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Unit I: Relations and Functions


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
  • Composition of function and invertible function

Inverse Trigonometric Functions

  • Definition
  • Range
  • Domain
  • Graphs of inverse trigonometric functions
  • Properties of inverse trignometric functions

UNIT II: ALGEBRA


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).

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
  • 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


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 interpretation.

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)
  • Maximum and minimum value of function in closed interval

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 of the following types and problems based on them.
  • Geometric interpretation of indefinite integral
  • Some properties of indefinite integrals
  • Comparison between differentiation and integration

Applications of the Integrals

  • Area under simple curves
  • Concept of exponential and logarithmic functions.
  • Area of a region bounded by curve and a line
  • Area between two curves

Differential Equations

  • Basic conceptn
  • Degree of a differential equation
  • General and particular solutions of differential equation
  • Formation of differential equation whose general solution is given
  • Methods of solving first order, first degree differential equation
  • Homogeneous differential equation
  • 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


Vectors

  • Some basic concepts
  • Types of vectors
  • Addition of vectors
  • Multiplication of vector by a scalar: components of vector
  • Vector joing two points
  • Section formula
  • Product of two vectors: scalar product and application
  • Scalar triple product
  • Cross product of two vectors

Three - dimensional Geometry

  • Direction cosines and direction ratios of a line
  • Equation of a line in space
  • Angle between two lines
  • Shortest distance between two lines
  • Plane: equation of plane in normal form
  • "Equation of a plane perpendicular to a given vector and passing through a given point"
  • "Equation of plane passing through three non collinear points"
  • "Intercept form of equation of line and a plane passing throug through the intersection of two given plane"
  • Coplanarity of two lines
  • Angle between two planes
  • Distance of a point from a plane
  • Angle between a line and a plane

Unit V: Linear Programming and Probability


Linear Programming

  • Linear programming problem and its mathematical formulation
  • Graphical method of solving linear programming problem
  • Different types of linear programming problem

Probability

  • Conditional 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