# NCERT Solutions for Class 12

## Introduction about NCERT Solutions for Class 12 Maths

NCERT Solutions for class 12 given here have been put together by our content analysts who have many years of experience of creating content for CBSE board, ICSE board and all Indian state boards. The solutions have been designed to simplify all class 12 Math problems, which are given in the textbooks. All the 13 chapters as prescribed by CBSE have been included in NCERT Maths class 12 solutions.

## Why students prefer NCERT Solutions for Class 12 Maths

Students prefer NCERT solutions to do their regular home assignment, mock tests and ace their final exams. In most of the common entrance tests conducted for admissions in engineering institutes, the questions are designed as per the syllabus of textbooks. Therefore, the students prefer to practice these NCERT Solution exercises on a regular basis to score better marks in exams.

## Topics Covered in NCERT Solutions for Class 12 Maths & Brief Detail About Them

NCERT Solutions of class 12 maths contains all chapter solutions in pdf. Solutions can be downloaded chapter wise. NCERT solutions for class 12 cover all the chapters including Relations and Functions, Inverse Trigonometric Functions, Matrices, Determinants, Continuity and Differentiability, Applications of Derivatives, Integrals, Applications of Integrals, Differential Equations, Vector Algebra, Three Dimensional Geometry, Linear Programming, Probability.

These all the chapters are fully comprehensive well explained. It can really help you with your exams, providing you with the easiest and fastest method to solve the question, additionally, the concept of solutions has kept so simple and easy to understand so that it can be remembered for a lifetime to students hence it will also help you cracking other higher-level exams.

Students can easily use the PDF of the chapter-wise solutions and gain conceptual knowledge to solve the problems according to the NCERT Maths textbook for Class 12 Part 1 or NCERT Maths textbook for Class 12 Part 2. This helps students enhance their confidence, which is required to master concepts and perform well in exams.

### NCERT Solutions for Class 12 Maths Chapter 1 – Relations and Functions

1. Types of Relations
• Empty relation is the relation R in X given by R = φ ⊂ X × X
• Universal relation is the relation R in X given by R = X × X
• Reflexive relation R in X is a relation with (a, a) ∈ R ∀a ∈ X.
• Symmetric relation R in X is a relation satisfying (a, b) ∈ R implies (b, a) ∈ R.
• Transitive relation R in X is a relation satisfying (a, b) ∈ R, and (b, c) ∈ R implies that (a, c) ∈ R.
• Equivalence relation R in X is a relation that is reflexive, symmetric, and transitive.
• Equivalence class [a] containing a ∈ X for an equivalence relation R in X is the subset of X containing all elements b related to a.
2. Types of Functions
3. 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.
4. Composition of Functions and Invertible Function:
• The composition of functions f : A → B and g : B → C is the function gof : A → C given by gof(x) = g(f(x))∀ x ∈ A. 
• A function f : X → Y is invertible if ∃ g : Y → X such that of = IX and fog = IY. 
• A function f : X → Y is invertible if and only if f is one-one and onto.
• Given a finite set X, a function f : X → X is one-one (respectively onto) if and only if f is onto (respectively one-one). This is the characteristic property of a finite set. This is not true for an infinite set
5. Binary Operations
• A binary operation ∗ on a set A is a function ∗ from A × A to A.
• An element e ∈ X is the identity element for binary operation ∗ : X × X → X, if a ∗ e = a = e ∗ a ∀a ∈ X. 
• An element a ∈ X is invertible for binary operation ∗ : X × X → X, if there exists b ∈ X such that a ∗ b = e = b ∗ a where e is the identity for the binary operation ∗. The element b is called the inverse of a and is denoted by a–1. 
• An operation ∗ on X is commutative if a ∗ b = b ∗ a ∀a, b in X. 
• An operation ∗ on X is associative if (a ∗ b) ∗ c = a ∗ (b ∗ c)∀a, b, c in X.

### NCERT Solutions for Class 12 Maths Chapter 2 – Inverse Trigonometric Functions

1. Basic Concepts
• The domains and ranges (principal value branches) of inverse trigonometric functions.
• The value of an inverse trigonometric functions which lies in its principal value branch is called the principal value of that inverse trigonometric functions
2. Properties of Inverse Trigonometric Functions
• Properties of Inverse trigonometric functions

### NCERT Solutions for Class 12 Maths Chapter 3 – Matrix

1. Matrix
2. Matrix: A matrix is an ordered rectangular array of numbers or functions.
3. Order of Matrix: A matrix having m rows and n columns are called a matrix of order m × n.
4. A point (x, y) in a plane can be represented by a matrix (column or row)
5. Vertices of a closed rectilinear figure can be represented in the form of a matrix.
6. Types of matrices
7. Column matrix: A matrix is said to be a column matrix if it has only one column.
8. Row matrix: A matrix is said to be a row matrix if it has only one row.
9. Square matrix: A matrix, in which the number of rows is equal to the number of columns, is said to be a square matrix.
10. Diagonal matrix: A square matrix B said to be a diagonal matrix if all its non-diagonal elements are zero.
11. Scalar matrix: A diagonal matrix is said to be a scalar matrix if its diagonal elements are equal.
12. Identity matrix: A square matrix in which elements in the diagonal are all 1 and rest are all zero is called an identity matrix
13. Zero matrix: A matrix is said to be a zero matrix or null matrix if all its elements are zero
14. Equality of matrices: Two matrices A and B are said to be equal if (i) they are of the same order (ii) each element of A is equal to the corresponding element of B.
15. Operations on Matrices
16. Addition of Matrices: The sum of two matrices is a matrix obtained by adding the corresponding elements of the given matrices.
17. Multiplication of a matrix by a scalar: if A = [aij]m × n is a matrix and k is a scalar, then kA is another matrix which is obtained by multiplying each element of A by the scalar k.
18. Negative of a matrix: –A = (–1) A
19. Difference of matrices: The difference between two matrices is a matrix obtained by adding the corresponding elements of the given matrices.

Commutative law

Associative law

Existence of identity

• Properties of scalar multiplication
• Multiplication of matrices
• Properties of multiplication of matrices
• Transpose of a Matrix
• Transpose of a matrix: If A is an m × n matrix, then the matrix obtained by interchanging the rows and columns of A is called the transpose of A.
• Symmetric and Skew Symmetric Matrix
• Symmetric Matrix: A square matrix A is said to be symmetric if A′ = A.
• Skew symmetric matrix: A square matrix A is said to be symmetric if A′ = -A.
• Elementary Operation of  a Matrix
• The interchanging of any two rows.
• The multiplication of the elements of any row by a non-zero scalar.
• The addition of the elements of any row, the corresponding elements of any other row multiplied by any non-zero number.
• Invertible Matrices
• Invertible matrices
• A rectangular matrix has no zero.
• If B is the inverse of A, then A is also the inverse of B.
• Inverse of a square matrix, if exists, is unique.
• Inverse of a matrix by elementary row transformations.

### NCERT Solutions for Class 12 Maths Chapter 4 – Determinants

1. Determinant
• Determinant: To every square matrix A = [aij] of order n, we can associate a number (real or complex) called the determinant of the square matrix A.
• Determinant of order 1
• Determinant of order 2
• Determinant of order 3
2. Properties of Determinant
• The value of the determinant remains unchanged if its rows and columns are interchanged.
• If any two rows or columns of a determinant are interchanged, then the sign of determinant changes.
• If any two rows or columns of a determinant are identical then the value of the determinant is 0.
• If each element of a row or column of a determinant is multiplied by a constant k then its value gets multiplied by k.
• If each element of a row or column of a determinant is expressed as a sum of two terms then the determinant can be expressed as the sum of two determinants.
• If each element of any row or column of a determinant, the equimultiplies of corresponding elements of another row (or column) are added, then the value of determinant remains the same.
• If all the elements of a row (or column) are zeros, then the value of the determinant is zero.
3. Area of a triangle:
• Area of a triangle
• Area is a positive quantity, so always consider the absolute value of the determinant.
• Area of the triangle formed by three collinear points is zero.
• Equation of line
4. Minors and Cofactors
• Minors
• Cofactors
• Minor of an element of a determinant of order n is a determinant of order n – 1.
• If elements of one row or column are multiplied with cofactors of elements of any other row or column, then their sum is zero.
5. Adjoint and Inverse of a Matrix
• Singular matrix
• Non-singular matrix
6. Applications of Determinants and Matrices
• Matrix method

### NCERT Solutions for Class 12 Maths Chapter 5 – Continuity and Differentiability

1. Continuity
• A real valued function is continuous at a point in its domain if the limit of the function at that point equals the value of the function at that point. A function is continuous if it is continuous on the whole of its domain.
• Algebra of continuous function: sum, difference, product, and quotient of continuous functions are continuous.
2. Differentiability
• Every differentiable function is continuous, but the converse is not true.
• Derivatives of composite functions
• Derivatives of implicit functions
• Derivatives of inverse trigonometric functions
3. Exponential and Logarithmic Functions
4. Logarithmic Differentiation: Logarithmic differentiation is a powerful technique to differentiate functions of the form f(x) = [u (x)]v(x). Here both f(x) and u(x) need to be positive for this technique to make sense.
5. Derivatives of Functions in Parametric Forms
6. Second Order Derivative
7. Mean Value Theorem
• Rolle’s Theorem
• Mean Value Theorem

### NCERT Solutions for Class 12 Maths Chapter 6 – Application of Derivatives

1. Rate of change of quantities
• Rate of change of quantities: the rate of change of y with respect to x can be calculated using the rate of change of y and that of x both with respect to t
2. Increasing and decreasing functions
• Increasing functions
• Strictly increasing functions
• Decreasing functions
• Strictly decreasing functions
• Neither increasing nor decreasing functions
3. Tangents and normal
• Slope and equation of tangent.
• Slope and equation of normal.
4. Approximations
• Approximations
5. Maxima and minima
• Without derivative test
• First derivative test
• Second derivative test
• Maximum and minimum values of a function in a closed interval.

### NCERT Solutions for Class 12 Maths Chapter 7 – Integrals

1. Integration as an Inverse Process of Differentiation
• Geometrical interpretation of indefinite integral: Integration is the inverse process of differentiation. In the differential calculus, we are given a function and we have to find the derivative or differential of this function, but in the integral calculus, we are to find a function whose differential is given. Thus, integration is a process that is the inverse of differentiation.
• Some properties of indefinite integral
• Comparison between differentiation and integration
2. Methods of Integration
• Integration by substitution: A change in the variable of integration often reduces an integral to one of the fundamental integrals. The method in which we change the variable to some other variable is called the method of substitution. When the integrand involves some trigonometric functions, we use some well-known identities to find the integrals
• Integration using trigonometric identities
3. Integrals of Some Particular Functions
• Integrals of some more types
4. Integration by Partial Fractions
5. Integration by Parts: integral of the product of two functions = first function × integral of the second function – integral of {differential coefficient of the first function × integral of the second function}.
6. Definite Integral
• Definite integral as the limit of a sum
7. Fundamental Theorem of Calculus
Area function
• First fundamental theorem of integral calculus:
• Second fundamental theorem of integral calculus

### NCERT Solutions for Class 12 Maths Chapter 8 – Application of Integrals

1. Area under simple curves
• The area of the region bounded by the curve y = f(x), x-axis and lines x = a and y = b.
• The area of the region bounded by the curve x = g(y), y-axis, and the lines y = c and y = d.
• If the position of the curve under the consideration is below the x-axis.
• If some portion of the curve is above the x-axis and some below the x-axis.
2. Area between two curves
• Area enclosed by curves y = f(x), y = g(x), and the lines x = a, x = b.

### NCERT Solutions for Class 12 Maths Chapter 9 – Differential Equations

1. Basic Concepts
• An equation involving derivatives of the dependent variable with respect to the independent variable (variables) is known as a differential equation.
• Order of a differential equation is the order of the highest order derivative occurring in the differential equation.
• Degree of a differential equation is defined if it is a polynomial equation in its derivatives.
• Degree (when defined) of a differential equation is the highest power (positive integer only) of the highest order derivative in it.
2. General and Particular Solutions of a Differential Equation
• A function that satisfies the given differential equation is called its solution. The solution which contains as many arbitrary constants as the order of the differential equation is called a general solution and the solution free from arbitrary constants is called a particular solution
3. Formation of a Differential Equation whose General Solution is given
• To form a differential equation from a given function we differentiate the function successively as many times as the number of arbitrary constants in the given function and then eliminate the arbitrary constants.
4. Methods of Solving First Order, First Degree Differential Equations
• Differential equations with variables separable
• Homogeneous differential equations
• Linear differential equations

### NCERT Solutions for Class 12 Maths Chapter 10 – Vector Algebra

1. Some Basic Concepts
A quantity that has magnitude as well as direction is called a vector
2. Types of Vectors
• Zero Vector: A vector whose initial and terminal points coincide, is called a zero vector.
• Unit Vector: A vector whose magnitude is unity (i.e., 1 unit) is called a unit vector.
• Coinitial Vector: Two or more vectors having the same initial point are called coinitial vectors.
• Collinear Vector: Two or more vectors are said to be collinear if they are parallel to the same line, irrespective of their magnitudes and directions
• Equal Vectors: Two vectors are said to be equal if they have the same magnitude and direction regardless of the positions of their initial points,
• Negative of a Vector: A vector whose magnitude is the same as that of a given vector, but the direction is opposite to that of it, is called negative of the given vector.
 Commutative property
 Associative property
4. Multiplication of a Vector by a Scalar
• Components of a vector
• Vector joining two point
• Section formula
5. Product of Two Vectors
• Scalar (or dot) product of two vectors
Properties of scalar product:
 Distributivity of scalar product over addition
• Projection of a vector on a line
• Vector (or cross) product of two vectors
6. Scalar Triple Product

### NCERT Solutions for Class 12 Maths Chapter 11 – Three Dimensional Geometry

1. Direction Cosines and Direction Ratios
• Direction cosine of a line: Direction cosines of a line are the cosines of the angles made by the line with the positive directions of the coordinate axes.
• Direction ratios of a line: Direction ratios of a line are the numbers that are proportional to the direction cosines of a line.
• Relation between the direction cosines of a line.
• Direction cosines of a line passing through two points.
2. Equation of a line in a space
• Equation of a line through a given point and parallel to a given vector.
• Equation of a line passing through two given points.
3. Angle between two lines
• Vector form
• Cartesian form
4. Shortest distance between two lines
• Distance between two skew lines
• Distance between two parallel lines
5. Plane
• Equation of a plane in normal form
• Equation of a plane perpendicular to a given vector and passing through a given point.
• Equation of a plane passing through three non-collinear points.
• Intercept form of the equation of the plane
• Plane passing through the intersection of two given planes
6. Coplanarity of two lines
• Co-planarity of two lines
7. Angle between two planes
• Vector form and Cartesian form
8. Distance of a point from a plane
• Vector form
• Cartesian form
9. Angle between a line and a plane
• Angle between a line and a plane

### NCERT Solutions for Class 12 Maths Chapter 12 – Linear Programming

1.  Linear programming problem and its mathematical formulation
• A linear programming problem is one that is concerned with finding the optimal value (maximum or minimum) of a linear function of several variables (called objective function) subject to the conditions that the variables are non-negative and satisfy a set of linear inequalities (called linear constraints). Variables are sometimes called decision variables and are non-negative.
• Graphical method of solving linear programming problems
2. Different types of linear programming problems
• Manufacturing problems
• Diet problems
• Transportation problems
• The common region determined by all the constraints including the non-negative constraints x ≥ 0, y ≥ 0 of a linear programming problem is called the feasible region (or solution region) for the problem.
• Points within and on the boundary of the feasible region represent feasible solutions to the constraints.
• Any point outside the feasible region is an infeasible solution.
• Any point in the feasible region that gives the optimal value (maximum or minimum) of the objective function is called an optimal solution.

### NCERT Solutions for Class 12 Maths Chapter 13 – Probability

1. Conditional Probability
• Conditional Probability
• Properties of conditional probability
• P(S|F) = P(F|F) =1
• If A and B are any two events of a sample space S and F is an event of S such that P(F) ≠ 0, then P((A ∪ B)|F) = P(A|F) + P(B|F) – P((A ∩ B)|F)
• P(E′|F) = 1 − P(E|F)
2. Multiplication Theorem on Probability
• Multiplication Theorem: Multiplication rule of probability for more than two events If E, F, and G are three events of sample space, we have P(E ∩ F ∩ G) = P(E) P(F|E) P(G|(E ∩ F)) = P(E) P(F|E) P(G|EF)
3. Independent Events
• Independent  events: Two events E and F are said to be independent, if P(F|E) = P (F)  provided P (E)  ≠ 0 and P(E|F) = P (E)  provided P (F)  ≠ 0
• Three events A, B, and C are said to be mutually independent, if P(A ∩ B) =P(A) P(B) P(A ∩ C) =P(A) P(C) P(B ∩ C) =P(B) P(C) and P(A ∩ B ∩ C) = P(A) P(B) P(C)
4. Baye’s Theorem
• Partition of a sample space: The events E1, E2, …, En represent a partition of the sample space S if they are pairwise disjoint, exhaustive, and have nonzero probabilities
• Theorem  of total probability
5. Random Variables and its Probability Distributions
• A random variable is a real-valued function whose domain is the sample space of a random experiment.
• Probability distribution of a random variable
• Mean  of a random variable
• Variance of a random variable
6. Bernoulli Trials and Binomial Distribution
• Bernoulli Trials: Trials of a random experiment are called Bernoulli trials, if they satisfy the following conditions: (i) There should be a finite number of trials. (ii) The trials should be independent. (iii) Each trial has exactly two outcomes: success or failure. (iv) The probability of success remains the same in each trial.
• Binomial Distribution

## Benefit of NCERT Solutions for Class 12 Maths

NCERT solutions for class 12 Maths will create better and effective learning for students. It will help in building a strong foundation of all the concepts for higher-level classes and also for competitive exams. NCERT class 12th Maths solutions will help students to clear their doubts by offering an in-depth understanding of the concepts. Through detailed explanations, students can learn the concepts which will enhance their thinking and learning abilities.

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### How Fliplearn’s NCERT solutions assist us in scoring good marks in the exam?

The content analysts at Fliplearn have designed the NCERT Solutions in accordance with the syllabus designed by the CBSE board. The essential explanation is provided for major points to make the concepts easier for the students while learning. NCERT solutions are designed with the aim of helping students to ace the exam without fear. The solutions mainly help students to improve their problem-solving abilities which are important for the exam.

### How many chapters are there in Class 12 Maths NCERT textbook?

The NCERT textbook of Class 12 Maths has 2 parts. Part 1 has chapters 1 to 6 whereas part 2 has chapters 7 to 13. So the chapters are – Matrices, Inverse Trigonometric Functions, Relations and Functions, Determinants, Applications of Derivatives, Continuity, and Differentiability, Applications of Integrals, Vector Algebra, Differential Equations, Three Dimensional Geometry, Probability, and Linear Programming.

### Is it required to solve each and every problem of NCERT Solutions for Class 12th Maths?

In the CBSE board, each and every problem irrespective of their understanding level are important for the exam. So the students are recommended to solve the NCERT textbook on a daily basis to gain a grip on the fundamental concepts. By regular practice, students will be able to analyze their areas of weakness and work on them for a better academic score. The step-wise explanation under each problem will help students to perform well in the annual exams.

### Can I get answers for miscellaneous exercises in the NCERT Solutions for Class 12 Maths?

The solutions for the miscellaneous exercise are available on Fliplearn’S website. Students can easily download the solutions PDF. The solutions are prepared in a systematic way based on the marks weightage for the concepts in accordance with the CBSE board. The level of these exercises is high to help the students prepare for the board exams with much confidence.

### NCERT Solutions for other Classes

NCERT Solutions is also available for other classes. Students preparing for class 6th, 7th, 8th, 9th, 10th, and 11th will find it the best way to score good marks. NCERT Solutions for class 6th, 7th, 8th, 9th, 10th, and 11th provides a plethora of study material and references that help students trust it the most.

## NCERT Solution for Class 12 Maths

 Addition 2020-21 Board CBSE Curriculum Subject Math PDF Download Yes Textbook Solutions Class 12 Total Chapters 13

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