## CBSE Class 12 Maths Syllabus - Free PDF Download

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

### CBSE Class 12 Mathematics Course Structure 2024-25 With Marking Scheme

# | Unit/Topic | Weightage |
---|---|---|

I | Relations and Functions | |

1 | Relations and Functions | |

2 | Inverse Trigonometric Functions | |

II | Algebra | |

3 | Matrices | |

4 | Determinants | |

III | Calculus | |

5 | Continuity and Differentiability | |

6 | Applications of Derivatives | |

7 | Integrals | |

8 | Applications of the Integrals | |

9 | Differential Equations | |

IV | Vectors and Three-dimensional Geometry | |

10 | Vectors | |

11 | Three - Dimensional Geometry | |

V | Linear Programming | |

12 | Linear Programming | |

VI | Probability | |

13 | Probability | |

VII | Sets | |

Total | - |

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## Syllabus

### CBSE Class 12 Mathematics Syllabus for Chapter 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

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

- 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

- Inverse Trigonometric Functions
- Introduction of Inverse Trigonometric Functions

- Graphs of Inverse Trigonometric Functions

- Inverse Trigonometric Functions - Principal Value Branch

- Inverse Trigonometric Functions (Simplification and Examples)
- Properties of Inverse Trigonometric Functions
Inverse of Sin, Inverse of cosin, Inverse of tan, Inverse of cot, Inverse of Sec, Inverse of Cosec

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

- 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
- Addition of Matrices

- Multiplication of a Matrix by a Scalar

- Properties of Matrix Addition
- Commutative Law
- Associative Law
- Existence of additive identity
- 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)

- 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 (R
_{1}) - Expansion along the second row (R
_{2}) - Expansion along the first Column (C
_{1})

- 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

- Concept of Continuity
- Algebra of Continuous Functions
- Concept of Differentiability
- Derivatives of Composite Functions - Chain Rule
- Derivatives of Implicit Functions
- Derivatives of Inverse Trigonometric Functions
- Exponential and Logarithmic Functions
- Logarithmic Differentiation
- Derivatives of Functions in Parametric Forms
- Second Order Derivative
- Derivative - Exponential and Log
- Proof Derivative X^n Sin Cos Tan
f(x) = xn

f(x) = sin x

f(x) = cos x

f(x) = tan x

- Infinite Series
- Higher Order Derivative
- Derivative of Functions Which Expressed in Higher Order Derivative Form

- Continuous Function of Point
Continuous left hand limit

Continuous right hand limit

- Mean Value Theorem
- Rolle’s and Lagrange’s Mean Value Theorems (without proof) and their geometric interpretations

- Introduction to Applications of Derivatives
- Rate of Change of Bodies or Quantities
- Increasing and Decreasing Functions
- Maxima and Minima
- 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

- Maximum and Minimum Values of a Function in a Closed Interval
- Simple Problems on Applications of Derivatives
Simple problems (that illustrate basic principles and understanding of the subject as well as real-life situations)

- Graph of Maxima and Minima
- Approximations
- Tangents and Normals

- 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 `int`sin x dx = -cos x +C `d/(dx)` (tan x) = sec ^{2}x`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 ∫ e
^{x}[ f(x) + f'(x)] dx = e^{x}f(x) + C - Integrals of some more types

- `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`
- `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`
- `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
- `int_a^a f(x) dx = 0`
- `int_a^b f(x) dx = - int_b^a f(x) dx`
- `int_a^b f(x) dx = int_a^b f(t) dt`
- `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] - `int_a^b f(x) dx = int_a^b f(a + b - x) dx`
- `int_0^a f(x) dx = int_0^a f(a - x) dx`
- `int_0^(2a) f(x) dx = int_0^a f(x) dx + int_0^a f(2a - x) dx`
- `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

- Area of the Region Bounded by a Curve and a Line
- 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

- Area Between Two Curves
- Area Under Simple Curves
- Simple curves: lines, parabolas, polynomial functions

- Differential Equations
- Order and Degree of a Differential Equation
- General and Particular Solutions of a Differential Equation
- Methods of Solving First Order, First Degree Differential Equations
- Linear Differential Equations

- Homogeneous Differential Equations

- Differential Equations with Variables Separable Method

- Solutions of Linear Differential Equation
- Solutions of linear differential equation of the type:

- `dy/dx` + py = q, where p and q are functions of x or constants.
- `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`.

- Formation of a Differential Equation Whose General Solution is Given
- Procedure to Form a Differential Equation that Will Represent a Given Family of Curves

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

- Introduction of Vector
- Basic Concepts of Vector Algebra
- Position Vector
- Direction Cosines and Direction Ratios of a Vector

- Direction Cosines
- Vectors and Their Types
- Zero Vector
- Unit Vector
- Co-initial and Co-terminus Vectors
- Equal Vectors
- Negative of a Vector
- Collinear Vectors
- Free Vectors
- Localised Vectors

- Addition of Vectors
- Properties of Vector Addition
- Multiplication of a Vector by a Scalar
- Components of Vector
- Vector addition using components
- Components of a vector in two dimensions space
- Components of a vector in three-dimensional space

- Vector Joining Two Points
- Section Formula
- Section formula for internal division
- Midpoint formula
- Section formula for external division

- Product of Two Vectors
- Vector (Or Cross) Product of Two Vectors

- Scalar (Or Dot) Product of Two Vectors

- Projection of a Vector on a Line

- Geometrical Interpretation of Scalar
- Scalar Triple Product of Vectors
- Position Vector of a Point Dividing a Line Segment in a Given Ratio
- Magnitude and Direction of a Vector
- Vectors Examples and Solutions
- Introduction of Product of Two Vectors

- Introduction of Three Dimensional Geometry
- Direction Cosines and Direction Ratios of a Line
- Direction cosines of a line passing through two points.

- Relation Between Direction Ratio and Direction Cosines
- Equation of a Line in Space
- 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

- Angle Between Two Lines
- Shortest Distance Between Two Lines
- Coplanar
- Skew lines
- Distance between two skew lines
- Distance between parallel lines

- Three - Dimensional Geometry Examples and Solutions
- Equation of a Plane
- Equation of a Plane Passing Through Three Non Collinear Points

- Intercept Form of the Equation of a Plane

- Equation of a Plane in Normal Form

- Equation of a Plane Perpendicular to a Given Vector and Passing Through a Given Point

- Plane Passing Through the Intersection of Two Given Planes

- Coplanarity of Two Lines
- Distance of a Point from a Plane
- Angle Between Line and a Plane
- Angle Between Two Planes
- Vector and Cartesian Equation of a Plane

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

- Introduction of Linear Programming
- Definition of related terminology such as constraints, objective function, optimization.

- Mathematical Formulation of Linear Programming Problem
- Different Types of Linear Programming Problems
**Different types of linear programming (L.P.) problems**

- Manufacturing problem
- Diet Problem
- Transportation problem

- Graphical Method of Solving Linear Programming Problems
- Graphical method of solution for problems in two variables
- Feasible and infeasible regions and bounded regions
- Feasible and infeasible solutions
- Optimum feasible solution

- Linear Programming Problem and Its Mathematical Formulation

### CBSE Class 12 Mathematics Syllabus for Chapter 6: Probability

- Introduction of Probability
- Random experiment
- Outcome
- Equally likely outcomes
- Sample space
- Event

- Conditional Probability
- Independent Events

- Properties of Conditional Probability
- Multiplication Theorem on Probability
- Independent Events
- Bayes’ Theorem
- Partition of a sample space
- Theorem of total probability

- Variance of a Random Variable
- Probability Examples and Solutions
- Random Variables and Its Probability Distributions
- Probability distribution of a random variable

- Mean of a Random Variable
- Bernoulli Trials and Binomial Distribution

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

- Sets
- Properties of Set Operations