MATHEMATICS  Syllabus for Class 12

Note : There will be one Question Paper which will contain Two Sections i.e. Section A and Section B [B1 and B2].

Section A will have 15 questions covering both i.e. Mathematics/Applied Mathematics which will be compulsory for all candidates

Section B1 will have 35 questions from Mathematics out of which 25 questions need to be attempted. Section B2 will have 35 questions purely from Applied Mathematics out of which 25 question will be attempted.

SECTION A

1. Algebra

(i) Matrices and types of Matrices

(ii) Equality of Matrices, transpose of a Matrix, Symmetric and Skew Symmetric Matrix 

(iii) Algebra of Matrices 

(iv) Determinants 

(v) Inverse of a Matrix

(vi) Solving of simultaneous equations using Matrix Method

2. Calculus  

(i) Higher order derivatives 

(ii) Tangents and Normals 

(iii) Increasing and Decreasing Functions 

(iv). Maxima and Minima

3. Integration and its Applications 

(i) Indefinite integrals of simple functions 

(ii) Evaluation of indefinite integrals 

(iii) Definite Integral 

(iv). Application of Integration as area under the curve

4. Differential Equations

(i) Order and degree of differential equations

(ii) Formulating and solving of differential equations with variable separable

5. Probability Distributions

(i) Random variables and its probability distribution

(ii) Expected value of a random variable

(iii) Variance and Standard Deviation of a random variable

(iv). Binomial Distribution

6. Linear Programming

(i) Mathematical formulation of Linear Programming Problem

(ii) Graphical method of solution for problems in two variables

(iii) Feasible and infeasible regions

(iv). Optimal feasible solution

Section B1: Mathematics

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. Binary operations.

2. Inverse Trigonometric Functions

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

UNIT II: ALGEBRA

1. Matrices

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

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

Continuity and differentiability, derivative of composite functions, chain rule, derivatives of inverse trigonometric functions, derivative of implicit function. Concepts of exponential, logarithmic functions. Derivativesoflog xandex.Logarithmicdifferentiation.Derivativeoffunctions expressedinparametric forms. Second-order derivatives. Rolle’s and Lagrange’s Mean Value Theorems (without proof) and their geometric interpretations.

2. Applications of Derivatives

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). Tangent and Normal.

3. Integrals

Integrationasinverseprocessofdifferentiation.Integration ofavarietyoffunctionsbysubstitution, by partial fractions and by parts, only simple integrals of the type –

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

Applications in finding the area under simple curves, especially lines, arcs of circles/parabolas/el- lipses (in standard form only), area between the two above said curves (the region should be cleraly 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, 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 constant dx

dx /dy + Px =  Q, where P and Q are functions of y or constant dy

UNIT IV: VECTORS AND THREE-DIMENSIONAL GEOMETRY

1. Vectors

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

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.

Unit V: 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, feasible and infeasible solutions, optimal feasible solutions (up to three non-trivial constrains).

Unit VI: Probability

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.

Section B2: Applied Mathematics 

Unit I: Numbers, Quantification and Numerical Applications

Modulo Arithmetic

• Define modulus of an integer
• Apply arithmetic operations using modular arithmetic rules

Congruence Modulo

• Define congruence modulo
• Apply the definition in variousproblems

Allegation and Mixture

• Understand the rule of allegation to produce a mixture at a givenprice
• Determine the mean price of amixture
• Apply rule of allegation

Numerical Problems

• Solve real life problems mathematically

Boats and Streams

• Distinguish between upstreamand downstream
• Express the problem in the formof an equation

Pipes and Cisterns

• Determine the time taken by two or more pipes to fill or

Races and Games

• Compare the performance of two players w.r.t. time,
• distance taken/distance covered/ Work done from the given data

Partnership

• Differentiate between active partner and sleeping partner
• Determine the gain or loss tobe divided among the partners in the ratio of their investment with due
• consideration of the time volume/surface area for solid formed using two or more shapes

Numerical Inequalities

• Describe the basic concepts of numerical inequalities
• Understand and write numerical inequalities

UNIT II: ALGEBRA

A. Matrices and types of matrices

• Define matrix
• Identify different kinds ofmatrices

B. Equality of matrices, Transpose of amatrix, Symmetric andSkew symmetric matrix

• Determine equality of two matrices
• Write transpose of given matrix
• Define symmetric and skewsymmetric matrix

UNIT III: CALCULUS

Higher Order Derivatives

• Determine second and higherorder derivatives
• Understand differentiation ofparametric functions and implicit functions Identify dependent andindependent variables

Marginal Cost and Marginal Revenue usingderivatives

• Define marginal cost and marginal revenue
• Find marginal cost and marginal revenue

Maxima and Minima

• Determine critical points of thefunction
• 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

UNIT IV: PROBABILITY DISTRIBUTIONS

Probability Distribution

• Understand the concept ofRandom Variables and its Probability Distributions
• Find probability distribution ofdiscrete random variable

Mathematical Expectation

• Apply arithmetic mean of frequency distribution to find the expected value of a random variable

Variance

• Calculate the Variance and S.D.of a random variable

UNIT V: INDEX NUMBERS AND TIME BASED DATA 

A. Index Numbers

•  Define Index numbers as aspecial type of average B. ConstructionofIndexnumbers
• Construct different type of indexnumbers C. TestofAdequacyofIndexNumbers
• Apply time reversal test

UNIT VI: UNIT V: INDEX NUMBERS AND TIME BASED DATA

Population and Sample

• Define Population and Sample
• Differentiate between population and sample
• Define a representative sample from a population

Parameter and Statistics and Statistical Interferences

• Define Parameter with reference to Population
• Define Statistics with referenceto Sample
• Explain the relation betweenParameter and Statistic
• Explain the limitation of Statisticto generalize the estimation for population
• Interpret the concept of Statistical Significance andStatistical Inferences
• State Central Limit Theorem
• Explain the relation betweenPopulation-Sampling Distribution-Sample

UNIT VII: INDEX NUMBERS AND TIME-BASED DATA

A. Time Series

• Identify time series aschronological data 

B. Components of Time Series

• Distinguish between different components of time series 

C. Time Series analysis for univariate data

• Solve practical problems basedon statistical data and Interpret 

UNIT VIII: FINANCIAL MATHEMATICS

Perpetuity,Sinking Funds

• Explain the concept of perpetuity and sinking fund
• Calculate perpetuity
• Differentiate between sinkingfund and saving account

Valuation of Bonds

• Define the concept of valuationof bond and related terms
• Calculate value of bond usingpresent value approach

Calculation of EMI

• Explain the concept of EMI
• Calculate EMI using various methods

Linear method of Depreciation

• Define the concept of linearmethod of Depreciation
• Interpret cost, residual value and useful life of an asset fromthe given information
• Calculate depreciation

UNIT IX: LINEAR PROGRAMMING

A. Introduction and related terminology

• Familiarize with terms related to Linear Programming Problem 

B. Mathematical formulation of Linear Programming Problem

• Formulate Linear Programming Problem

C. Different types of Linear Programming Problems

• Identify and formulate differenttypes of LPP

D. Graphical Method of Solution for problems in two Variables

• Draw the Graph for a system of linear inequalities involving two variables and to find its solution graphically

 E. Feasible and Infeasible Regions

•  Identify feasible, infeasible and bounded regions

F. Feasible and infeasible solutions, optimal feasible solution

• Understand feasible andinfeasible solutions
•  Find optimal feasible solution