electronics and communication engineering : mathematics, the gate academy

MATHEMATICS

for

EC / EE / IN / ME / CE

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Syllabus Mathematics

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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%

Contents Mathematics

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CCOONNTTEENNTTSS

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

Contents Mathematics

THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750, [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 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 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

Chapter-1 Mathematics

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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) - tr (AB) = tr (BA)



Symmetric Matrix 𝐀𝐓 = A Skew Symmetric Matrix 𝐀𝐓 = - 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 gh b fg f c

] [

h gh 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

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



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 = a ̅̅ ̅

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

(b) Each corresponding element in both the matrices are equal.



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

||



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 b

c c |

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

A2 = Cofactor of = (-1) |b b

c 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

|



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 e

a b c d e

a b c d e

| = |

a b c

a b c

a b c

| + |

a b d

a b d

a b d

| |

a b e

a b e

a 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 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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answer keys

mathematics linear algebra

partial differential

linear algebra linear

order equation linear

order of differential

matrix algebra

systems of linear equations