NCERT Solutions for Class 12 Maths - Get Free PDFs 2024-25

# NCERT Solutions for Class 12 Maths - Get Free PDFs 2024-25

Edited By Irshad Anwar | Updated on May 16, 2024 01:29 PM IST

NCERT Solutions Class 12 Maths (2024-2025) has been curated for students who are appearing CBSE board exams as well as competitive exams such as JEE. All the important concepts of the CBSE Class 12 Maths NCERT textbook are covered in these NCERT solutions by the expert team at Careers360 in accordance with the latest CBSE syllabus and pattern, comprehensively covering all the steps. So you should clear all your doubts and prepare for exams. In the NCERT 12 maths solution, there are 13 chapters that cover important theorems and formulas of NCERT textbook. You will find chapter-wise NCERT solutions for Class 12 will help you perform well in the board exams. These NCERT solutions are 100% accurate according to the Class 12 Maths syllabus.

## NCERT Solutions for Class 12 Math, All Chapters 2024–25

Chapter wise Class 12 Maths NCERT solutions are listed below. Select a chapter number of maths to view from the chapter-wise NCERT Solutions for Class 12 solutions.

 S. No. Chapter Chapter 1 Relations And Functions Chapter 2 Inverse Trigonometric Functions Chapter 3 Matrices Chapter 4 Determinants Chapter 5 Differentiability Chapter 6 Application of Derivatives Chapter 7 Integrals Chapter 8 Application of Integrals Chapter 9 Differential Equations Chapter 10 Vector Algebra Chapter 11 Three Dimensional Geometry Chapter 12 Linear Programming Chapter 13 Probability

Practising NCERT questions from class 12 maths chapters is very important in Maths, to understand the concepts. If you are not able to solve this, you can take the help of these Class 12 Maths NCERT solutions. In this article of NCERT solutions, you can click on the chapter you wish to study.

Here you will find all the class 12 maths NCERT solutions PDF downloads in a single place, which will be helpful when you are not able to solve the NCERT practice questions on your own. Also, you can use these solutions of NCERT as a quick revision of concepts before the exam to score more marks.

Also Read: NCERT Solutions for Class 12 Subject-wise

### CBSE Class 12 Maths Weightage Distribution

 Units Marks Relations & Functions 10 Algebra 13 Calculus 44 Vector & 3-Dimensional Geometry 17 Linear Programming 6 Probability 10 Total 100

## Class 12 Maths NCERT Solutions Exercises

### Chapter 1: Relations and Functions

NCERT Solutions for Class 12 Maths Chapter 1: Relations and Functions, are provided here. If two sets of elements are related in some ways, we say these two sets have a relation. Father, mother, sister and brother are real-life examples of relations. Relation becomes a function when, for every input, there is only one output. For different inputs, the output can be the same. Every function is a relation, but every relation doesn't represent a function. Students can study using relations and functions in the Class 12 PDF free download.

In the NCERT Class 12 Maths solutions, you will learn about types of relations, reflective, transactive and equivalence relation, functions, range, domain and co-domain of function, which have already been introduced in Maths of NCERT Class 11. Class 12 Maths NCERT solutions for chapter 1 has 4 exercises. Click on the above link to get Class 12 NCERT solutions exercise wise of this chapter.

### Chapter 2: Inverse Trigonometric Functions

In Class 11 Maths, you have already learned about trigonometric functions. The inverse trigonometric function is the inverse of a trigonometric function. If you are good at trigonometric function, it won't take much effort to get a command on the inverse trigonometric functions. Students can study using the inverse trigonometric function class 12 Solutions PDF.

In NCERT Class 12 Maths solutions of this chapter, you will learn concepts related to the definition of inverse trigonometric functions, domain, range, and principal value, graph, and properties of inverse trigonometric functions. In class 11, you have already learned that trigonometric functions are not one-to-one and onto in their natural ranges. We restrict domains and ranges of trigonometric functions to enable their inverse one-one and onto. This chapter contains 2 exercises with 35 questions. All the exercises, including Exercise 2.2 Class 12 Maths, are provided below.

### Chapter 3: Matrices

The array of numbers or mathematical objects called "Matrix" is used for operations like addition and multiplication. It was introduced to solve simultaneous linear equations. Matrix has a lot of applications in engineering, scientific studies, computer graphics (images or videos that you see are matrices of colour intensities), storage of big data, etc.

In this chapter of NCERT Class 12 Maths solutions, you will learn about matrices and their properties. Also, you are going to study the types, operation transpose, and inverse of the matrix, as well as the properties of matrices, which are one of the most powerful tools of mathematics. This tool simplifies our work when compared with other straight-forward methods. In Class 12 Maths NCERT Solutions Chapter 3, there are 4 exercises with 62 questions. Students can study using matrices class 12 Solutions.

### Chapter 4: Determinants

In the previous chapter, you have already learned about matrices and their properties. For every square matrix, you can compute a number called "Determinant". It shows the factor by which the area or volume of a linear transformation has changed. Students can study using the determinants class 12 Solutions PDF.

In Class 12 NCERT Maths solutions for this chapter, you will get questions about the determinants of a square matrix, properties of determinants, consistency, inconsistency, the number of solutions of linear equations, and solving systems of linear equations, which are important parts of matrices. So, we are dealing with determinants of order up to three only. In this chapter, there are 6 exercises. Practice all the exercises, including exercise 4.6 class 12, to get a good grasp on the concepts.

### Chapter 5: Continuity and Differentiability

In Class 11 NCERT, you have learned about the differentiation of certain functions. In Class 12 NCERT Solutions for this chapter, you will learn about differentiability, continuity and how differentiability is related to continuity. Students can study using continuity and differentiability class 12 solutions PDF.

The important concepts that you will study are continuity, differentiability, the derivative of a composite function, a derivative of a trigonometric function, the derivative of the inverse trigonometric functions, a derivative of the logarithm, the derivative of the exponential function, chain rule, role's theorem, Lagrange mean value theorem, etc. In this chapter, there are a total of eight exercises. Students can practice all exercises, including ex. 5.2 class 12, exercise 5.6 class 12, and exercise 5.7 class 12, to command the concepts discussed in this chapter that are listed below.

### Chapter 6: Application of Derivatives

In chapter 5, you have learned how to find the derivative of implicit functions, composite functions, exponential functions, inverse trigonometric functions, and logarithmic functions. In this chapter, you are going to study the applications of derivatives in various fields like science, social science, engineering, and many other fields. Students can study using the application of derivatives class 12 Solutions, a free PDF download.

This chapter of NCERT Class 12 Maths Solutions "Application of Derivatives" will cover the rate of change, maxima, minima of functions, equations of tangents and normals, etc. There are five exercises in this chapter and here you will find the solutions to all exercises, including exercise 6.5, class 12.

### Chapter 7: Integrals

Differential calculus is centred around the concepts of differentiation, defining tangent, and calculating the slope of a line. Integral calculus is centred around defining and calculating the area of the region bounded by the graph of a function. Integration means 'bringing together, which is the inverse process of differentiation (set-apart). Students can study using the integrals class 12 PDF.

The function is integrable only if it is already differentiated. Remember, every function is not integrable, which means you can't integrate every function. In this chapter, you will learn about the integrals of various functions and also discuss their applications, like the calculation of areas under graphs. There are 11 exercises in this chapter of NCERT 12th Maths solutions. All of them are listed below, including exercise 7.9 class 12. Practice them to score well in the exam.

### Chapter 8: Application of Integrals

In the previous chapter, you learned about the integration of different complicated functions. Well, it has a lot of applications, like calculating the area between curves, calculating distance, velocity, acceleration, finding the average value of a function, probability, surface area, etc. Students can study using the application of integrals Class 12 Solutions PDFs.

In this chapter, you will study a specific application of integrals to find the area between the lines and arcs of circles, ellipses and parabolas and area under simple curves. The two exercises of this chapter deal with finding the area bounded by the above-mentioned curves.

### Chapter 9: Differential Equations

You have learned about differentiation and integration. A differential equation is an equation of a function with one or more derivatives. In this chapter, you are going to learn some basic concepts related to the differential equation, particular and general solutions of a differential equation, different methods to solve a first-order-first degree differential equation, solutions of linear differential equations, the formation of differential equations and discuss some applications of differential equations in different areas. All the exercises, including exercise 9.4 class 12, exercise 9.6 class 12, are listed below. Practice them to get deeper understanding of the concepts. Students can study using differential equations in the CBSE Class 12 Solution Free PDF.

### Chapter 10: Vector Algebra

Think about the situation where you are applying force on a box. So it has two quantities: the magnitude that you are applying and the direction in which you are applying the force. Such quantities with both magnitude and direction are termed vector quantities. Students can study using the vector algebra class 12 Solutions PDF.

In Physics and engineering, you often come across these quantities. In the class 12 NCERT Maths solutions of this chapter, you will learn about some basic concepts of vectors, types of vectors, the component of vectors, algebraic properties addition, multiplication by a scalar, dot product, cross product, and their geometric properties. All the exercises, including exercise 10.2 class 12, exercise 10.3 class 12, are helpful for students. They can practice all of the exercise solutions listed below.

### Chapter 11: Three Dimensional Geometry

In the previous class, you have learned 2-D geometry. 3-D geometry is an extension of 2-D geometry that takes into consideration three orthogonal axes. Augustus De Morgan quotes "the moving power of mathematical invention is not reasoning but imagination". To visualise in 3-D geometry, it requires more of your imagination power. Students can study using three dimensional geometry Class 12 Solutions PDF.

This chapter makes the study of 3-D geometry simple. In the class 12 maths solutions of this chapter, "Three Dimensional Geometry," you will study the direction ratio and direction cosines of a line joining two points, equations of planes and lines in space under different conditions, the angle between two planes, two lines, a plane, and a line, and other important concepts.

### Chapter 12: Linear Programming

You often come across problems where you seek to maximise profit or minimise cost. Linear programming is useful to find the optimal solution for such problems. We formulate these real-life problems into a mathematical model. Students can study using linear programming Class 12 Solutions PDF.

In Class 12 NCERT Maths solutions of this chapter, we deal with some problems on linear programming, different types of linear programming and their solutions by the graphical method, feasible and infeasible region, feasible and infeasible solutions, optimal feasible solutions and also about many other methods to solve such problems. There are two exercises in Chapter 12 Linear programming. The exercises including exercise 12.2 class 12 are important for students, practice them to get benefit.

### Chapter 13: Probability

In previous classes, you have learned that probability means how likely something is to happen. In this chapter, you will learn the important concept of the conditional probability of an event given that another event has occurred, which will be helpful in understanding the multiplication rule of probability, Bayes' theorem and independence of events. Students can study using the probability class 12 solutions PDF.

In Class 12 Maths NCERT Solutions Chapter 13, you will also get questions on random variables and their probability distributions, as well as the mean and variance of a probability distribution.

### Key Features Of NCERT Solutions Class 12th Maths

1. Structured Content: The class 12 math ncert solution are well-structured and organised according to the chapters and topics in the NCERT Class 12 Mathematics textbook. This makes it easy for students to find relevant solutions.
2. Comprehensive Approach: The class 12 math ncert solution cover all the exercises and examples present in the NCERT textbook. This ensures that students have access to solutions for all types of problems, from basic to advanced. also includes deleted exercise of maths class 12 in these solutions.
3. Step-by-Step Solutions: Each cbse 12th maths solutions is presented in a step-by-step manner. This helps students grasp the logical sequence of solving mathematical problems, making it easier to follow and understand.

4. Clarity and Conciseness: The NCERT 12 maths solutions are written in a clear and concise language, avoiding unnecessary jargon. This ensures that students can easily comprehend the concepts and procedures involved in solving problems.

5. Conceptual Understanding: NCERT class 12th maths NCERT solutions focus on promoting a deep understanding of mathematical concepts rather than rote memorization. The solutions often include explanations and insights to help students grasp the underlying principles.

Also check:

### Benefits of Class 12 Maths NCERT Solutions

• Before solving the NCERT exercise question, understand the concept and solve the solved examples of the NCERT books for 12 Maths.

• You should try to solve every question, including examples, on your own, If you are not able to do so, you can take help with these Class 12 Maths NCERT solutions, in which detailed explanations using a step-by-step approach are provided to facilitate an easy and smooth understanding of concepts. It will give you a new insight to solving CBSE maths problems.

• These Class 12 NCERT Maths solutions will help you learn CBSE Class 12 Maths as well as crack JEE Main and other competitive exams like BITSAT, VITEEE, etc.

• With the help of these NCERT solutions for Class 12 Maths, you can grasp the concepts based on their application very easily.

### Check out the NCERT books and NCERT syllabus:

1. How are the NCERT solutions beneficial in the CBSE Class 12 board exams?

Only knowing the answer is not enough to perform well in the board exams. One should know how to write an answer in order to get good marks. CBSE class 12 maths solutions are provided by experts who know how best to answer in the board exam in order to get good marks. Follow NCERT syllabus for a good score. Students can practice questions from the NCERT exemplar for Class 12 Maths. Also, they can take help from class 12th cbse maths solutions. practice all exercise including exercise 5.2 class 12,

2. Are NCERT Solutions for maths class 12 chapters considered the best reference material for students?

NCERT books are considered a top resource for Class 12 students. Memorizing information just before exams is not an effective strategy for achieving high grades. It's crucial to have a thorough understanding of concepts. Daily practice with textbook problems will give students an idea of what to expect on exams and improve their performance. Also, students can use BCSE class 12 maths ncert solutions to get good hold in the concepts.

3. How many chapters are there in the Class 12 Maths?

There are 13 chapters starting from Relations and Functions to Probability in the CBSE Class 12 Maths. The chapters are from Algebra, Trigonometry, Calculus and Probability. Students can practice these ncert solutions class 12 maths to get deeper understanding of the concepts. Maths class 12 ncert solutions are very important as these are provide comprehensive explanation of all the problems.

Class 12 is a crucial year for students, as their grades will impact their future career aspirations. They can access class 12 maths ncert book solutions online or download them as PDFs and use them while working through textbook problems. The chapter-wise Maths Class 12 NCERT Solutions available for download at Careers360 enhance understanding of concepts and aid in scoring well on exams. Practice these ncert maths class 12 solutions to get best benefit during the exams.

5. Can I rely solely on 12th cbse maths solutions for exam preparation?

While NCERT Maths Solutions are an excellent resource, it's recommended to supplement them with additional study materials, reference books, and practice tests, especially if you're aiming for competitive exams. Athoug these class 12th maths ncert solutions are covered comprehensively, keeping in mid of latest syllabus of CBSE, practice additional resources with ncert maths solutions class 12.

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Get answers from students and experts

A block of mass 0.50 kg is moving with a speed of 2.00 ms-1 on a smooth surface. It strikes another mass of 1.00 kg and then they move together as a single body. The energy loss during the collision is

 Option 1) Option 2) Option 3) Option 4)

A person trying to lose weight by burning fat lifts a mass of 10 kg upto a height of 1 m 1000 times.  Assume that the potential energy lost each time he lowers the mass is dissipated.  How much fat will he use up considering the work done only when the weight is lifted up ?  Fat supplies 3.8×107 J of energy per kg which is converted to mechanical energy with a 20% efficiency rate.  Take g = 9.8 ms−2 :

 Option 1) 2.45×10−3 kg Option 2)  6.45×10−3 kg Option 3)  9.89×10−3 kg Option 4) 12.89×10−3 kg

An athlete in the olympic games covers a distance of 100 m in 10 s. His kinetic energy can be estimated to be in the range

 Option 1) Option 2) Option 3) Option 4)

A particle is projected at 600   to the horizontal with a kinetic energy . The kinetic energy at the highest point

 Option 1) Option 2) Option 3) Option 4)

In the reaction,

 Option 1)   at STP  is produced for every mole   consumed Option 2)   is consumed for ever      produced Option 3) is produced regardless of temperature and pressure for every mole Al that reacts Option 4) at STP is produced for every mole Al that reacts .

How many moles of magnesium phosphate, will contain 0.25 mole of oxygen atoms?

 Option 1) 0.02 Option 2) 3.125 × 10-2 Option 3) 1.25 × 10-2 Option 4) 2.5 × 10-2

If we consider that 1/6, in place of 1/12, mass of carbon atom is taken to be the relative atomic mass unit, the mass of one mole of a substance will

 Option 1) decrease twice Option 2) increase two fold Option 3) remain unchanged Option 4) be a function of the molecular mass of the substance.

With increase of temperature, which of these changes?

 Option 1) Molality Option 2) Weight fraction of solute Option 3) Fraction of solute present in water Option 4) Mole fraction.

Number of atoms in 558.5 gram Fe (at. wt.of Fe = 55.85 g mol-1) is

 Option 1) twice that in 60 g carbon Option 2) 6.023 × 1022 Option 3) half that in 8 g He Option 4) 558.5 × 6.023 × 1023

A pulley of radius 2 m is rotated about its axis by a force F = (20t - 5t2) newton (where t is measured in seconds) applied tangentially. If the moment of inertia of the pulley about its axis of rotation is 10 kg m2 , the number of rotations made by the pulley before its direction of motion if reversed, is

 Option 1) less than 3 Option 2) more than 3 but less than 6 Option 3) more than 6 but less than 9 Option 4) more than 9