NCERT Solutions Class 12 Maths
NCERT solutions for class 12 maths comprise all the answers to questions that are available in the NCERT class 12th textbook. These solutions can be used by kids as a reference if they get stuck while solving questions or can act as a source to clarify concepts. These NCERT solutions for class 12 maths are prepared with immense care by math experts to ensure that a student of any intelligence quota can easily understand the explanations.
Class 12 maths NCERT solutions are extremely helpful for students preparing for the 12th board exams as they give kids an idea of the type of questions that can appear in the examination. They are also very useful for competitive exams such as IIT-JEE, BITS Pilani, etc. CBSE NCERT solutions for class 12 maths help kids build an ironclad mathematical foundation that is also applicable to college-level questions.
NCERT Solutions Class 12 Maths PDF Download
The grades a student gets in the class 12 board exam act as a deciding factor for a child’s future and hence, following well-structured solutions is very important. The precise and brief explanation of class 12th NCERT solutions is available for free pdf download. The links are arranged according to chapters and are given below.
Class 12 Maths NCERT Solutions Chapter 1 to 13
By regularly practicing the solutions linked above a student can not only get an excellent score in the home and board exams but will also develop an in-depth knowledge of the subject that will be useful for him at the college level. Download the NCERT class 12 maths book given below and effectively prepare for any examination.
☛ Download Class 12 Maths NCERT Book
NCERT Solutions for Class 12 Maths - PART 1
NCERT Solutions for Class 12 Maths Chapter 1 - Relations and Functions
NCERT solutions for class 12 maths relations and numbers give a detailed and simplified explanation of types of relations and functions, binary functions, and certain operations that can be performed on the elements of a function.
| Class 12 Maths NCERT Solutions Chapter 1 Exercise PDFs |
|---|
NCERT Solutions Class 12 Maths Chapter 1 Study Tips
NCERT solutions class 12 chapter 1 can prove to be very confusing thus, kids can follow the tips to help with the learning process.
- Try to visualize mapping questions.
- Solve numerous problems till you do not get confident with them.
Class 12 Maths Chapter 1 Relations and Functions
Topics Covered: Class 12 Maths NCERT Solutions talks about empty, universal, reflexive, transitive relations along with injective, bijective, and surjective mapping of functions. Apart from this, questions based on binary operations, invertible elements, and commutative operations are also included in this chapter.
Total Questions: Class 12 maths chapter 1 relations and functions have a total of 74 exemplar problems out of which 45 are challenging sums, 20 are easy and 9 are medium.
NCERT Solutions Class 12 Maths Chapter 2 - Inverse Trigonometric Functions
NCERT solutions for class 12 maths inverse trigonometric functions introduce students to the domain and graphical representation of inverse trigonometric functions. Kids have the opportunity to learn about the properties of these functions and solve questions using the given concepts.
| Class 12 Maths NCERT Solutions Chapter 2 Exercise PDFs |
|---|
Important Formulas for NCERT Solutions Class 12 Maths Chapter 2
- sin–1 (–x) = – sin–1 x
- tan–1 x + cot–1 x = π/2
- sin–1 x + cos–1 x = π/2
- cos–1 (–x) = π – cos–1 x
- cot–1 (–x) = π – cot–1 x
Class 12 Maths Chapter 2 Inverse Trigonometric Functions
Topics Covered: Identifying the domain and finding the value of inverse trigonometric functions at a particular point, as well as simplifying trigonometric expressions using the properties of inverse trigonometry are very important topics under Class 12 Maths NCERT Solutions
Total Questions: Class 12 maths chapter 2 inverse trigonometric functions has a total of 52 questions out of which 20 are simple formula-based sums, 18 are moderate and 14 are tough problems.
NCERT Solutions Class 12 Maths Chapter 3 - Matrices
NCERT solutions for class 12 maths matrices help students amass the tools required to solve problems based on the elements of matrices. This chapter includes the various categories of matrices, how to add, subtract and multiply matrices, along with applying elementary operations to get the inverse of a matrix.
| Class 12 Maths NCERT Solutions Chapter 3 Exercise PDFs |
|---|
Important Formulas for NCERT Solutions Class 12 Maths Chapter 1
- – A = (–1)A
- A – B = A + (–1) B
- A + B = B + A
- (A + B) + C = A + (B + C)
- k(A + B) = kA + kB
where A, B and C are matrices.
Class 12 Maths Chapter 3 Matrices
Topics Covered: Class 12 Maths NCERT Solutions covers topics such as types of matrices, finding unknown quantities using equivalent matrices, performing arithmetic operations on matrices, as well as how to transpose matrices and subsequently find the inverse using different methods.
Total Questions: There are 78 questions in Class 12 maths chapter 3 matrices. 36 are complex sums, 14 are easy and 28 are intermediate problems.
NCERT Solutions Class 12 Maths Chapter 4 - Determinants
NCERT solutions for class 12 maths determinants clearly explain the concept of the determinant of a square matrix and how to find it. Properties of determinants, minors, cofactors, the inverse of a matrix, and linear equations are crucial sub-topics that kids need to have a good understanding of.
Class 12 Maths Chapter 4 Determinants
Topics Covered: Evaluating the determinants, proving expressions using the properties of determinants, finding the inverse of a matrix using minors, cofactors, and the adjoint of that matrix as well as finding the solution of a linear system of equations using matrices are topics covered under Class 12 Maths NCERT Solutions that are important for board exams.
Total Questions: Class 12 maths chapter 4 determinants have a pool of 87 fantastic questions. 17 are simple property-based questions, 30 are moderately difficult, and 40 conceptual sums meant to challenge a student’s thinking.
NCERT Solutions Class 12 Maths Chapter 4 Summary
The NCERT solutions class 12 maths chapter 4 builds on the previous chapter of matrices to include concepts of determinants, the inverse of a matrix, and other elements. The lesson then teaches children how to apply these properties learned to real-life examples.
NCERT Solutions Class 12 Maths Chapter 5 - Continuity and Differentiability
NCERT solutions for class 12 maths continuity and differentiability consist of well-structured explanations pertaining to establishing the continuity or differentiability of a particular function at a point. This topic is very important as it is required to improve a child’s understanding of calculus.
Important Formulas for NCERT Solutions Class 12 Maths Chapter 5
- Chain Rule: df/dv = (dt/dx) * (dt/dx)
- d/dx (sin-1x) = 1/√(1 - x2)
- d/dx (cos-1x) = - 1/√(1 - x2)
- d/dx (tan-1x) = 1/(1 + x2)
- d/dx (cot-1x) = 1/(1 + x2)
- d/dx (sec-1x) = 1/x √(1 - x2)
Class 12 Maths Chapter 5 Continuity and Differentiability
Topics Covered: Proving the continuity of a function, and performing algebraic operations on continuous functions, are topics covered under Class 12 Maths NCERT Solutions. Other subject matter included that is essential for exams are finding the derivatives of composite, implicit, trigonometric, exponential, and logarithmic functions, differentiation of functions in parametric form, second-order derivatives, mean value theorem as well as the chain rule for differentiation.
Total Questions: Class 12 maths chapter 5 continuity and differentiability has a total of 144 questions. 80 are easy formula-based sums, 35 are slightly more complicated, and 29 are difficult problems that require great use of a student’s acumen.
NCERT Solutions Class 12 Maths Chapter 6 - Application of Derivatives
NCERT solutions for class 12 maths applications of derivatives is a chapter that introduces the real-life utilization of derivatives in judging the rate of change of quantities, verifying the nature of a function, finding tangents to a function, and various other associated concepts.
Class 12 Maths Chapter 6 Application of Derivatives
Topics Covered: Class 12 Maths NCERT Solutions encapsulates topics such as finding the rate of change of a particular quantity, determining if a function is increasing or decreasing, detecting the slope of a tangent, and using differentiation to approximate a number. Additionally, this lesson also includes problems on finding the maxima and minima of a function along with establishing the results of the first and second derivative test.
Total Questions: There are 126 well-researched questions in the Class 12 maths chapter 6 application of derivatives. Out of these 40 are short answer type simple problems, 75 are of moderate complexity and 11 are tough sums.
NCERT Solutions for Class 12 Maths - PART 2
NCERT Solutions Class 12 Maths Chapter 7 - Integrals
NCERT solutions for class 12 maths integrals introduce students to the concept of the anti-derivative also known as an integral. Kids have the opportunity to learn about the geometric representation of integrals, and how to use various techniques with formulas to perform the integration of functions. In addition to this, children also study definite integrals.
Class 12 Maths Chapter 7 Integrals
Topics Covered: Class 12 Maths NCERT Solutions comprise topics such as finding the integral of a function using integration by substitution, partial fractions, parts, and by using trigonometric identities. Definite integrals, their limits, related properties, and the fundamental theorem of calculus are also part of this lesson.
Total Questions: Class 12 maths chapter 7 integrals has a pool of 271 questions. 115 are formula-based sums of a moderate level, 40 are easy and 116 are tough problems based on the mathematical skills of a student.
NCERT Solutions Class 12 Maths Chapter 7 Summary
The NCERT solutions class 12 maths chapter 7 gives a holistic view of integrals to students. There are several concepts such as integration by different methods, using integral formulas, and definite integrals that are very important from a board examination point of view. Thus, kids are advised to go through all concepts for a thorough understanding of integrals.
NCERT Solutions Class 12 Maths Chapter 8 - Application of Integrals
NCERT solutions for class 12 maths application of integrals aids students in understanding the practical use of integrals such as finding the area under a curve, or the area between a curve and a line segment. This chapter gives students a precise and comprehensive cognizance of calculus.
| Class 12 Maths NCERT Solutions Chapter 8 Exercise PDFs |
|---|
Class 12 Maths Chapter 8 Application of Integrals
Topics Covered: Class 12 Maths NCERT Solutions explains in detail topics such as area bound by the curve of a function, area between two curves, and other associated miscellaneous examples.
Total Questions: Class 12 maths chapter 8 application of integrals has 39 problems. 15 are easy, 8 are of intermediate level and 16 are tough sums.
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
NCERT solutions for class 12 maths differential equations help kids build crystal clear concepts of how to work with differential equations. Students learn about the order, degree, and solutions of a differential equation, as well as the procedure of forming such an equation.
NCERT Solutions Class 12 Maths Chapter 9 Study Tips
NCERT solutions class 12 maths chapter 9 can help kids combat confusion and strive to clear concepts of calculus. Students can follow the study tips mentioned below to streamline their learning process.
- Ensure that you are well versed with the previous chapters before studying differential equations.
- Learn formulas and make notes.
- Practice regularly.
Class 12 Maths Chapter 9 Differential Equations
Topics Covered: Determining the order and degree of an equation, verifying the solution of a differential equation, and forming such an equation from the general solution, are covered under the Class 12 Maths NCERT Solutions. Solving a homogeneous and linear differential equation are other topics in this lesson that are equally important to learn.
Total Questions: There are 113 sums in Class 12 maths chapter 9 differential equations of a wide-ranging variety. 72 are tough problems, 15 are simple and 26 are of a medium level.
NCERT Solutions Class 12 Maths Chapter 10 - Vector Algebra
NCERT solutions for class 12 maths vector algebra have a step-wise elucidation of problems based on vectors and their algebra. Students study how to perform dot and cross products, arithmetic vector operations, along with other vital topics that are important not only for Mathematics but also Physics.
| Class 12 Maths NCERT Solutions Chapter 10 Exercise PDFs |
|---|
Class 12 Maths Chapter 10 Vector Algebra
Topics Covered: Addition, multiplication, and laws of vector operations, section formula, computing the magnitude of vectors, vector projections, as well as dot and cross product, are topics that form the building blocks of Class 12 Maths NCERT Solutions.
Total Questions: Class 12 maths chapter 10 vector algebra has 63 sums divided into 20 straightforward questions, 37 intermediate, and 6 complicated problems.
NCERT Solutions Class 12 Maths Chapter 11 - Three Dimensional Geometry
NCERT solutions for class 12 maths three dimensional geometry builds on previous lessons for introducing kids to direction cosines, equations and other geometrical aspects related to a line. This chapter also helps to clarify concepts of a plane
| Class 12 Maths NCERT Solutions Chapter 11 Exercise PDFs |
|---|
Important Formulas for NCERT Solutions Class 12 Maths Chapter 11
- Direction cosines of a line joining two points A and B:
(x2 - x1)/AB,(y2 - y1)/AB,(z2 - z1)/AB
where AB = √[(x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2]
- cosθ = |[a1a2 + b1b2 + c1c2] / √(a1 + b1 + c1)2√(a2 + b2 + c2)2]|
where a1, b1, c1 and a2, b2, c2 are direction ratios and θ is the acute angle between the two lines.
Class 12 Maths Chapter 11 Three Dimensional Geometry
Topics Covered: Class 12 Maths NCERT Solutions includes topics pertaining to finding the direction cosines of a line, determining the distance and angle between two lines, as well as writing the equation of a plane when different constraints are given. Moreover, the distance and angle between two planes together with the coplanarity of two lines are also covered in this chapter.
Total Questions: Class 12 maths chapter 11 three dimensional geometry has 59 fantastic sums that can be split into 34 easy formula-based problems, 12 complex, and 13 moderate level questions.
NCERT Solutions Class 12 Maths Chapter 12 - Linear Programming
NCERT solutions for class 12 maths linear programming is one of the most essential chapters that is widely used in industrial applications. Students learn how to mathematically formulate a linear programming problem, and its real-life applications.
| Class 12 Maths NCERT Solutions Chapter 12 Exercise PDFs |
|---|
Class 12 Maths Chapter 12 Linear Programming
Topics Covered: Solving linear programming problems graphically, maximizing and minimizing a given equation, as well as using these concepts for various applications such as diet and manufacturing questions fall under the scope of Class 12 Maths NCERT Solutions.
Total Questions: There are 31 questions in Class 12 maths chapter 12 that are necessary for board examinations. Out of these 14 are simple, 10 are slightly tougher and 7 are higher-order thinking questions.
NCERT Solutions Class 12 Maths Chapter 12 Summary
NCERT solutions class 12 maths chapter 12 give various methods to solve and analyze linear equations. Problems related to maximizing or minimizing an equation and then marking the required area on a graph helps students to understand the practical implications of that question. Children also grasp the various techniques that are used to solve a linear equation.
NCERT Solutions Class 12 Maths Chapter 13 - Probability
NCERT solutions for class 12 maths probability enhances a student’s skills to solve probability sums by giving a well-detailed explanation of conditional probability, probability distribution, and many other concepts that are required by kids to successfully attempt their test paper.
Class 12 Maths Chapter 13 Probability
Topics Covered: Class 12 Maths NCERT Solutions encompass topics like conditional probability and its properties, applying the multiplication theorem on probability to sums, independent events, Bayes’ theorem, probability distribution, mean and variance of a random variable, Bernoulli trials, as well as binomial distribution.
Total Questions: Class 12 maths chapter 13 probability has 100 well-curated questions. 23 are simple sums based on formulas, 37 are moderate and 40 are tricky problems.
Important Formulas in NCERT Solutions Class 12 Maths
Formulas are a generalized representation of the relationship between two or more given quantities. It includes various theorems and expressions used to deliver information in a concise manner. They can be seen as vital keys that are required to solve questions. Some of the important formulas covered in NCERT solutions for class 12 maths that students must have knowledge of are given below.
Inverse Trigonometry
- sin-1(1/x) = cosec-1x
- cos-1(1/x) = sec-1x
- sin-1(-x) = -sin-1x
- cos-1(-x) = π - cos-1x
- sin-1x + cos-1x = π/2
Derivatives
- d/dx (sin-1x) = 1/√(1 - x2)
- d/dx (cos-1x) = - 1/√(1 - x2)
- d/dx (ex) = ex
Importance of NCERT Solutions Class 12 Maths
Class 12 Maths NCERT Solutions are extremely helpful for CBSE students because they provide a reliable way for kids to cross-check their answers and clear doubts. The following are the advantages of the NCERT solutions.
- Board exam preparation - While appearing for board exams it becomes essential for students to write their papers in a format that can be easily comprehended by the people who correct them. The NCERT solutions have a clear and step-wise explanation that gives kids an insight into how to solve any sum. As a certain set of marks are allotted to steps, hence if children follow the pattern of these NCERT solutions they are sure to get the best grade possible.
- Conceptual Understanding - The NCERT Class 12 Maths Solutions help students to clear their conceptual doubts. The topics in class 12 are not only important for passing exams but also lay the foundation for studies in college. Therefore, it is imperative that kids refer to these NCERT solutions when they hit roadblocks so as to develop the correct understanding of these topics.
- Revision - During the examination days, children have a very cramped up studying schedule and cannot solve all coursebook questions. In such a situation, the NCERT Class 12 Maths Solutions helps to give the students a quick overview of how to solve different kinds of problems and can help in saving a lot of time.
FAQs on NCERT Solutions for Class 12 Maths
Why are NCERT Solutions for Class 12 Maths Important?
Getting a good score in class 12th is crucial as this can be the deciding factor for a child’s career. Hence, it is important to consult the NCERT solutions class 12 maths as they give the most reliable, and to-the-point explanation of a question ensuring that kids do not have any doubts regarding a topic.
How CBSE Students can utilize Class 12 Maths NCERT Solutions effectively?
Before attempting any problem kids must be well versed with the theory and the formulas. They can then use the class 12 maths NCERT solutions to strengthen their understanding of that topic. Students must also structure their lesson plans in such a way that they can revise the entire syllabus at least twice so as to get the maximum benefit from these solutions.
Do I Need to Make Notes while Referring to NCERT Solutions Class 12 Maths?
When students solve a particular question they might come up with certain pointers detailed in notes that can supplement their learnings. Thus, it is always advised to make well-written notes for NCERT solutions class 12 maths to aid kids in recalling the concepts given in that specific sum.
Do I Need to Practice all Questions Provided in NCERT Solutions Class 12 Maths?
Constant practice and due diligence ensure success. Thus, kids must make it a point to practice all questions in the NCERT solutions class 12 maths. This not only helps them to build a robust mathematical foundation but also enables them to gain confidence and attempt questions with ease.
Where Can I Get Chapter-wise NCERT Solutions for Class 12 Maths?
The links provided above give students chapter-wise access to NCERT solutions for class 12 maths. Moreover, kids can also find the solutions to all the exercises that have been given by experts with certain tips and tricks to score the highest marks possible.
Why Should I Practice Class 12 Maths NCERT Solutions Regularly?
In order to master any subject, practice is key. Regularly revising NCERT solutions class 12 maths helps students to get a clearer perspective of a topic enabling them to apply concepts proficiently. As students will be well-versed with all the chapters they can maximize the probability of getting an excellent score in their board examination.
What are the Important Topics Covered in NCERT Solutions Class 12 Maths?
All topics such as probability, linear programming, application of derivatives, etc. given in the NCERT solutions class 12 maths are equally important for board examinations. The syllabus of these exams is based on the NCERT curriculum. Thus, it is vital for students to study each and every topic with immense care so as to successfully solve all problems given in the test papers.
Are NCERT Solutions Class 12 Maths Sufficient for CBSE Boards Exams?
NCERT solutions class 12 maths are sufficient for board exams as the paper pattern follows the structure set by these textbooks. However, it is best to consult more subject material based on the same topics to have a broader scope of learning. It is always better to be over-prepared as opposed to being under-prepared.
Why Choose Cuemath for Class 12 Maths NCERT Solutions?
Cuemath is the forerunner for class 12 maths NCERT solutions. These solutions are prepared by math geniuses who are IIT graduates. The methods used to solve questions are brief, accurate, and simple to understand, therefore, helping students to combat any doubts or confusions.
What are the Benefits of Class 12 Maths NCERT Solutions for CBSE Students?
The class 12 maths NCERT solutions give students an insight into how to give a step-by-step detailed solution while attempting a problem. This ensures that kids have written their examination in a clear manner thus, improving their chances of getting better marks.
visual curriculum