# Mathematics Class 12 Science (English Medium) CBSE Topics and Syllabus

CBSE Syllabus For Class 12 Mathematics: Knowing the Syllabus is very important for the students of Class 12. Shaalaa has also provided a list of topics that every student needs to understand.

The CBSE Class 12 Mathematics syllabus for the academic year 2023-2024 is based on the Board's guidelines. Students should read the Class 12 Mathematics Syllabus to learn about the subject's subjects and subtopics.

Students will discover the unit names, chapters under each unit, and subtopics under each chapter in the CBSE Class 12 Mathematics Syllabus pdf 2023-2024. They will also receive a complete practical syllabus for Class 12 Mathematics in addition to this.

## CBSE Class 12 Mathematics Revised Syllabus

CBSE Class 12 Mathematics and their Unit wise marks distribution

## Syllabus

### CBSE Class 12 Mathematics Syllabus for Relations and Functions

1 Relations and Functions
• Types of relations: reflexive, symmetric, transitive and equivalence relations.
• One to one and onto functions, composite functions, inverse of a function. Binary operations.
2 Inverse Trigonometric Functions
1. Definition, range, domain, principal value branch.
2. Graphs of inverse trigonometric functions.
3. Elementary properties of inverse trigonometric functions.

### CBSE Class 12 Mathematics Syllabus for Algebra

3 Matrices
• Concept, notation, order, equality, types of matrices, zero andidentity matrix, transpose of a matrix, symmetric and skew symmetric matrices.
• Operation on matrices: Addition and multiplication and multiplication with a scalar.
• Simple properties of addition, multiplication and scalar multiplication.
• Non­ commutativity of multiplication of matrices and existence of non-zero matrices whose product is the zero matrix (restrict to square matrices of order).
• Concept of elementary row and column operations.
• Invertible matrices and proof of the uniqueness of inverse, if it exists; (Here all matrices will have real entries).
4 Determinants
• 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)
• Inverse of Matrix
• 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

• Determinants of Matrix of Order One and Two
• Introduction of Determinant
• Area of a Triangle
• Minors and Co-factors
• 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)
• Rule A=KB
• Determinant of a square matrix (up to 3 x 3 matrices), properties of determinants, minors, co-factors and applications of determinants in finding the area of a triangle. Adjoint and inverse of a square matrix.
• Consistency, inconsistency and number of solutions of system of linear equations by examples, solving system of linear equations in two or three variables (having unique solution) using inverse of a matrix.

### CBSE Class 12 Mathematics Syllabus for Calculus

5 Continuity and Differentiability
• Continuity and differentiability, derivative of composite functions, chain rule, derivatives of inverse trigonometric functions, derivative of implicit functions. Concept of exponential and logarithmic functions.
• Derivatives of logarithmic and exponential functions. Logarithmic differentiation, derivative of functions expressed in parametric forms. Second order derivatives. Rolle's and Lagrange's Mean Value Theorems (without proof ) and their geometric interpretation.
6 Applications of Derivatives
• Applications of derivatives: rate of change of bodies, increasing/decreasing functions, tangents and normals, use of derivatives in approximation, maxima and minima (first derivative test motivated geometrically and second derivative test given as a provable tool).
• Simple problems (that illustrate basic principles and understanding of the subject as well as real-life situations).
7 Integrals
• Definite Integrals Problems
• Indefinite Integral Problems
• Comparison Between Differentiation and Integration
• Integration
• 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

• Indefinite Integral by Inspection
• Some Properties of Indefinite Integral
• Integration Using Trigonometric Identities
• Introduction of Integrals
• 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
• Fundamental Theorem of Calculus

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

• Definite Integral as the Limit of a Sum
• Evaluation of Simple Integrals of the Following Types and Problems
• 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
• 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 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 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
• Integration as inverse process of differentiation.
• Integration of a variety of functions by substitution, by partial fractions and by parts, Evaluation of simple integrals of the following types and problems based on them.
• Definite integrals as a limit of a sum, Fundamental Theorem of Calculus (without proof).
• Basic properties of definite integrals and evaluation of definite integrals.
8 Applications of the Integrals
• Applications in finding the area under simple curves, especially lines, circles/ parabolas/ellipses (in standard form only), Area between any of the two above said curves (the region should be clearly identifiable).
9 Differential Equations
• Definition, order and degree, general and particular solutions of a differential equation.
• Formation of differential equation whose general solution is given.
• Solution of differential equations by method of separation of variables solutions of homogeneous differential equations of first order and first degree.

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.

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

10 Vectors
• Vectors and scalars, magnitude and direction of a vector.
• Direction cosines and direction ratios of a vector.
• Types of vectors (equal, unit, zero, parallel and collinear vectors), position vector of a point, negative of a vector, components of a vector, addition of vectors, multiplication of a vector by a scalar, position vector of a point dividing a line segment in a given ratio.
• Definition, Geometrical Interpretation, properties and application of scalar (dot) product of vectors, vector (cross) product of vectors, scalar triple product of vectors.
11 Three - Dimensional Geometry
• Direction cosines and direction ratios of a line joining two points.
• Cartesian equation and vector equation of a line, coplanar and skew lines, shortest distance between two lines.
• Cartesian and vector equation of a plane.
• Angle between (i) two lines, (ii) two planes, (iii) a line and a plane.
• Distance of a point from a plane.

### CBSE Class 12 Mathematics Syllabus for Linear Programming

12 Linear Programming
• Introduction, related terminology such as constraints, objective function, optimization, different types of linear programming (LP.) problems, mathematical formulation of LP. problems, graphical method of solution for problems in two variables, feasible and infeasible regions (bounded or unbounded), feasible and infeasible solutions, optimal feasible solutions (up to three non-trivial constraints).

### CBSE Class 12 Mathematics Syllabus for Probability

13 Probability
• Conditional probability, multiplication theorem on probability, independent events, total probability, Bayes' theorem, Random variable and its probability distribution, mean and variance of random variable. Repeated independent (Bernoulli) trials and Binomial distribution.

### CBSE Class 12 Mathematics Syllabus for Sets

• Sets
• Properties of Set Operations