Mathematics and Statistics 12th Board Exam HSC Commerce (Marketing and Salesmanship) Maharashtra State Board Topics and Syllabus

• Mathematical Logic 
  • Statements - Introduction in Logic 
• Logical Connective, Simple and Compound Statements 
  • Concept of Statements

  1. Conjunction (∧)
  2. Disjunction (∨)
  3. Conditional statement (Implication) (→)
  4. Biconditional (Double implication) (↔) or (⇔)
  5. Negation (∼)
• Statement Patterns and Logical Equivalence 
• Algebra of Statements 
  • Idempotent law
  • Associative law
  • Commutative law
  • Distributive law
  • Identity law
  • Complement law
  • Involution law
  • DeMorgan’s laws
• Venn Diagrams 

• Statements
• Logical Connectives
• Statement patterns and logical equivalence
• Algebra of statements
• Venn diagrams

• Truth Value of Statement 
• Logical Connective, Simple and Compound Statements 
  • Concept of Statements

  1. Conjunction (∧)
  2. Disjunction (∨)
  3. Conditional statement (Implication) (→)
  4. Biconditional (Double implication) (↔) or (⇔)
  5. Negation (∼)
• Quantifier and Quantified Statements in Logic 
  • Universal quantifier (∀)
  • Existential quantifier (∃)
• Statement Patterns and Logical Equivalence 
• Algebra of Statements 
  • Idempotent law
  • Associative law
  • Commutative law
  • Distributive law
  • Identity law
  • Complement law
  • Involution law
  • DeMorgan’s laws
• Venn Diagrams 

• Determinant of a Matrix 
  • Introduction
  • Meaning
  • Definition
• Types of Matrices 
• Algebra of Matrices 
  • Transpose of a Matrix
  • Symmetric Matrix
  • Skew-Symmetric Matrix
  • Equality of Two matrices
  • Addition of Two Matrices
  • Scalar Multiplication of a Matrix
  • Multiplication of Two Matrices
• Properties of Matrices 
• Elementary Transformations 
  • Interchange of any two rows or any two columns
  • Multiplication of the elements of any row or column by a non-zero scalar
  • Adding the scalar multiples of all the elements of any row (column) to corresponding elements of any other row (column)
• Inverse of Matrix 
  •  Inverse of a nonsingular matrix by elementary transformation
  •  Inverse of a square matrix by adjoint method
• Application of Matrices 
  • Method of Inversion
  • Method of Reduction

• Derivatives of Composite Functions - Chain Rule 
• Derivatives of Inverse Functions 
• The Concept of Derivative 
  • Derivatives of Logarithmic Functions 
• Derivatives of Implicit Functions 
• Derivatives of Parametric Functions 
• Second Order Derivative 

• Introduction of Derivatives 
• Increasing and Decreasing Functions 
• Maxima and Minima 
  • First Derivative test 
  • Second Derivative test
• Application of Derivatives to Economics 

• Integration 
• Methods of Integration: Integration by Substitution 
• Methods of Integration: Integration by Parts 
  • `int(u.v) dx = u intv dx - int((du)/(dx)).(intvdx) dx`
• Methods of Integration: Integration Using Partial Fractions 

  No	From of the rational function	Form of the partial fraction
  1	`(px + q )/((x-a)(x-b))`a ≠ b	`A/(x-a) + B/(x-b)`
  2	`(px+q)/(x-a)^2`	`A/(x-a) + B/(x-a)^2`
  3	`((px)^2 + qx +r)/((x-a)(x-b)(x-c))`	`A/(x-a)+B/(x-b) + C /(x-c)`
  4	`((px)^2 + qx + r)/((x-a)^2 (x-b))`	` A/(x-a) + B/(x-a)^2 +C/(x-b)`
  5	`((px)^2 + qx +r)/((x-a)(x^2 + bx +c))`	`A/(x-a) + (Bx + C)/ (x^2 + bx +c)`,

• Fundamental Theorem of Integral Calculus 

  If ∫ f(x) dx = g(x) + c, then

  `int_a^b f(x) dx = [g(x) + c]_a^b = g(b) - g(a)`.

• Properties of Definite Integrals 
  1. `int_a^a f(x) dx = 0`
  2. `int_a^b f(x) dx = - int_b^a f(x) dx`
  3. `int_a^b f(x) dx = int_a^b f(t) dt`
  4. `int_a^b f(x) dx = int_a^c f(x) dx + int_c^b f(x) dx`
     where a < c < b, i.e., c ∈ [a, b]
  5. `int_a^b f(x) dx = int_a^b f(a + b - x) dx`
  6. `int_0^a f(x) dx = int_0^a f(a - x) dx`
  7. `int_0^(2a) f(x) dx = int_0^a f(x) dx + int_0^a f(2a - x) dx`
  8. `int_(-a)^a f(x) dx = 2. int_0^a f(x) dx`, if f(x) even function
     = 0,  if f(x) is odd function

• Standard Forms of Parabola and Their Shapes 
• Standard Forms of Ellipse 
• Area Under Simple Curves 

• Differential Equations 
• Order and Degree of a Differential Equation 
• Formation of Differential Equation by Eliminating Arbitary Constant 
• Methods of Solving First Order, First Degree Differential Equations 
  • Differential Equations with Variables Separable Method 
  • Homogeneous Differential Equations 
  • Linear Differential Equations 
• Application of Differential Equations 
  • Population Growth and Growth of bacteria
  • Ratio active Decay
  • Newton's Law of Cooling
  • Surface Area

• Introduction of Matrices 
• Algebra of Matrices 
  • Transpose of a Matrix
  • Symmetric Matrix
  • Skew-Symmetric Matrix
  • Equality of Two matrices
  • Addition of Two Matrices
  • Scalar Multiplication of a Matrix
  • Multiplication of Two Matrices
• Inverse of Matrix 
  •  Inverse of a nonsingular matrix by elementary transformation
  •  Inverse of a square matrix by adjoint method
• Solution of Equations 

• Definition and types of matrices
• Algebra of matrices
• Inverse of a matrix
• Solution of equations

• Commission and Brokerage Agent 
  • Principal
  • Commission
  • Commission Agents
  • Broker
  • Factor
  • Del Credere Agent
• Discount 
  • Drawer and Drawee
  • Date of bill and Face value
  • Nominal Due Date and Legal Due Date
  • Discounting a Bill
  • Banker’s Discount, Cash Value, Banker’s Gain

• Concept of Insurance 

  Meaning of Insurance

  Principles of Insurance : 

  • utmost good faith
  • insurable interest
  • indemnity
  • contribution
  • doctrine of subrogation
  • causa proxima
  • mitigation of loss

  Types of Insurance - Fire

  Types of Insurance - Life

  Types of Insurance - Marine

• Fire Insurance 
• Accident Insurance 
• Marine Insurance 
• Annuity 

  - Terminology of Annuity
  - Annuity Due
  - Sinking Fund

• Regression 
  • Evolution of Regression
  • Difference between Correlation and Regression
  • Two Regression lines

  1. X on Y => X = a + by
  2. Y on X => Y = a + bx
• Types of Linear Regression 
• Fitting Simple Linear Regression 
• The Method of Least Squares 
• Lines of Regression of X on Y and Y on X Or Equation of Line of Regression 
• Properties of Regression Coefficients 

• Introduction to Time Series 
• Uses of Time Series Analysis 
• Components of a Time Series 
  • Secular Trend
  • Seasonal Variation
  • Cyclical Variation
  • Irregular Variation
• Mathematical Models 
  • Additive Model
  • Multiplicative Model
• Measurement of Secular Trend 
  • Method of Freehand Curve ( Graphical Method)
  • Method of Moving Averages
  • Method of Least Squares

• Index Numbers 
  1. Meaning, Classifications and Uses 
  2. Weighted Index Number 
  3. Test of adequacy for an Index Number

  • Construction of Cost of Living Index Number

  1. Uses of Cost of Living Index Number
  2. Methods of constructing Cost of Living Index Number
  3. Aggregate Expenditure Method
  4. Family Budget Method

   

• Types of Index Numbers 
  • Price Index Number
  • Quantity Index Number
  • Value Index Number
• Index Numbers - Terminology and Notation 
• Construction of Index Numbers 
  • The Aggregative Method
  • Method of Averaging relatives
  • Simple Aggregate Method 
  • Weighted Aggregate Method 
    • Laspeyre's Price Index Number
    • Paasche’s Price Index Number
    • Dorbish-Bowley’s Price Index Number
    • Fisher’s Ideal Price Index Number
    • Marshall-Edgeworth’s Price Index Number
    • Walsh’s Price Index Number
• Cost of Living Index Number 

  Steps in Construction of Cost of Living Index Numbers

  • Choice of Base Year
  • Choice of Commodities
  • Collection of Prices
  • Choice of Average
  • Choice of Weights
  • Purpose of Index Numbers
• Method of Constructing Cost of Living Index Numbers - Aggregative Expenditure Method 
• Method of Constructing Cost of Living Index Numbers - Family Budget Method 
• Uses of Cost of Living Index Number 

• Introduction of Linear Programming 
• Linear Programming Problem (L.P.P.) 
  • Meaning of Linear Programming Problem
  • Mathematical formulation of a linear programming problem
• Mathematical Formulation of Linear Programming Problem 

• Assignment Problem 
  • Definition and formulation
  • Solution of assignment problems (Hungarian Method)
• Hungarian Method of Solving Assignment Problem 
• Special Cases of Assignment Problem 
  • Unbalanced assignment problem
  • Maximization Problem
  • Restricted assignment problem
  • Alternative optimal solutions
• Sequencing Problem 
• Types of Sequencing Problem 
  • Sequencing n jobs on Two Machine
  • Sequencing n jobs on three machine
• Finding an Optimal Sequence 

• Mean of a Random Variable 
• Types of Random Variables 
  • Discrete random variable
  • Continuous random variable
  • Probability Mass Function
  • Cumulative Distribution Function or Distribution Function
  • Cumulative Distribution Function from Probability Mass function
  • Probability Mass Function from Cumulative Distribution Function 
• Random Variables and Its Probability Distributions 
• Probability Distribution of Discrete Random Variables 
  • Probability mass function
  • Cumulative distribution function
  • Expected values and variance
• Probability Distribution of a Continuous Random Variable 
  • Probability density function
  • Cumulative distribution function
• Binomial Distribution 
• Bernoulli Trial 
• Mean of Binomial Distribution (P.M.F.) 
• Variance of Binomial Distribution (P.M.F.) 
• Poisson Distribution 

• Continuous Function of Point 

  Continuous left hand limit

  Continuous right hand limit
  

Continuity of a function at a point

• Derivatives of Implicit Functions 
• Logarithmic Differentiation 
• Derivatives of Functions in Parametric Forms 
• Second Order Derivative 
• The Concept of Derivative 
  • Derivative of Inverse Function 

• Derivative of Inverse function
• Logarithmic Differentiation
• Derivative of implicit function
• Derivative of parametric function
• Second order derivative

• Increasing and Decreasing Functions 
• Maxima and Minima 
  • Increasing and decreasing functions
  • Stationary Value of a function
  • Local and Global (Absolute) Maxima and Minima

• Increasing and decreasing functions
• Maxima and minima

• Indefinite Integration 
  • Definition of an Integral 
  • Integral of Standard Functions 
  • Rules of Integration 
  • Methods of Integration 
    • Decomposition method
    • Decomposition by Partial Fractions
    • Method of substitution or change of variable
    • Important Results 
    • Integration by parts
    • Bernoulli’s formula for Integration by Parts
    • Integrals of the form
    • Integration of Rational Algebraic Functions
  • Integration by Parts 

    Integral of the types

    `int e^x{f(x)+f^'(x)}dx`

    `intsqrt(x^2+a^2)dx, intsqrt(x^2-a^2)dx`

    `int (px+q) sqrt(ax^2+bx+c)" "dx`

• Definition of an integral
• Integral of standard functions
• Rules of Integration
• Methods of Integration
• Integration by parts

• Properties of Definite Integrals 
  1. `int_a^a f(x) dx = 0`
  2. `int_a^b f(x) dx = - int_b^a f(x) dx`
  3. `int_a^b f(x) dx = int_a^b f(t) dt`
  4. `int_a^b f(x) dx = int_a^c f(x) dx + int_c^b f(x) dx`
     where a < c < b, i.e., c ∈ [a, b]
  5. `int_a^b f(x) dx = int_a^b f(a + b - x) dx`
  6. `int_0^a f(x) dx = int_0^a f(a - x) dx`
  7. `int_0^(2a) f(x) dx = int_0^a f(x) dx + int_0^a f(2a - x) dx`
  8. `int_(-a)^a f(x) dx = 2. int_0^a f(x) dx`, if f(x) even function
     = 0,  if f(x) is odd function
• Applications of Definite Integrals 

• Definite Integral
• Properties
• Applications

• Ratio, Proportion and Partnership 

Ratio, proportion and partnership

• Commission, Brokerage and Discount 

• Commission and Brokerage
• Discount

• Insurance and Annuity 

Insurance and Annuity

• Concept of Demography 
• Uses of Vital Statistics in Demography 
• Measurements of Mortality 
• Life Tables 

• Introduction, Definition
• Uses of vital statistics
• Measurements of Mortality
• Life tables

• Statistics 
  • Bivariate Frequency Distribution 

    bivariate data, tabulation of bivariate data

  • Karl Pearson’s Coefficient of Correlation 
• Rank Correlation 
  • Spearman’s Rank Correlation Coefficient 

• Bivariate frequency distribution
• Karl Pearson’s coefficient of correlation
• Rank correlation

• Lines of Regression of X on Y and Y on X Or Equation of Line of Regression 
• Regression Coefficient of X on Y and Y on X 
• Regression Propertise 

• Equation of line of regression
• Regression coefficients and their properties

• Random Variables and Its Probability Distributions 
• Binomial Theorem and Binomial Distribution 
• Poisson Distribution 
• Normal Distribution 
• Probability Distribution of a Discrete Random Variable 
• Probability Distribution 
  • Distribution Function of a Continuous Random Variable 

• Definition and types of random variables
• Probability Distribution of a Discrete Random Variable
• Probability Distribution of a Continuous Random Variable
• Binomial Theorem
• Binomial Distribution
• Poisson Distribution
• Normal Distribution

• Inequations in Management Mathematics 
• Linear Programming Problem in Management Mathematics 
• Assignment Problem 
  • Definition and formulation
  • Solution of assignment problems (Hungarian Method)
• Sequencing in Management Mathematics 

• Inequations
• Linear Programming Problem
• Assignment Problem
• Sequencing