Syllabus: Applied Mathematics for Class 12 | Applied Mathematics for Class 12 - Commerce PDF Download

 Page 1


Grade XII (2023-24) 
 
Number of Paper:   1                                                        
Total number of Periods:  240 (35 Minutes Each) 
Time:    3 Hours                                                                           
Max Marks:    80 
No. Units No. of 
Periods 
Marks 
I Numbers, Quantification and Numerical 
Applications 
30 11 
II Algebra 20 10 
III Calculus 50 15 
IV Probability Distributions 35 10 
V Inferential Statistics 10 05 
VI Index Numbers and Time-based data 30 06 
VII Financial Mathematics 50 15 
VIII Linear Programming 15 08 
Total 240 80 
Internal Assessment                             20 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Page 2


Grade XII (2023-24) 
 
Number of Paper:   1                                                        
Total number of Periods:  240 (35 Minutes Each) 
Time:    3 Hours                                                                           
Max Marks:    80 
No. Units No. of 
Periods 
Marks 
I Numbers, Quantification and Numerical 
Applications 
30 11 
II Algebra 20 10 
III Calculus 50 15 
IV Probability Distributions 35 10 
V Inferential Statistics 10 05 
VI Index Numbers and Time-based data 30 06 
VII Financial Mathematics 50 15 
VIII Linear Programming 15 08 
Total 240 80 
Internal Assessment                             20 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
CLASS XII 
Sl. No. Contents Learning Outcomes: 
Students will  be able to 
Notes / Explanation 
UNIT-1  NUMBERS, QUANTIFICATION AND  NUMERICAL APPLICATIONS 
1.1 Modulo 
Arithmetic 
? Define modulus of an integer 
? Apply arithmetic operations 
using modular arithmetic rules 
? Definition and meaning 
? Introduction to modulo 
operator 
? Modular addition and 
subtraction 
1.2 Congruence 
Modulo 
? Define congruence modulo 
? Apply the definition in various 
problems 
? Definition and meaning 
? Solution using congruence 
modulo  
? Equivalence class 
1.4 Alligation and 
Mixture  
? Understand the rule of alligation 
to  produce a mixture at a given 
price 
? Determine the mean price of a 
mixture 
? Apply rule of alligation 
? Meaning and Application of 
rule of alligation 
? Mean price of a mixture 
1.5 Numerical 
Problems 
Solve real life problems mathematically 
Boats and 
Streams 
(upstream and 
downstream) 
? Distinguish between upstream 
and downstream 
? Express the problem in the form 
of an equation 
?  Problems based on speed of 
stream and the speed of boat 
in still water 
Pipes and 
Cisterns 
? Determine the time taken by 
two or more pipes to fill or 
empty the tank 
? Calculation of the portion of 
the tank filled or drained by 
the pipe(s) in unit time 
Races and 
Games 
? Compare the performance of 
two players w.r.t. time, 
distance 
? Calculation of the time 
taken/ distance covered / 
speed of each player  
1.6 Numerical 
Inequalities 
? Describe the basic concepts of 
numerical inequalities 
? Understand and write numerical 
inequalities 
? Comparison between two 
statements/situations which 
can be compared numerically 
? Application of the techniques 
of numerical solution of 
algebraic inequations 
UNIT-2     ALGEBRA 
2.1 Matrices and 
types of 
matrices 
? Define matrix  
? Identify different kinds of 
matrices 
? Find the size / order of matrices 
? The entries, rows and 
columns of matrices 
? Present a set of data in a 
matrix form  
2.2 Equality of 
matrices, 
Transpose of a 
matrix, 
Symmetric and 
Skew 
symmetric 
matrix 
? Determine equality of two 
matrices 
? Write transpose of given matrix 
? Define symmetric and skew 
symmetric matrix 
? Examples of transpose of 
matrix 
? A square matrix as a sum of 
symmetric and skew 
symmetric matrix 
? Observe that diagonal 
elements of skew symmetric 
matrices are always zero 
 
Page 3


Grade XII (2023-24) 
 
Number of Paper:   1                                                        
Total number of Periods:  240 (35 Minutes Each) 
Time:    3 Hours                                                                           
Max Marks:    80 
No. Units No. of 
Periods 
Marks 
I Numbers, Quantification and Numerical 
Applications 
30 11 
II Algebra 20 10 
III Calculus 50 15 
IV Probability Distributions 35 10 
V Inferential Statistics 10 05 
VI Index Numbers and Time-based data 30 06 
VII Financial Mathematics 50 15 
VIII Linear Programming 15 08 
Total 240 80 
Internal Assessment                             20 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
CLASS XII 
Sl. No. Contents Learning Outcomes: 
Students will  be able to 
Notes / Explanation 
UNIT-1  NUMBERS, QUANTIFICATION AND  NUMERICAL APPLICATIONS 
1.1 Modulo 
Arithmetic 
? Define modulus of an integer 
? Apply arithmetic operations 
using modular arithmetic rules 
? Definition and meaning 
? Introduction to modulo 
operator 
? Modular addition and 
subtraction 
1.2 Congruence 
Modulo 
? Define congruence modulo 
? Apply the definition in various 
problems 
? Definition and meaning 
? Solution using congruence 
modulo  
? Equivalence class 
1.4 Alligation and 
Mixture  
? Understand the rule of alligation 
to  produce a mixture at a given 
price 
? Determine the mean price of a 
mixture 
? Apply rule of alligation 
? Meaning and Application of 
rule of alligation 
? Mean price of a mixture 
1.5 Numerical 
Problems 
Solve real life problems mathematically 
Boats and 
Streams 
(upstream and 
downstream) 
? Distinguish between upstream 
and downstream 
? Express the problem in the form 
of an equation 
?  Problems based on speed of 
stream and the speed of boat 
in still water 
Pipes and 
Cisterns 
? Determine the time taken by 
two or more pipes to fill or 
empty the tank 
? Calculation of the portion of 
the tank filled or drained by 
the pipe(s) in unit time 
Races and 
Games 
? Compare the performance of 
two players w.r.t. time, 
distance 
? Calculation of the time 
taken/ distance covered / 
speed of each player  
1.6 Numerical 
Inequalities 
? Describe the basic concepts of 
numerical inequalities 
? Understand and write numerical 
inequalities 
? Comparison between two 
statements/situations which 
can be compared numerically 
? Application of the techniques 
of numerical solution of 
algebraic inequations 
UNIT-2     ALGEBRA 
2.1 Matrices and 
types of 
matrices 
? Define matrix  
? Identify different kinds of 
matrices 
? Find the size / order of matrices 
? The entries, rows and 
columns of matrices 
? Present a set of data in a 
matrix form  
2.2 Equality of 
matrices, 
Transpose of a 
matrix, 
Symmetric and 
Skew 
symmetric 
matrix 
? Determine equality of two 
matrices 
? Write transpose of given matrix 
? Define symmetric and skew 
symmetric matrix 
? Examples of transpose of 
matrix 
? A square matrix as a sum of 
symmetric and skew 
symmetric matrix 
? Observe that diagonal 
elements of skew symmetric 
matrices are always zero 
 
2.3 Algebra of 
Matrices 
? Perform operations like addition 
& subtraction on matrices of 
same order 
? Perform multiplication of two 
matrices of appropriate order 
? Perform multiplication of a 
scalar with matrix 
? Addition and Subtraction of 
matrices 
? Multiplication of matrices (It 
can be shown to the students 
that Matrix multiplication is 
similar to multiplication of two 
polynomials) 
? Multiplication of a matrix with 
a real number 
2.4 Determinants ? Find determinant of a square 
matrix 
? Use elementary properties of 
determinants  
? Singular matrix, Non-singular 
matrix 
? |AB|=|A||B| 
? Simple problems to find 
determinant value  
2.5 Inverse of a 
matrix 
? Define the inverse of a square 
matrix  
? Apply properties of inverse of 
matrices 
? Inverse of a matrix using:  
 a)  cofactors 
If A and B are invertible 
square matrices of same size,   
i)  (AB)
-1
=B 
-1
A 
–1
  
ii)  (A
-1
)
-1
 =A 
iii) (A
T
)
-1 
= (A
-1
)
T 
 
2.6 Solving system 
of 
simultaneous 
equations 
using matrix 
method, 
Cramer’s rule 
and   
? Solve the system of 
simultaneous equations using 
i) Cramer’s Rule 
ii) Inverse of coefficient  matrix 
? Formulate real life problems 
into a system of simultaneous 
linear equations and solve it 
using these methods 
? Solution of system of  
simultaneous equations upto 
three variables only 
(non- homogeneous 
equations) 
UNIT- 3   CALCULUS 
Differentiation and its Applications 
3.1 Higher Order 
Derivatives 
? Determine second and higher 
order derivatives 
? Understand differentiation of 
parametric functions and  
implicit functions 
? Simple problems based on 
higher order derivatives 
?  Differentiation of parametric 
functions 
and implicit functions (upto 
2
nd
 order) 
3.2 Application of 
Derivatives 
? Determine the rate of change of 
various quantities 
? Understand the gradient of 
tangent and normal to a curve at 
a given point 
? Write the equation of tangents 
and normal to a curve at a given 
point 
? To find the rate of change of 
quantities such as area and 
volume with respect to time or 
its dimension 
? Gradient / Slope of tangent 
and normal to the curve 
? The equation of the tangent 
and normal to the curve 
(simple problems only) 
3.3 Marginal Cost 
and Marginal 
Revenue using 
derivatives 
? Define marginal cost and 
marginal revenue 
? Find marginal cost and marginal 
revenue 
? Examples related to marginal 
cost, marginal revenue, etc. 
 
Page 4


Grade XII (2023-24) 
 
Number of Paper:   1                                                        
Total number of Periods:  240 (35 Minutes Each) 
Time:    3 Hours                                                                           
Max Marks:    80 
No. Units No. of 
Periods 
Marks 
I Numbers, Quantification and Numerical 
Applications 
30 11 
II Algebra 20 10 
III Calculus 50 15 
IV Probability Distributions 35 10 
V Inferential Statistics 10 05 
VI Index Numbers and Time-based data 30 06 
VII Financial Mathematics 50 15 
VIII Linear Programming 15 08 
Total 240 80 
Internal Assessment                             20 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
CLASS XII 
Sl. No. Contents Learning Outcomes: 
Students will  be able to 
Notes / Explanation 
UNIT-1  NUMBERS, QUANTIFICATION AND  NUMERICAL APPLICATIONS 
1.1 Modulo 
Arithmetic 
? Define modulus of an integer 
? Apply arithmetic operations 
using modular arithmetic rules 
? Definition and meaning 
? Introduction to modulo 
operator 
? Modular addition and 
subtraction 
1.2 Congruence 
Modulo 
? Define congruence modulo 
? Apply the definition in various 
problems 
? Definition and meaning 
? Solution using congruence 
modulo  
? Equivalence class 
1.4 Alligation and 
Mixture  
? Understand the rule of alligation 
to  produce a mixture at a given 
price 
? Determine the mean price of a 
mixture 
? Apply rule of alligation 
? Meaning and Application of 
rule of alligation 
? Mean price of a mixture 
1.5 Numerical 
Problems 
Solve real life problems mathematically 
Boats and 
Streams 
(upstream and 
downstream) 
? Distinguish between upstream 
and downstream 
? Express the problem in the form 
of an equation 
?  Problems based on speed of 
stream and the speed of boat 
in still water 
Pipes and 
Cisterns 
? Determine the time taken by 
two or more pipes to fill or 
empty the tank 
? Calculation of the portion of 
the tank filled or drained by 
the pipe(s) in unit time 
Races and 
Games 
? Compare the performance of 
two players w.r.t. time, 
distance 
? Calculation of the time 
taken/ distance covered / 
speed of each player  
1.6 Numerical 
Inequalities 
? Describe the basic concepts of 
numerical inequalities 
? Understand and write numerical 
inequalities 
? Comparison between two 
statements/situations which 
can be compared numerically 
? Application of the techniques 
of numerical solution of 
algebraic inequations 
UNIT-2     ALGEBRA 
2.1 Matrices and 
types of 
matrices 
? Define matrix  
? Identify different kinds of 
matrices 
? Find the size / order of matrices 
? The entries, rows and 
columns of matrices 
? Present a set of data in a 
matrix form  
2.2 Equality of 
matrices, 
Transpose of a 
matrix, 
Symmetric and 
Skew 
symmetric 
matrix 
? Determine equality of two 
matrices 
? Write transpose of given matrix 
? Define symmetric and skew 
symmetric matrix 
? Examples of transpose of 
matrix 
? A square matrix as a sum of 
symmetric and skew 
symmetric matrix 
? Observe that diagonal 
elements of skew symmetric 
matrices are always zero 
 
2.3 Algebra of 
Matrices 
? Perform operations like addition 
& subtraction on matrices of 
same order 
? Perform multiplication of two 
matrices of appropriate order 
? Perform multiplication of a 
scalar with matrix 
? Addition and Subtraction of 
matrices 
? Multiplication of matrices (It 
can be shown to the students 
that Matrix multiplication is 
similar to multiplication of two 
polynomials) 
? Multiplication of a matrix with 
a real number 
2.4 Determinants ? Find determinant of a square 
matrix 
? Use elementary properties of 
determinants  
? Singular matrix, Non-singular 
matrix 
? |AB|=|A||B| 
? Simple problems to find 
determinant value  
2.5 Inverse of a 
matrix 
? Define the inverse of a square 
matrix  
? Apply properties of inverse of 
matrices 
? Inverse of a matrix using:  
 a)  cofactors 
If A and B are invertible 
square matrices of same size,   
i)  (AB)
-1
=B 
-1
A 
–1
  
ii)  (A
-1
)
-1
 =A 
iii) (A
T
)
-1 
= (A
-1
)
T 
 
2.6 Solving system 
of 
simultaneous 
equations 
using matrix 
method, 
Cramer’s rule 
and   
? Solve the system of 
simultaneous equations using 
i) Cramer’s Rule 
ii) Inverse of coefficient  matrix 
? Formulate real life problems 
into a system of simultaneous 
linear equations and solve it 
using these methods 
? Solution of system of  
simultaneous equations upto 
three variables only 
(non- homogeneous 
equations) 
UNIT- 3   CALCULUS 
Differentiation and its Applications 
3.1 Higher Order 
Derivatives 
? Determine second and higher 
order derivatives 
? Understand differentiation of 
parametric functions and  
implicit functions 
? Simple problems based on 
higher order derivatives 
?  Differentiation of parametric 
functions 
and implicit functions (upto 
2
nd
 order) 
3.2 Application of 
Derivatives 
? Determine the rate of change of 
various quantities 
? Understand the gradient of 
tangent and normal to a curve at 
a given point 
? Write the equation of tangents 
and normal to a curve at a given 
point 
? To find the rate of change of 
quantities such as area and 
volume with respect to time or 
its dimension 
? Gradient / Slope of tangent 
and normal to the curve 
? The equation of the tangent 
and normal to the curve 
(simple problems only) 
3.3 Marginal Cost 
and Marginal 
Revenue using 
derivatives 
? Define marginal cost and 
marginal revenue 
? Find marginal cost and marginal 
revenue 
? Examples related to marginal 
cost, marginal revenue, etc. 
 
3.4 Increasing 
/Decreasing 
Functions 
? Determine whether a function is 
increasing or decreasing 
? Determine the conditions for a 
function to be increasing or 
decreasing 
? Simple problems related to 
increasing and decreasing 
behaviour of a function in the 
given interval 
3.5 Maxima and 
Minima 
? Determine critical points of the 
function 
? Find the point(s) of local maxima 
and local minima and 
corresponding  local maximum 
and local minimum values 
? Find the absolute maximum and 
absolute minimum value of a 
function  
?  Solve applied problems 
? A point x= c is called the 
critical point of f if 
   f is defined at c and 
   f
'
(c) =
0 or f is not differentiable 
   at c 
? To find local maxima and local 
minima by: 
i)  First Derivative Test  
ii) Second Derivative Test 
? Contextualized real life 
problems 
 
Integration and its Applications 
3.6 Integration 
? Understand and determine  
indefinite integrals  of simple 
functions as anti-derivative 
 
? Integration as a reverse 
process of differentiation 
? Vocabulary and Notations 
related to Integration 
3.7 Indefinite 
Integrals as 
family of 
curves 
? Evaluate indefinite integrals of 
simple algebraic functions by 
method of: 
i) substitution 
ii) partial fraction 
iii) by parts 
? Simple integrals based on 
each method (non-
trigonometric function) 
 
3.8 Definite 
Integrals as 
area under the 
curve 
? Define definite integral as area 
under the curve 
? Understand fundamental 
theorem of Integral calculus and 
apply it to evaluate the definite 
integral 
? Apply properties of definite 
integrals to solve the problems 
? Evaluation of definite integrals 
using properties 
3.9 Application of 
Integration 
? Identify the region representing 
C.S. and P.S. graphically  
? Apply the definite integral to find 
consumer surplus-producer 
surplus 
Problems based on finding  
? Total cost when Marginal Cost  
is given 
? Total Revenue when Marginal 
Revenue is given   
? Equilibrium price and 
equilibrium quantity and hence 
consumer and producer 
surplus 
 
Differential Equations and Modeling 
3.10 Differential 
Equations 
? Recognize a differential 
equation  
? Find the order and degree of a 
differential equation 
? Definition, order, degree and 
examples 
Page 5


Grade XII (2023-24) 
 
Number of Paper:   1                                                        
Total number of Periods:  240 (35 Minutes Each) 
Time:    3 Hours                                                                           
Max Marks:    80 
No. Units No. of 
Periods 
Marks 
I Numbers, Quantification and Numerical 
Applications 
30 11 
II Algebra 20 10 
III Calculus 50 15 
IV Probability Distributions 35 10 
V Inferential Statistics 10 05 
VI Index Numbers and Time-based data 30 06 
VII Financial Mathematics 50 15 
VIII Linear Programming 15 08 
Total 240 80 
Internal Assessment                             20 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
CLASS XII 
Sl. No. Contents Learning Outcomes: 
Students will  be able to 
Notes / Explanation 
UNIT-1  NUMBERS, QUANTIFICATION AND  NUMERICAL APPLICATIONS 
1.1 Modulo 
Arithmetic 
? Define modulus of an integer 
? Apply arithmetic operations 
using modular arithmetic rules 
? Definition and meaning 
? Introduction to modulo 
operator 
? Modular addition and 
subtraction 
1.2 Congruence 
Modulo 
? Define congruence modulo 
? Apply the definition in various 
problems 
? Definition and meaning 
? Solution using congruence 
modulo  
? Equivalence class 
1.4 Alligation and 
Mixture  
? Understand the rule of alligation 
to  produce a mixture at a given 
price 
? Determine the mean price of a 
mixture 
? Apply rule of alligation 
? Meaning and Application of 
rule of alligation 
? Mean price of a mixture 
1.5 Numerical 
Problems 
Solve real life problems mathematically 
Boats and 
Streams 
(upstream and 
downstream) 
? Distinguish between upstream 
and downstream 
? Express the problem in the form 
of an equation 
?  Problems based on speed of 
stream and the speed of boat 
in still water 
Pipes and 
Cisterns 
? Determine the time taken by 
two or more pipes to fill or 
empty the tank 
? Calculation of the portion of 
the tank filled or drained by 
the pipe(s) in unit time 
Races and 
Games 
? Compare the performance of 
two players w.r.t. time, 
distance 
? Calculation of the time 
taken/ distance covered / 
speed of each player  
1.6 Numerical 
Inequalities 
? Describe the basic concepts of 
numerical inequalities 
? Understand and write numerical 
inequalities 
? Comparison between two 
statements/situations which 
can be compared numerically 
? Application of the techniques 
of numerical solution of 
algebraic inequations 
UNIT-2     ALGEBRA 
2.1 Matrices and 
types of 
matrices 
? Define matrix  
? Identify different kinds of 
matrices 
? Find the size / order of matrices 
? The entries, rows and 
columns of matrices 
? Present a set of data in a 
matrix form  
2.2 Equality of 
matrices, 
Transpose of a 
matrix, 
Symmetric and 
Skew 
symmetric 
matrix 
? Determine equality of two 
matrices 
? Write transpose of given matrix 
? Define symmetric and skew 
symmetric matrix 
? Examples of transpose of 
matrix 
? A square matrix as a sum of 
symmetric and skew 
symmetric matrix 
? Observe that diagonal 
elements of skew symmetric 
matrices are always zero 
 
2.3 Algebra of 
Matrices 
? Perform operations like addition 
& subtraction on matrices of 
same order 
? Perform multiplication of two 
matrices of appropriate order 
? Perform multiplication of a 
scalar with matrix 
? Addition and Subtraction of 
matrices 
? Multiplication of matrices (It 
can be shown to the students 
that Matrix multiplication is 
similar to multiplication of two 
polynomials) 
? Multiplication of a matrix with 
a real number 
2.4 Determinants ? Find determinant of a square 
matrix 
? Use elementary properties of 
determinants  
? Singular matrix, Non-singular 
matrix 
? |AB|=|A||B| 
? Simple problems to find 
determinant value  
2.5 Inverse of a 
matrix 
? Define the inverse of a square 
matrix  
? Apply properties of inverse of 
matrices 
? Inverse of a matrix using:  
 a)  cofactors 
If A and B are invertible 
square matrices of same size,   
i)  (AB)
-1
=B 
-1
A 
–1
  
ii)  (A
-1
)
-1
 =A 
iii) (A
T
)
-1 
= (A
-1
)
T 
 
2.6 Solving system 
of 
simultaneous 
equations 
using matrix 
method, 
Cramer’s rule 
and   
? Solve the system of 
simultaneous equations using 
i) Cramer’s Rule 
ii) Inverse of coefficient  matrix 
? Formulate real life problems 
into a system of simultaneous 
linear equations and solve it 
using these methods 
? Solution of system of  
simultaneous equations upto 
three variables only 
(non- homogeneous 
equations) 
UNIT- 3   CALCULUS 
Differentiation and its Applications 
3.1 Higher Order 
Derivatives 
? Determine second and higher 
order derivatives 
? Understand differentiation of 
parametric functions and  
implicit functions 
? Simple problems based on 
higher order derivatives 
?  Differentiation of parametric 
functions 
and implicit functions (upto 
2
nd
 order) 
3.2 Application of 
Derivatives 
? Determine the rate of change of 
various quantities 
? Understand the gradient of 
tangent and normal to a curve at 
a given point 
? Write the equation of tangents 
and normal to a curve at a given 
point 
? To find the rate of change of 
quantities such as area and 
volume with respect to time or 
its dimension 
? Gradient / Slope of tangent 
and normal to the curve 
? The equation of the tangent 
and normal to the curve 
(simple problems only) 
3.3 Marginal Cost 
and Marginal 
Revenue using 
derivatives 
? Define marginal cost and 
marginal revenue 
? Find marginal cost and marginal 
revenue 
? Examples related to marginal 
cost, marginal revenue, etc. 
 
3.4 Increasing 
/Decreasing 
Functions 
? Determine whether a function is 
increasing or decreasing 
? Determine the conditions for a 
function to be increasing or 
decreasing 
? Simple problems related to 
increasing and decreasing 
behaviour of a function in the 
given interval 
3.5 Maxima and 
Minima 
? Determine critical points of the 
function 
? Find the point(s) of local maxima 
and local minima and 
corresponding  local maximum 
and local minimum values 
? Find the absolute maximum and 
absolute minimum value of a 
function  
?  Solve applied problems 
? A point x= c is called the 
critical point of f if 
   f is defined at c and 
   f
'
(c) =
0 or f is not differentiable 
   at c 
? To find local maxima and local 
minima by: 
i)  First Derivative Test  
ii) Second Derivative Test 
? Contextualized real life 
problems 
 
Integration and its Applications 
3.6 Integration 
? Understand and determine  
indefinite integrals  of simple 
functions as anti-derivative 
 
? Integration as a reverse 
process of differentiation 
? Vocabulary and Notations 
related to Integration 
3.7 Indefinite 
Integrals as 
family of 
curves 
? Evaluate indefinite integrals of 
simple algebraic functions by 
method of: 
i) substitution 
ii) partial fraction 
iii) by parts 
? Simple integrals based on 
each method (non-
trigonometric function) 
 
3.8 Definite 
Integrals as 
area under the 
curve 
? Define definite integral as area 
under the curve 
? Understand fundamental 
theorem of Integral calculus and 
apply it to evaluate the definite 
integral 
? Apply properties of definite 
integrals to solve the problems 
? Evaluation of definite integrals 
using properties 
3.9 Application of 
Integration 
? Identify the region representing 
C.S. and P.S. graphically  
? Apply the definite integral to find 
consumer surplus-producer 
surplus 
Problems based on finding  
? Total cost when Marginal Cost  
is given 
? Total Revenue when Marginal 
Revenue is given   
? Equilibrium price and 
equilibrium quantity and hence 
consumer and producer 
surplus 
 
Differential Equations and Modeling 
3.10 Differential 
Equations 
? Recognize a differential 
equation  
? Find the order and degree of a 
differential equation 
? Definition, order, degree and 
examples 
3.11 Formulating 
and Solving 
Differential 
Equations 
? Formulate differential equation 
? Verify the solution of differential 
equation 
? Solve simple differential 
equation 
? Formation of differential 
equation by eliminating 
arbitrary constants 
? Solution of simple differential 
equations (direct integration 
only) 
3.12 Application of 
Differential 
Equations 
? Define Growth and Decay Model 
? Apply the differential equations 
to solve Growth and Decay 
Models 
? Growth and Decay Model in 
Biological sciences, 
Economics and business, etc. 
UNIT- 4   PROBABILITY DISTRIBUTIONS 
4.1 Probability 
Distribution 
? Understand the concept of 
Random Variables and its 
Probability Distributions 
? Find probability distribution of 
discrete random variable 
? Definition and example of 
discrete and continuous 
random variable and their 
distribution 
 
4.2 Mathematical 
Expectation 
? Apply arithmetic mean of 
frequency distribution to  find the 
expected value of a random 
variable 
? The expected value of discrete 
random variable as summation 
of product of discrete random 
variable by the probability of its 
occurrence. 
4.3 Variance ? Calculate the Variance and S.D. 
of a random variable 
? Questions based on variance 
and standard deviation 
4.4 Binomial 
Distribution 
? Identify the Bernoulli Trials and 
apply Binomial Distribution 
? Evaluate Mean, Variance and 
S.D of a binomial distribution 
 
 
? Characteristics of the binomial 
distribution 
? Binomial formula: 
   P(r) = 
n
C r  p
r 
q
n-r
 
   Where n = number of trials 
                P = probability of 
success 
                q = probability of 
failure 
   Mean =np  
   Variance = npq 
   Standard Deviation = v?????? 
4.5 Poison 
Distribution 
? Understand the Conditions of 
Poisson Distribution 
? Evaluate the Mean and 
Variance of Poisson distribution 
? Characteristics of Poisson   
Probability distribution 
   Poisson formula: 
   P(x) = 
?? ?? .  ?? -?? ?? !
   
? Mean = Variance = ?? 
4.6 Normal 
Distribution 
? Understand normal distribution 
is a Continuous distribution  
? Evaluate value of Standard 
normal variate  
? Area relationship between Mean 
and Standard Deviation 
 
? Characteristics of a normal 
probability distribution 
? Total area under the curve = 
total probability = 1 
? Standard Normal Variate: 
 Z = 
?? - ?? ??  where  
 x = value of the random variable 
 ?? = mean 
?? = S.D.