Syllabus of Applied Mathematics 2020-21

Syllabus 

Course Title: Applied Mathematics 

(Code-241) 

Grade XI-XII 

Secondary School Education prepares students to explore future career options after graduating  from  the  school.  Mathematics  is  an  important  subject  helps  students  to choose various fields of their choices. Mathematics is widely used in higher studies in the  field  of  Economics,  Commerce,  Social  Sciences  and  many  other.  It  has  been observed  that  the  syllabus  of  Mathematics  meant  for  Science  subjects  may  not  be appropriate for the students pursuing Commerce or Social Science-based subjects in university  education.  By  keeping  this  in  mind,  one  more  elective  course  in Mathematics syllabus is developed for Sr. Secondary classes with an aim to provide students  relevant  experience  in  Mathematics  which  can  be  used  in  the  fields  other than Physical Sciences. This  course  is  designed  to  develop  substantial  mathematical  skills  and  methods needing in other subject areas. Topics covered in two years aim to enable students
to  use  mathematical  knowledge  in  the  field  of  business,  economic  and  social sciences. It aims to promote appreciation of mathematical power and simplicity for its countless  applications  in  diverse  fields.  The  course  continues  to  develop mathematical  language  and  symbolism  to  communicate  and  relate  everyday experiences mathematically.  In addition, it reinforces the  logical reasoning skills of formulating  and  validating  mathematical  arguments,  framing  examples,  finding counter examples. It encourages students to engage in mathematical investigations and to build connections within mathematical topics and with other disciplines. The course  prepares  students  to  use  algebraic  methods  as  a  means  of  representation and as a problem-solving tool. It also enables students to interpret two dimensional geometrical  figures  using  algebra  and  to  further  deduce  properties  of  geometrical figures in coordinate system. The course content will help students to develop sound understanding of descriptive and inferential statistics which they can use to describe and analysis a give set of data and to further make meaningful inferences out of it. Data  based  case  studies  from  the  field  of  business,  economics,  psychology, education, biology and census data will be used to appreciate the power of data in contemporary society. It is expected that the subject is taught connecting concepts to the application in various fields. The objectives of the course areas are as follows:
Objectives:
a)  To  develop  an  understanding  of  basic  mathematical  and  statistical  tools and  their applications  in  the  field  of  commerce  (business/ finance/economics) and social sciences;
b)  To model real world experiences/problems into mathematical expressions using numerical/algebraic / graphical representation;
c)  To  make  sense  from  the  data  by  organizing,  representing,  interpreting, analysing,  and  to  make  meaningful  inferences  from  the  real-world situations;
d)  To develop logical reasoning skills and apply the same in simple problem solving;
e)  To  reinforce  mathematical  communication  by  formulating  conjectures,
validating logical arguments and testing hypothesis;
f)  To make connections between Mathematics and other disciplines.

Unit I Numbers, Quantification and Numerical Applications 
 Prime Numbers, Encryptions using Prime Numbers
 Binary Numbers
 Complex Numbers (Preliminary idea only)
 Indices, Logarithm and Antilogarithm
 Laws and properties of logarithms
 Simple applications of logarithm and antilogarithm
 Numerical problems on averages, calendar, clock, time, work and distance,
mensuration, seating arrangement

Unit II Algebra 
 Sets
 Types of sets
 Venn diagram
  De Morgan's laws
 Problem solving using Venn diagram
 Relations and types of relations
 Introduction of Sequences, Series
 Arithmetic and Geometric progression
 Relationship between AM and GM
 Basic concepts of Permutations and Combinations
 Permutations, Circular Permutations, Permutations with restrictions
 Combinations with standard results

Unit III Mathematical and Logical Reasoning 

 Mathematically acceptable statements
 Connecting words/ phrases in Mathematical statement consolidating the
understanding of "if and only if (necessary and sufficient) condition", "implies",
"and/or", "implied by", "and", "or", "there exists" and their use through variety
of examples related to real life and Mathematics
 Problems based on logical reasoning (coding-decoding, odd man out, blood
relation, syllogism etc)

Unit IV Calculus 

 Introducing functions
 Domain and Range of a function
 Types of functions (Polynomial function; Rational function; Composite
function; Logarithm function; Exponential function; Modulus function; Greatest
Integer function, Signum function)
 Graphical representation of functions
 Concept of limits and continuity of a function
 Instantaneous rates of change
 Differentiation as a process of finding derivative
 Derivatives of algebraic functions using Chain rule
 Tangent line and equations of tangents

Unit V Probability 

 Random experiment, sample space, events, mutually exclusive events
 Independent and Dependent Events
 Law of Total Probability
 Bayes’ Theorem

Unit VI Descriptive Statistics 

 Types of data (raw data, univariate data, bivariate and multi-variate data)
 Data on various scales (nominal, ordinal, interval and ratio scale)
 Data representation and visualization
 Data interpretation (central tendency, dispersion, deviation, variance, skewness and kurtosis)
 Percentile rank and quartile rank
 Correlation (Pearson and Spearman method of correlation)
 Applications of descriptive statistics using real time data

Unit VII Basics of Financial Mathematics 

 Interest and interest rate
 Accumulation with simple and compound interest
 Simple and compound interest rates with equivalency
 Effective rate of interest
 Present value, net present value and future value
 Annuities, calculating value of regular annuity
 Simple applications of regular annuities (up to 3 period)
 Tax, calculation of tax and simple applications of tax calculation in Goods and
service tax, Income Tax
 Bills, tariff rates, fixed charge, surcharge, service charge
 Calculation and interpretation of electricity bill, water supply bill and other
supply bills
(Comparing interest rates on various types of savings; calculating income tax;
electricity bills, water bill; service surcharge using realistic data)

Unit VIII Coordinate Geometry 

 Straight Line
 Circles
 Parabola
(only standard forms and graphical representation on two-dimensional plane) 

Practical: Use of spread sheet 

Calculating average, interest (simple and compound), creating pictographs, drawing
pie chart, bar graphs, calculating central tendency; visualizing graphs (straight line,
circles and parabola using real time data)

Suggested practical using spread sheet 

1.  Plot the graph of functions on excel; study the nature of function at various
points, drawing lines of tangents;
2.  Create budget of income and spending;
3.  Create compare sheet of price, features to buy a product;
4.  Prepare best option plan to buy a product by comparing cost, shipping
charges, tax and other hidden cost;
5.  Smart purchasing during sale season;
6.  Prepare a report card using scores of last four exams and compare the
performance;
7.  Collect the data on weather, price, inflation, and pollution. Sketch different
types of graphs.

Unit I Numbers, Quantification and Numerical Applications 

 Modulo Arithmetic
 Congruence modulo
 Simple arithmetic functions
 Allegation or Mixture
 Numerical problems on boats and streams; partnership; pipes and cistern;races and games, scheduling
 Numerical inequalities

Unit II Algebra 

 Solution of simultaneous linear equations using elimination method (up to 3 variables)
 Matrices and types of matrices
 Algebra of matrices
 Determinants
 Inverse of a matrix
 Cramer’s rule and its application
 Simple applications of matrices and determinants including Leontiff input
output model for two variables

Unit III Calculus 

 Application of derivatives
 Increasing/Decreasing functions
 Maxima and Minima
 Marginal cost and marginal revenue using derivatives
 Integration
 Indefinite integral as family of curves
 Definite integral as area under the curve
 Integration of simple algebraic functions (primitive, by substitution, by parts)
 Application of Integration (consumer surplus-producer surplus)
 Differential equation (definition, order, degree)
 Formulating and solving linear differential equation
 Application of differential equation (Growth and Decay Model)

Unit IV Probability 

 Probability Distribution
 Mathematical Expectation
 Variance
 Binomial Distribution
 Poisson distribution
 Normal distribution
 Basic applications and inferences

Unit V Inferential Statistics 

 Population and sample
 Parameter, statistic and statistical inferences
 t-Test (one sample t-test and two independent groups t-test)

Unit VI Index numbers and Time-based data 

 Index numbers, uses of index numbers
 Construction of index numbers (simple and weighted)
 Tests of adequacy of index numbers (unit test and time reversal test)
 Time series, Time series analysis for univariant data sets
 Trend analysis by moving average method
 Trend analysis by fitting of linear trend line using least squares

Unit VII Financial Mathematics 

 Perpetuity, Sinking funds
 Valuation of Bonds (Present value approach and Relative price approach)
 Calculation of EMI
 Calculation of returns, nominal rate of return, effective rate of interest
 Compound annual growth rate
 Stock, shares and debentures
 Linear method of depreciation

Unit VIII Linear Programming 

 Introduction and related terminologies (constraints, objective function, optimization)
 Mathematical formulation of linear programming problems
 Different types of linear programming problems (Transportation and assignment problem)
 Graphical method of solution for problems in two variables
 Feasible and infeasible regions (bounded and unbounded)
 Feasible and infeasible solutions, optimal feasible solutions (up to three nontrivial constraints)

Practical: Use of spread sheet 

Graphs of exponential function, demand and supply functions on Excel and study the
nature of function at various points, maxima/minima
Matrix operations using Excel

Suggested practical using the spreadsheet 

1.  Plot the graphs of functions on excel and study the graph to find out point of maxima/minima;
2.  Probability and dice roll simulation;
3.  Matrix multiplication and inverse of a matrix;
4.  Stock Market data sheet on excel;
5.  Collect the data on weather, price, inflation, and pollution; analyze the data and make meaningful inferences;
6.  Collect data from newspapers on traffic, sports activities and on market trends and use excel to study future trends.

List of Suggested projects ( class XI /XII) 

Use of prime numbers in coding and decoding of messages;
Prime numbers and divisbility rules;
Logrithms  for  financial  calculations  such  as  interest,  present  value,  future  vale,
profit/loss etc with large values);
Cardinality of a set and orders of infinity;
Comparing sets of Natural numbers, rational numbers, real numbers and others;
Use of Venn Diagram in solving practical problems;
Fibonacci Sequence: Its’ history and presence in nature;
Testing the validity of mathematical statements and framing truth tables;
Investigating graphs of functions for their properties;
Visit the census site of India
http://www.censusindia.gov.in/Census_Data_2001/Census_Data_Online/Language/
State ment3.htm Depict the information given there in a pictorial form;
Prepare a questionnaire to collect information about money spent by your friends in
a month on activities like traveling, movies, recharging of the mobiles, etc. and draw
interesting conclusions;
Check  out  the  local  newspaper  and  cut  out  examples  of  information  depicted  by
graphs. Draw your own conclusions from the graph and compare it with the analysis
given in the report;
Analysis  of  population  migration  data  –  positive  and  negative  influence  on
urbanization;
Each  day  newspaper  tells  us  about  the  maximum  temperature,  minimum
temperature,  humidity.  Collect  the  data  for  a  period  of  30  days  and  represent  it
graphically.  Compare  it  with  the  data  available  for  the  same  time  period  for  the
previous year;
Analysis  career  graph  of  a  cricketer  (batting  average  for  a  batsman  and  bowling
average for a bowler). Conclude the best year of his career. It may be extended for
other players also – tennis, badminton, athlete;
Vehicle registration data – correlating with pollution and number of accidents;
Visit a village near Delhi and collect data of various crops over past few years from
the farmers. Also collect data about temperature variation and rain over the period
for  a  particular  crop.  Try  to  find  the  effect  of  temperature  and  rain  variations  on
various crops;
Choose any week of your ongoing semester. Collect data for the past 10 – 15 years
for  the  amount  of  rainfall  received  in  Delhi  during  that  week.  Predict  amount  of
rainfall for the current year;
Weather prediction (prediction of monsoon from past data);
Visit Kirana shops near your home and collect the data of sale of certain
commodities over a month. Try to figure out the stock of a particular commodity
which should be in the store in order to maximize the profit;
Stock price movement ;
Risk assessments by insurance firms from data;
Predicting stock market crash;
Predicting outcome of election – exit polls;
Predicting mortality of infants.
Assessment Plan
1.  Overall Assessment of the course is out of 100 marks.
2.  Assessment plan consists of External Exam and Internal Assessment.
3.  External Exam will be of 03 hours duration Paper/Pencil Test consisting of
80 marks.
4.  Weightage of Internal Assessment is of 20 marks. Internal Assessment can
be a combination of activities spread throughout semester/ academic year.
Internal  Assessment  activities  include,  projects  and  excel  based  practical.
Teachers  can  choose  activities  from  the  suggested  list  of  practical  or  they
can plan activities of similar nature. For data based practical, teachers are
encouraged  to  use  data  from  local  sources  to  make  it  more  relevant  for
students.
5.  Weightage for each area of internal assessment may be as under:
Sr.No.  Area and weightage  Assessment Area  Marks
allocated
1  Project work
(10 marks)
Project work and record  5
Year  End  Presentation/Viva  of  the
Project
5
2  Practical work
(10 marks)
Performance of practical and record  5
Yearend test of any one practical  5
Total  20