NCERT Solutions for Class 12 Maths provides a solid conceptual base for all the topics included in the CBSE Class 12 Maths Syllabus. It covers all the important theorems and formulae with detailed explanations to ensure better conceptual understanding in students. NCERT Solutions for Class 12 Maths will help students in preparation for undergraduate-level Engineering entrance exams like JEE Mains, BITSAT, VITEEE, etc., to be conducted for the year 2023-2024.

Chapter 1 of NCERT textbook deals with relations and functions, types of relation, types of functions, the composition of functions and invertible function, binary operations and miscellaneous examples. Here, you can find the exercise solution links for the topics covered in this chapter.

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Topics Covered in Class 12 Maths Chapter 1 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.

Chapter 2, Inverse Trigonometric Functions of NCERT Class 12 Maths, gives an account of various topics such as the remarks based on basic concepts of inverse trigonometric functions, properties of inverse trigonometric functions and miscellaneous examples. These concepts are explained very well through examples. Solutions to the exercises from the chapter can be accessed from here.

In Chapter 3 of NCERT textbook, we shall see the definition of a matrix, types of matrices, equality of matrices, operations on matrices such as the addition of matrices and multiplication of a matrix by a scalar, properties of matrix addition, properties of scalar multiplication, multiplication of matrices, properties of multiplication of matrices, transpose of a matrix, properties of the transpose of the matrix, symmetric and skew-symmetric matrices, elementary operation or transformation of a matrix, the inverse of a matrix by elementary operations and miscellaneous examples. Here, you can find the exercise solution links for the topics covered in this chapter.

Topics Covered in Class 12 Maths Chapter 3 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. 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).

Chapter 4 of 12 Class NCERT Maths Solution discusses the topic of determinants. Students will get to learn about the definition and meaning of determinants, remarks based on order of determinants, properties of determinants, finding area of triangle using determinants, minors and cofactors of determinants, adjoint of a matrix, inverse of matrix, applications of determinants and matrices, and miscellaneous examples. Below, we have links provided to each exercise solution covered in this chapter.

Topics Covered in Class 12 Maths Chapter 4 Determinants:

Determinant of a square matrix (up to 3 3 matrices), properties of determinants, 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.

Chapter 5 of NCERT textbook starts with the definition of continuity. Students go on to learn about continuity, algebra of continuous function, definition and meaning of differentiability, derivatives of composite functions, derivative of implicit functions, derivative of inverse trigonometric functions, exponential and logarithmic functions, logarithmic differentiation, derivatives of functions in parametric forms, second order derivatives, mean value theorem via miscellaneous examples. Here, students can find the exercises explaining these concepts properly with solutions.

Chapter 6 of NCERT textbook provides definition of derivatives, rate of change of quantities, increasing and decreasing functions, tangents and normals, approximations, maxima and minima, first derivative test, maximum and minimum values of a function in a closed interval, and miscellaneous examples. Here, students can find the exercises explaining these concepts properly.

Topics Covered in Class 12 Maths Chapter 6 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).

In Chapter 7, Integrals, we shall see the definition of indefinite integral, integration as an inverse process of differentiation, geometrical interpretation of indefinite integral, properties of indefinite integral, comparison between differentiation and integration, methods of integration such as integration by substitution, integration using partial fractions, integration by parts, integration using trigonometric identities, integration of some integral functions, definition and meaning of definite integrals, definite integral as the limit of a sum, fundamental theorem of calculus, evaluation of definite integrals by substitution, some properties of definite integrals and miscellaneous examples. Here, we have given exercises with solutions based on these topics from the chapter.

Topics Covered in Class 12 Maths Chapter 7 Integrals:

Integration as inverse process of differentiation. Integration of a variety of functions by substitution, by partial fractions and by parts, only simple integrals of the type

In Chapter 8 of NCERT textbook, we continue the discussions about integrals starting with definition, area under simple curves, area of the region bounded by a curve and a line, area between two curves and miscellaneous examples. From the below-given links, students can access the exercises with solutions explaining the concepts from this chapter.

In this chapter, students concentrate on the definition of differential equations, basic concepts related to differential equations, degree of a differential equation, order of a differential equation, general and particular solutions of a differential equation, formation of a differential equation whose general solution is given, procedure to form a differential equation that will represent a given family of curves, methods of solving first order, first degree differential equations, differential equations with variable separable, homogeneous differential equations, linear differential equations, procedure involved to solve first order linear differential equations, and miscellaneous examples. Solutions to the exercises from the chapter can be accessed from here.

Topics Covered in Class 12 Maths Chapter 9 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 equation of the type:

dy/dx + Py = Q, where P and Q are functions of x

In this chapter, students are introduced to vector algebra. The concepts in this chapter include how to find position vector, some basic concepts related to vector algebra, direction cosines, types of vectors such as zero vector, unit vector, collinear vector, equal vector, negative of a vector, addition of vectors, properties of vector addition, multiplication of a vector by a scalar, components of a vector, vector joining two points, section formula, product of two vectors, scalar or dot product of two vectors, properties of scalar product, projection of a vector on a line, vector or cross product of two vectors and miscellaneous examples. From the below-given links, students can access the exercises with solutions explaining the concepts from this chapter.

Topics Covered in Class 12 Maths Chapter 10 Vector 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 11 of NCERT Class 12 Maths constitutes topics related to three dimensional geometry, direction cosines and direction ratios of a line, relation between the direction cosines of a line, direction cosines of a line passing through two points, equation of a line in space, equation of a line through a given point and parallel to a given vector, derivation of cartesian form from vector form, equation of a line passing through two given points, angle between two lines, shortest distance between two lines, distance between two skew lines, distance between two parallel lines, plane in cartesian form, equation of a plane in a normal form, equation of a plane perpendicular to a given vector and passing through a given point, equation of a plane passing through three non collinear points, intercept form of the equation of a plane, plane passing through the intersection of two given planes, coplanarity of two lines, angle between two planes, distance of a point from a plane, angle between a line and a plane and miscellaneous examples. Below, we have listed links to the solutions for the exercises from this chapter. 152ee80cbc

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