gate mathematics book

8/10/2019 GATE Mathematics Book

1/12


2/12

MATHEMATICS

for

EC / EE / IN / ME / CE

By

wwwthegateacademy com
http://www.thegateacademy.com/http://www.thegateacademy.com/http://www.thegateacademy.com/http://www.thegateacademy.com/http://www.thegateacademy.com/http://www.thegateacademy.com/


3/12

Syllabus Mathematics

THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30th

Cross, 10th

Main, Jayanagar 4th

Block, Bangalore-11

080 65700750 i f @th t C i ht d W b th t d

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 Greens theorems.

Differential equations: First order equation (linear and nonlinear), Higher order linear differential

equations with constant coefficients, Method of variation of parameters, Cauchys and Eulers equations,

Initial and boundary value problems, Partial Differential Equations and variable separable method.

Complex variables: Analytic functions, Cauchys integral theorem and integral formula, Taylors 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%


4/12

Contents Mathematics


Cross, 10th

Main, Jayanagar 4th

Block, Bangalore-11

080 65700750 i f @th t d C i ht d W b th t d P I

CONTENTS

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



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-2 85

Answer Keys 86

Explanations 86 90

#4. Calculus 91 - 134 Indeterminate form 91 - 97

Derivative 97 - 98

Maxima-minima 98101

Integration 102 - 108

Vector calculus 108 - 115

Assignment-1 116 -119


Answer Keys 123

Explanations 123 134


5/12

Contents Mathematics


Cross, 10th

Main, Jayanagar 4th

Block, Bangalore-11

080 65700750 i f @th t d C i ht d W b th t d P II

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


Answer Keys 166


#6. Complex Variables 173 - 199

Complex Variables 173 - 175

Analytical Function 176 - 178

Cauchys Intergral Theorem 178 184

Residue Theorem 184 186



Answer Keys 192


#7. Laplace Transform 200 - 217

Introduction 200

Laplace Transform 200 - 205 Assignment-1 206 -208


Answer Keys 212


Module Test 218-235

Test Questions 218- 224

Answer Keys 225

Explanations 225 - 235

Reference Books 236


6/12

Chapter-1 Mathematics


Cross, 10th

Main, Jayanagar 4th

Block, Bangalore-11

080 65700750 i f @th t d C i ht d W b th t d P 1

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 columnsConventionally, single capital letter is used to denote matricesThus,

A = a a a a a a a aaa a a aaithrow, jthcolumn

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 columnse.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 MatrixThe sum of the diagonal elements of a square matrix.

-

tr ( A) = tr(A)

is scalar-

- tr ( A+B) = tr (A) + tr (B)

-

tr (AB) = tr (BA)


7/12



Cross, 10th

Main, Jayanagar 4th

Block, Bangalore-11


Symmetric Matrix = A Skew Symmetric Matrix = - A

3.

Rectangular Matrix

Number of rows

Number of columns

4. Diagonal MatrixA 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 18. Symmetric and Skew symmetric matrices

* Symmetric, when a= +afor all i and j. In other words = ANote :- Diagonal elements can be anything.

* Skew symmetric, when a= -a In other words = - ANote :- All the diagonal elements must be zero.

Symmetric Skew symmetric

a h gh b fg f c

h gh f g f

9.

Triangular matrixA matrix is said to be upper triangular if all the elements below its principal diagonalare zeros.A matrix is said to be lower triangular if all the elements above its principal diagonalare zeros.

a h g b f c

a g b f h c

Upper triangular matrix Lower triangular matrix

10. Orthogonal matrix:If A.A= , then matrix A is said to be Orthogonal matrix.


8/12



Cross, 10th

Main, Jayanagar 4th

Block, Bangalore-11


11.

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

12.

Unitary matrix

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

A = , A= A.A= , and Hence its a Unitary Matrix

13.

Hermitian matrixIt is a square matrix with complex entries which is equal to its own conjugate transpose.

A= A or a = a For example: 0 i i 1Note: 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 = a For example =

0 i i 1

Note: In Skew-Hermitian matrix , diagonal elements either zero or Pure Imaginary15. 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.


9/12



Cross, 10th

Main, Jayanagar 4th

Block, Bangalore-11


Addition and Subtraction of matrices

[a bc d] [a bc d] = [a a b bc c d d]ules

1.

Matrices of same order can be added

2. Addition is commutative A+B = B+A3. 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 isequal to the number of rows of the second matrix. Multiplication of (m n )and (n p)matrices results in matrix of (m p)dimension 0m nn 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 ofquestion to find rank.

Determinant

An norder determinant is an expression associated with n n square matrix.If A = [a] , Element awith ithrow, jthcolumn.For n = 2 , D = det A = a aa a= (aa- aa)Determinant of ordernD =

|A| =det A =

a a a aa a a a a


10/12



Cross, 10th

Main, Jayanagar 4th

Block, Bangalore-11


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 ab b bc c cMinor of a= b bc c

Cofactor is the minor with proper sign. The sign is given by ( -1) (where the elementbelongs to ithrow, jthcolumn).

A2 = Cofactor of

= (-1

)

b

b

c c

Cofactor matrix can be formed as A A A C C CIn general

= = j = 0 if ij* , , , , . +

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 ca b ca

b

c =

a a ab b bc c c2.

If two parallel lines of a determinant are inter-changed, the determinant retains itnumerical values but changes in sign. (In a general manner, a row or column is referred as

line).

a b ca b ca b c = a c ba c ba c b =

c a bc a bc a b3.

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 ca b ca b c=

a b ca b ca b c


11/12



Cross, 10th

Main, Jayanagar 4th

Block, Bangalore-11


5.

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

sum of the m determinants.

a b c d ea b c d ea b c d e = a b ca b ca b c+ a b da b da b d a b ea b ea b e 6. 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.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 thingand its technically unfair to even compare them. However just for readersconvenience, 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 Cant be reduced to one numberInterchanging 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 ofsecond matrix

Multiplication of the 2 matrices is done by

multiplying rows of first matrix & column ofsecond matrix


12/12

gate mathematics book

Documents