course structure

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 × 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 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

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 of the following types and problems based on them

  $\int \frac{dx}{x^2\pm {a^2}'}$, $\int \frac{dx}{\sqrt{x^2\pm {a^2}'}}$, $\int \frac{dx}{\sqrt{a^2-x^2}}$, $\int \frac{dx}{ax^2+bx+c} \int \frac{dx}{\sqrt{ax^2+bx+c}}$

  $\int \frac{px+q}{ax^2+bx+c}dx$, $\int \frac{px+q}{\sqrt{ax^2+bx+c}}dx$, $\int \sqrt{a^2\pm x^2}dx$, $\int \sqrt{x^2-a^2}dx$

  $\int \sqrt{ax^2+bx+c}dx$, $\int \left ( px+q \right )\sqrt{ax^2+bx+c}dx$

• 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 l.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

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