Math Tutoring for Grade 12-
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XII
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MLK Home and Online Tutoring Service 's Math Refresher course is designed to build a strong foundation in the basic math skills essential for success on the following tests:
Topics Covered by our Math Home Tutoring Program:
MATHEMATICS SYLLABUS FOR Grade XII
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
Inverse Trigonometric Functions
- Definition, range, domain, principal value branches
- Graphs of inverse trigonometric functions
- Elementary properties of inverse trigonometric functions
UNIT II: ALGEBRA
Matrices
- Concept, notation, order, equality
- types of matrices, 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)
Determinants
- Determinant of a square matrix (up to 3 × 3 matrices)
- properties of determinants, minors, cofactors
- 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 function
- Concepts of exponential, logarithmic functions
- Derivatives of loge x and ex
- 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 interpretations
Applications of Derivatives
- Applications of derivatives: Rate of change
- increasing/decreasing functions
- tangents and normals
- 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)
Integrals
- Integration as inverse process of differentiation
- Integration of a variety of functions by substitution, by partial fractions and by parts, only simple integrals to be evaluated
- Definite integrals as a limit of a sum
- Fundamental Theorem of Calculus (without proof)
- Basic properties of definite integrals and evaluation of definite integrals
Applications of the Integrals
- Applications in finding the area under simple curves, especially lines, arcs of circles/parabolas/ellipses(in standard form only)
- area between the two above said curves (the region should be cleraly identifiable)
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
- homogeneous differential equations of first order and first degree
- Solutions of linear differential equations
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UNIT IV: VECTORS 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
- scalar triple product
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
- 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 constrains)
Unit VI: Probability
- Multiplications theorem on probability
- Conditional probability
- independent events
- total probability
- Baye’s theorem
- Random variable and its probability distribution
- mean and variance of haphazard variable
- Repeated independent (Bernoulli) trials and Binomial distribution
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