Sets, Relations and Functions

Different kinds of sets and their basic properties

Relations, types of relations

Different types of real valued functions.

2 Limit, Continuity and Differentiability :

Limit, continuity and differentiability of algebraic functions,

trigonometric functions

exponential functions and logarithmic functions.

3 Complex and Vector Algebra :

Complex numbers and their algebraic properties

polar representation,

square root of a complex number

Vectors and scalars

types of vectors and their algebraic properties

scalar and vector product of two vectors

scalar triple product.

4 Differential calculus

Derivatives of sum, difference, product and quotient of functions

Derivatives of polynomial and trigonometric functions

Derivatives of implicit and explicit functions

Increasing and decreasing functions

Concept of second order derivative.

5 Integral calculus

Integration of functions by the method of substitution

partial fraction and by parts.

Basic properties of definite integrals and their uses to evaluate them.

6 Differential equations

Order and degree of a differential equation

solution of differential equations of first order and first degree.

7 Permutations and combinations

Derivation of formulae

their connections and simple applications.

Binomial theorem : Binomial theorem for positive integral indices

general and middle terms in binomial expansion.

8 Matrices

Various types of matrices

their basic operations and properties

Invertible matrices and their inverse.

Determinant : Determinant of a square matrix and their properties

Solution of system of linear equations in two or three variables using inverse of a matrix.

9 Two dimensional geometry

Straight line

standard equations and simple properties of circle, parabola, ellipse, hyperbola.

10 Applications of derivatives and integrals

Tangent and normals

maxima and minima of functions of one variable

Area under simple curves

area between the simple curves.

11 Statistics

Mean, Mode, Median for grouped data

measure of dispersion

Probability and their elementary laws

conditional probability.

Group Theory

Groups and their simple properties

order of an element

order of a group

permutation groups

cyclic groups and their properties

subgroups and their basic algebraic properties

cosets and their properties.

2 Normal subgroup and Rings

Normal subgroups and quotient groups

theorems on homomorphism and isomorphism.

Rings, ideals, integral domain and fields.

3 Theory of equations

Relation between the roots and coefficients of general polynomial equation in one variable

Transformation of equations

Descartes' rule of signs

solution of cubic equations by Cardon's method

Biquadratic equations by Ferari's method.

4 Calculus

Partial derivatives

curvature, asymptotes

envelopes and evolutes

maxima and minima of functions upto two variables

Beta and Gamma functions

double and triple integrals.

5 Advanced Calculus

Mean value theorems (Rolle's, Lagrange's, Taylor's theorems)

sequence and series with convergence properties.

6 Complex Analysis

Continuity and differentiability of complex functions

Analytic functions

Cauchy - Riemman equation

Harmonic functions

Conformal mappings.

7 Ordinary and Partial differential equations

Linear differential equations of first order and higher degree

Clairaut's form

Linear differential equations of constant coefficients

ordinary homogeneous differential equations

Linear differential equations of second order with variable coefficients

Partial differential equations of first order

solution by Lagrange's method.

8 Vector calculus

Gradient, divergence and curl, identities related to them

Line, surface and volume integrals

Applications of Gauss, Stoke's and Green's theorems.

9 Three dimensional geometry

Direction ratios and cosines

straight line, plane, sphere, cone and cylinder.

10 Statics

Equilibrium of co-planner forces, moments, friction, virtual work catenary.

11 Dynamics

Velocities and acceleration along radial and transverse directions and along tangential and normal directions

simple harmonic motion

Rectilinear motion under variable laws

Hook's law and problems

projectiles.

1 Linear Algebra and Metric Space

Vector spaces

linear dependence and independence

bases

dimensions

linear transformations

matrix representation

algebra of matrices

characteristic roots and vectors

determinants

Cayley – Hamilton theorem.

Metric Spaces : Bounded and unbounded metric spaces

Open and closed sets in a metric space

Cantor's ternary set, closure, bases, product spaces.

2 Integral transforms and special functions

Hyper-geometric functions

Legendre's polynomials,

Bessel's functions

Recurrence relations and orthogonal properties.

Laplace transform

inverse Laplace transform

Fourier sine and cosine transforms

Convolution theorem.

3 Differential Geometry and Tensors

Curves in spaces

Curvature

Torsion, Skew curvature

Serret - Frenet formulae

Helices Osculating circle and sphere.

Types of tensors and their algebraic properties

Christoffel's symbols

covariant and contravariant differentiation

Geodesics.

4 Numerical Analysis

Newton's formula for forward and backward interpolation for equal intervals,

Divided difference

Newton's Lagrange's, Starling's and Bessel's interpolation formulae.

5 Optimization Technique's

Convex set and its properties

Solution of a L.P.P. by using Simplex methods

Duality, Assignment

Transportation and Game theory.

1. Importance of Psychology in Teaching-Learning :

Learner,

Teacher,

Teaching-learning process,

School effectiveness.

2. Development of Learner :

Cognitive, Physical, Social, Emotional and Moral development patterns and characteristics among adolescent learner.

3. Teaching – Learning

Concept, Behavioural, Cognitive and constructivist principles of learning and its implication for senior secondary students.

Learning characteristics of adolescent and its implication for teaching.

4. Managing Adolescent Learner

Concept of mental health and adjustment problems.

Emotional Intelligence and its implication for mental health of adolescent.

Use of guidance techniques for nurturing mental health of adolescent.

5. Instructional Strategies for Adolescent Learner

Communication skills and its use.

Preparation and use of teaching-learning material during teaching.

Different teaching approaches: Teaching models- Advance organizer, Scientific enquiry, Information, processing, cooperative learning.

Constructivist principles based Teaching.

6. ICT Pedagogy Integration

Concept of ICT.

Concept of hardware and software.

System approach to instruction.

Computer assisted learning.

Computer aided instruction.

Factors facilitating ICT pedagogy integration.