A TEXTBOOK OF ENGINEERING MATHEMATICS

A TEXTBOOK OF

ENGINEERING MATHEMATICS For

B.E. II Year (III/IV Semester) M.D.U., G.J.U., K.U. and DCRUST, Haryana (Strictly According to Latest Syllabus)

By

N.P. BALI Formerly, Principal S.B. College, Gurgaon Haryana

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CONTENTS Preface

...

Syllabus Basic Concepts

... (viii)–(x) ... (xi)–(xix)

Chapters

(vii)

Pages SECTION-A

1.

Fourier Series

...

3

2.

Fourier Transforms

...

53

...

115

SECTION-B 3.

Functions of a Complex Variable SECTION-C

4.

Power Series and Contour Integration

...

223

5.

Probability Distributions

...

260

SECTION-D 6.

Hypothesis Testing

...

321

7.

Linear Programming

...

366

Appendix—I: Short Answer Type Questions

...

457

Appendix—II: Normal Table

...

471

Appendix—III: Some Important Curves

...

474

Examination Papers

...

477

PREFACE TO THE THIRTEENTH EDITION The rapid sale of the twelfth edition has encouraged me to bring in the thirteenth edition well in time. I express my sincerest thanks to the readers. I am grateful to all the professors who have patronised the book and pointed out certain missing topics for K.U.K. The needful has been done. The overwhelming response has prompted me to thoroughly revise the book again. In the present edition, many new types of questions have been added in examples as well as in exercises. A separate exercise has been provided for Fourier Integrals. The following topics have been added: (i) De Moivre’s Theorem (ii) Probability Distribution of a Continuous Variable (iii) Dual Simplex Method (iv) Conformal Transformation (For K.U.K. only) There are some changes in the syllabus also. Question 1 has been made compulsory. It is based on Short Answer Questions from all the four sections. Appendix-I contains 212 questions of this type. In the present form the book will be more useful. I thank my student Sandeep Yadav for checking all answers and pointing out some printing errors. All suggestions for the improvement of the book will be highly appreciated and gratefully acknowledged. —Author

PREFACE TO THE FIRST EDITION This book is a part of the original book (with 28 chapters and covering the syllabi of engineering courses of all semesters of all the Indian Universities) running its seventh edition and very well received by the teachers and students of all Indian Universities. The rapid sale of the seventh edition bears testimony to the overwhelming response. I thank them all for the appreciation. The present form of the book contains only five chapters and covers the entire syllabus for the students of third semester of M.D.U. and G.J.U. There is no dearth of books on Engineering Mathematics but the students find it difficult to solve most of the problems in the exercise in the absence of an adequate number of solved examples. An outstanding and distinguishing feature of the book is the large number of typical solved examples followed by well-graded problems. Many examples and problems have been selected from recent papers (2000 onwards) of various engineering examinations. I have endeavoured to present the fundamental concepts in a comprehensive and lucid manner. I am indebted to all authors, Indian and Foreign, whose works I have frequently consulted. All efforts have been made to keep the book free from errors. Answers to all exercises have been re-checked. All suggestions for improvement will be highly appreciated and gratefully acknowledged. (vii) —Author

SYLLABUS (M.D.U., Rohtak) MAT-201-F MATHEMATICS-III L T P 3 1 -

Class Work : Theory : Total : Duration of Exam. :

50 Marks 100 Marks 150 Marks 3 Hours

Note: Examiner will set 9 questions in total, with two questions from each section and one question covering all sections which will be Q.1. This Q.1 is compulsory and of short answers type. Each question carries equal mark (20 marks). Students have to attempt 5 questions in total at least one question from each section.

SECTION-A Fourier Series and Fourier Transforms: Euler’s formulae, conditions for a Fourier expansion, change of interval, Fourier expansion of odd and even functions, Fourier expansion of square wave, rectangular wave, saw-toothed wave, half and full rectified wave, half range sine and cosine series. Fourier integrals, Fourier transforms, Shifting theorem (both on time and frequency axes), Fourier transforms of derivatives, Fourier transforms of integrals, Convolution theorem, Fourier transform of Dirac-delta function. SECTION-B Functions of Complex Variable: Definition, Exponential function, Trigonometric and Hyperbolic functions, Logarithmic functions. Limit and Continuity of a function, Differentiability and Analyticity. Cauchy-Riemann equations, necessary and sufficient conditions for a function to be analytic, polar form of the Cauchy-Riemann equations. Harmonic functions, application to flow problems. Integration of complex functions. Cauchy-Integral theorem and formula. SECTION-C Power series, radius and circle of convergence, Taylor’s, Maclaurin’s and Laurent’s series. Zeroes and singularities of complex functions, Residues. Evaluation of real integrals using residues (around unit and semi circle only). Probability Distributions and Hypothesis Testing: Conditional probability, Baye’s theorem and its applications, expected value of a random variable. Properties and application of Binomial, Poisson and Normal distributions. SECTION-D Testing of a hypothesis, tests of significance for large samples, Student’s t-distribution (applications only), Chi-square test of goodness of fit. Linear Programming: Linear programming problems formulation, solving linear programming problems using (i) Graphical method (ii) Simplex method (iii) Dual simplex method. (viii)

SYLLABUS (K.U., Kurukshetra)

Theory : Sessional : Total : Duration of Exam. :

100 Marks 50 Marks 150 Marks 3 Hours

UNIT-I Fourier Series: Euler’s formulae. Conditions for Fourier expansions, Fourier expansion of functions having points of discontinuity, change of interval. Odd and even functions, Halfrange series. Fourier Transforms: Fourier integrals, Fourier transforms, Fourier cosine and sine transforms. Properties of Fourier transforms, Convolution theorem, Perseval’s identity, Relation between Fourier and Laplace transforms, Fourier transforms of the derivatives of a function, Application to boundary value problems. UNIT-II Functions of a Complex Variable: Functions of a complex variable, Exponential function, Trigonometric, Hyperbolic and Logarithmic functions, limit and continuity of a function, Differentiability and analyticity. Cauchy-Riemann Equations: Necessary and sufficient conditions for a function to be analytic, Polar Form of the Cauchy-Riemann equations, Harmonic functions, Application to flow problems, Conformal transformation, Standard transformations (Translation. Magnification and rotation, inversion and reflection, Bilinear) UNIT-III Probability Distributions: Probability, Baye’s theorem, Discrete and Continuous probability distributions, Moment generating function. Probability generating function, Properties and applications of Binomial, Poisson and normal distributions. UNIT-IV Linear Programming: Linear programming problems formulation, Solution of Linear Programming Problem using Graphical method. Simplex Method, Dual-Simplex Method.

(ix)

SYLLABUS (ECE DCRUST, Murthal) MATH - 201 MATHEMATICS-III B.Tech. Semester-III (Common for all Branches) L T P 3 2 -

Credits 5

Class Work : Exam. : Total : Duration of Exam. :

50 Marks 100 Marks 150 Marks 3 Hours

PART-A Fourier Series and Fourier Transforms: Euler’s formulae, conditions for a Fourier expansion, change of interval, Fourier expansion of odd and even functions, Fourier expansion of square wave, rectangular wave, saw-toothed wave, half and full rectified wave, half range sine and cosine series. Fourier integrals, Fourier transforms, Shifting theorem (both on time and frequency axes), Fourier transforms of derivatives, Fourier transforms of integrals, Convolution theorem, Fourier transform of Dirac-delta function. PART-B Functions of Complex Variable: Definition, Exponential function, Trigonometric and Hyperbolic functions, Logarithmic functions. Limit and Continuity of a function, Differentiability and Analyticity. Cauchy-Riemann equations, necessary and sufficient conditions for a function to be analytic, polar form of the Cauchy-Riemann equations. Harmonic functions, application to flow problems. Integration of complex functions. Cauchy-Integral theorem and formula. Power series, radius and circle of convergence, Taylor’s Maclaurin’s and Laurent’s series. Zeroes and singularities of complex functions, Residues. Evaluation of real integrals using residues (around unit and semi circle only). PART-C Probability Distributions and Hypothesis Testing: Conditional probability, Baye’s theorem and its applications, expected value of a random variable. Properties and application of Binomial, Poisson and Normal distributions. Testing of a hypothesis, tests of significance for large samples, Student’s t-distribution (applications only), Chi-square test of goodness of fit. Linear Programming: Linear programming problems formulation, Solving linear programming problems using (i) Graphical method (ii) Simplex method (iii) Dual simplex method.

(x)

Basic Concepts 1.

Greek Letters Used      i 

2.

      

kappa mu nu pi rho sigma cap. sigma

 iff  

belongs to does not belong to implies

1 = 0.3183  e = 2.7183 1 = 0.3679 e

3 = 1.732  = 3.1416

Quadratic Equation Roots of quadratic equation are

tau chi omega cap. delta

implies and implied by if and only if union intersection

b D , 2a

where

1 rad. = 57° 17 45 1° = 0.0174 rad. loge 10 = 2.3026

ax2 + bx + c = 0, a  0

D = b2 – 4ac is called discriminant

b c , Product of roots = a a If D > 0, roots are real and distinct. If D = 0, roots are equal. If D < 0, roots are imaginary. Progressions (i) For the A.P. (Arithmetic Progression) a, a + d, a + 2d, ... n n Tn = a + (n – 1)d, Sn = [2a + (n – 1)d ] = (a + l). 2 2 2 (ii) For the G.P. (Geometric Progression) a, ar, ar , ...

Sum of roots = –

5.

   

Useful Data

2 = 1.4142

4.

phi psi xi eta zeta lambda cap. gamma

Some Notations   

3.

      

alpha beta gamma delta epsilon iota theta

Tn =

arn –1,

Sn

%K a(1  r = & 1 r K' na,

n

( xi )

)

, when r  1 when r  1

log10e = 0.4343 loge2 = 0.6931 loge 3 = 1.0986

( xii )

a provided | r | < 1 i.e., – 1 < r < 1 1 r (iii) A sequence is said to be in H.P. (Harmonic Progression) if the reciprocals of its terms are in A.P. S =

1 1 1 1 , ..., Tn = , , a  ( n  1)d a a  d a  2d (iv) For two numbers a and b, For the H.P.

2ab ab , G.M. = ab , H.M. = 2 ab (v) For natural numbers 1, 2, 3, ..., n A.M. =

6.

8.

r

2

=

(iii) log (mn) = log m + log n, 9.

"# $

n n! Pr n!  , n Cr = (n  r) ! r! r ! (n  r ) ! n P = n ! , nC = n C nC = nC = 1 n r n–r , n 0 Binomial Theorem (i) When n is a positive integer (a + b)n = nC0 an + nC1 an–1 b + nC2 an – 2 b2 + ... + nCr an – r br + ... + nCn bn (1 + x) n = 1 + nC1 x + nC2 x2 + ... + nCn xn (ii) When n is a negative integer or a fraction n(n  1) 2 n(n  1) (n  2) 3 ... (1 + x)n = 1 + nx + x + x +  3! 2! provided | x | < 1. Logarithms (i) Natural logarithm of a positive real number x is denoted by loge x or simply log x or ln x. It is the inverse of ex. Common logarithm of a positive real number x is denoted by log10 x. Relation: (i) log10 x = 0.4343 loge x (ii) loga 1 = 0, loga a = 1, loga 0 = –  (a > 1) nP

7.

 !

n (n  1) n (n  1) (2n  1) n (n  1) , n2 = , n3 = 2 6 2 Permutations and Combinations

n =

log

 m = log m – log n  n

log (mn) = n log m, logn m × logm n = 1 Matrices and Determinants (i) Two matrices A = [aij] and B = [bij] are equal if they have same order and corresponding elements are equal. i.e., aij = bij for all i and j. (ii) If A = [aij] and B = [bij] are two matrices of the same order, then A + B is defined and A + B = [aij + bij], i.e., add corresponding elements. (iii) If A = [aij] is a matrix and k is a scalar, then kA is another matrix obtained by multiplying each element of A by the scalar k. Thus, kA = [kaij].

( xiii ) (iv) The product AB of two matrices A and B is defined if the number of columns in A is equal to the number of rows in B. If A is an m × n matrix and B is an n × p matrix then AB is a matrix of order m × p. The (i, j)th element of AB is obtained by multiplying the corresponding elements of ith row of A and jth column of B and adding all these products. Matrix multiplication is not commutative, i.e., AB  BA in general. If the product of two matrices is a zero matrix, it is not necessary that one of the matrices is a zero matrix. (AB)C = A(BC), A(B + C) = AB + AC whenever both sides of equality are defined. (v) For every square matrix A, there exists an identity matrix I of same order such that AI = IA = A. (vi) The matrix obtained by interchanging the rows and columns of a matrix A is called the transpose of A. Transpose of A is denoted by A or AT . If A = [aij]m × n , then A = [aji]n × m (A) = A, (kA) = kA, (A + B) = A + B, (AB) = BA. (vii) A square matrix A is called symmetric if A = A and skew symmetric if A = – A. All the diagonal elements of a skew symmetric matrix are zero. (viii) A square matrix A is said to be invertible if there exists a square matrix B of same order as A, such that AB = BA = I. The matrix B is called the inverse of A and it is denoted by A–1. Thus, AA–1 = A–1 A = I Also, (A–1)–1 = A, (AB)–1 = B–1 A–1. (ix) A determinant is a function which associates each square matrix with a unique number (real or complex). The determinant of a square matrix A is denoted by | A | or det. A or .

a b = ad – cb. c d

(x) a1 b1 c1

a2 b2 c2

a3 b b3  a1 2 c2 c3

b3 b b b  a2 1 3  a3 1 c3 c1 c3 c1

b2 c2

= a1(b2c3 – b3c2) – a2(b1c3 – b3 c1) + a3 (b1c2 – b2c1)

a1 0

a2 b2

a3 a1 b3  b1

0 b2

0 0

0

0

c3

c2

c3

c1

= a1 b2c3 = product of diagonal elements.

(xi) If A is a square matrix of order n, then | kA | = kn | A |. (xii) The value of a determinant remains unchanged if its rows and columns are interchanged, i.e., | A | = | A |. (xiii) If any two-rows (or columns) of a determinant are interchanged, then sign of determinant changes.

( xiv ) (xiv) If any two rows (or columns) of a determinant are identical or proportional, then value of determinant is zero. (xv) If each element of a row (or column) of a determinant is multiplied by a constant k, then its value gets multiplied by k. By this property, we can take out any common factor from any one row (or column) of a given determinant. (xvi) If each element of a row (or column) of a determinant is the sum of m terms, then the determinant can be expressed as the sum of m determinants. (xvii) If, to each element of a row (or column) of a determinant, be added equi-multiples of the corresponding elements of some other row (or column), then value of determinant remains the same.

a

##" #$

 !

0°

30°

45°

60°

90°

1 2

1

3 2

1

3 2 1

1 2

1 2

0

1

3



"# #$

 a13 A 11 A 12 A 13 a23 then adj. A = A 21 A 22 A 23 , where Aij is co-factor of aij. (xviii) If A = a21 A 31 A 32 A 33 a31 a32 a33 (xix) For a square matrix of order n, A (adj. A) = (adj. A) = | A | I where I is the identity matrix of order n. (xx) If A is a non-singular matrix of order n, then | adj. A | = | A |n – 1 (xxi) A square matrix A is invertible if and only if A is a non-singular matrix. 1 A–1 = adj. A |A| 10. Trigonometry (i) 11

!

x°

a12 a22

sin x

0

cos x

1

tan x

0

2

3

(ii) The figure shows a unit circle with centre at origin. If XOP = x, then P = (cos x, sin x).  cos 0 = 1, sin 0 = 0   cos = 0, sin =1 2 2 cos  = – 1, sin  = 0 3 3 cos = 0, sin =–1 2 2 cos 2 = 1, sin 2 = 0 (iii) Any t-ratio of (n. 90° ± x) = ± same t-ratio of x, when n is even = ± co-ratio of x, when n is odd.

Y B(0, 1) P(cos x, sin x) 1 X¢

C (– 1, 0)

x O

A(1, 0)

D(0, – 1) Y¢

Fig. 1

X

( xv ) where co-ratio of x is obtained by dropping co if present and adding co if absent. Thus, sin cos, tan cot, sec cosec. The sign ± or – is decided from the quadrant in which n. 90° ± x lies. (iv) Signs of t-ratios in different quadrants (Fig. 2) (v) cos2 x + sin2 x = 1, sec2 x – tan2 x = 1, cosec2 x – cot2 x = 1 (vi) sin (x ± y) = sin x cos y ± cos x sin y cos (x ± y) = cos x cos y B sin x sin y tan x  tan y tan (x ± y) = 1 B tan x tan y (vii)

sin 2x = 2 sin x cos x =

Y

sin

tan

cos

Y¢

Fig. 2

2 tan x

(ix)

sin 3x = 3 sin x – 4 sin3 x, cos 3x = 4 cos3 x – 3 cos x

1  tan 2 x 1  tan 2 x

3 tan x  tan 3 x

1 2 1 cos x sin y = 2 1 cos x cos y = 2 1 sin x sin y = 2

sin x cos y =

O

1  tan 2 x

cos 2x = cos2 x – sin2 x = 2 cos2 x – 1 = 1 – 2 sin2 x =

(x)

X

X¢

(viii)

tan 3x =

all

1  3 tan 2 x

[sin (x + y) + sin (x – y)] [ sin (x + y) – sin (x – y)] [cos (x + y) + cos (x – y)] [cos (x – y) – cos (x + y)]

x y x y cos 2 2 x y x y sin x – sin y = 2 cos sin 2 2 x y x y cos x + cos y = 2 cos cos 2 2 x y yx x y x y = 2 sin sin sin cos x – cos y = – 2 sin 2 2 2 2 (xii) a sin x + b cos x = r sin (x + ), a cos x + b sin x = r cos (x – ), where a = r cos , b = r sin 

(xi)

sin x + sin y = 2 sin

so that r =

a 2  b2 ,  = tan–1

(xiii) In any ABC:

 b  .  a

a b c   sin A sin B sin C

(sine formula)

: 395