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 also aid you in clearing the toughest engineering entrance exams such as JEE Main, JEE Advanced, BITSAT, etc. In this article, you will find chapter-wise NCERT Maths Solutions for Class 12 in PDF.
In this article, we will provide you with CBSE NCERT Solutions for Class 12 Maths for all the chapters. Read on to find more about Class 12 Maths NCERT Solutions.
SOLVE CBSE CLASS 12 MATHS CHAPTER-WISE QUESTIONS HERE
CBSE NCERT Solutions For Class 12 Maths PDF Download
Students can download all the CBSE NCERT Solutions for Class 12 Maths provided on this page for free without any cost. Working on NCERT Solutions for Class 12 Maths will not only help clear your basics but 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:
Class 12 Maths NCERT Chapter Wise Solutions
- -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 question of Class 12 Maths NCERT Solution comes with detailed step-wise solutions. Download Maths Class 12 NCERT Solutions for free and access them in offline mode as well.
Advantages Of Solving CBSE NCERT Solutions For Class 12 Maths
The advantages of solving CBSE NCERT Solutions for Class 12 Maths from Embibe are listed below:
- All the CBSE NCERT Solutions for Class 12 Maths are provided in the PDF form which can be downloaded for free by anyone, anywhere, and make use of them in offline mode.
- 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 & 12 Maths Syllabus. So having good command over the NCERT Solutions for Class 12 Maths will definitely help to clear the competitive exams easily.
- All the CBSE NCERT Solutions comes with a detailed step by step solution which will further 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
In Class 11, you have learned about the notion of relations and functions, domain, co-domain, and range along with different types of specific real-valued functions and their graphs. The concept of 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. Let A be the set of students of Class XII of a school and B be the set of students of Class XI of the same school. Then some of the examples of relations from A to B are:
- {(a, b) ∈ A × B: a is the brother of b},
- {(a, b) ∈ A × B: a is the sister of b},
- {(a, b) ∈ A × B: age of a is greater than the age of b},
- {(a, b) ∈ A × B: total marks obtained by A in the final examination is less than the total marks obtained by b in the final examination},
- {(a, b) ∈ A × B: a lives in the same locality as b}.
However, abstracting from this, we define mathematically a relation R from A to B as an arbitrary subset of A × B. If (a, b) ∈ R, we say that a is related to b under the relation R and we write as a R b. So as you learned in Class 11, the functions are special kind of relations whereas, in this 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 Chapter 1, you have studied that the inverse of a function f, denoted by f–1, exists if f is one-one and onto. There are many functions which are not one-one, onto or both and hence we can not talk of their inverses. In Class XI, you have learned that trigonometric functions are not one-one and onto over their natural domains and ranges and hence their inverses do not exist.
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. Also, the concepts of inverse trigonometric functions are also 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
The knowledge of matrices is necessary for various branches of mathematics. Matrices are one of the most powerful tools in mathematics. This mathematical tool simplifies our work to a great extent when compared with other straight forward methods. The evolution of the concept of matrices is the result of an attempt to obtain compact and simple methods of solving systems of linear equations.
Matrices are not only used as a representation of the coefficients in a system of linear equations, but the utility of matrices far exceeds that use. Matrix notation and operations are used in electronic spreadsheet programs for personal computers, which in turn is used in different areas of business and science like budgeting, sales projection, cost estimation, analyzing the results of an experiment, etc.
Also, many physical operations such as magnification, rotation, and reflection through a plane can be represented mathematically by matrices. Matrices are also used in cryptography. This mathematical tool is not only used in certain branches of sciences, but also in genetics, economics, sociology, modern psychology, and industrial management.
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|. Geometrically, it can be viewed as the volume scaling factor of the linear transformation described by the matrix. This is also the signed volume of the n-dimensional parallelepiped spanned by the column or row vectors of the matrix. The determinant is positive or negative according to whether the linear mapping preserves or reverses the orientation of n-space.
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. Also, you will study various properties of determinants, minors, cofactors and applications of determinants in finding the area of a triangle, adjoint and inverse of a square matrix, consistency and inconsistency of system of linear equations and solution of linear equations in two or three variables using the inverse of a matrix. 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
Continuity of a function is the characteristic of a function by virtue of which, the graphical form of that function is a continuous wave. A differentiable function is a function whose derivative exists at each point in its domain. 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. These functions lead to powerful techniques of differentiation. Also, this chapter illustrates certain geometrically obvious conditions through differential calculus. 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 NCERT Solutions
From 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. From this chapter, you will study applications of the derivative in various disciplines, e.g., in engineering, science, social science, and many other fields.
For instance, you will learn about how the derivative can be used
(i) to determine the rate of change of quantities
(ii) to find the equations of tangent and normal to a curve at a point
(iii) to find turning points on the graph of a function which in turn will help us to locate points at which the largest or smallest value (locally) of a function occurs.
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 CBSE CLASS 12 SYLLABUS HERE
NCERT Solutions For Class 12 Maths Chapter 7 – Integrals NCERT Solutions
Differential Calculus is centered on the concept of the derivative. The original motivation for the derivative was the problem of defining tangent lines to the graphs of functions and calculating the slope of such lines. Integral Calculus is motivated by the problem of defining and calculating the area of the region bounded by the graph of the functions.
The functions that could possibly have given function as a derivative are called anti-derivatives (or primitive) of the function. Further, the formula that gives all these anti-derivatives is called the indefinite integral of the function and such process of finding anti-derivatives is called integration. Such types of problems arise in many practical situations.
For instance, if we know the instantaneous velocity of an object at any instant, then there arises a natural question, i.e., can we determine the position of the object at any instant? There are several such practical and theoretical situations where the process of integration is involved. The development of integral calculus arises out of the efforts of solving the problems of the following types:
(a) the problem of finding a function whenever its derivative is given
(b) the problem of finding the area bounded by the graph of a function under certain conditions. These two problems lead to the two forms of the integrals, e.g., indefinite and definite integrals, which together constitute the Integral Calculus. 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 NCERT Solutions
In geometry, we have learned formulae to calculate areas of various geometrical figures including triangles, rectangles, trapeziums, and circles. Such formulae are fundamental in the applications of mathematics to many real-life problems. The formulae of elementary geometry allow us to calculate areas of many simple figures. However, they are inadequate for calculating the areas enclosed by curves.
For that, we shall need some concepts of Integral Calculus. In the previous chapter, we have studied to find the area bounded by the curve y = f (x), the ordinates x = a, x = b and x-axis, while calculating definite integral as the limit of a sum. Here, in this chapter, we shall 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 NCERT Solutions
An equation of the form (1) is known as a differential equation. A formal definition will be given later. These equations arise in a variety of applications, may it be in Physics, Chemistry, Biology, Anthropology, Geology, Economics, etc.
Hence, an in-depth study of differential equations has assumed prime importance in all modern scientific investigations. 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 NCERT Solutions
In our day to day life, we come across many queries such as – What is your height? How should a football player hit the ball to give a pass to another player of his team? Observe that a possible answer to the first query maybe 1.6 meters, a quantity that involves only one value (magnitude) which is a real number. Such quantities are called scalars. However, an answer to the second query is a quantity (called force) which involves muscular strength (magnitude) and direction (in which another player is positioned). Such quantities are known vectors.
In mathematics, physics, and engineering, we frequently come across with both types of quantities, namely, scalar quantities such as length, mass, time, distance, speed, area, volume, temperature, work, money, voltage, density, resistance etc. and vector quantities like displacement, velocity, acceleration, force, weight, momentum, electric field intensity, etc.
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 as mentioned above. 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 NCERT Solutions
In Class XI, while studying Analytical Geometry in two dimensions, and the introduction to three-dimensional geometry, we confined to the Cartesian methods only. In the previous chapter of this book, we have studied some basic concepts of vectors. We will now use vector algebra to three-dimensional geometry. The purpose of this approach to 3-dimensional geometry is that it makes the study simple and elegant.
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 obtained in vector form. Nevertheless, we shall also translate these results in the Cartesian form which, at times, presents a more clear geometric and analytic picture of the situation. 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 NCERT Solutions
In earlier classes, we have discussed systems of linear equations and their applications in day to day problems. In Class XI, we have studied linear inequalities and systems of linear inequalities in two variables and their solutions by graphical method. Many applications in mathematics involve systems of inequalities/equations. In this chapter, we shall apply the systems of linear inequalities/equations to solve some real-life problems of the type as given below:
A furniture dealer deals in only two items–tables and chairs. He has Rs 50,000 to invest and has storage space of at most 60 pieces. A table costs Rs 2500 and a chair Rs 500. He estimates that from the sale of one table, he can make a profit of Rs 250 and that from the sale of one chair a profit of Rs 75. He wants to know how many tables and chairs he should buy from the available money so as to maximize his total profit, assuming that he can sell all the items which he buys. Such type of problems which seek to maximize (or, minimize) profit (or, cost) form a general class of problems called optimization problems. Thus, an optimization problem may involve finding maximum profit, minimum cost, or minimum use of resources, etc.
A special but very important class of optimization problems is a linear programming problem. The above-stated optimization problem is an example of a linear programming problem. Linear programming problems are of much interest because of their wide applicability in industry, commerce, management science, etc. In this chapter, we shall study some linear programming problems and their solutions by graphical method only, though there are many other methods also 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 NCERT Solutions
In earlier classes, we have studied the probability as a measure of uncertainty of events in a random experiment. We discussed the axiomatic approach formulated by Russian Mathematician, A.N. Kolmogorov (1903-1987), and treated probability as a function of outcomes of the experiment. We have also established the equivalence between the axiomatic theory and the classical theory of probability in case of equally likely outcomes.
On the basis of this relationship, we obtained probabilities of events associated with discrete sample spaces. We have also studied the addition rule of 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. In the last section of the chapter, we shall study an important discrete probability distribution called Binomial distribution. Throughout this chapter, we shall take up the experiments having equally likely outcomes, unless stated otherwise. 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 at 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’ questions 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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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 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 exerts, 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.
NCERT Solutions for Class 12 Maths: List Of Deleted Portions
The list of topics you can exclude in 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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