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NCERT Solutions for Class 12 Maths - Download Chapter Wise Solution PDF

NCERT Solutions for Class 12 Maths - Download Chapter Wise Solution PDF

Edited By Komal Miglani | Updated on Apr 14, 2025 06:14 PM IST

Mathematics is one of the main pillars of science and technology, and class 12 is the time when students learn about the advanced concepts of mathematics that build a strong foundation for their higher education and many competitive exams. With concepts like Calculus, Matrices, Probability, and three-dimensional geometry, the students will require a clear understanding, practice, patience, and the right guidance, This is where the NCERT solutions for Class 12 Maths come into play. These NCERT solutions provide a detailed step-by-step explanation of every problem in the NCERT textbook, making these advanced concepts of class 12 very easy to understand.

This Story also Contains
  1. NCERT Solutions for Class 12 Math, All Chapters 2025–26
  2. Class 12 Maths NCERT Solutions Exercises
  3. NCERT Exemplar solutions for class 12 - subject-wise
  4. NCERT solutions for class 12 - subject-wise
  5. NCERT Books and NCERT Syllabus
NCERT Solutions for Class 12 Maths - Download Chapter Wise Solution PDF
NCERT Solutions for Class 12 Maths - Download Chapter Wise Solution PDF

NCERT Solutions for Class 12 Math, All Chapters 2025–26

Chapter-wise Class 12 Maths NCERT solutions are listed below.

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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 the NCERT as a quick revision of concepts.

CBSE Class 12 Maths Marks Distribution

Units

Marks

Relations & Functions

08

Algebra

10

Calculus

35

Vector & 3-Dimensional Geometry

14

Linear Programming

05

Probability

08

Total

80

Class 12 Maths NCERT Solutions Exercises

Students can download the NCERT Solution of Maths of Class 12, chapter-wise, from the links given below.

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 way, we say these two sets have a relation. Father, mother, sister, and brother are real-life examples of relations. A 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. For PDF and notes, refer to Class 12 Maths Chapter 1 Relations and Functions Notes. Also, for the exemplar solutions of Relations and functions, refer to the NCERT Exemplar Solutions for Class 12 Maths Chapter 1 Relations and Functions.

In the NCERT Class 12 Maths solutions, you will learn about types of relations, reflective, transitive, and equivalence relations, functions, range, domain, and co-domain of a function, which have already been introduced in Maths of NCERT Class 11. Class 12 Maths NCERT solutions for chapter 1 have 4 exercises. Click on the above link to get Class 12 NCERT solutions exercise-wise for 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 functions, it won't take much effort to get a command of the inverse trigonometric functions. Students can study using the inverse trigonometric function class 12 Solutions PDF. For PDF and notes, refer to Class 12 Maths Chapter 1 Inverse Trigonometric Functions Notes. Also, for the exemplar solutions of inverse trigonometric functions, refer to NCERT Exemplar Solutions for Class 12 Maths Chapter 1 Inverse Trigonometric Functions.

In NCERT Class 12 Maths solutions of this chapter, you will learn concepts related to the definition of inverse trigonometric functions, domain, range, 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 to be 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. For PDF and notes, refer to Class 12 Maths Chapter 3 Matrices Notes. Also, for the exemplar solutions of matrices, refer to NCERT Exemplar Solutions For Class 12 Maths Chapter 3 Matrices.

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, operations, 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 straightforward methods. In Class 12 Maths NCERT Solutions Chapter 3, there are 4 exercises with 62 questions. Students can study using the 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. For PDF and notes, refer to Class 12 Maths Chapter 4 Determinants Notes. Also, for the exemplar solutions of determinants, refer to the NCERT Exemplar Solutions for Class 12 Maths Chapter 4 Determinants.

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. For PDF and notes, refer to Class 12 Maths Chapter 5 Continuity and Differentiability Notes. Also, for the exemplar solutions of Continuity and Differentiability, refer to Ncert Exemplar Solutions For Class 12 Maths Chapter 5 Continuity and Differentiability

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 exercise 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. For PDF and notes, refer to Class 12 Maths Chapter 6 Application of derivatives Notes. Also, for the exemplar solutions of Application of Derivatives, refer to Ncert Exemplar Solutions For Class 12 Maths Chapter 6 Application of Derivatives.

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). For PDF and notes, refer to Class 12 Maths Chapter 7 Integrals Notes. Also, for the exemplar solutions of Integrals, refer to Ncert Exemplar Solutions For Class 12 Maths Chapter 7 Integrals.

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 the 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. For PDF and notes, refer to Class 12 Maths Chapter 9 Applications of Integrals Notes. Also, for the exemplar solutions of Application of Integrals, refer to NCERT Exemplar Solutions for Class 12 Maths Chapter 8 Applications of Integrals.

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, as well as the 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 differential equations, 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 a deeper understanding of the concepts. For PDF and notes, refer to Class 12 Maths Chapter 9 Differential Equations Notes. Also, for the exemplar solutions of Differential Equations, refer to NCERT Exemplar Solutions for Class 12 Maths Chapter 9 Differential Equations.

Chapter 10: Vector Algebra

Think about the situation where you are applying force to 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. For PDF and notes, refer to NCERT Notes for Class 12 Maths Chapter 10 Vector Algebra. Also, for the exemplar solutions of Vector Algebra, refer to NCERT Exemplar Solutions for Class 12 Maths Chapter 10 Vector Algebra.

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 of 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 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. For PDF and notes, refer to Class 12 Maths Chapter 11 Three Dimensional Geometry Notes. Also, for the exemplar solutions of Three Dimensional Geometry, refer to the NCERT Exemplar Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry

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. For PDF and notes, refer to Class 12 Maths Chapter 12 Linear Programming Notes. Also, for the exemplar solutions of Linear Programming, refer to Ncert Exemplar Solutions For Class 12 Maths Chapter 12 Linear Programming.

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 benefits.

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. For PDF and notes, refer to Class 12 Maths Chapter 13 Probability Notes. Also, for the exemplar solutions of Probability, refer to Ncert Exemplar Solutions For Class 12 Maths Chapter 13 Probability.

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.

NCERT Exemplar solutions for class 12 - subject-wise

Given below are the subject-wise links for the NCERT exemplar solutions of class 12:

NCERT solutions for class 12 - subject-wise

Here are the subject-wise links for the NCERT solutions of class 12:

NCERT Books and NCERT Syllabus

Here are some useful links for NCERT books and the NCERT syllabus for class 12:

Frequently Asked Questions (FAQs)

1. Where can I find the NCERT Solutions for Class 12 Maths?

To find the NCERT Solutions for Class 12 Maths refer to the link given below:
NCERT Solutions for Class 12 Maths.

2. Are NCERT Solutions for Class 12 Maths enough for exams?

NCERT Solutions for Class 12 Maths are enough for most of the exams, as they contain various types of problems to enhance the problem-solving ability of the students.

3. Is Class 12 NCERT Maths tough?

Students will learn new concepts, theorems, and formulae of advanced mathematics in class 12 NCERT Maths. It will be challenging for sure, but with consistent practice and enough conceptual clarity, students can ace this subject. Students should not stop after solvingthe NCERT textbook exercises. They should also try to solve questions from other important books and Examplar questions.

4. How to understand difficult topics in Class 12 Maths?

There are some challenging chapters in Class 12 Maths and students need to handle those chapters strategically to achieve good marks. Here are some tips to deal with hard chapters.

  • Strengthen the core concepts of those chapters from NCERT textbooks. If it is still not clear, then take references from other books.
  • Watch video tutorials from decent edu tech sites like Careers360.
  • Take help from experienced teachers if you get stuck in a chapter.
  • Break the chapter into small, manageable parts and analyze each part separately.
  • Take mock tests on those chapters to check your progress.
  • Practice the NCERT Exemplar for a better understanding of the topics.
5. Is NCERT enough for Class 12 Maths numerical questions?

Yes. For the class 12 Maths final exam, questions and solved examples of NCERT textbooks and examplars solutions are more than enough. But for other high-level competitive exams like JEE, Olympiads or BITSAT, students should also practice questions from other books and online materials. Frequent practice of mock tests can be helpful in both circumstances.

Articles

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)

0.34\; J

Option 2)

0.16\; J

Option 3)

1.00\; J

Option 4)

0.67\; J

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)

2,000 \; J - 5,000\; J

Option 2)

200 \, \, J - 500 \, \, J

Option 3)

2\times 10^{5}J-3\times 10^{5}J

Option 4)

20,000 \, \, J - 50,000 \, \, J

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

Option 1)

K/2\,

Option 2)

\; K\;

Option 3)

zero\;

Option 4)

K/4

In the reaction,

2Al_{(s)}+6HCL_{(aq)}\rightarrow 2Al^{3+}\, _{(aq)}+6Cl^{-}\, _{(aq)}+3H_{2(g)}

Option 1)

11.2\, L\, H_{2(g)}  at STP  is produced for every mole HCL_{(aq)}  consumed

Option 2)

6L\, HCl_{(aq)}  is consumed for ever 3L\, H_{2(g)}      produced

Option 3)

33.6 L\, H_{2(g)} is produced regardless of temperature and pressure for every mole Al that reacts

Option 4)

67.2\, L\, H_{2(g)} at STP is produced for every mole Al that reacts .

How many moles of magnesium phosphate, Mg_{3}(PO_{4})_{2} 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

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