: Mechanical Engineering, THE GATE ACADEMY by THE GATE ACADEMY

MATHEMATICS for

EC / EE / IN / ME / CE By

www.thegateacademy.com

Syllabus

Mathematics

Syllabus for Mathematics Linear Algebra: Matrix Algebra, Systems of linear equations, Eigen values and eigen vectors. 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. 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. Transform Theory: Fourier transform, Laplace transform, Z-transform.

Analysis of GATE Papers (Mathematics) Year

ECE

2013

N.A

2012

2011

2010

14.5%

14.33%

Over All Percentage

11.25%

11.00% 11.75%

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Contents

Mathematics

CONTENTS

#1.

#2.

#3.

#4.

Chapter

Page No.

Linear Algebra

1-37

      

1 - 4 4 - 17 17 - 20 21 - 24 24- 27 28 28 - 37

Matrix Determinant Eigen Vectors Assignment-1 Assignment-2 Answer Keys Explanations

Probability and Distribution

38-70

      

38 44 50 58 60 64 64

Permutation & Combination Random Variable Normal Distribution Assignment-1 Assignment-2 Answer Keys Explanations

– 44 - 50 - 57 - 60 - 63 – 70

Numerical Methods

71-90

      

71 – 77 77 - 80 80 - 82 83 - 84 85 86 86 – 90

Solution of algebraic & Transcendental equation Numerical Integration Differential Equation Assignment-1 Assignment-2 Answer Keys Explanations

Calculus

91 - 134

        

91 - 97 97 - 98 98– 101 102 - 108 108 - 115 116 -119 120 - 122 123 123 – 134

Indeterminate form Derivative Maxima-minima Integration Vector calculus Assignment-1 Assignment-2 Answer Keys Explanations

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Contents

#5.

#6.

#7.

Mathematics

Differential Equations

135 - 172

      

135 - 145 145 - 153 153 – 160 161 -163 164 - 165 166 166 – 172

Order of differential equation Higher order differential equation Partial Differential Equation Assignment-1 Assignment-2 Answer Keys Explanations

Complex Variables

173 - 199

       

173 - 175 176 - 178 178 – 184 184 – 186 187 - 189 189 - 191 192 192 – 199

Complex Variables Analytical Function Cauchy’s Intergral Theorem Residue Theorem Assignment-1 Assignment-2 Answer Keys Explanations

Laplace Transform

200 - 217

     

200 200 - 205 206 -208 209 - 211 212 212 – 217

Introduction Laplace Transform Assignment-1 Assignment-2 Answer Keys Explanations

Module Test

218-235



Test Questions

218- 224



Answer Keys

225



Explanations

225 - 235

Reference Books

236

Chapter-1

Mathematics

CHAPTER 1 Linear Algebra Linear algebra comprises of the theory and applications of linear system of equations, linear transformations and Eigen-value problems.

Matrix Definition A system of “m n” numbers arranged along m rows and n columns Conventionally, single capital letter is used to denote matrices Thus,

A=[

a a

a a a

a a a a

]

ith row, jth column

Types of Matrices 1. Row and Column matrices  Row Matrices [ 2, 7, 8, 9] 

Column Matrices

[

]

single row ( or row vector) single column (or column vector)

2. Square matrix  Same number of rows and columns.  Order of Square matrix no. of rows or columns e.g. A = [ 



] ; order of this matrix is 3

Principal Diagonal (or main diagonal or leading diagonal) The diagonal of a square matrix (from the top left to the bottom right) is called as principal diagonal. Trace of the Matrix The sum of the diagonal elements of a square matrix. - tr (λ A) = λ tr(A) , λ is scalar- tr ( A+B) = tr (A) + tr (B) - tr (AB) = tr (BA)

Chapter-1

Mathematics

3. Rectangular Matrix Number of rows Number of columns 4. Diagonal Matrix A Square matrix in which all the elements except those in leading diagonal are zero. e.g. [

]

5. Scalar Matrix A Diagonal matrix in which all the leading diagonal elements are same. e.g [

]

6. Unit Matrix (or Identity Matrix) A Diagonal matrix in which all the leading diagonal elements are â&#x20AC;&#x2DC; â&#x20AC;&#x2122;. e.g.

[

]

7. Null Matrix (or Zero Matrix) A matrix is said to be Null Matrix if all the elements are zero. e.g.

8. Symmetric and Skew symmetric matrices * Symmetric, when a = +a for all i and j. In other words =A Note :- Diagonal elements can be anything. * Skew symmetric, when a = - a In other words = -A Note :- All the diagonal elements must be zero. Symmetric Skew symmetric a h g h g f] [h b f ] [h g f c g f

Symmetric Matrix

đ??&#x20AC;đ??&#x201C; = A

Skew Symmetric Matrix

đ??&#x20AC;đ??&#x201C; = - A

9. Triangular matrix ď&#x201A;ˇ A matrix is said to be â&#x20AC;&#x153;upper triangularâ&#x20AC;? if all the elements below its principal diagonal are zeros. ď&#x201A;ˇ A matrix is said to be â&#x20AC;&#x153;lower triangularâ&#x20AC;? if all the elements above its principal diagonal are zeros. a a h g [ [g b ] b f] f h c c Upper triangular matrix Lower triangular matrix 10.

Orthogonal matrix: If A. A = , then matrix A is said to be Orthogonal matrix. THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 ď&#x20AC;Š: 080-65700750, ď&#x20AC;Ş info@thegateacademy.com ÂŠ Copyright reserved. Web: www.thegateacademy.com Page 2

Chapter-1

Mathematics

11.

Singular matrix: If |A| = 0, then A is called a singular matrix.

12.

Unitary matrix ̅ ) = transpose of a conjugate of matrix A If we define, A = (A Then the matrix is unitary if A . A = For example A=[

13.

],A =[

]

A. A = , and Hence it’s a Unitary Matrix

Hermitian matrix It is a square matrix with complex entries which is equal to its own conjugate transpose. A = A or a = a̅ i For example: 0 1 i Note: In Hermitian matrix , diagonal elements

14.

always real

Skew Hermitian matrix : It is a square matrix with complex entries which is equal to the negative of conjugate transpose. A = A or a = a̅̅̅ i For example = 0 1 i Note: In Skew-Hermitian matrix , diagonal elements

either zero or Pure Imaginary

15.

Idempotent Matrix : If A = A, then the matrix A is called idempotent matrix.

16.

Nilpotent Matrix : If A = 0 (null matrix), then A is called Nilpotent matrix (where K is a +ve integer).

17.

Periodic Matrix : If A

= A (where k is a +ve integer), then A is called Periodic matrix.

If k =1 , then it is an idempotent matrix. 18.

Proper Matrix : If |A| = 1, matrix A is called Proper Matrix.

Equality of matrices Two matrices can be equal if they are of (a) Same order (b) Each corresponding element in both the matrices are equal. THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  info@thegateacademy.com © Copyright reserved. Web: www.thegateacademy.com Page 3

Chapter-1

Mathematics

Addition and Subtraction of matrices [

a c

b ] d

[

a c

b ] = d

[

a c

b d

b ] d

Rules 1. Matrices of same order can be added 2. Addition is commutative A+B = B+A 3. Addition is associative (A+B) +C = A+ (B+C) = B + (C+A) Multiplication of matrix by a Scalar Every element of the matrix gets multiplied by that scalar.

Multiplication of matrices Condition: Two matrices can be multiplied only when number of columns of the first matrix is equal to the number of rows of the second matrix. Multiplication of (m n) and (n m n p) matrices results in matrix of (m p)dimension 0 n p = m p1. To find Rank always remember to make matrix a triangular matrix (Upper triangular) or (lower triangular) [

] or

[

]

Try to make I, zero then II and then III in any Matrix to find Rank of Matrix, for easy execution of question to find rank.

Determinant An n order determinant is an expression associated with n

n square matrix.

If A = [a ] , Element a with ith row, jth column. For n = 2 ,

a D = det A = |a

a a | = (a

-a

a )

Determinant of “order n”

D = |A| = det A = ||

a a

| |

Chapter-1

Mathematics

Minors & Co-factors 

The minor of an element in a determinant is the determinant obtained by deleting the row and the column which intersect that element. a a a b b | D=| b c c c Minor of a = |



b c

b | c

Cofactor is the minor with “proper sign”. The sign is given by (-1) belongs to ith row, jth column). A2 = Cofactor of

= (-1)

b | c

(where the element

b | c A

Cofactor matrix can be formed as |

In general  

= =0 *

, ,

if i

=j j ,

Drill Problem Can the determinant be expanded about the diagonal?

Properties of Determinants 1.

A determinant remains unaltered by changing its rows into columns and columns into rows. a a a a b c b b | b c | = | b | a c c c a b c

If two parallel lines of a determinant are inter-changed, the determinant retains it numerical values but changes in sign. (In a general manner, a row or column is referred as line). a b c a c b c a b b c |= c b | =| c a b | | a | a a b c a c b c a b

3. 4.

Determinant vanishes if two parallel lines are identical. If each element of a line be multiplied by the same factor, the whole determinant is multiplied by that factor. [Note the difference with matrix]. a b c a b c a b c a b c | | =| | a b c a b c THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  info@thegateacademy.com © Copyright reserved. Web: www.thegateacademy.com Page 5

Chapter-1

If each element of a line consists of sum of the m determinants. a b c d e a b c d e | =| a | a a b c d e a

Mathematics

the m terms, then determinant can be expressed as b b b

c a c |+| a c a

b b b

d d | d

a | a a

b b b

e e | e

If each element of a line be added equi-multiple of the corresponding elements of one or more parallel lines, determinant is unaffected. e.g. By the operation, + p +q , determinant is unaffected. Determinant of an upper triangular/ lower triangular/diagonal/scalar matrix is equal to the product of the leading diagonal elements of the matrix. If A & B are square matrix of the same order, then |AB|=|BA|=|A||B|.

If A is non singular matrix, then |A |=| | (as a result of previous).

10. 11. 12. 13.

Determinant of a skew symmetric matrix (i.e. A =-A) of odd order is zero. If A is a unitary matrix or orthogonal matrix (i.e. A = A ) then |A|= ±1. If A is a square matrix of order n then |k A| = k |A|. | | = 1 ( is the identity matrix of order n).

Multiplication of determinants  

The product of two determinants of same order is itself a determinant of that order. In determinants we multiply row to row (instead of row to column which is done for matrix).

Comparison of Determinants & Matrices 

Although looks similar, but actually determinant and matrix is totally different thing and its technically unfair to even compare them. However just for reader’s convenience, following comparative table has been prepared.

Determinant

Matrix

No of rows and columns are always equal

No of rows and column need not be same (square/rectangle)

Scalar multiplication: elements of one line (i.e. one row and column) is multiplied by the constant

Scalar multiplication: all elements of matrix is multiplied by the constant

Can be reduced to one number

Can’t be reduced to one number

Interchanging rows and column has no effect

Interchanging rows and columns changes the meaning all together

Multiplication of 2 determinants is done by multiplying rows of first matrix & rows of second matrix

Multiplication of the 2 matrices is done by multiplying rows of first matrix & column of second matrix