CONTENTS
APPLIED MATHEMATICS
CHAPTER 1. ALGEBRA
1.1. Permutation and combination 3
1.2. Arithmetic, Geometric and Harmonic progression 8
1.3. Matrix Algebra and evaluation of determinants 14
1.4. Inverse of a matrix by adjoint method 29
1.5. Solution of simultaneous equations by Cramer’s rule & 31
inverse method
Review Questions 34
CHAPTER 2. DIFFERENTIAL CALCULUS I
2.1. Differentiation – definition – methods of differentiation 39
2.2. Higher order derivatives- Applications of differentiation 51
2.3. Partial differentiation –Homogeneous functions and 56
Euler’s Theorem
Review Questions 60
CHAPTER 3. DIFFERENTIAL CALCULUS II
3.1. Increasing and decreasing function- Maxima and 65
minima of single variables
3.2. Maxima and minima of several variables without constraints 70
3.3. Maxima and minima of several variables with constraints 72
Review Questions 74
1
,CHAPTER 4. INTEGRAL CALCULUS
4.1. Integration – methods of integration and definite integrals 79
4.2. Integration by parts -Application of integration in area 86
and volume.
Review Questions 90
CHAPTER 5. MATHEMATICAL MODELS
5.1. Agricultural systems - Mathematical models - 95
Classification of mathematical models
5.2. Linear, Quadratic, Exponential and Logistic models - 99
their applications in agriculture.
Review Questions 108
FORMULAE
2
, CHAPTER 1
ALGEBRA
1.1. Permutation and Combination
Permutation (or) Arrangement:
The different arrangements which can be made out of a given number of
things by taking some or all of them at a time are called permutations.
Meaning of n Pr:
The number of permutations (or arrangements) of ‘n’ things taken ‘r’ a time
(r ≤ n) is denoted by nPr (or) P (n,r)
(ie) Suppose we have ‘n’ objects, at each time we select ‘r’ objects and arrange
the selected ‘r’ objects, the total number of such arrangements is denoted by nP r.
Note:
The permutations of three letters a,b,c taken two at a time are ab,ba,bc,cb,ac,ca.
(ie) The number of these three letters taken two at a time is denoted by 3P 2 = 6.
The permutations of a,b,c taken all (three) at a time are abc, acb, bca, bac, cab,
cba.
(ie) The number of permutations of three letters taken three at a time is denoted
by 3P3 = 6.
Fundamental principle of counting:
(a) Addition Principle:
If an event can occur in ‘m’ different ways and a second event in ‘n’ different
3
, ways, then either of the two events can occur in ‘(m+n)’ ways provided only one
event can occur at a time.
(b) Multiplication Principle :
If an event can occur in ‘m’ different ways and a second event in ‘n’ different
ways, then both the events can occur simultaneously in ‘m x n’ different ways.
Factorial Notation:
The product 1.2.3...... n of the first n consecutive positive integers is usually
denoted by the symbol n! which is read as factorial n.
(ie) n! = n (n-1).......3.2.1.
Example: 4! = 1.2.3.4 = 24
5! = 1.2.3.4.5 = 120.
Note: 0! = 1.
Example: Consider 5P3
This stands for selection of ‘3’ from ‘5’ and then the arrangements of the
selected ‘3’ at each time.
Let the ‘5’ objects be a,b,c,d,e. The different selections with ‘3’ objects at a
time are
a b c
a b d
a b e
a c d
a c e
a d e
b c d
b d e
c d e
b c e
Each set can be arranged in 3! Ways.
Hence, the total number of arrangements is = 10 x 3! = 10 x 6 = 60.
(ie) 5 P3 = 60.
4
APPLIED MATHEMATICS
CHAPTER 1. ALGEBRA
1.1. Permutation and combination 3
1.2. Arithmetic, Geometric and Harmonic progression 8
1.3. Matrix Algebra and evaluation of determinants 14
1.4. Inverse of a matrix by adjoint method 29
1.5. Solution of simultaneous equations by Cramer’s rule & 31
inverse method
Review Questions 34
CHAPTER 2. DIFFERENTIAL CALCULUS I
2.1. Differentiation – definition – methods of differentiation 39
2.2. Higher order derivatives- Applications of differentiation 51
2.3. Partial differentiation –Homogeneous functions and 56
Euler’s Theorem
Review Questions 60
CHAPTER 3. DIFFERENTIAL CALCULUS II
3.1. Increasing and decreasing function- Maxima and 65
minima of single variables
3.2. Maxima and minima of several variables without constraints 70
3.3. Maxima and minima of several variables with constraints 72
Review Questions 74
1
,CHAPTER 4. INTEGRAL CALCULUS
4.1. Integration – methods of integration and definite integrals 79
4.2. Integration by parts -Application of integration in area 86
and volume.
Review Questions 90
CHAPTER 5. MATHEMATICAL MODELS
5.1. Agricultural systems - Mathematical models - 95
Classification of mathematical models
5.2. Linear, Quadratic, Exponential and Logistic models - 99
their applications in agriculture.
Review Questions 108
FORMULAE
2
, CHAPTER 1
ALGEBRA
1.1. Permutation and Combination
Permutation (or) Arrangement:
The different arrangements which can be made out of a given number of
things by taking some or all of them at a time are called permutations.
Meaning of n Pr:
The number of permutations (or arrangements) of ‘n’ things taken ‘r’ a time
(r ≤ n) is denoted by nPr (or) P (n,r)
(ie) Suppose we have ‘n’ objects, at each time we select ‘r’ objects and arrange
the selected ‘r’ objects, the total number of such arrangements is denoted by nP r.
Note:
The permutations of three letters a,b,c taken two at a time are ab,ba,bc,cb,ac,ca.
(ie) The number of these three letters taken two at a time is denoted by 3P 2 = 6.
The permutations of a,b,c taken all (three) at a time are abc, acb, bca, bac, cab,
cba.
(ie) The number of permutations of three letters taken three at a time is denoted
by 3P3 = 6.
Fundamental principle of counting:
(a) Addition Principle:
If an event can occur in ‘m’ different ways and a second event in ‘n’ different
3
, ways, then either of the two events can occur in ‘(m+n)’ ways provided only one
event can occur at a time.
(b) Multiplication Principle :
If an event can occur in ‘m’ different ways and a second event in ‘n’ different
ways, then both the events can occur simultaneously in ‘m x n’ different ways.
Factorial Notation:
The product 1.2.3...... n of the first n consecutive positive integers is usually
denoted by the symbol n! which is read as factorial n.
(ie) n! = n (n-1).......3.2.1.
Example: 4! = 1.2.3.4 = 24
5! = 1.2.3.4.5 = 120.
Note: 0! = 1.
Example: Consider 5P3
This stands for selection of ‘3’ from ‘5’ and then the arrangements of the
selected ‘3’ at each time.
Let the ‘5’ objects be a,b,c,d,e. The different selections with ‘3’ objects at a
time are
a b c
a b d
a b e
a c d
a c e
a d e
b c d
b d e
c d e
b c e
Each set can be arranged in 3! Ways.
Hence, the total number of arrangements is = 10 x 3! = 10 x 6 = 60.
(ie) 5 P3 = 60.
4
