1 / V1 / R11 / AD [GATE – IN – 2009]
(A) 2e2
(B) 1 2e2
(C) 3e 2
(D) 1 3e2
(06) If three coins are tossed simultaneously, the probability of getting at least one head is
eE1 / T5 / K6 / L2 / V2 / R11 / AD [GATE – CS – 2011]
(02) If the difference between the expectation of the (A) 1/8
square of a random variable | E(X 2 ) | and the
(B) 3/8
square of the expectation of the random variable (C) ½
(D) 7/8
E(X 2 ) is denoted by R, then, 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
(A) R = 0
(B) R < 0
(C) R 0
(D) R > 0
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Page 83
ENGINEERING MATHEMATICS eE1 / T5 / K6 / L2 / V2 / R11 / AA [GATE – PI – 2007]
eE1 / T5 / K6 / L3 / V2 / R11 / AB [GATE – EC – 2009]
(03) The random variable X taken on the values 1, 2
(03) A discrete random variable X takes value from 1
2 5P 1 3P , and 5 5
to 5 with probabilities as shown in the table. A
1.5 2 P respectively the values of P and E(X) 5
student calculates the mean of X as 3.5 and her
(or) 3 with probabilities
teacher calculates the variance to X as 1.5. Which
are respectively of the following statements is true? (A) 0.05, 1.87
(B) 1.90, 5.87
(C) 0.05, 1.10
(D) 0.25, 1.40
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
-----00000-----
(B) Both the student and the teacher are wrong
Question Level – 03 eE1 / T5 / K6 / L3 / V2 / R11 / AD [GATE – CE – 2009]
(C) The student is wrong but the teacher is right
(01) The standard normal probability function can be (D) The student is right but the teacher is wrong
approximated as F(XN) =
1
1 exp 1.7255 X N | X N |0.012
where
eE1 / T5 / K6 / L3 / V2 / R11 / AC [GATE – IN – 2009]
(04) A screening test is carried out to detect a certain XN = standard normal deviate. If mean and standard deviation of annual precipitation are 102
disease. It is found that 12% of the positive
cm and 27 cm respectively, the probability that
reports and 15% of the negative reports are
the annual precipitation will be b/w 90 cm and
incorrect. Assuming that the probability of a
102 cm is person getting positive report is 0.01, the (A) 66.7%
(B) 50.0%
(C) 33.3%
(D) 16.7%
probability that a person tested gets an incorrect report is
eE1 / T5 / K6 / L3 / V2 / R11 / AC [GATE – EC – 2009]
(A) 0.0027
(B) 0.0173
(C) 0.1497
(D) 0.2100
(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)?
eE1 / T5 / K6 / L3 / V2 / R11 / AD [GATE – CS – 2011]
(05) Consider a finite sequence of random values X =
{x1 ,x 2 ,x3 ,...........xn }. Let μx be the mean and (A) 0
(B) 1/16
x be the standard deviation of X. Let another (C) 1/6
Page 84
(D) 1
TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
TOPIC. 05 – PROBABILITY AND STATISTICS finite sequence Y of equal length be derived from this yi a.xi b , where a and b are positive constants. Let μy be the mean and y be the
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
standard deviation of this sequence. Which one
is true?
of the following statements is incorrect? (A) There is a sample point at which X has the (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
value = 5
(B) There is a sample point at which X has the value > 5
as the index position of median of Y in Y. (C) There is a sample point at which X has a (C) μ y aμ x b
value 5
(D) y a x b (D) None of the above -----00000-----
-----00000-----
Question Level – 03
5.7 EXPECTION
eE1 / T5 / K2 / L3 / V2 / R11 / AD [GATE – CS – 2004]
(01) An exam paper has 150 multiple choice questions
Question Level – 01
of 1 mark each, with each question having four choices. Each incorrect answer fetches – 0.25
eE1 / T5 / K2 / L1 / V1 / R11 / AA [GATE – EC – 2007]
marks. Suppose 1000 students choose all their
(01) If E denotes expectation, the variance of a
answers randomly with uniform probability. The
random variable X is given by
sum total of the expected marks obtained by all the students is
2
2
(A) E ( X ) E ( X )
(C) E ( X 2 )
2
2
(B) E ( X ) E ( X ) (A) 0
(B) 2550
(C) 7525
(D) 9375
(D) E 2 ( X )
-----00000---------00000-----
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Page 85
ENGINEERING MATHEMATICS
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 have taken 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-----
Page 86
TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
06 Numerical Methods Complete subtopic in this chapter, is in the scope of “GATE-CS/ ME/EC/EE SYLLABUS”
eE1 / T6 / K / L2 / V2 / R11 / AC [GATE – –]
6.1 Clubbed problem
(02) The polynomial p ( x) x 5 x 2 has
Question Level – 01 (A) all real roots eE1 / T6 / K / L1 / V1 / R11 / A [GATE – –]
(01) In the interval [0, π ] the equation x cos x has
(B) 3 real and 2 complex roots
(A) No solution
(C) 1 real and 4 complex roots
(B) Exactly one solution
(D) all complex roots
(C) Exactly 2 solutions (D) An infinite number of solutions
eE1 / T6 / K / L2 / V2 / R11 / AC [GATE – –]
(03) It is known that two roots of the non-linear 3
2
equation x 6x 11x 6 0 are 1 and 3. The -----00000-----
Question Level – 02 eE1 / T6 / K / L2 / V2 / R11 / AB [GATE – – ]
third root will be
(A) j
(B) j
(C) 2
(D) 4
(01) For solving algebraic and transcendental equation which one of the following is used?
(A) Coulomb’s theorem
-----00000-----
(B) Newton-Raphson method (C) Euler’s method (D) Stoke’s theorem
Page 87
TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
ENGINEERING MATHEMATICS eE1 / T6 / K / L3 / V2 / R11 / AC [GATE – – ]
Question Level – 03
(03) Matching exercise choose the correct one out of eE1 / T6 / K / L3 / V2 / R11 / AC [GATE – – ]
the alternatives A, B, C, D
(01) Match the following and choose the correct Group – I
combination P. E.
Newton –
(1)
Solving non-linear
Raphso
equations
n
nd
2 order
Group – II (1)
Runge –
differe
Kutta
ntial
method
equatio
method
ns Q. F.
Runge-Kutta
(2)
Solving linear
method
simultaneous equations
Non-linear
(2)
Newton –
algebra
Raphso
ic
n
equatio
method
ns G.
Simpson’s Rule
(3)
Solving ordinary differential
R.
equations H.
(A)
Gauss
(4)
Numerical
elimina
intergration
tion
method
E – 6, F – 1, G
Interpolation
(B)
E – 1, F – 6, G – 4, H
– 5, H –
(3)
Gauss
algebra
Elimin
ic
ation
equatio ns S.
(5)
Linear
Numerical
(4)
integrat
Simpson’s Rule
ion
–3
3 (C)
E – 1, F – 3, G
(D)
E – 5, F – 3, G – 4, H
– 4, H –
eE1 / T6 / K / L3 / V2 / R11 / AA [GATE – – ]
that
(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
–1
2
(02) Given
(A) P-3, Q-2, R-4, S-1
one
root
of
the
equation
eE1 / T6 / K2 / L1 / V1 / R11 / AB [GATE – – ]
(04) Back ward Euler method for solving the
x3 10x2 31x 30 0 is 5 then other roots
differential equation
arc
by
dy f ( x, y ) is specified dx
(A) yn 1 yn h f ( xn , yn ) (A) 2 and 3
(B) 2 and 4 (B) yn 1 yn h f ( xn 1 , yn1 )
(C) 3 and 4
(D) 2 and 3 (C) yn1 yn1 2h f ( xn , yn ) (D) yn1 (1 h) f ( xn 1 , yn1 )
Page 88
TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
TOPIC. 06 – NUMERICAL METHODS eE1 / T6 / K3 / L2 / V2 / R11 / AA [GATE – – ]
Question Level – 01
(05) The following equation needs to be numerically eE1 / T6 / K4 / L1 / V1 / R11 / AB [GATE – – ]
solved using the Newton – Raphson method
x3 4x 9 0. The iterative equation for this purpose is ( k indicates the iteration level)
(01) The iteration formula to find the square root of a positive real number by using the NewtonRaphson method is
2 xk3 9 (A) xk 1 2 3xk 4
3xk3 9 (B) xk 1 2 2 xk 9
(A) xk 1
3( xk b) 2 xk
x22 b (B) xk 1 2 xk
(C) xk 1 xk 3k2 4
(D) xk 1
4 xk2 3 9 xk2 2
(C) xk 1
xk 2 xk 1 xk2 b
(D) None
-----00000-----
eE1 / T6 / K4 / L1 / V1 / R11 / AC [GATE – – ]
(02) Given a > 0, we wish to calculate it reciprocal value
6.2 Newton-Rap son
1 by using Newton – Raphson method for a
f ( x) 0. The Newton-Raphson algorithm for Question Level – 00 (Basic Problem)
the function will be
eE1 / T6 / K4 / L0 / V1 / R11 / AD [GATE – – ]
(01) The Newton-Raphson method is to be used to
(A) xk 1
1 a xk 2 xk
(B) xk 1 xk
a 2 xk 2
(D) xk 1 xk
a 2 xk 2
find the root of the equation and f '( x) is the derivative of f . the method converges
(C) xk 1 2 xk axk2
(A) Always eE1 / T6 / K4 / L1 / V1 / R11 / AA [GATE – – ]
(B) Only is f is a polynomial
(03) Identify the Newton – Raphson iteration scheme for the finding the square root of 2
(C) Only if f ( x0 ) 0
(A) xn 1
1 2 xn 2 xn
(B) xn 1
1 2 xn 2 xn
(C) xn 1
2 xn
(D) None of the above
-----00000-----
(D) xn 1 2 xn
www.targate.org
Page 89
ENGINEERING MATHEMATICS eE1 / T6 / K4 / L1 / V1 / R11 / A [GATE – – ]
(04) The
Newton
–
Raphson
iteration
1 R xn can be used to compute the 2 xn
xn 1
eE1 / T6 / K4 / L1 / V1 / R11 / AC [GATE – – ] x
(07) The recursion relation to solve x e Newton – Raphson method is (A) xn 1 e
(A) square or R
using
xn
(B) reciprocal of R
(C) square root of R
(D) logarithm of R
eE1 / T6 / K4 / L1 / V1 / R11 / A [GATE – – ]
(B) xn 1 xn e
xn
(1 xn )e xn (C) xn 1 (1 e xn )
2
(05) Let x 117 0. The iterative steps for the solution using Newton – Raphson’s method given by
(A) xk 1
1 117 xk 2 xk
117 (B) xk 1 xk xk (C) xk 1 xk
xk 117
xn2 e xn (1 xn ) 1 (D) xn 1 xn e xn
eE1 / T6 / K4 / L1 / V1 / R11 / A [GATE – – ]
(08) The integral
3
1
rd
simpson’s 1/ 3
1 dx when evaluated by using x rule on two equal sub intervals
each of length 1, equal to
(A) 1.000
(B) 1.008
(C) 1.1111
(D) 1.120
1 117 (D) xk 1 xk xk 2 xk -----00000----eE1 / T6 / K4 / L1 / V1 / R11 / AA [GATE – – ]
(06) Newton-Raphson formula to find the roots of an equation f ( x) 0 is given by
(A) xn 1 xn
f ( xn ) f 1 ( xn )
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
(B) xn 1 xn
(C) xn1
f ( xn ) f 1 ( xn )
point x0 is
(A)
f ( x0 h) f ( x0 h) 2
(B)
f ( x0 h) f ( x0 h) 2h
f ( xn ) xn f 1 ( xn )
(D) none of the above
Page 90
TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
TOPIC. 06 – NUMERICAL METHODS (C)
(D)
eE1 / T6 / K4 / L2 / V2 / R11 / AB [GATE – – ]
f ( x0 h) 2 f ( x0 ) f ( x0 h) h2
3
2
(05) The equation x x 4 x 4 0 is to be solved using the Newton – Raphson method using x 2
f ( x0 h) 2 f ( x0 ) f ( x0 h) h2
taken as the initial approximation of the solution eE1 / T6 / K4 / L2 / V2 / R11 / AC [GATE – – ]
then the next approximation using this method,
(02) The Newton-Raphson iteration formula for finding
3
will be
c , where c > 0 is ,
(A) xn 1
(C) xx 1
2 xn3 3 c 3 xn2
2 xn3 c 3 xn2
(B) xn 1
(D) xn 1
2 xn3 3 c 3xn2
(A) 2/3
(B) 4/3
(C) 1
(D) 3/2
2 xn3 c 3xn2
eE1 / T6 / K4 / L2 / V2 / R11 / AA [GATE – – ] x
(06) Equation e 1 0 is required to be solved eE1 / T6 / K4 / L2 / V1 / R11 / AC [GATE – – ]
using Newton’s method with an initial guess
(03) Starting from x0 1 , one step of Newton –
x0 1. Then after one step of Newton’s Raphson
method in solving the
equation method estimate x1 of the solution will be given
3
x 3x 7 0 gives the next value x1 as by (A) x1 0.5
(B) x1 1.406
(C) x1 1.5
(D) x1 2
(A) 0.71828
(B) 0.36784
(C) 0.20587
(D) 0.0000
eE1 / T6 / K4 / L2 / V2 / R11 / A [GATE – – ]
eE1 / T6 / K4 / L2 / V2 / R11 / AB [GATE – – ]
(04) The real root of the equation xe 2 is
(07) Newton – Raphson method is used to compute a
evaluated using Newton – Raphson’s method. If
root of the equation x 13 0 with 3.5 as the
x
2
the first approximation of the value of x is initial value. The approximation after one 0.8679, the 2 nd approximation of the value of x correct to three decimal places is
iteration is
(A) 0.865
(B) 0.853
(A) 3.575
(B) 3.677
(C) 0.849
(D) 0.838
(C) 3.667
(D) 3.607
www.targate.org
Page 91
ENGINEERING MATHEMATICS eE1 / T6 / K4 / L2 / V2 / R11 / A [GATE – – ]
eE1 / T6 / K4 / L3 / V2 / R11 / A [GATE – – ]
(08) The square root of a number N is to be obtained
(02) Solution, the variable x1 and x2 for the following
by applying the Newton – Raphson iteration to equations is to be obtained by employing the
2
the equation x N 0. If i denotes the iteration index, the correct iterative scheme will be
Newton – Raphson iteration method Equation (i) 10 x2 sin x1 0.8 0
(A) xi 1
10 x22 10 x2 cos x1 0.6 0
1 N xi 2 xi
Assuming
the
initial
values
x1 0.0 and
x2 1.0 the Jacobian matrix is (B) xi 1
1 2 N xi 2 2 xi
10 0.8 0 0.6
(B)
0 0.8 10 0.6
(D)
(A) (C) xi 1
(D) xn1
10 0 0 10
1 N2 x i 2 xi (C)
xn f ( xn ) f 1 ( xn )
10 0 10 10
eE1 / T6 / K4 / L3 / V2 / R11 / AB [GATE – – ]
-----00000-----
(03) Give a > 0, we wish to calculate its reciprocal value
Question Level – 03 eE1 / T6 / K4 / L3 / V2 / R11 / A [GATE – – ]
(01) A
numerical
solution
of
the
equation
1 by using Newton – Raphson method for a
f ( x) = 0. For a 7 and starting with x0 0.2 the first two iteration will be
f ( x) x x 3 0 can be obtained using (A) 0.11, 0.1299
(B) 0.12, 0.1392
(C) 0.12, 0.1416
(D) 0.13, 0.1428
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 -----00000----(A) 0.306
(B) 0.739
(C) 1.694
(D) 2.306
Page 92
TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
TOPIC. 06 – NUMERICAL METHODS
6.3 Differential
6.4 Integration
Question Level – 00 (Basic Problem)
Question Level – 00 (Basic Problem)
eE1 / T6 / K5 / L0 / V1 / R11 / A [GATE – – ]
eE1 / T6 / K6 / L0 / V1 / R11 / AC [GATE – – ]
(01) During the numerical solution of a first order
(01) The trapezoidal rule for integration give exact
differential equation using the Euler (also known
result when the integrand is a polynomial of degree
as Euler Cauchy) method with step size h, the local truncation error is of the order of
2
(B) h
3
4
(D) h
(A) h
(A) but not 1
(B) 1 but not 0
(C) 0 (or) 1
(D)2
-----00000-----
5
(C) h
Question Level – 01 -----00000----eE1 / T6 / K6 / L1 / V1 / R11 / AC [GATE – – ]
(01) The Newton – Raphson method is used to find
Question Level – 02
2
the root of the equation x 2. if the iterations are started from 1, then the iteration will
eE1 / T6 / K5 / L2 / V1 / R11 / A [GATE – – ]
(01) Consider
a
dy ( x ) y ( x) x dx
differential with
initial
equation
(A) Converge to – 1
(B) Converge to
(C) Converge to 2
(D) not converge
condition
y(0) 0. 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
2
-----00000-----
Question Level – 03 eE1 / T6 / K6 / L3 / V2 / R11 / A [GATE – – ]
(01) The following algorithm computes the integral J = -----00000-----
b
a
at
f ( x )dx from the given values f j f ( x j ) equidistant
points
x0 a, x1 x0 h,
x2 x0 2h, .........x2 m x0 2mh b Compute S0 f 0 f 2m
www.targate.org
Page 93
ENGINEERING MATHEMATICS S1 f1 f 3 ...... f 2 m 1 S 2 f 2 f 4 ......... f 2 m 2 J=
h S0 4(S1 ) 2(S2 ) 3
The rule of numerical integration, which uses the above algorithm is
(A) Rectangle rule
(B) Trapezoidal rule
(C) Four – point rule
(D) Simpson’s rule
-----00000-----
Page 94
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)
(A)
a s a2
(B)
a s a2
(C)
s s a2
(D)
s s a2
2
2
eE1 / T7 / K2 / L0 / V1 / R11 / AB [GATE – EE – 1995]
(01) The Laplace transform of f(t) is F(s). Given F(s) =
, the final value of f(t) is ________. s 2
2
2
2
eE1 / T7 / K2 / L0 / V1 / R11 / AB [GATE – – 2004]
(A) Initially
(B) Zero
(04) A delayed unit step function is defined as
u(t a) = (C) One
(D) None
Its Laplace transform is __________ .
eE1 / T7 / K2 / L0 / V1 / R11 / AC [GATE – – ]
(02) Let Y(s) be the laplace transform of function y(t),
as
(A) a e
(B) e
as
/s
then the final value of the function is as
as
(C) e / s (B) LimY ( s)
(A) LimY ( s )
s
s0
(D) e / a
eE1 / T7 / K2 / L0 / V1 / R11 / AA [GATE – EC – 2006]
(05) Consider the function f(t) having Laplace (C) LimsY (s) s 0
(D) LimsY (s) s
eE1 / T7 / K2 / L0 / V1 / R11 / AB [GATE – – ]
transform F(s) =
0 , Re(s) > 0. The final s 20 2
value of f(t) would be _________
(03) If L denotes the Laplace transform of a function. L{sin at} will be equal to
Page 95
(A) 0
(B) 1
(C) 1 f () 1
(D)
TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
ENGINEERING MATHEMATICS eE1 / T7 / K2 / L0 / V1 / R11 / AA [GATE – – 2007]
eE1 / T7 / K2 / L0 / V1 / R11 / AC [GATE – – 2010]
(06) If F(s) is the Laplace transform of the function
(10) u(t) represents the unit function. The Laplace
f(t) than Laplace transform of
t
0
transform of u(t τ ) is
f (t ) dx is
(A) (A)
1 F ( s) s
(B)
(C) sF (s) f (0)
(D)
1 F (s ) f (0) s
1 sτ
(B)
e sτ (C) s
F ( s ) ds
1 sτ
sτ
(D) e
-----00000----eE1 / T7 / K2 / L0 / V1 / R11 / AD [GATE – – 2008]
(07) Laplace transform of 8t 3 is
(A)
8 s4
Question Level – 01
(B)
24 (C) 4 s
16 s4
eE1 / T7 / K2 / L1 / V1 / R11 / A [GATE – IN – 1995] at
(01) Find L { e cos t } when L{ cos t } =
s s 2
48 (D) 4 s
2
eE1 / T7 / K2 / L1 / V1 / R11 / AD [GATE – – ] eE1 / T7 / K2 / L0 / V1 / R11 / AA [GATE – – 2008]
(02)
(08) Laplace transform of sin ht is
1 (A) 2 s 1
(s 1)2 is the Laplace transform of
(A) t
1 (B) 1 s2
2
(B) t
2t
(D) te
(C) e (C)
s s 1
(D)
2
s 1 s2
eE1 / T7 / K2 / L0 / V1 / R11 / AB [GATE – EC – 1998]
(09) If L
f (t )
=
w s w2 2
then the value of
3
t
\ eE1 / T7 / K2 / L1 / V1 / R11 / AB [GATE – – ]
(03) If L{f(t)} = h(t)=
t
0
s2 1 s2 , L { g ( t )} , s2 1 ( s 3)( s 2)
f (T ) g (t T )dT
Then L{h(t)} is ___________
Lim f (t ) ________. t
(A)
s2 1 s 3
(C)
s2 1 s2 2 (D) None (s 3)( s 2) s 1
(A) can not be determined (B) Zero
(C) unity
Page 96
(D) Infinite
(B)
TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
1 s3
TOPIC. 07 – TRANSFORM THEORY eE1 / T7 / K2 / L1 / V1 / R11 / AD [GATE – EC – 2005]
(04) The Dirac delta Function (t ) is defined as
1, 0,
t 0 other wise
, 0,
t 0 other wise
(A) (t )
(B) (t )
eE1 / T7 / K2 / L1 / V1 / R11 / AC [GATE – – 2009]
(08) The inverse Laplace transform of
t
(B) 1 e
t
t
(D) 1 e
(A) 1 e
(C) 1 e
1, t 0 and 0, other wise
, t 0 and 0, other wise
(C) (t )
1 is (s s) 2
t
(t )dt 1 -----00000-----
(D) (t )
(t )dt 1
eE1 / T7 / K2 / L1 / V1 / R11 / AA [GATE – EC – 1997]
(05) The inverse Laplace transform of the function
Question Level – 02 eE1 / T7 / K2 / L2 / V2 / R11 / A [GATE – – 1994]
(01) If f(t) is a finite and continuous Function for
t 0 the Laplace transformation is given by
s 5 is _______ ( s 1)( s 3)
F =
0
e st f (t ), then for f (t ) cos h mt , the
Laplace Transformation is _____________ (A) 2e e
t
3t
(B) 2e e
t
3t
t
3t
t
3t
eE1 / T7 / K2 / L2 / V2 / R11 / AD [GATE – – 1999]
(C) e 2e
(D) e 2e
(02) The Laplace transform of the function
f (t ) k, 0 t c.
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) (k / s )e sc
(B) (k / s )e sc
(D) (k / s )(1 e sc )
(A) e sT F ( s )
(B) e sT F ( s )
(C) k e sc
F ( s) (C) 1 e sT
F (s) (D) 1 e sT
eE1 / T7 / K2 / L2 / V2 / R11 / AC [GATE – – 1999]
(03) Laplace transform of (a bt ) 2 where ‘a’ and ‘b’ are constants is given by:
eE1 / T7 / K2 / L1 / V1 / R11 / AA [GATE – EC – 1997] αt
(07) The Laplace transform of e cos αt is equal to
(A) (a bs )2
________ (B) 1/ (a bs ) 2 (A)
(C)
sα ( s α )2 α 2
(B)
sα ( s α) 2 α 2
1 ( s α) 2
(D) None
(C) (a 2 / s ) (2ab / s 2 ) (2b2 / s 3 )
(D) (a 2 / s ) (2ab / s 2 ) (b 2 / s 3 )
www.targate.org
Page 97
ENGINEERING MATHEMATICS eE1 / T7 / K2 / L2 / V2 / R11 / AD [GATE – – 2001]
eE1 / T7 / K2 / L2 / V2 / R11 / AA [GATE – – 2005]
(04) The inverse Laplace transforms of 1/ (s 2 2s ) is
(08) The Laplace transform of a function f(t) is F(s) =
(A) (1 e 2t )
(B) (1 e2t ) / 2
(C) (1 e 2t ) / 2
(D) (1 e 2t ) / 2
5s 2 23s 6 . As t , f(t) approaches s ( s 2 2 s 2) (A) 3
(B) 5
(C) 17/2
(D)
eE1 / T7 / K2 / L2 / V2 / R11 / AC [GATE – – 2002]
(05) The Laplace transform of the following function
eE1 / T7 / K2 / L2 / V2 / R11 / AD [GATE – – 2010]
is
sin t f (t ) 0
(09) Given
for 0 t π for t π
f(t)
=
3s 1 L1 3 . 2 s 4s (k 3)s
If
Lt f (t ) = 1 then value of k is
t
(A) 1 (1 s 2 ) for all x > 0
(B) 1/ (1 s 2 ) for all s < π
(C) (1 e πs ) / (1 s 2 ) for all s > 0
(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 (D) e πs / (1 s 2 ) for all s > 0 (A)
a s a2
(B)
s s a2
(C)
a s a2
(D)
s s a2
eE1 / T7 / K2 / L2 / V2 / R11 / A [GATE – EE – 2002]
(06) Using 2
Laplace
transforms,
2
2
solve
2
(d y / dt ) 4 y 12t
2
2
eE1 / T7 / K2 / L2 / V2 / R11 / AB [GATE – – 2009] eE1 / T7 / K2 / L2 / V2 / R11 / AC [GATE – EC – 2003]
(11) Given that F(s) is the one-sided Laplace
(07) The Laplace transform of i(t) is given by I(s) = transform of f(t), the Laplace transform of
2 As t , the value of i(t) tends to s(1 s) ____ .
(A) 0
(C) 2
Page 98
t
0
f ( τ ) dτ is
(A) sF (s) f (0)
(B) 1
(D)
(C)
s
0
f ( τ ) dτ
(B)
1 F ( s) s
(D)
1 [ F ( s ) f (0)] s
TARGATE EDUCATION: GATE, IES, PSU (EE/EC/ME/CS)
TOPIC. 07 – TRANSFORM THEORY eE1 / T7 / K2 / L2 / V2 / R11 / AA [GATE – EC – 2005]
(12) In what range should Re(s) remain so that the ( a 2)t 5
laplace transform of the function e
(A) Re(s) > a + 2
exists?
(B) Re (s) > a + 7
sin t , if (2n 1)π t 2nπ (n 1, 2, 3,..) f (t ) other wise 0 eE1 / T7 / K2 / L3 / V2 / R11 / AD [GATE – EE – 1995]
(02) The (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( s 1) then the initial s 4s 7
inverse
Laplace
transform
of
( s 9) / ( s 2 6s 13) is
(A) cos 2t 9 sin 2 t
2
3t
cos2t 3e3t sin 2t
(C) e
3t
sin 2t 3e3t cos 2t
(D) e
3t
cos2t 3e3t sin 2t
(B) e
and final values of f(t) are respectively
(A) 0,2
(C) 0,
(B) 2, 0
2 7
(D)
2 ,0 7
eE1 / T7 / K2 / L3 / V2 / R11 / AB [GATE – EC – 1995]
eE1 / T7 / K2 / L2 / V2 / R11 / AA [GATE – – 1998]
(14) The Laplace Transform of a unit step function
(03) If L{f(t)} =
2( s 1) then f(0 ) and f( ) are s 2s s 2
given by _______
ua (t ), defined as u a (t ) 0 for t < a is = 1 for t > a,
(A) e
as
/s
(B) se
(A) 0, 2 respectively
(B) 2, 0 respectively
(C) 0, 1 respectively
(D)
2 , 0 respectively 5
as
eE1 / T7 / K2 / L3 / V2 / R11 / A [GATE – – 1996]
(04) Using Laplace Transform, solve the initial value (C) s u (0)
(D) se
as
1
problem
9 y11 6 y1 y 0
y(0) 3 and
y1 (0) 1, where prime denotes derivative with
-----00000-----
respect to t.
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. – 1993)
(Gate
eE1 / T7 / K2 / L3 / V2 / R11 / A [GATE – ME – 1997]
(05) Solve
the
initial
value
problem
d2y dy dy 4 3 y 0 with y = 3 and 7 at 2 dx dx dt x0
www.targate.org
Page 99
ENGINEERING MATHEMATICS eE1 / T7 / K2 / L3 / V2 / R11 / AC [GATE – EC – 1998]
(06) The laplace transform of (t 2 2t )u (t 1) is ________ .
(A)
eE1 / T7 / K2 / L3 / V2 / R11 / AA [GATE – – 2010]
(09) The Laplace transform of f(t) is
2 s 2 s e 2e s3 s
(B)
2 2 s 2 s e 2e s3 s
function
(A) t 1 e (C)
2 s 2 s e e s3 s
1 . The s (s 1) 2
t
(B) t 1 e
t
(D) None t
(C) 1 e
t
(D) 2t e
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?
eE1 / T7 / K2 / L3 / V2 / R11 / AA [GATE – – 2011]
(10) Given two continuous time signals x(t) = e y(t) = e
2t
t
which exists for t > 0 then the
convolution z(t) = f(t) * y(t) is ___________ . (A) £[df / dt ] 1/ s F (s); £
f (τ (dτ t
0
t
(A) e e
2t
(B) e
2t
= sF(s) f(0) (B) £[df / dt ] = sF(s) – F(0). £
f (τ )dτ t
0
(C) e
t
t
(D) e e
3t
(C) £[df / dt ] = s F(s) – F(0);
£
f (τ )dτ F (s a) t
------THE END ------
0
(D) £[df / dt ] = s F(s) – F(0);
£
f (τ )dτ 1/ s F (s) t
0
eE1 / T7 / K2 / L3 / V2 / R11 / AA [GATE – – 2005]
(08) Laplace transform of f(t) = cos( pt q) is
(A)
s cos q p sin q s 2 p2
(B)
s cos q p sin q s2 p2
(C)
s sin q p cos q s 2 p2
(D)
s sin q p cos q s2 p2
Page 100
and
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