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 the 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
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
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).
7. Integrals
Integration as the 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.
8. Applications of the Integrals
Applications in finding the area under simple curves, especially lines, arcs of circles/parabolas/ellipses (in standard form only), and the area between the two above said curves (the region should be clearly identifiable).
9. 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.
10. 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.
11. Three-dimensional Geometry
Direction cosines/ratios of a line joining two points. Cartesian and vector equations of a line, coplanar and skew lines, and the shortest distance between two lines. Cartesian and vector equations of a plane. Angle between (i) two lines, (ii) two planes, (iii) a line and a plane. Distance of a point from a plane.
12. Linear Programming
Introduction, related terminology such as constraints, objective function, optimization, and 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 constraints).
13. Probability
Multiplication theorem on probability. Conditional probability, independent events, total probability,
Bayes’s theorem. Random variable and its probability distribution, mean, and variance of haphazard
variable. Repeated independent (Bernoulli) trials and binomial distribution.
