Maths Commerce (English Medium) Class 12 CBSE Syllabus 2025-26

• Empty Relation
• Universal Relation
• Trivial Relations
• Identity relation
• Symmetric relation
• Transitive relation
• Equivalence Relation
• Antisymmetric relation
• Inverse relation
• One-One Relation (Injective)
• Many-one relation
• Into relation
• Onto relation (Surjective)

• Types of Function based on Elements:
  1) One One Function (or injective)
  2) Many One Function
  3) Onto Function (or surjective)
  4) One One and Onto Function (or bijective)
  5) Into Function
  6) Constant Function
• Types of Function based on Equation:
  1) Identity Function
  2) Linear Function
  3) Quadratic Function
  4) Cubic Function
  5) Polynomial Functions
• Types of Function based on the Range:
  1) Modulus Function
  2) Rational Function
  3) Signum Function
  4) Even and Odd Functions
  5) Periodic Functions
  6) Greatest Integer Function
  7) Inverse Function
  8) Composite Functions
• Types of Function based on the Domain:
  1) Algebraic Functions
  2) Trigonometric Functions
  3) Logarithmic Functions
• Explicit and Implicit Functions
• Value of a Function
• Equal Functions

• Commutative Binary Operations
• Associative Binary Operations
• Identity Binary Operation,
• Invertible Binary Operation

• Introduction of Inverse Trigonometric Functions

Inverse of Sin, Inverse of cosin, Inverse of tan, Inverse of cot, Inverse of Sec, Inverse of Cosec

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

• 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

• Determine equality of two matrices

• Write transpose of given matrix

• Define symmetric and skew symmetric matrix

• 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)

• 1st, 2nd and 3rd Row
• 1st, 2nd and 3rd Columns
• Expansion along the first Row (R₁)
• Expansion along the second row (R₂)
• Expansion along the first Column (C₁)

• Consistent System
• Inconsistent System
• Solution of a system of linear equations using the inverse of a matrix

• 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)

• Property 1 - The value of the determinant remains unchanged if its rows are turned into columns and columns are turned into rows.
• Property 2 -  If any two rows  (or columns)  of a determinant are interchanged then the value of the determinant changes only in sign.
• Property 3 - If any two rows ( or columns) of a  determinant are identical then the value of the determinant is zero.
• Property  4  -  If each element of a row (or column)  of a determinant is multiplied by a  constant k then the value of the new determinant is k times the value of the original determinant.
• Property  5  -  If each element of a row (or column) is expressed as the sum of two numbers then the determinant can be expressed as the sum of two determinants
• Property  6  -  If a constant multiple of all elements of any row  (or column)  is added to the corresponding elements of any other row  (or column  )  then the value of the new determinant so obtained is the same as that of the original determinant. 
• Property 7 -  (Triangle property) - If all the elements of a  determinant above or below the diagonal are zero then the value of the determinant is equal to the product of its diagonal elements.

up to 3 x 3 matrices

f(x) = xn

f(x) = sin x

f(x) = cos x

f(x) = tan x

• Derivative of Functions Which Expressed in Higher Order Derivative Form

Continuous left hand limit

Continuous right hand limit

• Rolle’s Theorem
• Lagrange’s Mean Value Theorem
• Applications

• 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

Simple problems (that illustrate basic principles and understanding of the subject as well as real-life situations)

• ∫ 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

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

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

3) `int (dx)/(x^2 - a^2) = 1/a  tan^(-1) (x/a) + C`

4) `int (dx)/sqrt (x^2 - a^2) = log |x + sqrt (x^2-a^2)| + C`

5) `int (dx)/sqrt (a^2 - x^2) = sin ^(-1) (x/a) +C`

6)  `int (dx)/sqrt (x^2 + a^2) = log |x + sqrt (x^2 + a^2)| + C`

7) To find the integral `int (dx)/(ax^2 + bx + c)`

8) To find the integral of the type `int (dx)/sqrt(ax^2 + bx + c)`

9) To find the integral of the type `int (px + q)/(ax^2 + bx + c) dx`

10) For the evaluation of the integral of the type `int (px + q)/sqrt(ax^2 + bx + c) dx`

• `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`

Area function, First fundamental theorem of integral calculus and Second fundamental theorem of integral calculus

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

• Area of the Region Bounded by a Curve & X-axis Between two Ordinates
• Area of the Region Bounded by a Curve & Y-axis Between two Abscissa 
• Circle-line, elipse-line, parabola-line

• Simple curves: lines, parabolas, polynomial functions

• Solutions of linear differential equation of the type:

1. `dy/dx` + py = q, where p and q are functions of x or constants.
2. `dx/dy` + px = q, where p and q are functions of y or constants.

• Differential equations, order and degree.
• Solution of differential equations.
• Variable separable.
• Homogeneous equations
• Linear form `dy/dx` + Py = Q where P and Q are functions of x only. Similarly, for `dx/dy`.

• Position Vector
• Direction Cosines and Direction Ratios of a Vector

• Introduction: Vector Operations
• Statement: Multiplication of a Vector by a Scalar
• Example

• Vector addition using components
• Components of a vector in two dimensions space
• Components of a vector in three-dimensional space

• Section formula for internal division
• Midpoint formula
• Section formula for external division

• Direction cosines of a line passing through two points.

• Equation of a line through a given point and parallel to a given vector `vec b`
• Equation of a line passing through two given points

• Coplanar
• Skew lines
• Distance between two skew lines
• Distance between parallel lines

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

• Different types of linear programming (L.P.) problems

1. Manufacturing problem
2. Diet Problem
3. Transportation problem

• Graphical method of solution for problems in two variables
• Feasible and infeasible regions and bounded regions
• Feasible and infeasible solutions
• Optimum feasible solution

• Random experiment
• Outcome
• Equally likely outcomes
• Sample space
• Event

• Independent Events

• Partition of a sample space
• Theorem of total probability

• Probability distribution of a random variable