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

## Syllabus

1 Relations and Functions

Prescribed Books:

1. Mathematics Textbook for Class XI, NCERT Publications
2. Mathematics Part I- Textbook for Class XII, NCERT Publication
3. Mathematics Part II - Textbook for Class XII, NCERT Publication
2 Algebra
3 Calculus
4 Vectors and Three-dimensional Geometry
5 Linear Programming
6 Probability
7 Sets
101 Inverse Trigonometric Functions
1. Definition, range, domain, principal value branch.
2. Graphs of inverse trigonometric functions.
3. Elementary properties of inverse trigonometric functions.
102 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.
201 Determinants
• Applications of Determinants and Matrices
• Elementary Transformations
• Inverse of a 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 first Row (R1), Expansion along second row (R2),Expansion along 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.
202 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).
301 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.
302 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).
303 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).
304 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.
305 Integrals
• 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.
401 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.
402 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.
501 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).
601 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.