Chapter – Relations and Functions
Types of relations: reflexive, symmetric, transitive and equivalence relations. One to one and onto functions, composite functions, inverse of a function.
Chapter – Inverse Trigonometric Function
Definition, range, domain, principal value branch, graphs of inverse trigonometric
functions. Elementary properties of inverse trigonometric function.
Chapter – Matrices
Concept, notation, order equality, types of matrices,zero and identity matrix,
transpose of a matrix, symmetric and skew symmetric matrices: Addition and
Multiplication and Multiplication with a scalar. Simple properties of addition,
multiplication and scalar multiplication. Non-commutability of multiplication of matrices and existence of non-zero matrices whose product in null matrix (restrict to square matrices of order 2). Concept of elementary row and column operations. Invertible matrices and proof of uniqueness of inverse, if it exists; (Here all matrices will have real entries).
Chapter – Determinants:
Determinants of square matrix (upto 3 x 3 order), properties of determinants, minors, cofactors and applications of determinants in finding the area of triangle. Adjoint and inverse of a square matrix. Properties of inverse of a matrix. Consistency, inconsistency and number of solutions of system of linear equations in two or three variables (having unique solution ) using inverse of matrix.
Chapter – 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
Chapter – 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).
Chapter – Integrals
Integration as inverse process of differentiation. Integration of a variety of
functions by substitution, by partial fractions and by parts.Definite integrals as a limit of a sum , Fundamental Theorem of Calculus (without proof).Basic properties of definite integrals and evaluation of definite integrals.
Chapter – 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).
Chapter – 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.
Chapter – 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 an application of scalar (dot) product of vectors, vector (cross) product of vectors, scalar triple product of vectors
Chapter – 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.
Chapter – 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 or unbounded), feasible and infeasible solutions, optimal feasible solutions (up to three non-trivial constraints)
Chapter – 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.