GATE Mathematics Book

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MATHEMATICS

for

EC / EE / IN / ME / CE

By

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Syllabus Mathematics

THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30th_{Cross, 10}th_{Main, Jayanagar 4}th_{Block, Bangalore-11}

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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 EE IN ME CE 2013 10 12 10 15 N.A 2012 14 10 15 15 15 2011 9 10 10 13 13 2010 12 12 12 15 15 Over All Percentage 11.25% 11.00% 11.75% 14.5% 14.33%

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Contents Mathematics

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Chapter

Page No.

#1. Linear Algebra

1-37



Matrix

1 - 4



Determinant

4 - 17



Eigen Vectors

17 - 20



Assignment-1

21 - 24



Assignment-2

24- 27



Answer Keys

28 

Explanations

28 - 37

#2. Probability and Distribution

38-70



Permutation & Combination

38 – 44



Random Variable

44 - 50



Normal Distribution

50 - 57



Assignment-1

58 - 60



Assignment-2

60 - 63



Answer Keys

64 

Explanations

64 – 70

#3. Numerical Methods

71-90



Solution of algebraic & Transcendental equation

71 – 77



Numerical Integration

77 - 80



Differential Equation

80 - 82



Assignment-1

83 - 84



Assignment-2

85 

Answer Keys

86 

Explanations

86 – 90

#4. Calculus

91 - 134



Indeterminate form

91 - 97



Derivative

97 - 98



Maxima-minima

98– 101



Integration

102 - 108



Vector calculus

108 - 115



Assignment-1

116 -119



Assignment-2

120 - 122



Answer Keys

123 

Explanations

123 – 134

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Contents Mathematics

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#5. Differential Equations

135 - 172



Order of differential equation

135 - 145



Higher order differential equation

145 - 153



Partial Differential Equation

153 – 160



Assignment-1

161 -163



Assignment-2

164 - 165



Answer Keys

166 

Explanations

166 – 172

#6. Complex Variables

173 - 199



Complex Variables

173 - 175



Analytical Function

176 - 178



Cauchy’s Intergral Theorem

178 – 184



Residue Theorem

184 – 186



Assignment-1

187 - 189



Assignment-2

189 - 191



Answer Keys

192 

Explanations

192 – 199

#7. Laplace Transform

200 - 217



Introduction

200 

Laplace Transform

200 - 205



Assignment-1

206 -208



Assignment-2

209 - 211



Answer Keys

212 

Explanations

212 – 217

Module Test

218-235



Test Questions

218- 224



Answer Keys

225 

Explanations

225 - 235

Reference Books

236

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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 a a a a ] a ith row, jth column

Types of Matrices

1. Row and Column matrices

 Row Matrices [ 2, 7, 8, 9] single row ( or row vector)

 Column Matrices [

] 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)

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Symmetric Matrix 𝐀

𝐓

= A

Skew Symmetric Matrix 𝐀

𝐓

= - A

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 ‘ ’. e.g. = [

]

7. Null Matrix (or Zero Matrix)

A matrix is said to be Null Matrix if all the elements are zero. e.g. 0

1

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 b f g f c ] [ h g h f g f ] 9. Triangular matrix

 A matrix is said to be “upper triangular” if all the elements below its principal diagonal are zeros.

 A matrix is said to be “lower triangular” if all the elements above its principal diagonal are zeros. [ a h g b f c ] [ a g b f h c ]

Upper triangular matrix Lower triangular matrix

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Chapter-1 Mathematics 11. Singular matrix: If |A| = 0, then A is called a singular matrix.

12. Unitary matrix

If we define, A = (A̅) = transpose of a conjugate of matrix A Then the matrix is unitary if A . A =

For example A = [

] , A = [

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

13. Hermitian matrix

It is a square matrix with complex entries which is equal to its own conjugate transpose. A = A or a = a̅

For example: 0 i i 1

Note: In Hermitian matrix , diagonal elements always real

14. 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

For example = 0 i i 1

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

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Addition and Subtraction of matrices [a b c d ] [ a b c d ] = [ a a b b c c d 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

p) matrices results in matrix of (m p)dimension 0 m n _{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 , D = det A = |a_a a a | = (a a - a a ) Determinant of “order n” D = |A| = det A = || a a a a a a a a a | |

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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.

D = |

a a a b b b c c c | Minor of a = |b_c b_c |

 Cofactor is the minor with “proper sign”. The sign is given by (-1) (where the element belongs to ith_{row, j}th_column).

A2 = Cofactor of = (-1) |b_c b_c | Cofactor matrix can be formed as |

A A A C C C | In general  = = j  = 0 if i 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 b c a b c a b c | = | a a a b b b c c c |

2. 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 b c a b c | = | a c b a c b a c b | = | c a b c a b c a b | 3. Determinant vanishes if two parallel lines are identical.

4. 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 |

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5. If each element of a line consists of the m terms, then determinant can be expressed as sum of the m determinants.

more parallel lines, determinant is unaffected.

e.g. By the operation, + p +q , determinant is unaffected.

7. Determinant of an upper triangular/ lower triangular/diagonal/scalar matrix is equal to the product of the leading diagonal elements of the matrix.

8. If A & B are square matrix of the same order, then |AB|=|BA|=|A||B|. 9. If A is non singular matrix, then |A _|=

| | (as a result of previous).

10. Determinant of a skew symmetric matrix (i.e. A =-A) of odd order is zero. 11. If A is a unitary matrix or orthogonal matrix (i.e. A = A _{) then |A|= ±1.}

12. If A is a square matrix of order n then |k A| = k |A|. 13. | | = 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

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