Mathematics engineering GATE exam preparation for computer science and engineering, Study notes of Engineering Mathematics

Download Mathematics engineering GATE exam preparation for computer science and engineering and more Study notes Engineering Mathematics in PDF only on Docsity!

Dear GATE Aspirant,

The GATE program will help you to overcome the common feeling of confusion about the exact scope of the syllabus. The course content will enable you to cover the course smoothly in the limited time at your disposal without wasting time on unnecessary details, without omitting any useful part.

The program provides graded questions in every topic leading you to deeper and more intricate and basic concepts. This will help you to stimulate your own thinking and also makes the process of learning, enjoyable and highly efficient.

The quick methods suggested to tackle questions and the practice that one has to put in while solving the problems develops in you the required skill of tackling tricky problems independently and confidently. We are sure you will be in a position to deal with every and any problem most successfully.

I wish you all the best for your GATE.

Thank You

Rashmi Deshpande Director Vidyalankar Group of Educational Institutes

EC / EE / IN / CS / ME / CE

INDEX

Contents Topics PageNo.

Chapter  1 : Linear Algebra

Contents Topics PageNo.

Chapter  2 : Calculus

• Pearl Centre, S.B. Marg, Dadar (W), Mumbai  400 028. Tel. - MODULE  ENGINEERING MATHEMATICS
  • 1.1 Determinants Notes
  • 1.2 Matrices
  • 1.3 Rank of a Matrix
  • 1.4 System of Linear Equations
  • 1.5 Eigen Values and Eigen Vectors
  • 1.6 Vectors
  • Assignment  Assignments
  • Assignment 
  • Assignment 
  • Assignment 
  • Assignment 
  • Assignment 
  • Assignment 
  • Assignment 
  • Assignment 
  • Assignment 
  • Assignment 
  • Test Paper  Test Papers
  • Test Paper 
  • Test Paper 
  • Test Paper 
  • Test Paper 
  • Test Paper 
• 2.1 Function of single variable Notes
• 2.2 Limit of a function
• 2.3 Continuity
• 2.4 Differentiability
• 2.5 Mean Value Theorems
• 2.6 Maxima and Minima
• 2.7 Integration
• 2.8 Definite Integration
• 2.9 Double Integrals
• 2.10 Triple Integrals
• 2.11 Change of Variables in Double and Triple Integralsand Jacobians
• 2.12 Application of Integration
• 2.13 Partial and Total Derivatives
• 2.14 Taylor's Series and Maclaurin's Series
• 2.15 Fourier Series
• List of Formulae
• Assignment  Assignments
• Assignment 
• Assignment 
• Assignment 
• Assignment 
• Assignment 
• Assignment 
• Assignment 
• Assignment 
• Test Paper  Test Papers
• Test Paper 
• Test Paper 
• Test Paper 
• Test Paper 
• Test Paper 

Contents Topics PageNo.

Solutions  Probability and Statistics

Assignment

Answer Key 308 Model Solutions 310

Test Paper

Answer Key 332 Model Solutions 333

1

Chapter - 1 : Linear Algebra

1.1 Determinants

If a, b, c and d are any four terms then the representation a^ b c d

  is called a determinant

and is denoted by D.

A determinant of order 2 is evaluated as follows:

D = ^ a^ b  c d

= ad  bc

A determinant of order 3 can be evaluated as follows:

D =

1 2 3 1 2 3 1 2 3

a a a b b b c c c

2 3 1 3 1 2

a b^ b^ a b^ b^ a b^ b c c c c c c

For example:

  =^

Properties of Determinants

Interchange of rows and columns (R (^) i  Ci)

The value of determinant is not affected by changing the rows into the corresponding

columns, and the columns into the corresponding rows. Thus

1 2 3 1 1 1 1 2 3 2 2 2 1 2 3 3 3 3

a a a a b c b b b a b c c c c a b c

Identical rows and columns (R (^) i = R (^) j , C (^) i= C (^) j).

If two rows or two columns of a determinant are identical, the determinant has the value

zero. Thus,

1 2 3 1 1 3 1 2 3 1 1 3 1 2 3 1 1 3

a a a a a a a a a 0 b b b 0 c c c c c c

Notes on Linear Algebra

3

Note : 1. Area of a triangle whose vertices are (x 1 , y 1 ) (x 2 , y 2 ) and (x 3 , y 3 ) is

given by the absolute value of 1 2

1 1 2 2 3 3

x y 1 x y 1 x y 1

**2. Area of a quadrilateral can be found by dividing it into two triangles.

3. If the area of a triangle obtained from the three given points is zero,** then the three points lie on a line.  The condition for three points to be collinear is

1 1 2 2 3 3

x y 1 (^1) x y 1 (^2) x y 1

Solved Example 1 :

Find the area of a triangle whose vertices

are (2, 1), (4, 3), (2, 5).

Solution :

The area of the triangle is

A =

1 1 2 2 3 3

x y 1 (^1) x y 1 2 x y 1

× abs [2( 3  5)  4(1  5)  2(1 + 3)]

× abs [16 + 16  8] = 4 sq. units

Solved Example 2 : Find if the three points (1, 1), (5, 7) and (8, 11) are collinear. Solution : If the points are collinear, area of the triangle formed by the three given points should be zero.

Area A = 1 2

[1(7  11)  5( 1  11)

+ 8 ( 1  7)]

[4 + 60  64] = 0

The given points are collinear.

Vidyalankar : GATE – Engineering Mathematics

4

1.2 Matrices

A Matrix is a rectangular array of elements written as

A =

1 2 n

11 12 1n 21 22 2n

m m m

a a.... a a a.... a

.... .... .... .... a a.... a

The above matrix A has m rows and n columns. So it is a m × n matrix or it is said that

the size of the matrix is m × n.

Types of Matrices

Square Matrix :

It is a Matrix in which number of rows = number of columns

For example:

is a square matrix of order 3.

Diagonal Matrix :

It is a square matrix in which all non diagonal elements are zero.

For example:

Scalar Matrix :

It is a diagonal matrix in which all diagonal elements are equal

For example:

Unit Matrix :

It is a scalar matrix with diagonal elements as unity. It is also called Identity Matrix.

Identity matrix of order 2 is I 2 = 1 0 0 1

Vidyalankar : GATE – Engineering Mathematics

6

Symmetric Matrix :

If for a square matrix A, A = A then A is symmetric

For example:

Skew Symmetric matrix :

If for a square matrix A, A = A then it is skew  symmetric matrix.

For example:

Note : For a skew symmetric matrix, diagonal elements are zero.

Orthogonal Matrix :

A square matrix A is orthogonal if AA = AA = I

For example: A = cos^ sin sin cos

 ^ 

Here AA = I

Note : For orthogonal matrix A, A ^1 = A 

Conjugate of a Matrix :

Let A be a complex matrix of order m  n. Then conjugate of A is the matrix obtained by

taking conjugate of every element in the matrix and denoted by A

For example: if A = 7 i^3 3i^4 9 2i i 8 4i

 ^  

then conjugate of A = A 7 i^3 3i^4 9 2i i 8 4i

 ^ ^  

Matrix A ^ :

The transpose of the conjugate of a matrix A is denoted by A

For example: Let A = 7 i^2 4i^4 3 2i i 1 2i

 ^  

then A 7 i^2 4i^4 3 2i i 1 2i

 ^ ^  

Notes on Linear Algebra

7

and A^ = (^)  

7 i 3 2i A 2 4i i 4 1 2i

 ^  

Unitary Matrix :

A square matrix A is said to be unitary if A^ A = I

For example: A =

1 i 1 i 2 2 1 i 1 i 2 2

 ^ ^  

Here AA = 1 0 0 1

 ^ = I

Hermitian Matrix :

A square matrix A is called Hermitian matrix if a (^) ij = aji

For example: A =

4 1 i 2 5i 1 i 3 1 2i 2 5i 1 2i 8

 ^  

The necessary and sufficient condition for a matrix A to be Hermitian is that A = A .

Skew  Hermitian Matrix :

A square matrix A is skew Hermitian matrix if a (^) ij =  aji

For example:

2i 2 8i 1 2i (2 8i) 0 2i (1 2i) 2i 4i

 ^  

The necessary and sufficient condition for a matrix A to be skewHermitian is that A ^ =  A

Note : All the diagonal elements of a skew Hermitian matrix are either zeroes or pure imaginary.

Idempotent Matrix :

Matrix A is called idempotent matrix if A 2 = A

For example: A =

 ^  

Here A^2 = A

Notes on Linear Algebra

9

Adjoint and Inverse of a Square Matrix

Minor : Consider the determinant

11 12 13 21 22 23 31 32 33

a a a a a a a a a

To find minor leave the row and column passing through the element a (^) ij.

The minor of the element a 21 = M 21 = 12 13 32 33

a a a (^) a 

The minor of the element a 32 = M 32 = 11 13 21 23

a a a (^) a 

The minor of the element a 11 = M 11 = 22 23 32 33

a a a (^) a 

Cofactor : The minor Mij multiplied by (1) i+j^ is called the cofactor of the element a (^) ij.

The cofactor of the element a 21 = A 21 = (1) 2+1^ M 21 =  12 13 32 33

a a a (^) a 

The cofactor of the element a 32 = A 32 = (1) 3+2^ M 32 =  11 13 21 23

a a a a

The cofactor of the element a 11 = A 11 = (1) 1+1^ M 11 = 22 23 32 33

a a a a

And so on.

Adjoint of a Matrix :

Adjoint of a square matrix A is the transpose of the matrix formed by the cofactors of the

elements of the given matrix A.

If A =

11 12 13 21 22 23 31 32 33

a a a a a a a a a

Then adj (A) =

11 21 31 12 22 32 13 23 33

A A A

A A A

A A A

Vidyalankar : GATE – Engineering Mathematics

10

Inverse of a Square Matrix :

For a nonsingular square matrix A

A^1 = 1 adj(A) | A |

where A^1 is called the inverse of square matrix.

Note : A A ^1 = A ^1 A = I

Solved Example 3 :

Calculate the adjoint of A,

where A =

Solution :

A 11 = the cofactor of a 11 in

| A | = 2 3 1 3

A 12 = the cofactor of a 12 in

| A | =  1 3 2 3

A 13 = the cofactor of a 13 in

| A | = 1 2  2  1 

A 21 = the cofactor of a 21 in

| A | =  1 1 1 3

A 22 = the cofactor of a 22 in

| A | = 1 1  2 3 

A 23 = the cofactor of a 23 in

| A | =  1 1  2  1 

A 31 = the cofactor of a (^31)

in | A | =

A 32 = the cofactor of a 32 in

| A | =  1 1 1 3

A 33 = the cofactor of a 33 in

| A | = 1 1 1 2

Adj (A) = transpose of the matrix formed by cofactor

11 21 31 12 22 32 13 23 33

A A A 3 4 5

A A A 9 1 4

A A A 5 3 1

Solved Example 4 : Find the inverse of the matrix by finding its

adjoint where A =

Solution : |A| = 1  0 A^1 exists

Now A =