Course Syllabus

Unit I: Relations and Functions

Chapter 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

Chapter 2: Inverse Trigonometric Functions

• Definition, range, domain, principal value branch
• Graphs of inverse trigonometric functions

• Elementary properties of inverse trigonometric functions

Unit II: Algebra

Chapter 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
• Noncommutativity 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).

Chapter 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
• Ad joint 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

Chapter 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
• 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

Chapter 2: Applications of Derivatives

• Applications of derivatives: rate of change of bodies, increasing/decreasing functions, tangents and normal, 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 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 and problems based on them

𝑑𝑥, ∫ √𝑎² ± 𝑥² dx, ∫ √𝑥² − 𝑎² dx

∫ √𝑎𝑥² + 𝑏𝑥 + 𝑐 dx, ʃ (px + q) √𝑎𝑥² + 𝑏𝑥 + 𝑐 𝑑𝑥

• 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 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)

Chapter 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

Chapter 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

Chapter 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:
  • Two lines
  • Two planes
  • A line and a plane
• Distance of a point from a plane

Unit V: Linear Programming

Chapter 1: Linear Programming

• Introduction
• Related terminology such as:
  • Constraints
  • Objective function
  • Optimization
  • Different types of linear programming (L.P.) Problems
  • Mathematical formulation of 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

Chapter 1: Probability

• Conditional probability
• Multiplication theorem on probability
• Independent events, total probability
• Baye's theorem
• Random variable and its probability distribution
• Mean and variance of random variable
• Repeated independent (Bernoulli) trials and Binomial distribution