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 nonlinear algebraic equations, single and multistep 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, Ztransform.
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
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
137
1  4 4  17 17  20 21  24 24 27 28 28  37
Matrix Determinant Eigen Vectors Assignment1 Assignment2 Answer Keys Explanations
Probability and Distribution
3870
38 44 50 58 60 64 64
Permutation & Combination Random Variable Normal Distribution Assignment1 Assignment2 Answer Keys Explanations
– 44  50  57  60  63 – 70
Numerical Methods
7190
71 – 77 77  80 80  82 83  84 85 86 86 – 90
Solution of algebraic & Transcendental equation Numerical Integration Differential Equation Assignment1 Assignment2 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 Maximaminima Integration Vector calculus Assignment1 Assignment2 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 Assignment1 Assignment2 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 Assignment1 Assignment2 Answer Keys Explanations
Laplace Transform
200  217
200 200  205 206 208 209  211 212 212 – 217
Introduction Laplace Transform Assignment1 Assignment2 Answer Keys Explanations
Module Test
218235
Test Questions
218 224
Answer Keys
225
Explanations
225  235
Reference Books
236
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Chapter1
Mathematics
CHAPTER 1 Linear Algebra Linear algebra comprises of the theory and applications of linear system of equations, linear transformations and Eigenvalue 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]
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)
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Chapter1
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€˜ â€™. 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 g f] [h b f ] [h g f c g f
Symmetric Matrix
đ??€đ??“ = A
Skew Symmetric Matrix
đ??€đ??“ =  A
9. Triangular matrix ď‚ˇ A matrix is said to be â€œupper triangularâ€? if all the elements below its principal diagonal are zeros. ď‚ˇ A matrix is said to be â€œlower triangularâ€? 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, Bangalore11 ď€Š: 08065700750, ď€Ş info@thegateacademy.com ÂŠ Copyright reserved. Web: www.thegateacademy.com Page 2
Chapter1
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 SkewHermitian 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, Bangalore11 : 08065700750, info@thegateacademy.com © Copyright reserved. Web: www.thegateacademy.com Page 3
Chapter1
Mathematics
Addition and Subtraction of matrices [
a c
b ] d
[
a c
b ] = d
[
a c
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
a )
Determinant of “order n”
D = A = det A = 
a a
a
a
a
a
a a
 
a
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Chapter1
Mathematics
Minors & Cofactors
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
A
A
C
C
C
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
2.
If two parallel lines of a determinant are interchanged, 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, Bangalore11 : 08065700750, info@thegateacademy.com © Copyright reserved. Web: www.thegateacademy.com Page 5
Chapter1
5.
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
6.
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
8.
If each element of a line be added equimultiple 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=AB.
9.
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).
7.
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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