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Mathematics 12 Mathematics 12
UNIT I: Relations and Functions
- reflexive
- symmetric
- transitive and equivalence relations
- composite functions
- inverse of a function. Binary operations
UNIT II: Inverse Trigonometric Functions
- Definition
- range
- domain
- principal value branch
- Graphs of inverse trigonometric functions
- Elementary properties of inverse trigonometric functions
UNIT III: Matrices
- Concept
- notation
- order
- equality
- zero 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
- 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)
UNIT IV: Determinants
- Determinant of a square matrix (up to 3 × 3 matrices)
- minors
- cofactors and applications of determinants in finding the area of a triangle
- inverse of a square matrix
UNIT V: Continuity and Differentiability
- derivative of composite functions
- chain rule
- Concept of exponential and logarithmic functions
UNIT VI: Application of Derivatives
- Rate of change of bodies
- 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)
UNIT VII: 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
UNIT VIII: Applications of 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)
UNIT IX: 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
UNIT X: Vectors
- Vectors and scalars
- addition of vectors
- components of a vector in two dimensions and three-dimensional space
- scalar and vector products
- scalar and vector triple product
UNIT XI: Three-dimensional Geometry
- Direction cosines and direction ratios of a line joining two points
- Cartesian equation and vector equation of a line
- coplanar lines
- shortest distance between two lines
- Cartesian and vector equation of a plane
- distance of a point from a plane
UNIT XII: 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)
- feasible and infeasible solutions
- optimal feasible solutions (up to three non-trivial constraints)
UNIT XIII: Probability
- Conditional probability
- multiplication theorem on probability
- independent events
- total probability
- mean and variance of random variable
BETA
