Maths HSC Commerce (English Medium) 12th Standard Board Exam Maharashtra State Board Syllabus 2025-26

• Concept of Statements

1. Conjunction (∧)
2. Disjunction (∨)
3. Conditional statement (Implication) (→)
4. Biconditional (Double implication) (↔) or (⇔)
5. Negation (∼)

• Idempotent law
• Associative law
• Commutative law
• Distributive law
• Identity law
• Complement law
• Involution law
• DeMorgan’s laws

• Concept of Statements

1. Conjunction (∧)
2. Disjunction (∨)
3. Conditional statement (Implication) (→)
4. Biconditional (Double implication) (↔) or (⇔)
5. Negation (∼)

• Universal quantifier (∀)
• Existential quantifier (∃)

• Idempotent law
• Associative law
• Commutative law
• Distributive law
• Identity law
• Complement law
• Involution law
• DeMorgan’s laws

• Introduction
• Meaning
• Definition

• Row Matrix
• Column Matrix
• Zero or Null matrix
• Square Matrix
• Diagonal Matrix
• Scalar Matrix
• Unit or Identity Matrix
• Upper Triangular Matrix
• Lower Triangular Matrix
• Triangular Matrix
• Symmetric Matrix
• Skew-Symmetric Matrix
• Determinant of a Matrix
• Singular Matrix
• Transpose of a Matrix

• 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

• 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 a nonsingular matrix by elementary transformation
•  Inverse of a square matrix by adjoint method

• Method of Inversion
• Method of Reduction

• First and Second Derivative test
• 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

• ∫ tan x dx = log | sec x |  + C
• ∫ cot x dx = log | sin x | + C
• ∫ sec x dx = log | sec x + tan x | + C
• ∫ cosec x dx = log | cosec x – cot x | + C

• `int(u.v) dx = u intv dx - int((du)/(dx)).(intvdx) dx`
• Integral of the type ∫ e^x [ f(x) + f'(x)] dx = e^xf(x) + C
• Integrals of some more types

1. `I = int sqrt(x^2 - a^2) dx = x/2 sqrt(x^2 - a^2) - a^2/2 log | x + sqrt(x^2 - a^2 )| + C`
2. `I = int sqrt(x^2 - a^2) dx = x/2 sqrt(x^2 - a^2) - a^2/2 log | x + sqrt(x^2 - a^2 )| + C`
3. `I = int sqrt(x^2 - a^2) dx = x/2 sqrt(x^2 - a^2) - a^2/2 log | x + sqrt(x^2 - a^2 )| + C`

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

• Simple curves: lines, parabolas, polynomial functions

• Population Growth and Growth of bacteria
• Ratio active Decay
• Newton's Law of Cooling
• Surface Area

• Matrices
• Determinants
• Cramer’s Rule
• Application in Economics

• 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 a nonsingular matrix by elementary transformation
•  Inverse of a square matrix by adjoint method

• Principal
• Commission
• Commission Agents
• Broker
• Factor
• Del Credere Agent

- Terminology of Annuity
- Annuity Due
- Sinking Fund

• 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

• Secular Trend
• Seasonal Variation
• Cyclical Variation
• Irregular Variation

• Additive Model
• Multiplicative Model

• Method of Freehand Curve ( Graphical Method)
• Method of Moving Averages
• Method of Least Squares

• Flowchart
• Types

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

• Definition of related terminology such as constraints, objective function, optimization.

• Meaning of Linear Programming Problem
• Mathematical formulation of a linear programming problem
• Familiarize with terms related to Linear Programming Problem

• Definition and formulation
• Solution of assignment problems (Hungarian Method)

• Unbalanced assignment problem
• Maximization Problem
• Restricted assignment problem
• Alternative optimal solutions

• Sequencing n jobs on Two Machine
• Sequencing n jobs on three machine

• 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 

• Probability distribution of a random variable

• Probability density function
• Cumulative distribution function

Continuous left hand limit

Continuous right hand limit

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

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

• Spearman’s Rank Correlation Coefficient 

b_xy × b_yx = r², 0 ≤ b_xy × b_yx ≤ 1

• Probability distribution of a random variable

• Definition and formulation
• Solution of assignment problems (Hungarian Method)