Class 12 Mathematics : Master the Logic…
Complete step-by-step Understanding
Unit-I: Relations and Functions
Relations and Functions
Types of relations: reflexive, symmetric, transitive and equivalence relations. One to one and onto functions.
Unit-II: Algebra
Determinants
Determinant of a square matrix (up to 3 x 3 matrices), 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, chain rule, derivative of composite functions, derivatives of inverse trigonometric functions like $sin^{-1} x$, $cos^{−1}𝑥$ and $tan^{−1}𝑥$, 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.
Applications of Derivatives
Applications of derivatives: rate of change of quantities, increasing/decreasing functions, 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, Evaluation of simple integrals of the following types and problems based on them.
$\int \frac{d x}{x^2 \pm a^2}$, $\int \frac{d x}{\sqrt{x^2 \pm a^2}}$, $\int \frac{d x}{\sqrt{a^2-x^2}}$, $\int \frac{d x}{a x^2+b x+c}$, $\int \frac{d x}{\sqrt{a x^2+b x+c}}$, $\int \frac{p x+q}{a x^2+b x+c} dx$, $\int \frac{p x+q}{\sqrt{a x^2+b x+c}} dx$, $\int \sqrt{a^2 \pm x^2} d x$, $\int \sqrt{x^2-a^2} dx$, $\int \sqrt{a x^2+b x+c} d x$
Fundamental Theorem of Calculus (without proof). Basic properties of definite integrals and evaluation of definite integrals.
Application of the Integrals
Applications in finding the area under simple curves, especially lines, circles/ parabolas/ellipses (in standard form only)
Differential Equations
Definition, order and degree, general and particular solutions of a differential equation. Solution of differential equations by method of separation of variables, solutions of homogeneous differential equations of first order and first degree. Solutions of linear differential equation of the type: $ \frac{dy}{dx}+py=q $, where p and q are functions of x or constants. $\frac{dx}{dy}+px=q$, where where p and q are functions of y or constants
Unit-IV: Vectors and Three-dimensional Geometry
Vectors
Vectors and scalars, magnitude and direction of a vector. Direction cosines and direction ratios of a vector. 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. Definition, Geometrical Interpretation, properties and application of scalar (dot) product of vectors, vector (cross) product of vectors.
Three-dimensional Geometry
Direction cosines and direction ratios of a line joining two points. Cartesian equation and vector equation of a line, skew lines, shortest distance between two lines. Angle between two lines.
Unit-V: Linear Programming Problem
Linear Programming
Introduction, related terminology such as constraints, objective function, optimization, graphical method of solution for problems in two variables, feasible and infeasible regions (bounded or unbounded), feasible and infeasible solutions, optimal feasible solutions (up to three non-trivial constraints).
Unit-VI: Probability
Probability
Conditional probability, multiplication theorem on probability, independent events, total probability, Bayes’ theorem.