CBSE Class 12 Maths Solutions 2022-23 | Orchids

# CBSE Class 12 Mathematics Syllabus

Mathematics as a subject is a part of the core subjects for some XII students who have opted for a set of subjects, and an elective for some who have opted for another set of subjects. In both instances, the subject has the same syllabus. It covers mostly calculus and algebra with arithematics, vectors, geometry, etc included in minority. It is much more intense in terms of difficulty as compared to the previous years.

### Syllabus

The syllabus for the year 2022-2023 is divided into two terms. The detailed curriculum as well as the marking scheme is described  below.

### NCERT books

1. Mathematics Part I - Textbook for Class XII

2. Mathematics Part II - Textbook for Class XII

3. Mathematics Exemplar Problem for Class XII

4. Mathematics Lab Manual class XII

#### General Guidelines

1. All concepts/identities must be illustrated by situational examples.

2. The language of ‘word problems’ must be clear, simple and unambiguous.

3. Problems given should be testing the understanding of the subject.

4. All proofs to be produced in a manner that allow the learner to see flow of reasons. Wherever possible, give more than one proof.

5. Motivate results, wherever possible. Prove explicitly those results where a short and clear argument reinforces mathematical thinking and reasoning. There must be emphasis on correct way of expressing the arguments

### UNIT I: RELATIONS AND FUNCTIONS  (Total Periods 180)

1. Relations and Functions (Periods 10)

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 (Periods 12)

Definition, range, domain, principal value branches. Graphs of inverse trigonometric functions. Elementary properties of inverse trigonometric functions.

### UNIT II: ALGEBRA

1. Matrices (Periods 18)

Concept, notation, order, equality, types of matrices, zero matrix, transpose of a matrix, symmetric and skew symmetric matrices. Addition, multiplication and scalar multiplication of matrices, 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 2). 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).

2. Determinants (Periods 20)

Determinant of a square matrix (up to 3 × 3 matrices), properties of determinants, minors, cofactors 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.

### UNIT III: CALCULUS

#### 1. Continuity and Differentiability (Periods 18)

Continuity and differentiability, derivative of composite functions, chain rule, derivatives of inverse trigonometric functions, derivative of implicit function. Concepts of exponential, logarithmic functions. Derivatives of loge x and ex . 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 interpretations.

#### 2. Applications of Derivatives (Periods 10)

Applications of derivatives: Rate of change, increasing/decreasing functions, tangents and normals, 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). 6

#### 3. Integrals (Periods 20)

Integration as inverse process of differentiation. Integration of a variety of functions by substitution, by partial fractions and by parts, only simple integrals of the type and to be evaluated. Definite integrals as a limit of a sum. Fundamental Theorem of Calculus (without proof). Basic properties of definite integrals and evaluation of definite integrals.

#### 4. Applications of the Integrals (Periods 10)

Applications in finding the area under simple curves, especially lines, arcs of circles/parabolas/ellipses (in standard form only), area between the two above said curves (the region should be cleraly identifiable).

#### 5. Differential Equations (Periods 10)

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, homogeneous differential equations of first order and first degree. Solutions of linear differential equation of the type – , dy Py Q dx + = where P and Q are functions of x or constant + = , dx Px Q dy where P and Q are functions of y or constant

### UNIT IV: VECTORS AND THREE-DIMENSIONAL GEOMETRY

#### 1. Vectors (Periods 10)

Vectors and scalars, magnitude and direction of a vector. Direction cosines/ratios of vectors. 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. Scalar (dot) product of vectors, projection of a vector on a line. Vector (cross) product of vectors, scalar triple product.

#### 2. Three-dimensional Geometry (Periods 12)

Direction cosines/ratios of a line joining two points. Cartesian 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. 7

### Unit V: Linear Programming (Periods 12)

Introduction, related terminology such as constraints, objective function, optimization, different types of linear programming (L.P.) problems, mathematical formulation of L.P. problems, graphical method of solution for problems in two variables, feasible and infeasible regions, feasible and infeasible solutions, optimal feasible solutions (up to three non-trivial constrains).

### Unit VI: Probability (Periods 18)

Multiplications theorem on probability. Conditional probability, independent events, total probability, Baye’s theorem. Random variable and its probability distribution, mean and variance of haphazard variable. Repeated independent (Bernoulli) trials and Binomial distribution.

### Curriculum Distribution and Weightage Term I

 No. Units Marks I. Relations and Functions 08 II. Algebra 10 III. Calculus 17 V. Linear Programming 05 Total 40 Internal Assessment 10
 INTERNAL ASSESSMENT 10 MARKS Periodic Test 5 Marks Mathematics Activities: Activity file record +Term end assessment of one activity & Viva 5  Marks

### Term II

 No. Units Marks III. Calculus 18 IV. Vectors and Three-Dimensional Geometry 14 VI. Probability 8 Total 40 Internal Assessment 10
 INTERNAL ASSESSMENT 10 MARKS Periodic Test 5 Marks Mathematics Activities: Activity file record +Term end assessment of one activity & Viva 5  Marks