# Maths Commerce (English Medium) Class 12 CBSE Syllabus 2024-25

CBSE Syllabus 2024-25 Class 12: The CBSE Class 12 Maths Syllabus for the examination year 2024-25 has been released by the Central Board of Secondary Education, CBSE. The board will hold the final examination at the end of the year following the annual assessment scheme, which has led to the release of the syllabus. The 2024-25 CBSE Class 12 Maths Board Exam will entirely be based on the most recent syllabus. Therefore, students must thoroughly understand the new CBSE syllabus to prepare for their annual exam properly.

The detailed CBSE Class 12 Maths Syllabus for 2024-25 is below.

## CBSE Class 12 Mathematics Revised Syllabus

CBSE Class 12 Mathematics and their Unit wise marks distribution

## Syllabus

### CBSE Class 12 Mathematics Syllabus for Chapter 1: Relations and Functions

1 Relations and Functions
• Introduction of Relations and Functions
• Types of Relations
• 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 Functions
• 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
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
• Composition of Functions and Invertible Function
• Concept of Binary Operations
• Commutative Binary Operations
• Associative Binary Operations
• Identity Binary Operation,
• Invertible Binary Operation
• Inverse of a Function
2 Inverse Trigonometric Functions

### CBSE Class 12 Mathematics Syllabus for Chapter 2: Algebra

3 Matrices
• Introduction of Matrices
• Matrices
• Determinants
• Cramer’s Rule
• Application in Economics
• Order of a Matrix
• Types of Matrices
• 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
• Equality of Matrices
• Determine equality of two matrices
• Introduction of Operations on Matrices
• Algebraic Operations on Matrices
• Multiplication of a Matrix by a Scalar
• Commutative Law
• Associative Law
• The existence of additive inverse
• Properties of Scalar Multiplication of a Matrix
• Multiplication of Matrices
• Non-commutativity of multiplication of matrices
• Zero matrix as the product of two non zero matrices
• Properties of Multiplication of Matrices
• The associative law
• The distributive law
• The existence of multiplicative identity
• Transpose of a Matrix
• Write transpose of given matrix
• Properties of Transpose of the Matrices
• Symmetric and Skew Symmetric Matrices
• Define symmetric and skew symmetric matrix
• Invertible Matrices
• Inverse of Matrix
• Inverse of a Matrix by Elementary Transformation
• Multiplication of Two Matrices
• Negative of Matrix
• Subtraction of Matrices
• Matrices
• Proof of the Uniqueness of Inverse
• Matrices Notation

Matrices Notation

• 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)
4 Determinants
• Introduction of Determinant
• Determinants of Matrix of Order One and Two
• Determinant of a Matrix of Order 3 × 3
• 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)
• Area of a Triangle
• Minors and Co-factors
• Inverse of Matrix
• Inverse of a Square Matrix by the Adjoint Method
• Applications of Determinants and Matrices
• Consistent System
• Inconsistent System
• Solution of a system of linear equations using the inverse of a matrix
• 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)
• Properties of Determinants
• 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.
• Determinant of a Square Matrix

up to 3 x 3 matrices

• Rule A=KB

### CBSE Class 12 Mathematics Syllabus for Chapter 3: Calculus

5 Continuity and Differentiability
6 Applications of Derivatives
7 Integrals
• Introduction of Integrals
• Integration as an Inverse Process of Differentiation
 Derivatives Integrals(Anti derivatives) d/(dx) (x^(n+1)/(n+1)) = x^n int x^n dx = x^(n+1)/(n+1) + "C", n ≠ –1 d/(dx)(x) = 1 int dx = x + C d/(dx)(sin x) = cos x int cos x dx = sin x +C d/(dx) (-cos x) = sin x intsin x dx = -cos x +C d/(dx) (tan x) = sec2x int sec^2 x dx = tanx + C d/(dx)(-cot x) = cosec^2x  int cosec^2x dx = -cot x +C d/(dx) (sec x) = sec x tan x int sec x tan x dx = sec x +C d/(dx) (-cosecx) = cosec x cot x int cosec x cot x dx = -cosec x +C d/(dx) (sin^-1) = 1/(sqrt(1-x^2)) int (dx)/(sqrt(1-x^2))= sin^(-1) x +C  d/(dx) (-cos^(-1)) = 1/(sqrt (1-x^2)) int (dx)/(sqrt (1-x^2))= -cos^(-1) x + C  d/(dx) (tan^(-1) x) = 1/(1+x^2) int (dx)/(1+x^2)= tan^(-1) x + C  d/(dx) (-cot^(-1) x) = 1/(1+x^2) int (dx)/(1+x^2)= -cot^(-1) x + C  d/(dx) (sec^(-1) x) = 1/(x sqrt (x^2 - 1)) int (dx)/(x sqrt (x^2 - 1))= sec^(-1) x + C d/(dx) (-cosec^(-1) x) = 1/(x sqrt (x^2 - 1)) int (dx)/(x sqrt (x^2 - 1))=-cosec^(-1) x + C  d/(dx)(e^x) = e^x int e^x dx = e^x + C d/(dx) (log|x|) = 1/x int 1/x dx = log|x| +C d/(dx) ((a^x)/(log a)) = a^x int a^x dx = a^x/log a +C
• Some Properties of Indefinite Integral
• Methods of Integration: Integration by Substitution
• ∫ 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
• Integration Using Trigonometric Identities
• Integrals of Some Particular Functions

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

• 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),
• Methods of Integration: Integration by Parts
• int(u.v) dx = u intv dx - int((du)/(dx)).(intvdx) dx
• Integral of the type ∫ ex [ f(x) + f'(x)] dx = exf(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
• Fundamental Theorem of Calculus

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

• Evaluation of Definite Integrals by Substitution
• 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
• Definite Integrals Problems
• Indefinite Integral Problems
• Comparison Between Differentiation and Integration
• Integration
• Geometrical Interpretation of Indefinite Integrals
• Indefinite Integral by Inspection
• Definite Integral as the Limit of a Sum
• Evaluation of Simple Integrals of the Following Types and Problems
8 Applications of the Integrals
9 Differential Equations

### CBSE Class 12 Mathematics Syllabus for Chapter 4: Vectors and Three-dimensional Geometry

10 Vectors
11 Three - Dimensional Geometry

### CBSE Class 12 Mathematics Syllabus for Chapter 5: Linear Programming

12 Linear Programming

13 Probability

### CBSE Class 12 Mathematics Syllabus for Chapter 7: Sets

• Sets
• Properties of Set Operations