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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
• 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