# Subject Details

At kaushal Acedemy, Mathematics is not only about problem solving and achieving marks, Our academy provides the students with :

• Constructing Logical Arguments
• Presenting Ideas Intricately
• Overcoming Cognitive Biases
• Periodic Tests
• Extra Classes & Doubt Classes
• Research Based Assignments
• Semester Wise Exams
• Online Test
• Motivational Classes
• Ability to manipulate precisely
• Developing critical thinking
Course Structure
UNITS UNIT NAME MARKS
I. RELATIONS AND FUNCTIONS 8
II. ALGEBRA 10
III. CALCULUS 35
IV. VECTORS AND THREE DIMENSIONAL GEOMETRY 14
V. LINEAR PROGRAMMING 05
VI. PROBABILITY 8
Total 80
Internal Assessment 20
Grand Total 100

# CBSE MATHS SYLLABUS

## Unit I: 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.

### 2. Inverse Trigonometric Functions

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

## Unit II: Algebra

### 1. Matrices

Concept, notation, order, equality, types of matrices, zero and identity 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 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

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.

## Unit III: Calculus

### 1. 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.

### 2. 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).

### 3. 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.

### 4. 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).

### 5. 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.

## Unit IV: Vectors and Three-Dimensional Geometry

### 1. 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.

### 2. 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.

## Unit V: Linear Programming

### 1. Linear Programming

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 (bounded and unbounded), feasible and infeasible solutions, optimal feasible solutions (up to three non-trivial constraints).

## Unit VI: Probability

### 1. 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.

## CLASS-XII(2019-20)

Time : 3 Hours

S. No Typolgy of Questions Very Short Answer-Objective Type(VSA) (1 Mark) Short Answer-I(SA) (2 Marks) Short Answer-II(SA) (3 Marks) Long Answer (LA) (4 Marks) Total Marks % Weightage(approx.)
1 Remembering: Exhibit memory
of previously learned material
by recalling facts, terms, basic
4 1 1 1 16 20
2 Understanding: Demonstrate
understanding of facts and
ideas by organizing, comparing,
translating, interpreting, giving
descriptions, and stating
6 2 3 1 28 35
3 Applying: Solve problems to
new situations by applying
acquired knowledge, facts,
techniques and rules in a
different way.
6 2 1 1 20 25
4

Analysing: Examine and break information into parts by identifying motives or causes. Make inferences and find evidence to support generalizations
Evaluating: Present and defend opinions by making judgments about information, validity of ideas, or quality of work based on a set of criteria.
Creating: Compile information together in a different way by combining elements in a new pattern or proposing alternative solutions

4 1 1 1 16 20
Total 20x1=20 6x2=12 8x3=24 6x4=24 80 100

INTERNAL ASSESMENT 20 MARKS
Periodic Test(Best 2 out of 3 tests conducted) 10 Marks
Mathematics Activities 10 Marks