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Mathematics 12 Mathematics 12
CHAPTER I: RELATIONS AND FUNCTIONS
- Inverse of a function
- Binary operations
CHAPTER II: Inverse Trigonometrical Functions
- Definition, range, domain, principal value branches
- Graphs of inverse trigonometric functions
CHAPTER III: Matrices
- 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)
CHAPTER IV: Determinants
- 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
CHAPTER V: CONTINUITY AND DIFFERENTIABILITY
- Algebra of functions
- Rational
CHAPTER VI: APPLICATIONS OF DERIVATIVE
- Geometrical application-tangent and normal at a point
- Rolle’s theorem and Mean value theorem and their geometrical interpretation (without proof)
- Derivative as a rate measure-introduction
- Approximation (without proof)
- Maxima and minima
- Introduction of extrema and extreme values
- First derivative test
- Second derivative test
CHAPTER VII: INTEGRATION
- Indefinite integrals-methods of integration
- Substitution method
- Integrals of the various types
- Integration by parts (reduction formulae are not expected)
- Integration by partial fraction-factors involving repeated and non-repeated linear factors
- Non-repeated quadratic factors
- Definite integral-definite integral as a limit of sum
- Fundamental theorem of integral calculus (without proof)
- Evaluation of definite integral 1) by substitution, 2) integration by parts
- Properties of definite integrals
CHAPTER VIII: APPLICATIONS OF DEFINITE INTEGRAL
- Area under the curve
- Area bounded by curve and axis (simple problems)
- Area bounded by two curves
- Volume of solid of revolution-volume of solid obtained by revolving the area under the curve about the axis (simple problems)
CHAPTER IX: DIFFERENTIAL EQUATION
- Definition-differential equation
- Order
- Degree
- General solution
- Particular solution of differential equation
- Formation of differential equation-formation of differential equation by eliminating arbitary constants (at most two constants)
- Solution of first order and first degree differential equation-variable separable method
- Homogeneous differential equation (equation reducible to homogeneous form are not expected)
- Linear differential equation
- Applications : population growth, bacterial colony growth, surface area, Newton’s laws of cooling, radioactive decay
CHAPTER X: VECTORS ALGEBRA
- 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
CHAPTER XI: 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
CHAPTER XII: 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)
CHAPTER XIII: 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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