NCERT Solutions for Class 12 Maths PDF: Going through CBSE Class 12 Maths NCERT Solutions is a significant part of your preparation for Class 12th board exams. Working on CBSE NCERT Solutions for Class 12 Maths will clear your doubts in regard to any question and improve your application skills as well. This will not only help you secure good marks in board exams but will also aid you in clearing the toughest engineering entrance exams such as JEE Main, JEE Advanced, BITSAT, etc. In this article, you will get Maths NCERT Solutions Class 12 for all the chapters in PDF format.
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CBSE NCERT Solutions For Class 12 Maths PDF Download
Students can download all the NCERT Solutions of Maths Class 12 provided on this page for free. Working on NCERT Class 12 Maths solutions will not only help clear your basics but will also help in improving your problem-solving abilities. Just click on the Class 12 Maths NCERT Chapter-wise links listed below to check the solutions and important questions for that chapter. To download the solutions as PDF, you can visit the chapter-wise link and download it from there for free.
Maths Class 12 NCERT Solutions (Chapter-wise)
- -1st Chapter – Relations and Functions
- -2nd Chapter – Inverse Trigonometric Functions
- -3rd Chapter – Matrices
- -4th Chapter – Determinants
- -5th Chapter – Continuity and Differentiability
- -6th Chapter – Application of Derivatives
- -7th Chapter – Integrals
- -8th Chapter – Application of Integrals
- -9th Chapter – Differential Equations
- -10th Chapter – Vector Algebra
- -11th Chapter – Three Dimensional Geometry
- -12th Chapter – Linear Programming
- -13th Chapter – Probability
All the NCERT Solutions for Class 12 Maths provided on this page are solved by the top Maths teachers of Embibe. Each Class 12 Maths NCERT Solution comes with a detailed step-wise explanation so that students get clarity on the topic. Download Maths Class 12 NCERT Solutions PDF for free and access them in offline mode as well.
Advantages Of Solving CBSE NCERT Solutions For Class 12 Maths
The advantages of using CBSE NCERT Solutions for Class 12 Maths by Embibe are listed below:
- All CBSE NCERT Solutions for Class 12 Maths are provided in PDF form which can be downloaded for free by anyone, anywhere.
- Referring to the Class 12 Maths NCERT Solutions will help students understand all basic and fundamental concepts.
- As most of the engineering competitive exam syllabus such as JEE Main, JEE Advanced, BITSAT is almost the same as CBSE Class 11 and 12 Maths Syllabus, so, having a good command over the NCERT Solutions for Class 12 Maths will definitely help to clear the competitive exams easily.
- The CBSE NCERT Solutions help students to solve their homework and assignments on time without any difficulty.
CBSE NCERT Solutions For Class 12 Maths: Chapter Descriptions
Students can go through the description of each chapter to get an overview of the contents of the chapters. The chapter-wise description has been included in the following sections. To download the NCERT Solutions PDF for each chapter, click on the link given at the end of the following sections.
NCERT Solutions For Class 12 Maths Chapter 1: Relations And Functions
The term ‘relation’ in mathematics has been drawn from the meaning of relationships in the English language, according to which two objects or quantities are related if there is a recognizable connection or link between the two objects or quantities.
In Class 12 Chapter 1, you will learn about different types of relations and functions, the composition of functions, invertible functions, and binary operations. The sub-topics in this chapter are listed below:
- – 1.1 Introduction
- – 1.2 Types of Relations
- – 1.3 Types of Functions
- – 1.4 Composition of Functions and Invertible Function
- – 1.5 Binary Operations
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Exercise 1.1 |
16 Questions |
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Exercise 1.2 |
12 Questions |
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Exercise 1.3 |
14 Questions |
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Exercise 1.4 |
13 Questions |
Chapter 1: Relations and Functions – Download NCERT Solutions PDF
NCERT Solutions For Class 12 Maths Chapter 2: Inverse Trigonometric Functions
In this chapter, you will study the restrictions on domains and ranges of trigonometric functions that ensure the existence of their inverses and observe their behavior through graphical representations. Besides, some elementary properties will also be discussed. The inverse trigonometric functions play an important role in calculus for they serve to define many integrals. The concepts of inverse trigonometric functions are used in science and engineering. The sub-topics in this chapter are listed below:
- – 2.1 Introduction
- – 2.2 Basic Concepts
- – 2.3 Properties of Inverse Trigonometric Functions
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Exercise 2.1 |
14 Questions |
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Exercise 2.2 |
21 Questions |
Chapter 2: Inverse Trigonometric Functions – Download NCERT Solutions PDF
NCERT Solutions For Class 12 Maths Chapter 3: Matrices
In this chapter, you will learn about the fundamentals of matrix and matrix algebra. The sub-topics in this chapter are listed below:
- – 3.1 Introduction
- – 3.2 Matrix
- – 3.3 Types of Matrices
- – 3.4 Operation on Matrices
- – 3.5 Transpose of a Matrix
- – 3.6 Symmetric and Skew Symmetric Matrices
- – 3.7 Elementary Operation (Transformation) of a Matrix
- – 3.8 Invertible Matrices
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Exercise 3.1 |
10 Questions |
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Exercise 3.2 |
22 Questions |
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Exercise 3.3 |
12 Questions |
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Exercise 3.4 |
18 Questions |
Chapter 3: Matrices – Download NCERT Solutions PDF
NCERT Solutions For Class 12 Maths Chapter 4: Determinants
The determinant is a scalar value that can be measured from the elements of a square matrix and encodes certain properties of the linear transformation described by the matrix. The determinant of a matrix A is denoted det(A), det A, or |A|. In the case of a 2 × 2 matrix, the determinant may be defined as
Similarly, for a 3 × 3 matrix A, its determinant is:
![{\displaystyle {\begin{aligned}|A|={\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}}&=a\,{\begin{vmatrix}\Box &\Box &\Box \\\Box &e&f\\\Box &h&i\end{vmatrix}}-b\,{\begin{vmatrix}\Box &\Box &\Box \\d&\Box &f\\g&\Box &i\end{vmatrix}}+c\,{\begin{vmatrix}\Box &\Box &\Box \\d&e&\Box \\g&h&\Box \end{vmatrix}}\\[3pt]&=a\,{\begin{vmatrix}e&f\\h&i\end{vmatrix}}-b\,{\begin{vmatrix}d&f\\g&i\end{vmatrix}}+c\,{\begin{vmatrix}d&e\\g&h\end{vmatrix}}\\[3pt]&=aei+bfg+cdh-ceg-bdi-afh.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/14f2f2a449d6d152ee71261e47551aa0a31c801e)
So in this chapter, you will study determinants up to order three only with real entries. The sub-topics in this chapter are listed below:
- – 4.1 Introduction
- – 4.2 Determinant
- – 4.3 Properties of Determinants
- – 4.4 Area of a Triangle
- – 4.5 Minors and Cofactors
- – 4.6 Adjoint and Inverse of a Matrix
- – 4.7 Applications of Determinants and Matrices
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Exercise 4.1 |
8 Questions |
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Exercise 4.2 |
16 Questions |
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Exercise 4.3 |
5 Questions |
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Exercise 4.4 |
5 Questions |
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Exercise 4.5 |
18 Questions |
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Exercise 4.6 |
16 Questions |
Chapter 4: Determinants – Download NCERT Solutions PDF
NCERT Solutions For Class 12 Maths Chapter 5: Continuity And Differentiability
In this chapter, we will learn everything about Continuity and Differentiability of a function. In this chapter, students will learn about the very important concepts of continuity, differentiability, and relations between them.
We will also learn the differentiation of inverse trigonometric functions. Further, this chapter introduces a new class of functions called exponential and logarithmic functions. The sub-topics in this chapter are listed below:
- – 5.1 Introduction to Continuity and Differentiability
- – 5.2 Exponential and Logarithmic Functions
- – 5.3 Logarithmic Differentiation
- – 5.4 Derivatives of Functions in Parametric Forms
- – 5.5 Second Order Derivative
- – 5.6 Mean Value Theorem
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Exercise 5.1 |
34 Questions |
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Exercise 5.2 |
10 Questions |
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Exercise 5.3 |
15 Questions |
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Exercise 5.4 |
10 Questions |
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Exercise 5.5 |
18 Questions |
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Exercise 5.6 |
11 Questions |
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Exercise 5.7 |
17 Questions |
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Exercise 5.8 |
6 Questions |
Chapter 5: Continuity and Differentiability – Download NCERT Solutions PDF
NCERT Solutions For Class 12 Maths Chapter 6: Application Of Derivatives
In the chapter Continuity and Differentiability, you have learned how to find a derivative of composite functions, inverse trigonometric functions, implicit functions, exponential functions, and logarithmic functions. In this chapter, you will study applications of the derivative in various disciplines, e.g., in engineering, science, social science, and many other fields.
You will also use the derivative to find intervals on which a function is increasing or decreasing. Finally, here you will use the derivative to find the approximate value of certain quantities. The sub-topics in this chapter are listed below:
- – 6.1 Introduction
- – 6.2 Rate of Change of Quantities
- – 6.3 Increasing and Decreasing Functions
- – 6.4 Tangents and Normals
- – 6.5 Approximations
- – 6.6 Maxima and Minima
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Exercise 6.1 |
18 Questions |
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Exercise 6.2 |
19 Questions |
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Exercise 6.3 |
27 Questions |
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Exercise 6.4 |
9 Questions |
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Exercise 6.5 |
29 Questions |
Chapter 6: Application of Derivatives – Download NCERT Solutions PDF
CHECK OUT CBSE CLASS 12 SYLLABUS FROM HERE
NCERT Solutions For Class 12 Maths Chapter 7: Integrals
Differential Calculus is centered on the concept of the derivative. Integral Calculus is motivated by the problem of defining and calculating the area of the region bounded by the graph of the functions. The sub-topics in this chapter are listed below:
- – 7.1 Introduction to Integral Calculus
- – 7.2 Integration as an Inverse Process of Differentiation
- – 7.3 Methods of Integration
- – 7.4 Integrals of Some Particular Functions
- – 7.5 Integration by Partial Fractions
- – 7.6 Integration by Parts
- – 7.7 Definite Integral
- – 7.8 Fundamental Theorem of Calculus
- – 7.9 Evolution of Definite Integrals by Substitution
- – 7.10 Some Properties of Definite Integrals
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Exercise 7.1 |
22 Questions |
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Exercise 7.2 |
39 Questions |
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Exercise 7.3 |
24 Questions |
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Exercise 7.4 |
25 Questions |
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Exercise 7.5 |
23 Questions |
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Exercise 7.6 |
24 Questions |
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Exercise 7.7 |
11 Questions |
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Exercise 7.8 |
6 Questions |
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Exercise 7.9 |
22 Questions |
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Exercise 7.10 |
10 Questions |
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Exercise 7.11 |
21 Questions |
Chapter 7: Integrals – Download NCERT Solutions PDF
NCERT Solutions For Class 12 Maths Chapter 8: Application Of Integrals
Here, in this chapter, we will study a specific application of integrals, to find the area under simple curves, area between lines and arcs of circles, parabola, and ellipses (standard forms only). We shall also deal with finding the area bounded by the above-said curves. The sub-topics in this chapter are listed below:
- – 8.1 Introduction
- – 8.2 Area under Simple Curves
- – 8.3 Area Between Two Curves
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Exercise 8.1 |
13 Questions |
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Exercise 8.2 |
7 Questions |
Chapter 8: Application of Integrals – Download NCERT Solutions PDF
NCERT Solutions For Class 12 Maths Chapter 9: Differential Equations
In this chapter, we will study some basic concepts related to the differential equation, general and particular solutions of a differential equation, formation of differential equations, some methods to solve a first-order – first-degree differential equation, and some applications of differential equations in different areas. The sub-topics in this chapter are listed below:
- – 9.1 Introduction to Differential Equations
- – 9.2 Basic Concepts
- – 9.3 General and Particular Solutions of a Differential Equation
- – 9.4 Formation of a Differential Equation whose General Solution is given
- – 9.5 Methods of Solving First Order, First Degree Differential Equations
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Exercise 9.1 |
12 Questions |
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Exercise 9.2 |
12 Questions |
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Exercise 9.3 |
12 Questions |
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Exercise 9.4 |
23 Questions |
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Exercise 9.5 |
17 Questions |
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Exercise 9.6 |
19 Questions |
Chapter 9: Differential Equations – Download NCERT Solutions PDF
NCERT Solutions For Class 12 Maths Chapter 10: Vector Algebra
In this chapter, we will study some of the basic concepts about vectors, various operations on vectors, and their algebraic and geometric properties. These two types of properties, when considered together give a full realization to the concept of vectors and lead to their vital applicability in various areas. The sub-topics in this chapter are listed below:
- – 10.1 Introduction to Vectors
- – 10.2 Some Basic Concepts
- – 10.3 Types of Vectors
- – 10.4 Addition of Vectors
- – 10.5 Multiplication of a Vector by a Scalar
- – 10.6 Product of Two Vectors
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Exercise 10.1 |
5 Questions |
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Exercise 10.2 |
19 Questions |
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Exercise 10.3 |
18 Questions |
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Exercise 10.4 |
12 Questions |
Chapter 10: Vector Algebra – Download NCERT Solutions PDF
NCERT Solutions For Class 12 Maths Chapter 11: Three Dimensional Geometry
In this chapter, we shall study the direction cosines and direction ratios of a line joining two points and also discuss the equations of lines and planes in space under different conditions, the angle between two lines, two planes, a line and a plane, the shortest distance between two skew lines and distance of a point from a plane. Most of the above results are obtained in vector form. The sub-topics in this chapter are listed below:
- – 11.1 Introduction
- – 11.2 Direction Cosines and Direction Ratios of a Line
- – 11.3 Equation of a Line in Space
- – 11.4 Angle between Two Lines
- – 11.5 Short Distance between Two Lines
- – 11.6 Plane
- – 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
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Exercise 11.1 |
5 Questions |
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Exercise 11.2 |
17 Questions |
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Exercise 11.3 |
14 Questions |
Chapter 11: Three Dimensional Geometry – Download NCERT Solutions PDF
NCERT Solutions For Class 12 Maths Chapter 12: Linear Programming
In this chapter, we shall apply the systems of linear inequalities/equations to solve some real-life problems. Also, we shall study some linear programming problems and their solutions by graphical method only, though there are many other methods to solve such problems. The sub-topics in this chapter are listed below:
- – 12.1 Introduction to Linear Programming
- – 12.2 Linear Programming Problem and its Mathematical Formulation
- – 12.3 Different Types of Linear Programming Problems
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Exercise 12.1 |
10 Questions |
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Exercise 12.2 |
11 Questions |
Chapter 12: Linear Programming – Download NCERT Solutions PDF
NCERT Solutions For Class 12 Maths Chapter 13: Probability
In this chapter, we shall discuss the important concept of conditional probability of an event given that another event has occurred, which will be helpful in understanding the Bayes’ theorem, multiplication rule of probability, and independence of events.
We shall also learn an important concept of a random variable and its probability distribution and also the mean and variance of a probability distribution. The sub-topics in this chapter are listed below:
- – 13.1 Probability Introduction
- – 13.2 Conditional Probability
- – 13.3 Multiplication Theorem on Probability
- – 13.4 Independent Events
- – 13.5 Bayes’ Theorem
- – 13.6 Random Variables and its Probability Distributions
- – 13.7 Bernoulli Trials and Binomial Distribution
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Exercise 13.1 |
17 Questions |
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Exercise 13.2 |
18 Questions |
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Exercise 13.3 |
14 Question |
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Exercise 13.4 |
17 Questions |
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Exercise 13.5 |
15 Questions |
Chapter 13: Probability – Download NCERT Solutions PDF
How To Get Good Marks In Class 12 Maths?
Students can follow the tips listed below to get better at the subject and improve their score:
- First and most important is the regular practice of problems and exercises. You cannot get good marks in Maths if you do not practice regularly.
- You should be thorough with the syllabus of CBSE Class 12 Maths. You should be familiar with all the concepts and topics in every chapter.
- Start your preparation with the NCERT Class 12 Maths textbook. Once you are done with the NCERT textbook, you can pick up a reference book of your choice for practice.
- Solve all the exercises at the end of each chapter. If you are stuck on a problem, refer to the NCERT solutions for Class 12 Maths.
- Give special attention to the chapters/concepts that you find difficult. Do not skip any chapter or topic.
- Take timed mock tests to improve your speed and accuracy.
- Solve previous years’ question papers to understand what kind of questions are asked in the final exam.
Some Important Questions For 12th Class Maths
Below we have provided some important questions that will aid you in understanding the chapters as well as help you in scoring better:
| 1. Let f : N → Y be a function defined as f (x) = 5x + 2, where, Y = {y ∈ N: y = 5x + 2 for some x ∈ N}. Show that f is invertible. Find the inverse. |
| 2. Find the value of cos-1 (1/2) + 2 sin-1 (1/2) |
| 3. Verify the mean value theorem for the following function f (x) = (x – 4) (x – 8) (x – 12) in [4, 6] |
| 4. Using matrix method, solve the system of equations 2x + 3y – 2z = 3, x + 3y + 2z = 6 and 3x – 2y + z = 2. |
| 5. A circular disc of radius 4 cm is being heated. Due to expansion, its radius increases at a rate of 0.05 cm per second. Find the rate at which its area is increasing if the radius is 5.2 cm. |
| 6. Find the value of ∫tan8x sec4 x dx |
| 7. Determine the area of the region bounded by y2 = 8x, x = 3, x = 2 and the x-axis in the first quadrant. |
| 8. Find the differential equation representing the family of curves y = a sin (x + b), where a, b are arbitrary constants. |
| 9. Find the vector joining the points P(3, 4, 0) and Q(– 2, – 4, – 5) directed from P to Q. |
| 10. If a line makes an angle of 80°, 120°, 65° with the x, y, and z-axes respectively, determine its direction cosines. |
| 11. There are 5 cards numbered 1 to 5 with one number on each card. Two cards are drawn at random without replacement. Let X denote the sum of the numbers on two cards drawn. Find the mean and variance of X. |
| 12. The probability of solving a specific problem independently by persons A and B are 1/2 and 1/3, respectively. If both of them try to solve the problem independently, then calculate the probability that the problem is solved. |
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NCERT Maths Solution Class 12: List Of Deleted Topics For 2020-21
Due to COVID-19, the overall syllabus for CBSE Class 12 Maths is reduced by up to 30%. The list of topics you can exclude from your Class 12 Maths NCERT Solutions for your board exam preparation are given below:
| Unit 1: Relations and Functions 1. Relations and Functions – Composite functions, inverse of a function. 2. Inverse Trigonometric Functions – Graphs of inverse trigonometric functions – Elementary properties of inverse trigonometric functions Unit 2: Algebra 1. Matrices – Existence of non-zero matrices whose product is the zero matrix. – Concept of elementary row and column operations. – Proof of the uniqueness of inverse, if it exists. 2. Determinants – Properties of determinants – Consistency, inconsistency and number of solutions of system of linear equations by examples Unit 3: Calculus 1. Continuity and Differentiability – Rolle’s and Lagrange’s Mean Value Theorems (without proof) and their geometric interpretation. 2. Applications of Derivatives – Rate of change of bodies, use of derivatives in approximation 4. Applications of the Integrals – Area between any of the two above said curves 5. Differential Equations – Formation of differential equations whose general solution is given. – Solutions of linear differential equation of the type: dx/dy + px = q, where p and q are function of y or constant. Unit-IV: Vectors and Three- Dimensional Geometry 1. Vectors – scalar triple product of vectors. 2. Three – dimensional Geometry Angle between (i) two lines, (ii) two planes, (iii) a line and a plane Unit-V: Linear Programming 1. Linear Programming – mathematical formulation of L.P. problems – (unbounded) Unit-VI: Probability 1. Probability – mean and variance of random variable. Binomial probability distribution. |
Students preparing for upcoming CBSE board exams can also take the help of other important resources such as sample papers, previous years papers, etc.
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Embibe offers you many resources to learn and practice lessons online. You can solve JEE Main Practice Questions and JEE Advanced Practice Questions at Embibe for free. You can also take free JEE Main Mock Tests and JEE Advanced Mock Tests. These will definitely help you in getting better marks.
Candidates who wish to appear for medical entrance exams can also utilize the resources offered by Embibe. You can Practice NEET Questions online for free. You can also attempt free NEET Mock Tests.
You can follow the steps provided below to take a mock test or practice questions:
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FAQs On NCERT Solutions For Class 12 Maths
Some of the frequently asked questions on NCERT Solutions for Class 12 Maths are listed below:
A1. Many experts consider the NCERT Mathematics textbook the best for the Class 12 exam preparation.
A2. There are a total of 13 chapters in Class 12 Maths.
A3. According to many teachers and experts, the NCERT Mathematics textbook is sufficient to clear Class 12 Maths exams. However, students can also refer to other textbooks if they are aiming for the top marks.
A4. The NCERT solutions include all the details in an easy and step-wise manner. Hence, you will easily understand the problems. With the help of these solutions, you can revise the syllabus quickly. They are available both offline and online and thus, you can access them anytime you want.
Download CBSE NCERT Books For Class 12 From Here
We hope this article has been helpful to you. If you have any queries/doubts, leave them in the comment section below and we will get back to you at the earliest. Stay tuned to embibe.com for more on CBSE NCERT Solutions For Class 12.
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