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BCA 1ST YEAR MATHS NOTES...

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Karnataka State Open University

Study Material for BCA / IMCA Mathematics - Code - BCA 21 / IMCA 21 by

K.S. Srinivasa Retd. Principal & Professor of Mathematics Bangalore

Published by

Sharada Vikas Trust (R) Bangalore

BCA 21 / IMCA 21 / MATHEMATICS Syllabus 1.

Matrix Theory : Review of the fundamentals. Solution of linear equations by Cramers' Rule and by Matrix method, Eigen values and Eigen vectors, Cayley Hamilton's Theorem, Diagonalization of matrices, simple problems.

2.

Algebraic Structures Definition of a group, properties of groups, sub groups, permutation groups, simple problems, scalars & vectors, algebra of vectors, scalar & vector products, scalar triple product, simple problems.

3.

Analytical Geometry in three dimensions : Direction cosines and direction ratios, distance formula, section formula, equations to planes and straight lines, angles between two planes & two lines, problems, equation of a sphere, right circular cone and right circular cylinder, simple problems.

4.

Differential Calculus : Limits, continuity and differentiability (definition only), standard derivatives, rules for differentiation, derivatives of function of a function and parametric functions, problems. Successive differentiation, nth derivative of standard functions, statement of Leibnitz's Theorem, problems, statements of Rolle's, Lagrange's, Cauchy's and Taylor's Mean Value Theorems and simple problems, Indeterminate forms, L' Hospital's rule, partial derivatives, definition and simple problems.

5.

Integral Calculus Introduction, standard integrals, integration by substitution and by parts, integration of rational, irrational and trigonometric functions, definite integrals, properties (no proof), simple problems, reduction formulae and simple problems.

6.

Differential Equations of first order Introduction, solution by separation of variables, homogeneous equations, reducible to homogeneous linear equation, Bernoulli's equation, exact differential equations and simple problems.

Text Books 1.

Elementary Engineering Mathematics by Dr. B.S. Grewal, Khanna Publications

2.

Higher Engineering Mathematics by B.S. Grewal, Khanna Publications

Reference Books 1.

Differential Calculus by Shanti Narayan, Publishers S. Chand & Co.

2.

Integral Calculus by Shanti Narayan, Publishers S. Chand & Co.

3.

Modern Abstract Algebra by Shanti Narayan, Publishers S. Chand & Co.

CONTENTS Page Nos. 1.

Matrix Theory

01

2.

Algebraic Structures

21

3.

Analytical Geometry in three dimensions

43

4.

Differential Calculus

65

5.

Integral Calculus

96

6.

Differential Equations

118

MATRIX THEORY Review of the fundamentals A rectangular array of mn elements arranged in m rows & n columns is called a 'Matrix' of a order m by capital letters of The English Alphabet.

n matrices are denoted

Examples

Matrix of order 3

a1 b1 2 is a2 b2 a3 b3 a1 a2

Matrix of order 4

3 is

a3

b1 b2 b3 c1 c2 c3 d1 d 2 d3

Matrix of order 3

a1 b1 c1 3 is a 2 b 2 c 2 a3 b3 c3

Note :- Elements of Matrices are written in rows and columns with in the bracket ( ) or [ ].

Types of Matrices (1)

Equivalent Matrices : Two matrices are said to be equivalent if the order is the same.

(2)

Equal Matrices : Two matrices are said to be equal if the corresponding elements are equal.

(3)

Rectangular & Square Matrices : A matrix of order m

(4)

Row Matrix : A matrix having only one row is called Row Matrix.

(5)

Column Matrix : A matrix having only one column is called Column Matrix.

(6)

Null Matrix or Zero Matrix : A matrix in which all the elements are zeros is called Null Matrix or Zero Matrix denoted as O. [English alphabet O not zero where as elements are zeros]

(7)

Diagonal Matrix : A diagonal matrix is a square matrix in which all elements except the elements in the principal diagonal are zeros.

4 0 Example 0 6

n is said to be rectangular if m

n, square if m = n.

2 0 0 0 1 0 0 0 4

are diagonal matrices of order 2 & 3. (8)

Scalar Matrix : A diagonal matrix in which all the elements in the principal diagonal are same.

2 KSOU

Matrix Theory

8 0 0 0 8 0 0 0 8

4 0 0 4

Example

are Scalar Matrices of order 2 & 3. (9)

Unit Matrix or Identity Matrix : A diagonal matrix in which all the elements in the principal diagonal is 1 is called Unit Matrix or Identity Matrix denoted by I. 1 0 0 0 1 0

Example :

0 1

0 1 0 0

,

0 0 1 0 0 0 0 1

are unit matrices of order 2 & 4. (10) Transpose of a Matrix : If A is any matrix then the matrix obtained by interchanging the rows & columns of A is called 'Transpose of A and it is written as A' or AT.

a c e

Example : If A

A is of order 3

b d f

then A is

a

c

e

b

d

f

2 but A' is of order 2

3.

Matrix addition Two matrices can be added or subtracted if their orders are same. Example : If A

A B

A B

a1

b1

c1

a2

b2

c2

& B

a1 c1

c1

d1

e1

c2

d2

e2

b1 d 1

c 1 e1

a2 c 2 b2 d 2

c 2 e2

a1

c1

c1

b1 d1

a2 c 2 b2 d 2

e1

c2 e2

Matrix Multiplication If A is a matrix of order m p and B is matrix of order p n, then the product AB is defined and its order is m to be defined number of columns of A must be same as number of rows of B)

Example : Let A

then AB

a1 a2 a1 a2

b1 b2

1

b1

1

b2

which is of order 2

c1 & B c2

2

c1

2

c2

3

a1

3

a2

1

1

2

2

3

3

1

b1

1

b2

2

c1

3

2

c2

3

2.

Note :- If A is multiplied by A then AA is denoted as A2, AAA.... as A3 etc.

n. (ie. for AB

BCA 21 / IMCA 21 / Mathematics

SVT 3

Scalar Multiplication of a Matrix If A is a matrix of any order and K is a scalar (a constant), then KA represent a matrix in which every element of A is multiplied by K.

a1 a2 a3

Example : If A

b1 b2 b3

c1 c2 c3

then KA

Ka 1 Kb 1 Ka 2 Kb 2 Ka 3 Kb 3

Kc 1 Kc 2 Kc 3

Symmetric and Skew Symmetric Matrices Let A be a matrix of order n n an element in ith row and j th column can be denoted as aij. Hence a matrix of order n be denoted as (aij) or [aij] where i = 1, 2, .......n, j = 1, 2, .......n

n can

A matrix of order n n is said to be Symmetric if aij = aji and Skew Symmetric if aij = –aji or A is symmetric if A = AT or A = A', skew symmetric if A = –AT or A = –A' also A + A' is symmetric & A – A' is skew symmetric. Note :- In a skew symmetric matrix the elements in principal diagonal are all zeros.

Example : A

2 3 5 3 7 6 5 6 8 0 2 7

B

is symmetric where A

A

2 7 0 6 is skew symmetric where B 6 0

B

Determinant A determinant is defined as a mapping (function) from the set of square matrices to the set of real numbers. If A is a square matrix its determinant is denoted as A .

Example : Let A

a1

b1

c1

a1

b1

c1

a2 a3

b2 b3

c 2 then det. A or A c3

a 2 b2 a3 b3

c2 c3

Minors and Co-factors Let

A ( aij)

ie A

A

i 1, 2, 3

j 1, 2, 3

a11 a12 a13 a 21 a 22 a 23 a 31 a 32 a 33 a11 a21 a31

Consider

a12 a22 a32

a13 a23 a33

a22

a23

a32

a33

which is a determinant formed by leaning all the elements of row and column in which all lies. This

determinant is called Minor of a11. Thus we can form nine minors. In general if A is matrix of order n

n then minor of aij is

4 KSOU

Matrix Theory

obtained by leaning all the elements in the row and column in which aij lies in A .The order of this minor is n – 1where as the order of given determinant is n if this minor is multiplied by (–1)i + j then it is called Co-factors of aij.

Example : Let A

a11 a21 a31

a12 a22 a32

a13 a23 a33

a22 a32

Minor of a11

( 1) 1

Co - factor of a11

Minor of a21 is

a23 a33

a12

a13

a32

a33

Co - factor of a 21 is ( 1) 2

1 a 22

a 23

a 22 a 23

a 32 a 33

a32

1 a12

a13

a32

a33

a33

a12 a32

a13 a33

Value of a determinant Consider a matrix A of order n n. Consider all the elements of any row or column and multiply each element by its corresponding co-factor. Then the algebraic sum of the product is the value of the determinant. Example : Let A

a1 a2

b1 b2

Co-factor of a1 is b2 Co-factor of b1 is –a2

A

Let A

a1b2 b1a 2 a 1 b1 c 1 a 2 b2 c 2 a 3 b3 c 3

Co - factor of a1 is ( 1)1

1 b2

c2

b3

c3

a2

c2

a2 c2

a3

c3

a3

a2

b2

a2 b2

a3

b3

a3

Co - factor of b1 is ( 1) 1

2

1 Co - factor of c 1 is ( 1)

3

A

a b2 1 b3

c2 c3

a1( b2 c3 a1 b2 c3

b1

a2

c2

a3

c3

b3 c2 )

c1

b2 c 2 b3 c3

c3

b3

a2

b2

a3

b3

b1( a2 c3

a3 c2 )

c1( a2 b3

a3b2 )

a1 b3 c2 ) a2 b1 c3

a3 b1 c2

a2 b3 c1

a3 b2 c1

BCA 21 / IMCA 21 / Mathematics

SVT 5

Properties of determinants (1)

If the elements of any two rows or columns are interchanged then value of the determinant changes only in sign.

(2)

If the elements of two rows or columns are identical then the value of the determinant is zero.

(3)

If all the elements of any row or column is multipled by a constant K, then the value of the determinant is multipled by K.

(4)

If all the elements of any row or column are written as sum of two elements then the determinant can be written as sum of two determinants.

(5)

If all the elements of any row or column are multiplied by a constant and added to the corresponding elements of any other row or column then the value of the determinant donot alter.

Adjoint of a Matrix

Let A

a1 a2 a3

b1 b2 b3

c1 c2 c3

Let us denoted the co-factors of a1, b1, c1, a2, b2, c2, a3, b3, c3 as A1, B1, C1, A2, B2, C2, A3, B3, C3 transpose of matrix of co-factors is called Adjoint of the Matrix.

A1 A2 A3

Matrix of Co - factors

A1 Adjoint of A

A2

A3

B1 B2 C1 C2

B3 C3

Theorem A. adj.A

AI

a1 a2 a3

b1 b2 b3

A. adj.A

B1 B2 B3

C1 C2 C3

adj.A. A c 1 A1 c 2 B1 c 3 C1

a1 A1 b1 B1 a2 A1 b2 B1 a3 A1 b3 B1

c1 C1 c2 C1 c3 C1

Now a1 A1 b1B1 c1C1 Similarly a 2A 2 b 2B

A2 B2 C2

2

a1 c 2C

A3 B3 C3

a1 A2 b1 B2 c1C2 a2 A2 b2 B2 c2 C2 a3 A2 b3 B2 c3 C2 b2

c2

b3

c3

b1

a2

c2

a3

c3

a1 A3 b1 B3 c1C3 a2 A3 b2 B3 c2 C3 a3 A3 b3 B3 c3C3 c1

a2

b2

a3

b3

The value of the det. A.

2

a3 A3 b3 B3 c3C 3 a1 A2

b1 B2

c1 C2

a1

b1

c1

b3

c3

b1

a1

c1

a3

c3

c1

a1

b1

a 3 b3

a1( b1 c3

b3 c1)

b1( a1 c3

a3 c1 )

c1( a1 b3

a3 b1)

1a1 b3c

1a3b1c

1a1b3c

3a 1b1c

1a 3b 1c

3a 1b 1c

0

6 KSOU

Matrix Theory Similarly the other five elements of A adj.A is zero.

0 A.adj. A

0 0

0 0 where

A

0

1 0 0 .I

0 1 0 0 0 1

A. adj.A

adj. A

A I

Singular and Non-singular Matrices 0 and non-singular if A

A square matrix A is said to be singular if A

0.

Inverse of a Matrix Two non-singular matrices A & B of the same order is said to be inverse of each other if AB = I = BA. Inverse of A is denoted as A–1. Inverse of B is denoted as B–1 and further (AB)–1 = B–1A–1.

To find the inverse of A A.adj.A

A I

multiply by A 1 , AA 1 adj . .A ie adj. A

AA

1

A

1

Example : Find the inverse of

Let A

1 2 1

4 5 2

A A1

adj. A A

1 2 1

4 5 2

2 4 1

2 4 1 5 4

Matrix of Co - factors

2 1 4 2 2

1 2

4 5

4

( 5 8) (4 4 ) (16 10 ) 3 6 9 0 3 6 6 0 3

2 4 1 1

1 2

1 1 1 2 2

( 2 4) (1 2) (4 4 )

4

2

5

1 1

2 4

1 1

2 4

2

5

(4 5 ) ( 2 4) ( 5 8)

BCA 21 / IMCA 21 / Mathematics

3 0 6 6 3 0 9 6 3

adj.A

1 2 1

A

1

A

A

4 5 2

2 4 1

2

3 3

1

1( 5 8) 4( 2 4) 2( 4 5)

3 0 6 1 6 3 0 9 9 6 3

1 adj. A A 1

1

SVT 7

0

2

1 3 2 3

0 1

3 6 9

9 9 9

0

6

3 9 6 9

0 3

3 24 18 9

9

9

3

3

Solutions of Linear equations Cramer's Rule To solve the equations a1 x b1 y c1 z d1 a2 x b2 y c2 z

d2

a3 x b3 y c3 z

d3

Consider

a1

b1

c1

a2 a3

b2 b3

c2 c3

(1)

first evaluate & if it is not zero then multiply both sides of (1) by x.

x

a1 x a2 a3

b1 b2 b3

c1 c2 c3

a1 x b1 a 2 x b2 a 3x b3

c1 c2 c3

multiply the elements of columns 2 & 3 by y & z and add to elements of column 1.

then

x

a1 x b1 y c1 z b1 a2 x b2 y c2 z b2 a3 x b3 y c3 z b3

c1 c2 c3

d1 b1 c1 d 2 b2 c 2 d3 b3 c3

1 (say)

multiply both sides of (1) by y

y

a1 b1 ya b 2 2

c1 c2

a1 a2

b1 y c1 b2 y c2

a 3 b3

c3

a3

b3 y c3

(2)

8 KSOU

Matrix Theory

multiply the elements of columns 1 & 3 by x & z and add to the elements of column 2.

a1 a2 a3

a1x b1y c1z c 1 a2 x b2 y c 2 z c 2 a3 x b3 y c 3 z c 3

a1 d1 c1 a2 d 2 c2 a3 d 3 c3

2 (say)

(3)

multiply both sides of (1) by z

z

a1 z a2 a3

b1 b2 b3

a1 b1 c1 z a 2 b2 c 2 z a3 b3 c3 z

c1 c2 c3

multiply the elements of columns 1 & 2 by x & y and add to the elements of column 3.

a1 a2 a3

b1 b2 b3

a1 x b1 y c1 z a2 x b2 y c2 z a3 x b3 y c3 z

a1 a2 a3

b1 b2 b3

d1 d2 d3

1

then x

(4)

from (3)

3

z

(say)

from (2)

2

y

3

from (4)

Note :- Verification of values of x, y, z can be done by substituting in the given equations. Example - 1 Solve 2 x y z 3

x y z 1 x 2 y 3 y 4 2 1 1

Let

1 1 2

-1 1 3

(1)

2( 3 2) 1( 3 1) 1( 2 1)

2 4 3 5

multiply both sides of (1) by x

then

x

2 x1 1

-1 1 3

1 1 2

2x x x

1 1 2

-1 1 3

multiply the elements of columns 2 & 3 by y and z and add to the elements of column 1.

x

2x y z x y z x 2 y 3y

1 1 2

-1 1 3

BCA 21 / IMCA 21 / Mathematics

x

3 1 4

1 1 2

-1 1 3

10

10 5

2

3( 3 2) 1( 3 4) 1( 2 4)

SVT 9

3 7 6 10

multiply both sides of (1) by y

y

-1 1 3

2 3 y1 1 1 4

2 1 1

y y 2y

-1 1 3

multiply the elements of column 1 by x & 3 by z and to the corresponding elements of column 2.

then

2x y z

2 1

y

x y z x 2 y 3z

1

-1 1 3

2 3 1 1 1 4 5

y

5 5

-1 1 3

2( 3 4) 3( 3 1) 1( 4 1)

14 12 3

5

1

multiply both sides of (1) by z

...