Ncert solutions class 12 maths

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

● NCERT Solutions For Class 12 Maths Chapter 1
NCERT Solutions for Class 12 Maths Chapter 1 Relations
and Functions
NCERT Solutions for Class 12 Maths Chapter 1 Relations
and Functions

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:

● Exercise 1.1 Solutions : 16 Questions (14 Short Answers, 2 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.

● 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.

● Exercise 3.2 Solutions : 22 Questions(14 Long, 6 Short, 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
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

(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.
● 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.

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)

● 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 5.5 Solutions : 18 Questions ( Short Answers)

● Miscellaneous Exercise Solutions: 23 Questions (6 Long, 17 Short)
NCERT Solutions for Class 12 Maths Chapter 6 Application
of Derivatives
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

● 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 6.4 Solutions: 9 Questions (7 Short, 2 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

● 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
12. Miscellaneous Exercise On Chapter 7 Solutions 44 Questions

NCERT Solutions for Class 12 Maths Chapter 8 Application
of Integrals
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
Get a detailed solution for all the questions listed under the below exercises:

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

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
● 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

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:
● 10.2 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

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:
● 11.2 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.

● 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
● 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

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:
● 12.2 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
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.

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

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.

Ncert solutions class 12 maths

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