Ncert solutions class 12 maths

Are you looking for the most accurate Class 12 Maths NCERT Solutions, then CBSTTuts.com is the perfect website to download it in PDF format for absolutely free of cost. You might be aware of that the curriculum for all schools that follow the Central Board of Secondary Education (CBSE) is set by The National Council of Education Research and Training (NCERT) across the nation.
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1. NCERT Solutions Class 12 Maths 
NCERT Solutions Class 12 Maths 
Class 12 Maths NCERT Solutions​ provides a solid conceptual base for all the topics 
included in CBSE class 12 maths syllabus (2018-2019). It covers all the important 
theorems and formulas with detailed explanations to promote a better conceptual 
understanding in students. NCERT Maths solutions for Class 12 will help students in the 
preparation of undergraduate level engineering entrance exams like JEE Mains, BITSAT, 
VITEEE, etc, going to be conducted this year, in 2019.CBSETuts brings you NCERT 
Solutions class 12 Maths , designed by our subject experts to facilitate smooth and easy 
understanding of concepts. These CBSE class 12 maths solutions of NCERT have a 
detailed step-by-step explanMaths, action of problems given in the textbooks. Apart 
from solutions for each chapter of CBSE Class 12 Maths, we are also providing here 
class 12 maths notes, solved exemplary 
problems and previous year questions papers, worksheets as study materials for 
students who are preparing for their Class 12th board exams. 
Maths NCERT Solutions for Class 12 Maths Chapterwise 

2.  
●NCERT Solutions For Class 12 Maths Chapter 1 
●NCERT Solutions For Class 12 Maths Chapter 2 
●NCERT Solutions For Class 12 Maths Chapter 3 
●NCERT Solutions For Class 12 Maths Chapter 4 
●NCERT Solutions For Class 12 Maths Chapter 5 
●NCERT Solutions For Class 12 Maths Chapter 6 
●NCERT Solutions For Class 12 Maths Chapter 7 
●NCERT Solutions For Class 12 Maths Chapter 8 
●NCERT Solutions For Class 12 Maths Chapter 9 
●NCERT Solutions For Class 12 Maths Chapter 10 
●NCERT Solutions For Class 12 Maths Chapter 11 
●NCERT Solutions For Class 12 Maths Chapter 12 
●NCERT Solutions For Class 12 Maths Chapter 13 
NCERT Solutions for Class 12 Maths Chapter 1 Relations 
and Functions 
NCERT Solutions for Class 12 Maths Chapter 1 Relations 
and Functions 

3. NCERT solutions for class 12 Maths Chapter 1​- Relations and Functions, provides 
solutions for all the questions enlisted under the chapter. All exercise questions are 
solved using step by step approach. Students can practice these NCERT Solutions for 
Class 12 Maths to grasp the concepts perfectly. Students can download these NCERT 
Solutions of Class 12 mathematics to sharpen their maths skills. Get your free PDF now 
and start practicing. 
In NCERT Class 12 maths chapter 1, students will study different types of relations and 
equivalence relation, the composition of functions, invertible functions, and binary 
operations. 
The main features of this chapter are as follows: 
1.Empty relation 
2.Universal relation 
3.Reflexive relation 
4.Symmetric relation 
5.Transitive relation 
6.Equivalence relation 
●A function f : X → Y is one-one (or injective) if f (x1) = f(x2) ⇒ x1 = x2 ∀ x1, x2 ∈ 
X. 
●A function f : X → Y is onto (or surjective) if given any y ∈ Y, ∃ x ∈ X such that 
f(x) = y. 
●A function f: X → Y is one-one and onto (or bijective), if f is both one-one and 
onto. 
●A function f: X → Y is invertible if and only if f is one-one and onto. 
NCERT Solutions For Class 12 Maths Chapter 1 Exercises: 
Get detailed solution for all the questions listed under below exercises: 

4. ●Exercise 1.1 Solutions : 16 Questions (14 Short Answers, 2 MCQ) 
●Exercise 1.2 Solutions : 12 Questions (10 Short Answers, 2 MCQ) 
●Exercise 1.3 Solutions : 14 Questions (12 Short Answers, 2 MCQ) 
●Exercise 1.4 Solutions : 13 Questions (12 Short Answers, 1 MCQ) 
●Miscellaneous Exercise Solutions : 19 Questions (7 Long answers, 9 Short 
answer type, 3 MCQ) 
NCERT Solutions for Class 12, chapter 1-Relations and Functions, concepts of relation 
and functions given here in a detailed and easy manner to understand 
NCERT Solutions for Class 12 Maths Chapter 2 Inverse 
Trigonometric Functions 
NCERT Solutions for Class 12 Maths Chapter 2 Inverse 
Trigonometric Functions 
NCERT Solutions for Class 12 Maths Chapter 2​- Inverse Trigonometric Functions, 
provides solutions for all the questions enlisted under the chapter. These NCERT 
Solutions for maths are prepared by subject experts keeping in mind the student's 
requirements of CBSE Class 12. Solving all NCERT questions will give students the 
advantage of preparing themselves better before the final CBSE exams. Students can 
download these NCERT Solutions and get ready for their exams. 
In Class 12 maths chapter 2, students will deal with inverse trigonometric function and 
their properties. Get handy with the domains and ranges (principal value branches) of 
inverse trigonometric functions. While solving problems on this topic, students should 
keep in mind that, the value of an inverse trigonometric function which lies in its 
principal value branch is called the principal value of that inverse trigonometric 
functions. 

5. NCERT Solutions For Class 12 Maths Chapter 2 Exercises: 
Get detailed solution for all the questions listed under below exercises:
●Exercise 2.1 Solutions : 14 Questions (12 Short Answers, 2 MCQ) 
●Exercise 2.2 Solutions : 21 Questions (18 Short Answers, 3 MCQ)a  
●Miscellaneous Exercise Solutions: 17 Questions (14 Short Answers, 3 MCQ) 

NCERT Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Functions. NCERT 
Solutions for Class 12, chapter 2- Inverse Trigonometric Functions, deals with domain, 
range, values and properties of inverse trigonometric functions. 
NCERT Solutions for Class 12 Maths Chapter 3 Matrices 
NCERT Solutions for Class 12 Maths Chapter 3 Matrices 
NCERT Solutions for class 12 Maths Chapter 3​ – Matrices, provides solutions for all the 
questions enlisted under the chapter (All Exercises and Miscellaneous Exercise 
solutions). Class 12 maths NCERT Solutions are solved by maths experts using step by 
step approach under the guidelines of NCERT to assists students in their examination. 
This topic is extremely important for both CBSE board exam and for competitive exams. 
Students can practice these questions and improve their skills. Download free NCERT 
Solutions pdf now and excel your knowledge. 
In Class 12 Maths Chapter 3, students will deal with matrices and their types, properties, 
order of a matrix , operations on matrices (addition, multiplication), transpose of matrix, 
symmetric and skew-symmetric matrices, elementary operation (transformation) of a 
matrix, invertible matrices and inverse of a matrix by elementary operations. Students 
can practice problems on these topics and get ready for the exam. 
NCERT Solutions For Class 12 Maths Chapter 3 Exercises: 
Get detailed solution for all the questions listed under below exercises: 

6. ●Exercise 3.1 Solutions : 10 Questions (7 Short Answers, 3 MCQ) 
●Exercise 3.2 Solutions : 22 Questions(14 Long, 6 Short, 2 MCQ) 
●Exercise 3.3 Solutions : 12 Questions (10 Short Answers, 2 MCQ) 
●Exercise 3.4 Solutions : 18 Questions (4 Long, 13 Short, 1 MCQ) 
●Miscellaneous Exercise Solutions: 15 Questions (7 Long, 5 Short, 3 MCQ) 
NCERT Solutions for Class 12 Maths Chapter 4 
Determinants 
NCERT Solutions for Class 12 Maths Chapter 4 
Determinants 
NCERT solutions for class 12 Maths Chapter 4​ - Determinants, provides solutions for all 
the questions enlisted under the chapter (All Exercises and Miscellaneous Exercise 
solutions). Maths Class 12 NCERT Solutions are solved by maths experts. CBSETuts is 
to bring out the very best in students through additional skill-building exercises that are 
tailored to their grade levels, abilities and interests. Students can download these NCERT 
Solutions of Class 12 mathematics and improve their skills. 
In Class 12 Maths Chapter 4, students will study, how to calculate determinants, Solve 
matrix using different properties of determinants, area of the triangle, adjoint and the 
inverse of a matrix, applications of determinants and matrix and many more. Below are 
some important facts on determinants: 
If any two rows or any two columns are identical or proportional, then the value of the 
determinant is zero. A square matrix A has inverse if and only if A is non-singular. Unique 
solution of equation AX = B is given by X = A–1 B, where A ≠ 0. A system of equation is 
consistent or inconsistent according to as its solution exists or not. For a square matrix 
A in matrix equation AX = B 

7. (i) | A| ≠ 0, there exists unique solution 
(ii) | A| = 0 and (adj A) B ≠ 0, ta hen there exists no solution 
(iii) | A| = 0 and (adj A) B = 0, then system may or may not be consistent. 
NCERT Solutions For Class 12 Maths Chapter 4 Exercises: 
Get detailed solution for all the questions listed under below exercises: 
●Exercise 4.1 Solutions : 8 Questions (2 Long, 5 Short Answers, 1 MCQ) 
●Exercise 4.2 Solutions : 16 Questions(7 Long, 7 Short, 2 MCQ) 
●Exercise 4.3 Solutions : 5 Questions ( 4 Short Answers, 2 MCQ) 
●Exercise 4.4 Solutions : 5 Questions (4 Long, 1 MCQ) 
●Exercise 4.5 Solutions : 18 Questions (11 Long, 5 Short, 2 MCQ) 
●Exercise 4.6 Solutions : 16 Questions (13 Long, 3 Short) 
●Miscellaneous Exercise Solutions : 19 Questions (15 Long, 1 Short, 3 MCQ) 
NCERT Solutions for Class 12 Maths Chapter 5 Continuity 
and Differentiability 
NCERT Solutions for Class 12 Maths Chapter 5 Continuity 
and Differentiability 
NCERT solutions for class 12 Maths Chapter 5​- Continuity and Differentiability, provides 
solutions for all the questions enlisted under the chapter (All Exercises and 
Miscellaneous Exercise solutions). These NCERT Solutions have been carefully 
compiled and developed keeping into consideration the latest CBSE syllabus. All 
questions are solved by subject experts. Students can download these NCERT Solutions 
of Class 12 maths and strengthen their fundamentals. 

8. In this chapter, the student will deal with continuity and differentiability problems 
solutions, that contains questions based on proving an equation is continuous if given 
with different values of ‘x’. Also, we will examine certain functions of continuity and 
generate conclusions. You will come across explanations for the different types of 
questions, where you will provide the reasons if one quantity is replaced with different 
values. 
Some of the important concepts from the chapter are listed below: 
●Sum, difference, product, and quotient of continuous functions are continuous. 
●Every differentiable function is continuous, but the converse is not true. 
●Rolle’s Theorem: If f : [a, b] → R is continuous on [a, b] and differentiable on (a, b) 
such that f(a) = f(b), then there exists some c in (a, b) such that f ′(c) = 0. 
●Mean Value Theorem: If f : [a, b] → R is continuous on [a, b] and 
●differentiable on (a, b). Then there exists some c in (a, b) such that f'(c) = (f(b) – 
f(a))/(b-a) 
 
NCERT Solutions For Class 12 Maths Chapter 5 Exercises: 
Get detailed solution for all the questions listed under below exercises: 
●Exercise 5.1 Solutions : 34 Questions (Short Answers) 
●Exercise 5.2 Solutions : 10 Questions(Short Answers) 
●Exercise 5.3 Solutions : 15 Questions ( Short Answers) 
●Exercise 5.4 Solutions : 10 Questions (Short Answers) 
●Exercise 5.5 Solutions : 18 Questions ( Short Answers) 
●Exercise 5.6 Solutions : 11 Questions (Short Answers) 
●Exercise 5.7 Solutions : 17 Questions (Short Answers) 
●Exercise 5.8 Solutions : 6 Questions (Short Answers) 

9. ●Miscellaneous Exercise Solutions: 23 Questions (6 Long, 17 Short) 
NCERT Solutions for Class 12 Maths Chapter 6 Application 
of Derivatives 
NCERT Solutions for Class 12 Maths Chapter 6 
Application of Derivatives 
NCERT solutions for class 12 Maths Chapter 6​-Application of Derivatives, provides 
solutions for all the questions enlisted under the chapter. NCERT Solutions include a 
complete set of questions and answers organized with an advanced level of difficulty 
which provide students sufficient opportunity to apply their skills. Students can 
download these NCERT Solutions of Class 12 maths and strengthen their skills. 
This chapter mainly a set of topics like the rate of change of quantities, Increasing and 
decreasing functions, Tangents and normals, Approximations, Maxima and minima and 
many more. Some of the important concepts are: 
1. A function f is said to be 
(a) increasing on an interval (a, b) if x1 < x2 in (a, b) ⇒ f(x1) < f(x2) for all x1, x2 ∈ (a, b) 
(b) decreasing on (a,b) if x1 < x 2 in (a, b) ⇒ f(x1 ) > f(x2 ) for all x1 , x2 ∈ (a, b). 
(c) constant in (a, b), if f (x) = c for all x ∈ (a, b), where c is a constant. 
2. First Derivative Test 
3. Second Derivative Test 

10. NCERT Solutions For Class 12 Maths Chapter 6 Exercises: 
Get detailed solution for all the questions listed under below exercises: 
●Exercise 6.1 Solutions: 18 Questions (10 Long, 6 Short, 2 MCQ) 
●Exercise 6.2 Solutions: 19 Questions(10 Long, 7 Short, 2 MCQ) 
●Exercise 6.3 Solutions: 27 Questions (14 Long, 11 Short, 2 MCQ) 
●Exercise 6.4 Solutions: 9 Questions (7 Short, 2 MCQ) 
●Exercise 6.5 Solutions: 29 Questions (15 Long, 11 Short, 3 MCQ) 
●Miscellaneous Exercise Solutions: 24 Questions (14 Long, 4 Short, 6 MCQ) 
NCERT Solutions for Class 12 Maths Chapter 7- Integrals 
NCERT Solutions for Class 12 Maths Chapter 7- Integrals 
The NCERT Solutions for Class 12 maths chapter 7​ - Integrals, is available here. The 
NCERT maths book of class 12 includes the concept of integrals in chapter 7. In this 
chapter, students learn about integral calculus (definite and indefinite), their properties 
and much more. This topic is extremely important for both CBSE board exam and for 
competitive exams. 
The concepts of integrals are given in this chapter in a detailed and easy to understand 
manner. These class 12 NCERT solutions for integrals are very simple and can help the 
students in understanding the problem-solving method very easily. Students can reach 
for these NCERT solutions and download it for free to practice them offline as well. 
These solved questions make the students understand the method of applying the 
concepts of Integrals in each problem. 
The major concepts of Maths covered in Chapter 7- Integrals of NCERT Solutions for 
Class 12 include: 
●7.1 Introduction 

11. ●7.2 Integration as an Inverse Process of Differentiation 
●7.2.1 Geometrical interpretation of indefinite integral 
●7.2.2 Some properties of indefinite integral 
●7.2.3 Comparison between differentiation and integration 
●7.3 Methods of Integration 
●7.3.1 Integration by substitution 
●7.3.2 Integration using trigonometric identities 
●7.4 Integrals of Some Particular Functions 
●7.5 Integration by Partial Fractions 
●7.6 Integration by Parts 
●7.7 Definite Integral 
●7.7.1 Definite integral as the limit of a sum 
●7.8 Fundamental Theorem of Calculus 
●7.8.1 Area function 
●7.8.2 First fundamental theorem of integral calculus 
●7.8.3 Second fundamental theorem of integral calculus 
●7.9 Evaluation of Definite Integrals by Substitution 
●7.10 Some Properties of Definite Integrals 
1.Exercise 7.1 Solutions 22 Questions 
2.Exercise 7.2 Solutions 39 Questions 
3.Exercise 7.3 Solutions 24 Questions 
4.Exercise 7.4 Solutions 25 Questions 
5.Exercise 7.5 Solutions 23 Questions 
6.Exercise 7.6 Solutions 24 Questions 
7.Exercise 7.7 Solutions 11 Questions 
8.Exercise 7.8 Solutions 6 Questions 
9.Exercise 7.9 Solutions 22 Questions 
10.Exercise 7.10 Solutions 10 Questions 
11.Exercise 7.11 Solutions 21 Questions 
12.Miscellaneous Exercise On Chapter 7 Solutions 44 Questions 

12. NCERT Solutions for Class 12 Maths Chapter 8 Application 
of Integrals 
NCERT Solutions for Class 12 Maths Chapter 8 
Application of Integrals 
NCERT solutions for class 12 Maths Chapter 8​-Application of Integrals, provides 
solutions for all the questions enlisted under the chapter. This chapter will help the 
students to strengthen their foundation on the application of Integrals. Students can 
easily download and practice all the exercise-wise NCERT problems with solutions to 
score good marks in the final exams and get ready for any competitive exams. 
Download these NCERT Solutions of Class 12 maths and start practice offline. 
In this chapter students will deal with various Integrals applications. Some of the 
important concepts are listed below: 
1. The area of the region bounded by the curve y = f (x), x-axis and the lines x = a and x = 
b (b > a) is given by the formula: Area ∫ba y dx = ∫baf(x) dx 
2. The area of the region enclosed between two curves y = f (x), y = g (x) and the lines x = 
a, x = b is given by the formula, 
Area = ∫ba ; where f(x) ≥ g(x) in [a, b] 
3. If f (x) ≥ g (x) in [a, c] and f (x) ≤ g (x) in [c, b], a < c < b, then 
Area = ∫ca + ∫bc 
NCERT Solutions For Class 12 Maths Chapter 8 Exercises: 
Get a detailed solution for all the questions listed under the below exercises: 

13. Exercise 8.1 Solutions: 13 Questions (8 Long, 3 Short, 2 MCQ) 
Exercise 8.2 Solutions: 7 Questions(5 Long, 2 MCQ) 
Miscellaneous Exercise Solutions: 19 Questions (8 Long, 7 Short, 4 MCQ) 
NCERT Solution Class 12 Maths Chapter 9- Differential 
Equations 
NCERT Solution Class 12 Maths Chapter 9- Differential 
Equations 
The NCERT Solutions for Class 12 Maths Chapter 9​-Differential Equations, is available 
here. In this chapter, students learn about order and degree of differential equations, 
method of solving a differential equation, their properties and much more. This topic is 
extremely important for scoring high in both CBSE board exams and competitive exams. 
The concepts of Differential Equations are given in this chapter in a detailed and easy to 
understand manner. These class 12 NCERT solutions for Differential Equations are very 
simple and can help the students in understanding the problem-solving method very 
easily. Students can reach for these NCERT solutions and download it for free to 
practice them offline as well. These solved questions make the students understand the 
method of applying the concepts of Differential Equations in each problem 
The major concepts of Maths covered in Chapter 9- Differential Equations of NCERT 
Solutions for Class 12 includes: 
1.9.1 Introduction 
2.9.2 Basic Concepts 
3.9.2.1 Order of a differential equation 

14. 4.9.2.2 Degree of a differential equation 
5.9.3 General and Particular Solutions of a Differential Equation 
6.9.4 Formation of a Differential Equation whose General Solution is given 
7.9.4.1 Procedure to form a differential equation that will represent a given family 
of curves 
8.9.5 Methods of Solving First Order, First Degree Differential Equations 
9.9.5.1 Differential equations with variables separable 
10.9.5.2 Homogeneous differential equations 
11.9.5.3 Linear differential equations 
●Exercise 9.1 Solutions 12 Questions 
●Exercise 9.2 Solutions 12 Questions 
●Exercise 9.3 Solutions 12 Questions 
●Exercise 9.4 Solutions 23 Questions 
●Exercise 9.5 Solutions 17 Questions 
●Exercise 9.6 Solutions 19 Questions 
●Miscellaneous Exercise On Chapter 9 Solutions 18 Questions 
NCERT Solution Class 12 Maths Chapter 10- Vector Algebra 
NCERT Solution Class 12 Maths Chapter 10- Vector 
Algebra 
The ​NCERT Solutions For Class 12 Maths Chapter 10​ -Vector Algebra, of the Class 12 
NCERT textbooks are given here. In this chapter, students learn about the difference 
between a scalar and a vector quantity, their properties, operations of vectors, etc,. The 
topic has an important role in helping the students score high marks not only on the 
board exams but also the competitive exams. 
It is important to be prepared for the various problems asked during the 12th board 
examination. Solving through different exercises and problem sets give students the 
confidence to write the exams better. These class 12 NCERT solutions for Vector 

15. Algebra are very easy to understand. Students can also avail these NCERT solutions and 
download it for free to practice them offline as well. 
 
The major concepts of Maths covered in Chapter 10- Vector Algebra of NCERT Solutions 
for Class 12 includes: 
●10.1 Introduction 
●10.2 Basic Concepts 
●Position Vector 
●Direction Cosines 
●10.3 Types of Vectors 
●Zero Vector 
●Unit Vector 
●Coinitial Vectors 
●Collinear Vectors 
●Equal Vectors 
●Negative of a Vector 
●10.4 Addition of Vectors 
●Properties of vector addition 
●10.5 Multiplication of a Vector by a Scalar 
●10.5.1 Components of a vector 
●10.5.2 Vector joining two points 
●10.5.3 Section formula 
●10.6 Product of Two Vectors 
●10.6.1 Scalar (or dot) product of two vectors 
●10.6.2 Projection of a vector on a line 
●10.6.3 Vector (or cross) product of two vectors 
1.Exercise 10.1 Solutions 5 Questions 
2.Exercise 10.2 Solutions 19 Questions 

16. 3.Exercise 10.3 Solutions 18 Questions 
4.Exercise 10.4 Solutions 12 Questions 
5.Miscellaneous Exercise On Chapter 10 Solutions 19 Questions 
NCERT Solution Class 12 Maths Chapter 11- Three 
Dimensional Geometry 
NCERT Solution Class 12 Maths Chapter 11- Three 
Dimensional Geometry 
The ​NCERT Solutions For Class 12 Maths Chapter 11​ - Three Dimensional Geometry, of 
the Class 12 NCERT textbooks are available here. Looking around, we can observe that 
everything in the real world is in a three-dimensional shape. In this chapter, students 
learn about three-dimensional geometry in detail. Students can easily score full marks in 
the questions from this chapter by solving all the questions present in the NCERT 
textbook. 
The class 12 NCERT solutions for Three Dimensional Geometry are very easy to 
understand. These solutions cover all the exercise questions included in the book and 
are according to the latest guidelines of CBSE. Here, the PDF of the class 12 maths 
chapter 11 NCERT solutions is available which can be downloaded and referred in online 
mode. Students can also avail these NCERT solutions and download it for free to 
practice them offline as well. 
The major concepts of Maths covered in Chapter 11- Three Dimensional Geometry of 
NCERT Solutions for Class 12 includes: 
●11.1 Introduction 
●11.2 Direction Cosines and Direction Ratios of a Line 
●11.2.1 Relation between the direction cosines of a line 
●11.2.2 Direction cosines of a line passing through two points 
●11.3 Equation of a Line in Space 
●11.3.1 Equation of a line through a given point and parallel to a given vector b. 

17. ●11.3.2 Equation of a line passing through two given points 
●11.4 Angle between Two Lines 
●11.5 Shortest Distance between Two Lines 
●11.5.1 Distance between two skew lines 
●11.5.2 Distance between parallel lines 
●11.6 Plane 
●11.6.1 Equation of a plane in normal form 
●11.6.2 Equation of a plane perpendicular to a given vector, passing through a 
given point 
●11.6.3 Equation of a plane passing through three non-collinear points 
●11.6.4 Intercept form of the equation of a plane 
●11.6.5 Plane passing through the intersection of two given planes 
●11.7 Coplanarity of Two Lines 
●11.8 Angle between Two Planes 
●11.9 Distance of a Point from a Plane 
●11.10 Angle between a Line and a Plane 
●Exercise 11.1 Solutions 5 Questions 
●Exercise 11.2 Solutions 17 Questions 
●Exercise 11.3 Solutions 14 Questions 
●Miscellaneous Exercise On Chapter 11 Solutions 23 Questions 
NCERT Solution Class 12 Maths Chapter 12- Linear 
Programming 
NCERT Solution Class 12 Maths Chapter 12- Linear 
Programming 
The ​NCERT Solutions For Class 12 Maths Chapter 12​- Linear Programming, of the Class 
12 NCERT textbooks are available here. We can say that linear programming is a method 
to achieve the best outcome in a mathematical model whose requirements are 
represented by linear relationships. In this chapter, students learn about Linear 

18. Programming in detail. Solving all the questions from this chapter of the NCERT book of 
class 12 maths, repeatedly, will help the students in understanding the methods of 
solving the questions. 
The class 12 NCERT solutions of the chapter Linear Programming are very easy to 
understand as the subject experts at CBSETuts ensure that the solutions are given in the 
simplest form. These solutions cover all the exercise questions included in the book and 
are in accordance with the latest CBSE guidelines. Here, the PDF of the NCERT Solutions 
class 12 maths of chapter 12 is available which can be downloaded and referred in 
online mode. Students can also avail these NCERT solutions and download it for free. 
The major concepts of Maths covered in Chapter 12- Linear Programming of NCERT 
Solutions for Class 12 includes: 
●12.1 Introduction 
●12.2 Linear Programming Problem and its Mathematical Formulation 
●12.2.1 Mathematical formulation of the problem 
●12.2.2 Graphical method of solving linear programming problems 
●12.3 Different Types of Linear Programming Problems 
●12.3.1 Manufacturing problems 
●12.3.2 Diet problems 
●12.3.3 Transportation problems 
1.Exercise 12.1 Solutions 10 Questions 
2.Exercise 12.2 Solutions 11 Questions 
3.Miscellaneous Exercise On Chapter 12 Solutions 10 Questions 
NCERT Solution Class 12 Maths Chapter 13 - Probability 
NCERT Solution Class 12 Maths Chapter 13 - Probability 
The ​NCERT Solutions For Class 12 Maths Chapter 13​ - Probability, of the Class 12 
NCERT textbooks are available here. Probability is the chance of occurrence of an event. 

19. In this chapter, students learn about Probability in detail. Solving all the questions from 
this chapter of the NCERT book of class 12 maths, repeatedly, will help the students in 
understanding the problem-solving method. 
The class 12 NCERT solutions of the chapter Probability are very easy to understand as 
the subject experts at CBSETuts ensure that the solutions are given in the simplest form. 
These solutions cover all the exercise questions included in the book and are in 
accordance with the latest CBSE guidelines. Students can also avail these NCERT 
solutions and download it for free to practice them offline as well. 
The chapter Probability itself makes up to a whole unit, Unit Six- Probability, that carries 
8 marks of the total 80 marks. There are 2 exercises along with a miscellaneous 
exercise in this chapter to help students understand the concepts related to Probability 
thoroughly. Some of the topics discussed in Chapter 13 of NCERT Solutions for Class 12 
Maths are as follows: 
0 ≤ P (E|F) ≤ 1, P (E′|F) = 1 – P (E|F) 
P ((E ∪ F)|G) = P (E|G) + P (F|G) – P ((E ∩ F)|G) 
P (E ∩ F) = P (E) P (F|E), P (E) ≠ 0 
P (E ∩ F) = P (F) P (E|F), P (F) ≠ 0 
If E and F are independent, then 
P (E ∩ F) = P (E) P (F) 
P (E|F) = P (E), P (F) ≠ 0 

20. P (F|E) = P (F), P(E) ≠ 0 
The theorem of total probability 
Bayes’ theorem 
A random variable is a real-valued function whose domain is the sample space of a 
random experiment. 
Var (X) = E (X2) – [E(X)]2 
Trials of a random experiment are called Bernoulli trials if they satisfy the 
following conditions : 
There should be a finite number of trials. 
The trials should be independent. 
Each trial has exactly two outcomes: success or failure. 
The probability of success remains the same in each trial. 
Go through Chapter 13, Probability, of the NCERT textbook for maths to know more 
about topics related to Probability.