NCERT Solutions for Class 12 Maths

NCERT Solutions for Class 12 Maths can help you solve challenging problems during homework or while preparing for Class 12 Board exams. At Goprep, you can find solutions to all the questions for all exercises present in the NCERT textbook. Our Class 12 Maths NCERT Solutions Part-1 have been solved by subject matter experts who have years of experience in teaching mathematics.

Our NCERT Solutions for Class 12 are not only easy but also provide a step-by-step procedure to solve the questions quickly. With the help of these solutions, problems from topics including, Integration, Vectors, Differentiation, Limits & Derivatives will become much simpler.

So, to give your Maths exam preparation a competitive edge, click the links given below to browse CBSE Class 12 Maths book questions and their solutions for each chapter.

NCERT Solutions for Class 12 Maths - All Chapters

• NCERT Class 12 Maths Solutions (Part 2)

Developed by some of the best teachers, our NCERT Maths Solutions for Class 12 Part-1 help students grasp complex Mathematical problems and concepts better and faster. NCERT solutions can help you develop a deeper understanding of the concepts. Moreover, these solutions give in-depth and detailed answers to problems that are part of the Class 12th syllabus.

The Class 12th board exams play an important role in helping students secure admission in a good college. Further, these solutions have been prepared to enable students to fare well in board exams as well as JEE, BITSAT, VITEEE, and others.

While preparing for Class 12 Maths Board exam, you must practise Maths previous year question papers. Check out the links for CBSE Class 12 Maths Solved Question Papers below:

NCERT Solutions for Class 12 Maths (Chapter-wise description) 

Chapter 1: Relations and Functions

You have already studied the basic concepts of relations and functions, domain, co-domain, and range previously in class 9 NCERT maths book. In class 12, we will delve even deeper into the chapter. This year, the main features of the chapter will comprise different types of relations such as empty relation, universal relation, reflexive relation, to name a few. 

Further, the chapter will entail several functions such as injective (one-one), surjective (onto), and bijective (one-one and onto). Finally, you will also learn about the composition of functions. With a total of 51 examples, four exercises, and a miscellaneous exercise, you will get to solve a variety of questions. 

Topics

• Types of Relations
• Types of Functions
• Composition of Functions and Invertible Function
• Binary Operations

Chapter 2: Inverse Trigonometric Functions

Having studied the inverse of a function and one-one and onto functions in chapter 1, you will have to implement these concepts in inverse trigonometric functions. These trigonometric functions will come to play in the future when you solve the questions of integration. 

Finding the domains and ranges of inverse trigonometric functions will be crucial to get its principal values. Next, you will learn the properties of these functions, which will help you in simplifying the complex inverse trigonometric expressions. This chapter includes a total of 13 examples, two exercises and a miscellaneous exercise. 

Topics

• Basic Concepts
• Properties of Inverse Trigonometric Functions

Important formulas of Inverse Trigonometric Functions

For suitable values of domain, we have 

• y = sin^–1 x ⇒ x = sin y
• x = sin y ⇒ y = sin^–1 x
• sin (sin^–1 x) = x
• sin^–1 (sin x) = x
• sin^–1 (1/x) = cosec^–1 x
• cos^–1 (-x) = π – cos^–1 x
• cos^–1 (1/x) = sec^–1 x
• cot^–1 (-x) = π – cot^–1 x
• tan^–1 (1/x) = cot^–1 x
• sec^–1 (-x) = π – sec^–1 x
• sin^–1 (-x) =– sin^–1 x
• tan^–1 (-x) =– tan^–1 x
• tan^–1 x + cot^–1 x = π/2
• cosec^–1 (-x) =– cosec^–1 x
• tan^–1 x + tan^–1 y = tan^–1 [(x + y)/ (1-xy)]
• sin^–1 x + cos^–1 x = π/2
• cosec^–1 x + sec^–1 x = π/2
• 2tan^–1 x = sin^–1 [(2x/ 1+x²)] = cos^–1[(1 - x²/ 1 + x²)
• 2tan^–1 x = tan^–1 [2x/ (1-x²)]
• tan^–1 x + tan^–1 y = π + tan^–1 [(x + y)/ (1 - xy)], xy>1; x, y> 0

Chapter 3: Matrices

Matrices are a powerful tool of mathematics as it can help you represent the coefficients in a system of linear equations and utility of matrices. The knowledge of matrices is necessary as it will be required in various branches of mathematics in higher classes. Also, it will play a pivotal role in helping you solve questions in chapter 4 of class 12 maths NCERT book. 

You will be surprised to know that this mathematical tool has found its applications in the area of economics, genetics, industrial management, modern psychology, and sociology. In class 12, you will get acquainted with the core concepts of matrix and matrix algebra. 

Topics

• Matrix
• Types of Matrices
• Operations on Matrices 
• Transpose of a Matrix 
• Symmetric and Skew Symmetric Matrices
• Elementary Operation (Transformation) of a Matrix
• Invertible Matrices

Important formulas of Matrices

• A = [a_ij]_{m × m} is a diagonal matrix if a_ij = 0, when i ≠ j

• A = [a_ij]_{n × n} is a scalar matrix if a_ij = 0, when i ≠ j, a_ij = k, (k is some constant), when i = j

• A = [a_ij]_{n × n} is a scalar matrix if a_ij = 0, when i ≠ j, a_ij = k, (k is some constant), when i = j

• A = [a_ij] = [b_ij] = B if (i) A and B are of the same order, (ii) a_ij = b_ij for all possible values of i and j

• kA = k[a_ij]_{m × n} = [k(a_ij)]_{m × n}

• – A = (–1)A

• A + B = B + A

• A – B = A + (–1) B

• (A + B) + C = A + (B + C), where A, B and C are of the same order

• k(A + B) = kA + kB, where A and B are of the same order, k is constant

• (k + l) A = kA + lA, where k and l are constant

• If A = [a_ij]_{m × n} and B = [b_jk]_{n × p}, then AB = C = [c_ik]_{m × p}

Some more properties

• A(BC) = (AB)C
• A(B + C) = AB + AC
• (A + B)C = AC + BC
• (A′)′ = A
• (kA)′ = kA′
• (A + B)′ = A′ + B′
• (AB)′ = B′A′

Chapter 4: Determinants

In the previous chapter, we found that we can express the system of algebraic equations in the form of matrices. Proceeding further, we will learn about determinants having an order of 1, 2 and 3. 

Later, we will delve deeper into this chapter and discuss various properties of determinants. The key learning of this chapter will include topics such as minors, cofactors, and applications of determinants. Nevertheless, determinant is one of the most scoring chapters of NCERT class 12 maths. 

Topics

• Determinant
• Properties of Determinants
• Area of a Triangle
• Minors and Cofactors 
• Adjoint and Inverse of a Matrix
• Applications of Determinants and Matrices

Important formulas of Determinants

1. Determinant of a matrix A = [a₁₁ ]_{1× 1} is given by |a₁₁ | = a₁₁

2. The value of determinant of a matrix A can be obtained by adding product of elements of a row with corresponding cofactors. 

    A = a₁₁ A₁₁ + a₁₂ A₁₂ + a₁₃ A₁₃

3. A (adj A) = (adj A) A = |A| I, where A is a square matrix of order n

4. A^-1 = 1 (adj A) / |A|

5. If a₁ x + b₁ y + c₁z = d₁

    a₂x + b₂y + c₂z = d₂,

    a₃x + b₃y + c₃z = d₃,

Then these equations can be written as AX = B

6. Unique solution of equation AX = B is given by X = A^–1 B, where |A| ≠ 0.

Chapter 5: Continuity and Differentiability 

We have already studied the differentiation of several certain functions, including polynomial and trigonometric functions. The chapter-5 continuity and differentiability will introduce you to its various core concepts. 

Further, you will learn to differentiate inverse trigonometric, exponential, and logarithmic functions. There are two important theorems based on experiments performed by mathematicians- Rolle’s theorem and the mean value theorem.&nbsp

Topics

• Continuity
• Differentiability
• Exponential and Logarithmic Functions
• Logarithmic Differentiation
• Derivatives of Functions in Parametric Forms
• Second Order Derivative
• Mean Value Theorem

Important formulas of Continuity and Differentiability 

1. Sum/ Difference of continuous functions

If f and g are two continuous functions, then

(f ± g) (x) = f(x) ± g(x) is continuous

2. Product of continuous functions

If f and g are two continuous functions, then

(f . g) (x) = f(x) . g (x) is continuous

3. Quotient of continuous functions

If f and g are two continuous functions, then

(f/ g) (x) = f (x)/ g (x), (wherever g(x) ≠ 0) is continuous

4. Standard derivatives

• d/dx (sin^-1 x) = 1/ √1 - x² 
• d/dx (cos^-1 x) = - 1/ √1 - x² 
• d/dx (tan^-1 x) =  1/ √1 + x² 
• d/dx (cot^-1 x) =  - 1/ √1 + x²
• d/dx (sec^-1 x) =  1/ x√1 - x² 
• d/dx (cosec^-1 x) = - 1/ x√1 - x² 
• d/dx (e^x)= ex
• d/dx (log x) = 1/x

Chapter 6: Application of Derivatives

The concept of differentiation has its applications in various disciplines, for example, in engineering, science, social science etc. Having a prior knowledge on how to find the derivative of composite, exponential, inverse trigonometric, implicit, and logarithmic functions will allow you to solve questions of this chapter easily. 

In this chapter, you will learn how the derivatives can be used to determine the rate of change of quantities, to find turning points on the graph etc. Finally, you will be able to find the approximate value of certain quantities using the derivatives. 

Application of Derivatives is considered one of the most challenging chapters of this book. At times, you may get stuck in a few questions, but we have you covered with our NCERT class 12 maths solutions. 

Topics

• Rate of Change of Quantities
• Increasing and Decreasing Functions
• Tangents and Normals
• Approximations
• Maxima and Minima

Important formulas of Application of Derivatives

1. If two variables x and y are varying with respect to another variable t, i.e., if x = f (t) and y = g (t), then by the Chain Rule

dy/dx = (dy/ dt) / (dx/dt), if dx/ dt ≠ 0


2. 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 an interval (a,b) if

x₁ < x₂ in (a, b) ⇒ f(x₁) > f(x₂ ) for all x₁ , x₂ ∈ (a, b)

c). Constant on an interval (a,b) if

F (x) = c for all x ∈ (a, b), where c is a constant


3. The equation of the tangent at (x₀, y₀) to the curve y = f (x) is given by

y - y₀ = dy/dx]_{x0, y0} (x - x₀)


4. Equation of the normal to the curve y = f (x) at a point (x₀, y₀) is given by

y - y₀ = -1 (x- x₀) / (dy/ dx)] _{x0, y0}

5. Assume y = f (x), ∆x be a small increment in x and ∆y be the increment in y corresponding to the increment in x, i.e.,

 ∆y = f (x + ∆x) - f (x)

Then dy is given by

dy = f’ (x)dx or

dy = (dy/ dx) ∆x

Chapter 7: Integrals

One can say that chapter-7 integration is centred on the concept of differentiation. Integration, also known as integral calculus, will allow you to define and calculate the area of the region bounded by the graphs of the functions. 

Further, you will also learn a few standard integral formulas and formula that gives the anti-derivatives. In a few situations, where the degree of one polynomial is greater than the degree of another polynomial, then you will have to find the integral using partial fractions. While in questions where two functions are given, you will have to integrate by parts. 

As per the past trend, this chapter continued to have the highest weightage among all the chapters. It is recommended to solve as many questions as you can from this chapter to excel in the class 12 maths board exam.  

Topics

• Integration as an Inverse Process of Differentiation
• Methods of Integration
• Integration by Partial Fractions
• Integration by Parts
• Definite Integral
• Fundamental Theorem of Calculus
• Evaluation of Definite Integrals by Substitution
• Some Properties of Definite Integrals

Important formulas of Integrals

1. Indefinite integrals

Assume d/ dx F(x) = f (x), then

∫ f (x) dx = F (x) + C

2. Properties of indefinite integrals

• ∫ [f (x) + g(x)] dx = ∫ f (x) dx + ∫ g (x) dx 
• For any real number k, ∫ k f (x) dx = k ∫ f (x) dx

3. Some standard integrals

• ∫ xⁿ dx = {x^{(n +1)}/ (n + 1)} + C, n ≠ 1
• ∫ cos x dx = sin x + C
• ∫ sin x dx = - cos x + C
• ∫ sec² x dx = tan x + C
• ∫ cosec² x dx = - cot x + C
• ∫ sec x tan x dx = sec x + C
• ∫ cosec x cot x dx = - cosec x + C
• ∫ dx/ √1 - x² = - sin^-1 x + C
• ∫ dx/ √1 + x² = - cot^-1 x + C
• ∫ dx/ 1 + x² = tan^-1x + C
• ∫ dx/ 1 + x² = - cot^-1 x + C
• ∫ a^x dx = a^x / log a + C
• ∫ e^x dx = e^x + C
• ∫ dx/ (x √x² - 1) = sec^-1 x + C
• ∫ dx/ (x √x² - 1) = - cosec^-1 x + C
• ∫ (1/x) dx = log |x| + C

4. Integration by partial fractions

• (px + q) / (x - a) (x - b) = [A/ (x-a)] + [B/ (x-b)],  a ≠ b
• (px + q) / (x - a)² = [A/ (x-a)] + [B/ (x-a)²]
• (px² + qx + r)/ (x - a) (x - b) (x - c) = [A/ (x-a)] + [B/ (x-b)] + [C/ (x-c)]
• (px² + qx + r)/ (x - a)² (x - b) = [A/ (x-a)] + [B/ (x-a)²] + [C/ (x-b)]
• (px² + qx + r)/ (x - a) (x² + bx + c) =  [A/ (x-a)] + [(Bx + C)/ (x² + bx + c)]

5. Integration by substitution

• ∫ tan x dx = log | sec x | + C
• ∫ cot x dx = log | sin x | + C
• ∫ sec x dx = log | sec x + tan x | + C
• ∫ cosec x dx = log | cosec x - cot x | + C

6. Integrals of some special functions

• ∫dx/ x²- a² = (1/2a) log |(x - a)/ (x + a)| + C
• ∫dx/ (a²- x²) = (1/2a) log |(a + x)/ (a - x)| + C
• ∫dx/ (a² + x²) = (1/ a) (tan^-1 x/ a) + C
• ∫dx/ √(x² - a²) = log | x + √ x² - a²| + C 
• ∫dx/ (a²- x²) = sin^-1 (x/a) + C
• ∫dx/ (a² + x²) = log | x + √ x² + a²| + C 

7. Integration by parts

• ∫ f₁ (x). f₂ (x) dx = f₁ (x) ∫ f₂ (x) dx - ∫ [d/ dx (f₁ (x). ∫ f₂ (x) dx] dx
• ∫ e^x [f (x) + f’ (x)] dx = ∫ e^x f(x) dx + C   

Chapter 8: Application of Integrals

In previous classes, you learnt to calculate areas of various geometrical figures, including circles, rectangles, triangles, and trapeziums. However, those cannot be applied to calculate the areas enclosed by curves. To find the areas of such shapes, you will need some concepts of integral calculus. 

This chapter becomes automatically easier when you have already mastered the previous chapter. You can look into our NCERT class 12 maths solutions if you fail to find an accurate answer for any question of the textbook. 

Topics

• Area under Simple Curves
• Area between Two Curves

Important formulas of Application of Integrals

1. The area of the region covered by the curve y = f (x), x-axis and the lines x = a and x = b (b > a) is represented by the following formula

Area = ∫_a^b ydx = ∫_a^b f (x) dx

2. The area of the region covered by the curve x = ɸ (y), y-axis and the lines y = c and y = d is represented by the following formula

Area = ∫_c^d x dy = ∫_c^d ɸ (y) dy

Chapter 9: Differential Equations

Chapter-9 Differential Equations involves the combined use of formulas from differentiation and integration. Previously, you learnt to differentiate a given function and find the antiderivatives using Integral Calculus. 

In this chapter, you will learn some basic concepts of differential equations, the formation of differential equations etc. 

Topics

• Basic Concepts
• General and Particular Solutions of a Differential Equation
• Formation of a Differential Equation 
• Methods of Solving First Order, First Degree Differential Equations

Chapter 10: Vector Algebra

In physics, we first heard of the concept called vectors. We remember that a vector quantity represents magnitude as well as the direction. Examples of vector quantities are displacement, acceleration, velocity, momentum, force, etc. 

In this chapter, you will learn the concept of vector quantities from scratch. During your study, you will learn the basics of vectors, various operations on vectors, and their algebraic and geometric properties. 

Topics

• Some Basic Concepts
• Types of Vectors
• Addition of Vectors
• Multiplication of a Vector by a Scalar
• Product of Two Vectors

Chapter 11: Three Dimensional Geometry

3-D Geometry is to vectors what application of integrals is to integration. Having said that the concepts that you learned in the previous chapter will come to play in this chapter. 

In three-dimensional geometry, you will study the direction cosines and direction ratios of a line joining two points, angles between two lines, etc. Other important topics include study about a line and a plane, shortest distance between two skew lines and distance of a point from a plane. 

Topics

• Direction Cosines and Direction Ratios of a Line
• Equation of a Line in Space
• Angle between Two Lines
• Shortest Distance between Two Lines
• Plane
• Co-planarity of Two Lines
• Angle between Two Planes
• Distance of a Point from a Plane
• Angle between a Line and a Plane

Important formulas of Three Dimensional Geometry

1. Direction cosines of a line joining two points P (x₁, y₁, z₁) and Q (x₂, y₂, z₂) are

(x₂ - x₁)/ PQ, (y₂ - y₁)/ PQ, (z₂ - z₁)/ PQ

Where PQ = √ (x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²

2. If a, b and c are the direction ratios and l, m and n are the direction cosines of a line then

• l = a/ √ a² + b² + c²
• m = b/ √ a² + b² + c²
• n = c/ √ a² + b² + c²

3. Angle between the two lines whose direction cosines are (l₁, m₁, n₁) and (l₂, m₂, n₂) is given by 

 Cos θ = | (a₁ a₂ + b₁ b₂ + c₁ c₂) / (√ a₁² + b₁² + c₁². √ a₂² + b₂² + c₂²)|

4. Equation of a line through a point (x₁, y₁, z₁) and have direction cosines l, m and n can be represented as

(x - x₁)/ l  = (y - y₁)/ m  = (z - z₁)/ n

5. Cartesian equation of a line passing through two points (x₁, y₁, z₁) and (x₂, y₂, z₂) can be represented as 

(x - x₁)/ (x₂ - x₁)  = (y - y₁)/ (y₂ - y₁)   = (z - z₁)/ (z₂ - z₁) 

6. If equations of two lines are given as

(x - x₁)/ l₁  = (y - y₁)/ m₁  = (z - z₁)/ n₁ and (x - x₁)/ l₂  = (y - y₂)/ m₂  = (z - z₂)/ n₂ then

Acute angle between the two lines is Cos  θ = |l₁ l₂ + m₁m₂ + n₁n₂|

Chapter 12: Linear Programming

You should be well-versed with the systems of linear equations and their applications in day-to-day problems before you start this chapter. In class 12, you will learn to apply the systems of linear inequalities to solve real-life problems. Linear programming problems has its applications in commerce, industry, management science etc. 

Topics

• Linear Programming Problem and its Mathematical Formulation
• Different Types of Linear Programming Problems

Chapter 13: Probability

In earlier classes, you have learnt the fundamental concepts of probability. Also, you came to know about how to establish equivalence between the axiomatic theory and the classical theory of probability. In class 12, you will get to know the important concept of the conditional probability of an event given that another event has occurred. 

Further learning of the chapter will be centred on Bayes’ theorem, multiplication rule of probability, and independence of events. 

Topics

• Conditional Probability
• Multiplication Theorem on Probability
• Independent Events
• Bayes' Theorem
• Random Variables and its Probability Distributions
• Bernoulli Trials and Binomial Distribution

Important formulas of Probability

1. Theory of total probability

P(A) = P(E1 ) P (A|E1 ) + P (E2 ) P (A|E2 ) + ... + P (E_n ) P(A|E_n )

2. Bayes’ theorem

P (E_i | A) = [P (E_i) P(A|E_i)] / ∑ P (E_j) P(A|E_j)

3. Standard deviation of the random variable X

𝛔_x = √ Var (X) = √ ∑ (x_i - 𝜇)² p (x_i)

CBSE Class 12 Maths Syllabus 2019-20

As per the latest 12th Class Maths Syllabus, below we have provided the list of units and their chapters. 

CBSE Paper Pattern for Class 12 Maths 2020

The Central Board of Secondary Education has introduced a few major changes in the CBSE Paper Pattern for Class 12 Maths. As per the revised CBSE Syllabus, students will now face questions having analytical and reasoning perspective. This step will allow students to focus on understanding the concepts instead of rote learning. 

Best Reference Books for CBSE Class 12 Maths

Starting your preparation from Maths NCERT Class 12 book is a common practice for all. It can help you score over 70 marks easily without putting extra efforts. However, if you wish to score over 80 marks then we highly recommend you to practice questions from Class 12 Maths reference books. The list of best books for Class 12 Maths is tabulated below. 

CBSE Class 12 Maths Marking Scheme (Chapter-wise)

Before starting the Maths exam preparation, you should go through the CBSE marking scheme for Class 12 Maths, which is tabulated below. 

List of chapters related to Algebra in Class 12 NCERT Maths

The use of algebraic expressions and equations is prevalent in Class 12 Maths textbook. Check the list of chapters along with a brief explanation that involves the use of algebra in solving questions.

Benefits of CBSE Class 12 Maths NCERT Solutions

Students often panic before the Maths exam and emphasize more on reference books instead of NCERT Books. They are unaware that the most effective study resource is lying in their cupboard. Coupled with our free NCERT Solutions, you have all it takes to score high marks in CBSE Class 12 Maths exam. 

Apart from it, there are various other benefits associated with Class 12 Maths NCERT Solutions that we have listed below. 

• Whether a pending assignment, an upcoming unit test or CBSE Class 12 Maths Board exam, our NCERT Solutions for Class 12 Maths can help you do a quick revision.
• These solutions are explained in a step-by-step manner that makes them easier to understand. 
• You can rely on Goprep’s NCERT Solutions as our experts have years of experience as subject mentors. 
• Ask doubts from our experts by uploading the screenshot of the question by clicking on the ‘camera’ icon present on the homepage. 
• When you read these solutions, you will end up covering all the core concepts related to a topic.