Mathematics 12 Mathematics 12
UNIT I: RELATIONS AND FUNCTIONS
- inverse of a function, Binary operations
- Inverse Trigonometrical Functions
- Definition, range, domain, principal value branches. Graphs of inverse trigonometric functions
UNIT II: ALGEBRA
- Concept, notation, order, equality
- 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 inverse, if it exists; (Here all matrices will have real entries)
- Determinant of a square matrix (up to 3 x 3 matrices)
- 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
- Cramer’s Rule and its applications
UNIT III: CALCULAS
- Continuity and Differentiability
- Continuity and differentialiabity
- derivative of composite functions
- chain rule
- derivates of inverse trigonometric functions
- derivate of implicit functions
- concept of exponential and logarithmic functions to the base e
- Logarithmic functions as inverse of exponential functions
- lim 1/x, lim 1/x, lim (1+1/x)x, lim (1+x)1/x, lim log(1+x), lim ex-1 x->0 x-> à x-> à x->0 x->0 x x->0 x Derivatives of logarithmic and exponential functions
- Logarithmic differentiation, derivative of functions expressed in parametric forms
- Second order derivatives
- Rolle’s and Lagranges’s Mean value theorems (without proof) and their geometric interpretation and simple applications
- Derivatives, continuity, and differentiability
- algebra of functions, rational
UNIT IV: VECTOR AND THREE-DIMENSIONAL GEOMETRY
- 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
- Three - dimensional Geometry
- Direction cosines/ratios of a line joining two points
- Cartesian and vector equation of a line
- coplanar and skew lines
- shortest distance between two lines
- Cartesian and vector equation of a plane
- Angle between (i) two lines (ii) two planes (iii) a line and a plane
- Distance of a point from a plane
UNIT-V: LINEAR PROGRAMMING
- Introduction
- definition of 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
- feasible and infeasible solutions
- optimal feasible solutions (up to three non-trivial constraints)
UNIT VI: PROBABILITY
- Multiplication theorem on probability Conditional probability
- independent events
- total probability
- Random variable and its probability distribution mean and variance of random variable
- Repeated independent (Bernoulli) trials and Binomial distribution
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