Class 12 Maths NCERT Solutions [2024-2025]

Excelling in Class 12 Math: Your Ultimate Guide to NCERT Solutions

Welcome to your ultimate resource for Class 12 Math NCERT solutions! In this comprehensive guide, we break down complex mathematical concepts, providing step-by-step solutions that are tailored to your academic success. Let's dive into the world of 12th-grade mathematics and ensure you're well-prepared for your exams.

Ncert Solutions For Class 12 Mathematics Chapter - 1: Relations & Functions

Topics Covered in Class 12 Maths Chapter 1 Relations and Functions: 

1. Types of relations: reflexive, symmetric, transitive and equivalence relations.

2. One to one and onto functions.

Ncert Solutions For Class 12 Mathematics Chapter - 2: Inverse Trigonometry

Topics Covered in Class 12 Maths Chapter 2 Inverse Trigonometric Functions:

1. Definition, range, domain, principal value branch.

2. Graphs of inverse trigonometric functions.

Ncert Solutions For Class 12 Mathematics Chapter - 3: Matrices

Topics Covered in Class 12 Maths Chapter 3 Matrices:

1. Concept, notation, order, equality, types of matrices, zero and identity matrix, transpose of a matrix, symmetric and skew symmetric matrices.

2. Operations on matrices: Addition and multiplication and multiplication with a scalar. Simple properties of addition, multiplication and scalar multiplication.

3. Noncommutativity of multiplication of matrices and existence of non-zero matrices whose product is the zero matrix (restrict to square matrices of order 2).

4. Invertible matrices and proof of the uniqueness of inverse, if it exists; (Here all matrices will have real entries).

Ncert Solutions For Class 12 Mathematics Chapter - 4: Determinants

Topics Covered in Class 12 Maths Chapter 4 Determinants:

1. 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.

2. 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.

Ncert Solutions For Class 12 Mathematics Chapter - 5: Continuity & Differentiability

Topics Covered in Class 12 Maths Chapter 5 Continuity and Differentiability:

1. Continuity and differentiability, derivative of composite functions, chain rule, derivatives of inverse trigonometric functions, derivative of implicit function.
2. Concept of exponential and logarithmic functions and their derivatives. Logarithmic differentiation.
3. Derivative of functions expressed in parametric forms.
4. Second order derivatives. Rolle’s and Lagrange’s Mean Value Theorems (without proof) and their geometric interpretations.

Class 12 Maths Ncert Solutions Chapter 6 Applications of Derivatives

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:

1. 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).

Class 12 Maths Ncert Solutions Chapter 7 Integrals

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:

1. 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
2. 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.

1. Fundamental Theorem of Calculus (without proof). Basic properties of definite integrals and evaluation of definite integrals.
2. Definite integrals as a limit of a sum. Fundamental Theorem of Calculus (without proof). Basic properties of definite integrals and evaluation of definite integrals.

Class 12 Maths Ncert Solutions Chapter 8 Applications of Integrals:

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.

Topics Covered in Class 12 Maths Chapter 8 Applications of Integrals:

1. 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 clearly identifiable).

Class 12 Maths Ncert Solutions Chapter 9 Differential Equations:

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:

1. Definition, order and degree, general and particular solutions of a differential equation.
2. 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.
3. Solutions of linear differential equation of the type: dy/dx + Py = Q, where P and Q are functions of x 

Class 12 Maths Ncert Solutions Chapter 10 Vector Algebra:

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:

1. Vectors and scalars, magnitude and direction of a vector. Direction cosines/ratios of vectors.
2. 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.
3. Scalar (dot) product of vectors, projection of a vector on a line. Vector (cross) product of vectors.

Class 12 Maths Ncert Solutions Chapter 11 Three Dimensional Geometry

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.

Topics Covered in Class 12 Maths Chapter 11 Three Dimensional Geometry:

1. Direction cosines/ratios of a line joining two points.
2. Cartesian and vector equation of a line, coplanar and skew lines, shortest distance between two lines.
3. 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.

Class 12 Maths Ncert Solutions Chapter 12 Linear Programming

In this chapter, students are introduced to linear programming. The concepts covered in this chapter are linear programming problem and its mathematical formulation, mathematical formulation of the problem, graphical method of solving linear programming problems, different types of linear programming problems with miscellaneous examples. Students can find the exercises explaining these concepts properly with solutions.

Topics Covered in Class 12 Maths Chapter 12 Linear Programming:

1. 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).

Class 12 Maths Ncert Solutions Chapter 13 Probability

This chapter deals with probability. By reviewing the basic facts and formulae studied earlier in Class 11, we shall implement those formulas again in this chapter along with the advance concepts mentioned, i.e., conditional probability, properties of conditional probability, multiplication theorem on probability, multiplication rule of probability for more than two events, independent events, Bayes’ theorem, partition of a sample space, theorem on total probability, random variables and its probability distributions, probability distribution of random variable, mean of random variable, variance of random variable, Bernoulli trials and binomial distribution with miscellaneous examples. We have compiled here the solutions to exercises from this chapter that explain the concept thoroughly.

Topics Covered in Class 12 Maths Chapter 13 Probability:

1. Multiplication theorem on probability.
2. Conditional probability, independent events, total probability, Baye’s theorem.
3. Random variable and its probability distribution, mean and variance of haphazard variable.
4. Repeated independent (Bernoulli) trials and Binomial distribution.

Why Class 12 Math NCERT Solutions Matter:

Class 12 Math is a crucial milestone, and NCERT solutions play a pivotal role in your preparation. Our meticulously crafted solutions are designed to align with the latest curriculum, giving you a competitive edge in exams and a solid foundation for future academic pursuits.

Key Features of Our 12th Class Math NCERT Solutions:

1. Complete Coverage: Our NCERT solutions leave no stone unturned, offering a comprehensive solution set for every chapter and topic in the Class 12 Math curriculum.

2. Step-by-Step Clarity: Dive into problem-solving with ease. Our solutions break down complex problems into simple, step-by-step procedures, guiding you through each concept and ensuring complete understanding.

3. Precision in Presentation: Clear and concise presentations make learning efficient. Our solutions are crafted with precision, focusing on clarity to enhance your learning experience.

4. Chapter-wise Mastery: Nail each chapter with our chapter-wise practice exercises. Reinforce your knowledge systematically and hone your skills through targeted practice.

5. Exam-Ready Approach: Our NCERT solutions are geared towards exam success. Practice with confidence, knowing that our solutions are aligned with the patterns and question types commonly found in Class 12 Math exams.

Maximizing Your Learning Experience:

1. Consistent Practice: Dedicate regular study sessions to solving problems using our NCERT solutions. Consistency is key to mastering Class 12 Math.

2. Conceptual Grasp: Move beyond rote memorization. Understand the underlying concepts through our solutions, enabling you to tackle a variety of problems with confidence.

3. Complement with Resources: While our NCERT solutions are comprehensive, consider using additional resources such as reference books and online tutorials to deepen your understanding.

4. Seek Support: If you encounter challenges, don't hesitate to seek help from teachers, classmates, or online communities. Understanding and overcoming difficulties is an integral part of the learning process.

Conclusion: With our Class 12 Math NCERT solutions, you're not just studying – you're preparing for success. Elevate your understanding, build confidence, and excel in your exams with our expertly crafted solutions. Your journey to mathematical excellence starts here! Happy studying!