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# NCERT Solutions class 12 Maths PDF

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## NCERT Solutions Class 12 Maths PDF

### Chapter 13: Probability

Buy Books & Solutions Online Mathematics 12 by R D Sharma is a very good book for the concepts and practice material including a lots of solved questions with proper explanation. If you have done a chapter, the Together With Maths is very good to get proper practice. Just like R D Sharma, All in One for Mathematics is also very good book with ample practice questions. CBSE Mathematics ( 15 Sample Question Papers ) is very useful after going through whole syllabus. Buy books according to your need. Download NCERT Solutions class 12 Maths PDF.

### Main points to be recovered

#### Relations and Functions

Chapter 1 Relations and Functions – summary: 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

Chapter 2 Inverse Trigonometric Functions – summary: Definition, range, domain, principal value branch. Graphs of inverse trigonometric functions. Elementary properties of inverse trigonometric functions.

#### Matrices

Chapter 3 Matrices – summary: Concept, notation, order, equality, types of matrices, zero and identity 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. Noncommutativity 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

Chapter 4 Determinants – summary: Determinant of a square matrix (up to 3 x 3 matrices), properties of determinants, 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.

#### Continuity and Differentiability

Chapter 5 Continuity and Differentiability – summary: Continuity and differentiability, derivative of composite functions, chain rule, derivatives of inverse trigonometric functions, derivative of implicit functions. Concept of exponential and logarithmic functions. Derivatives of logarithmic and exponential functions. Logarithmicdifferentiation, derivative of functions expressed in parametric forms. Second order derivatives. Rolle’s and Lagrange’s Mean Value Theorems (without proof) and their geometric interpretation.

#### Applications of Derivatives

Chapter 6 Applications of Derivatives – summary: rate of change of bodies, increasing/decreasing functions, 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).

#### Integrals

Chapter 7 Integrals – summary: 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 types given in the syllabus and problems based on them. 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

Chapter 8 Applications of the Integrals – summary: 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).

#### Differential Equations

Chapter 9 Differential Equations – summary: 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. Solutions of linear differential equation of the type given in the syllabus.

#### Vectors

Chapter 10 Vectors – summary: 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, scalar triple product of vectors.

#### Three – dimensional Geometry

Chapter 11 Three – Dimensional Geometry – summary: Direction cosines and direction ratios of a line joining two points.Cartesian equation 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.

#### Linear Programming

Chapter 12 Linear Programming – summary: 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 and unbounded), feasible and infeasible solutions, optimal feasible solutions (up to three non-trivial constraints).

#### Probability

Chapter 12 Probability – summary: Conditional probability, multiplication theorem on probability, independent events, total probability, Bayes’ theorem, Random variable and its probability distribution, mean and variance of random variable. Repeated independent (Bernoulli) trials and Binomial distribution.

## Related Pages

Also see the complete solutions for NCERT (all classes and all subjects).