NCERT solutions for class 12 Maths
Class 12^{th} maths is already considered one of the advanced mathematics. The team at Physics Wallah has come forward to provide NCERT Solutions for class 12 maths. The complete study material has been prepared as per the latest CBSE guidelines. This study material in overall assists your progress to attain desired marks. Do solve all questions given in Physics Wallah Important questions for class 12 Maths.
Scoring optimum marks is crucial is this stage of life. Since marks scored in class 12^{th} maths would forge your future steps. On the other hand, maths holds a significant position and marks in all the competitive examination. Students looking forward to progress in maths area would require maths as hand in hand relation with NCERT Solutions for class 12 maths.
NCERT Solutions for Class 12 Maths Chapters (Updated 2022-23)
NCERT Solutions for Class 12 Maths Free PDF Download
Chapter 1 Relations and Functions
NCERT textbook Chapter 1 deals with relations and functions, types of relations, types of functions, the composition of functions and invertible functions, binary operations, and various examples. You can find links to exercise Solutions for the topics covered in this chapter.
Topics Covered in Chapter- 1 Relations and Functions for Class 12 Maths:
Types of relations: symmetric, Reflexive, transitive, and equivalence relations. The inverse of a function, Binary operations, composite functions, and One to one and onto functions.
NCERT Solutions for Class 12 Maths Chapter 1 Relations and Functions Exercises |
Exercise 1.1 |
Exercise 1.2 |
Exercise 1.3 |
Exercise 1.4 |
Miscellaneous Exercise |
Also, access the helpful resources for Class 12 Chapter 1 Relations and Functions at Physics Wallah
- Important questions for class 12 Maths Chapter 1 Relations and Functions
- RD Sharma Solution for Class 12 Maths Chapter 1 Relations and Functions
Chapter 2 Inverse Trigonometric Functions
Chapter-2 Inverse Trigonometric Functions of NCERT Class 12 Mathematics, provides an overview of various topics, such as notes based on basic concepts of inverse trigonometric functions, properties of inverse trigonometric functions, and various examples. These concepts are very well explained with examples. Solutions to exercises from this chapter can be accessed here.
Topics Covered in Chapter- 2 Inverse Trigonometric Functions of Class 12 Maths:
Definition, range, domain, principal value branches. Graphs of inverse trigonometric functions. Elementary properties of inverse trigonometric functions.
NCERT Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Functions Exercises |
Exercise 2.1 |
Exercise 2.2 |
Miscellaneous Exercise |
Also, access the helpful resources for Class 12 Chapter 2 Inverse Trigonometric Functions at Physics Wallah
- Important questions for class 12 Maths Chapter 2 Inverse Trigonometric Functions
- RD Sharma Solution for Class 12 Maths Chapter 2 Inverse Trigonometric Functions
Chapter 3 Matrices
In chapter 3 of the NCERT textbook, we will see the definition of the matrix, types of matrices, equality of matrices, matrix operations like the addition of matrices and multiplication of a matrix by a scalar, properties of addition of matrices, properties of scalar. Multiplication, matrix multiplication, matrix multiplication properties, matrix transpose, matrix transpose properties, symmetric and skew-symmetric matrices, elementary operations or matrix transformations, matrix inversions by elementary operations, and various examples. You can find links to exercise Solutions for the topics covered in this chapter.
Topics Covered in Chapter- 3 Matrices of Class 12 Maths:
Notation, order, equality, Concept, types of matrices, transpose of a matrix, zero matrix, symmetric and skew-symmetric matrices. Scalar multiplication of matrices and addition multiplication, multiplication, and scalar multiplication, simple properties of addition Existence of non-zero matrices, and Noncommutativity of multiplication of matrices whose product is the zero matrix (restrict to square matrices of order 2). Concept of elementary row and column operations. Proof of the uniqueness of inverse and Invertible matrices, if it exists; (Here, all matrices will have real entries).
NCERT Solutions for Class 12 Maths Chapter 3 Matrices Exercises |
Exercise 3.1 |
Exercise 3.2 |
Exercise 3.3 |
Exercise 3.4 |
Miscellaneous Exercise |
Also, access the helpful resources for Class 12 Chapter 3 Matrices at Physics Wallah
- Important questions for class 12 Maths Chapter 3 Matrices
- RD Sharma Solution for Class 12 Maths Chapter 3 Matrices
Chapter 4 Determinants
Chapter 4 Class 12 NCERT Solutions for Maths deals with the topic of Determinants. Students will be introduced to the definition and meaning of determinants, notations based on the order of determinants, properties of determinants, finding the area of a triangle using determinants, minor and cofactors of determinants, adjoint of a matrix, the inverse of matrix, determinants and matrix applications, and various examples. Below are links to the individual Solutions to the exercises covered in this chapter.
Topics Covered in Chapter- 4 Determinants of Class 12 Maths:
Determinants of a square matrix (3 × 3 matrices), minors, properties of determinants, cofactors, and applications of determinants in finding the area of a triangle. Adjoint and inverse of a square matrix. Inconsistency, consistency, and number of Solutions of a system of linear equations, for example, a system of linear equations in two or three variables (having a unique Solutions) using the inverse of a matrix.
NCERT Solutions for Class 12 Maths Chapter 4 Determinants Exercises |
Exercise 4.1 |
Exercise 4.2 |
Exercise 4.3 |
Exercise 4.4 |
Exercise 4.5 |
Exercise 4.6 |
Miscellaneous Exercise |
Also, access the helpful resources for Class 12 Chapter 4 Determinants at Physics Wallah
- Important questions for class 12 Maths Chapter 4 Determinants
- RD Sharma Solution for Class 12 Maths Chapter 4 Determinants
Chapter 5 Continuity and Differentiability
NCERT textbook of Chapter 5 begins with the definition of continuity. Students will also learn about continuity, the algebra of a continuous function, the definition and meaning of differentiability, derivatives of complex functions, derivatives of implicit functions, derivatives of inverse trigonometric functions, exponential and logarithmic functions, logarithmic derivatives, derivatives of functions in parametric forms, derivatives of the second order, the mean theorem value through various examples. Here students will find exercises that correctly explain these concepts with Solutions.
Topics Covered in Chapter- 5 Continuity and Differentiability of Class 12 Maths:
Continuity and differentiability, chain rule, a derivative of composite functions, derivatives of the inverse derivative of an implicit function, trigonometric functions. Concept of exponential and logarithmic functions and their derivatives. Logarithmic differentiation. Derivative of functions expressed in parametric forms. Second order derivatives, Rolle’s and Lagrange’s Mean Value Theorems (without proof), and their geometric interpretations.
NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Exercises |
Exercise 5.1 |
Exercise 5.2 |
Exercise 5.3 |
Exercise 5.4 |
Exercise 5.5 |
Exercise 5.6 |
Exercise 5.7 |
Exercise 5.8 |
Miscellaneous Exercise |
Also, access the helpful resources for Class 12 Chapter 5 Continuity and Differentiability at Physics Wallah
- Important questions for class 12 Maths Chapter 5 Continuity and Differentiability
- RD Sharma Solution for Class 12 Maths Chapter 5 Continuity and Differentiability
Chapter 6 Applications of Derivatives
NCERT textbook of Chapter 6 provides a definition of derivatives, increasing and decreasing functions, rate of change of quantities, first derivative test, tangents and normals, approximations, maximum and minimum values of a function in a closed interval, maxima and minima, and various examples. Here, students will find exercises that properly explain these concepts.
Topics Covered in Chapter- 6 Applications of Derivatives of Class 12 Maths:
Applications of derivatives: Rate of change, increasing/decreasing functions, tangents and normals, approximation, maxima, and minima. Simple problems (that illustrate the basic principles and understanding of the subject and real-life situations).
NCERT Solutions for Class 12 Maths Chapter 6 Applications of Derivatives Exercises |
Exercise 6.1 |
Exercise 6.2 |
Exercise 6.3 |
Exercise 6.4 |
Exercise 6.5 |
Miscellaneous Exercise |
Also, access the helpful resources for Class 12 Chapter 6 Application of Derivatives at Physics Wallah
- Important questions for class 12 Maths Chapter 6 Application of Derivatives
- RD Sharma Solution for Class 12 Maths Chapter 6 Application of Derivatives
Chapter 7 Integrals
In Chapter 7, Integrals, we will see the definition of the indefinite integral, integration as the inverse process of differentiation, the geometric interpretation of the indefinite integral, properties of the indefinite integral, comparison of differentiation and integration, methods of integration such as integration by substitution, integration by partial fractions, integration by parts, integration by of trigonometric identities, integration of some integral functions, definition and meaning of definite integrals, definite integral as limit of sum, basic theorem of calculus, calculation of definite integrals by substitution, some properties of definite integrals and various examples. Here we have provided exercises with Solutions based on these topics from the chapter.
Topics Covered in Chapter- 7 Integrals of Class 12 Maths:
Integration is the inverse process of differentiation. Integration of a variety of functions by substitution, partial fractions, and by parts, only simple integrals of the type
Basic properties of evaluation of definite integrals and definite integrals. Fundamental Theorem of Calculus (without proof).
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Exercises |
Exercise 7.1 |
Exercise 7.2 |
Exercise 7.3 |
Exercise 7.4 |
Exercise 7.5 |
Exercise 7.6 |
Exercise 7.7 |
Exercise 7.8 |
Exercise 7.9 |
Miscellaneous Exercise |
Also, access the helpful resources for Class 12 Chapter 7 Integrals at Physics Wallah
- Important questions for class 12 Maths Chapter 7 Integrals
- RD Sharma Solution for Class 12 Maths Chapter 7 Integrals
Chapter 8 Applications of Integrals
NCERT textbook of In chapter 8, we continue the discussions on integrals, area under simple curves, starting with definition, area of the region bounded by a curve and a straight line, area between two curves, and various examples. From the links below, students can access exercises with Solutions explaining the concepts from this chapter.
Topics Covered in Chapter- 8 Applications of Integrals of Class 12:
Applications in finding the area under simple curves, lines, arcs of parabolas/circles/ellipses, area between the two above curves (the region should be clearly identifiable).
NCERT Solutions for Class 12 Maths Chapter 8 Applications of Integrals Exercises |
Exercise 8.1 |
Exercise 8.2 |
Miscellaneous Exercise |
Also, access the helpful resources for Class 12 Chapter 8 Application of Integrals at Physics Wallah
- Important questions for class 12 Maths Chapter 8 Application of Integrals
- RD Sharma Solution for Class 12 Maths Chapter 8 Application of Integrals
Chapter 9 Differential Equations
In this chapter 9, students focus on the definition of differential equations, degree of a differential equation, basic concepts related to differential equations, general and particular Solutions of a differential equation, order of a differential equation, formation of a differential equation whose general Solutions is given, the procedure for formation of a differential equation that will represent a family of curves, first-order Solutions methods, first-degree differential equations, differential equations with a separable variable, homogeneous differential equations, linear differential equations, the procedure for solving a first-order linear differential equation, and various examples. Solutions to exercises from this chapter can be accessed here.
Topics Covered in Chapter- 9 Differential Equations of Class 12 Maths:
Definition, general and particular Solutions of a differential equation, order, and degree. Formation of differential equation whose available Solutions is mentioned. Solutions of differential equations by separating variables, first degree and homogeneous differential equations of the first order. Solutions of linear differential equation: dy/dx + Py = Q, where P and Q are the functions of x.
NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations Exercises |
Exercise 9.1 |
Exercise 9.2 |
Exercise 9.3 |
Exercise 9.4 |
Exercise 9.5 |
Exercise 9.6 |
Miscellaneous Exercise |
Also, access the helpful resources for Class 12 Chapter 9 Differential Equations at Physics Wallah
- Important questions for class 12 Maths Chapter 9 Differential Equations
- RD Sharma Solution for Class 12 Maths Chapter 9 Differential Equations
Chapter 10 Vector Algebra
In this chapter- 10, students will be introduced to vector algebra. The concepts in this chapter cover how to find the position vector, some basic concepts related to vector algebra, direction cosines, types of vectors such as zero vector, unit vector, collinear vector, equal vector, negative of a vector, vector addition, properties of vector addition, multiplication of a vector by a scalar, components of a vector, joining vectors of two points, cut formula, scalar or dot product of two vectors, scalar product properties, product of two vectors, projection of a vector onto a line vector or cross product of two vectors and various examples. From the links below, students can access exercises with Solutions explaining the concepts from this chapter.
Topics Covered in Chapter- 10 Vector Algebra of Class 12 Maths:
Vectors and Scalars, magnitude and direction of a vector. Ratios of vectors/Direction cosines. Types of vectors (unit, equal, zero, parallel, and collinear vectors), position vector of a point, negative of a vector, components of a vector, multiplication of a vector by a scalar, the addition of vectors, position vector of a point dividing a line segment in a given ratio. Scalar (dot) product of vectors, projection of a vector on a line. Vector (cross) product of vectors.
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Exercises |
Exercise 10.1 |
Exercise 10.2 |
Exercise 10.3 |
Exercise 10.4 |
Miscellaneous Exercise |
Also, access the helpful resources for Chapter 10 Vector Algebra at Physics Wallah
- Important questions for class 12 Maths Chapter 10 Vector Algebra
- RD Sharma Solution for Class 12 Maths Chapter 10 Vector Algebra
Chapter 11 Three dimensional Geometry
NCERT Class 12 Mathematics Chapter 11 contains topics related to three dimensional geometry, direction cosines and direction ratios of a line, relationship between direction cosines of a line, direction cosines of a line passing through two points, equation of lines in space, equation of a line passing through a given point and parallel to a given vector, derivation Cartesian form from vector form, the equation of a line passing through the two given points, angle between two lines, shortest distance between two lines, distance between two oblique lines, distance between two parallel lines, the equation of a plane in normal form, plane in Cartesian form, equation of a perpendicular plane to a given vector and passing through the given point, equation of a plane passing through three non-collinear points, intersection of the equation of a plane, plane passing through the intersection of two given planes, coplanarity of two lines, angle between two planes, distance of a point from a plane, angle between a line and a plane and various examples. Below are links to Solutions to exercises from this chapter.
Topics Covered in Chapter- 11 Three Dimensional Geometry of Class 12 Maths:
Direction cosines/ratios of a line joining two points. The shortest distance between the two lines, Cartesian and vector equation of a line, coplanar and skew lines. Cartesian and vector equation of a plane. Angle between (i) two lines, (ii)a line and a plane, (iii) two planes. Distance of a point from a plane.
NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry Exercises |
Exercise 11.1 |
Exercise 11.2 |
Exercise 11.3 |
Miscellaneous Exercise |
Also, access the helpful resources for Class 12 Chapter 11 Three Dimensional Geometry at Physics Wallah
- Important questions for class 12 Maths Chapter 11 Three Dimensional Geometry
- RD Sharma Solution for Class 12 Maths Chapter 11 Three Dimensional Geometry
Chapter 12 Linear Programming
In this chapter, students will discuss about the linear programming. Concepts discussed in this chapter are linear programming problem and its mathematical formulation, graphical method of solving linear programming problems, mathematical formulation of the problem, different types of linear programming problems with different examples. students can find exercises that correctly explain these concepts with Solutions.
Topics Covered in Chapter- 12 Linear Programming of Class 12 Maths:
Introduction, related terminology such as constraints, objective function, optimization, different types of linear programming (L.P.) problems, mathematical formulation of L.P. problems, graphical method of Solutions for problems in two variables, feasible and infeasible regions, feasible and infeasible Solutions, optimal feasible Solutions (up to three non-trivial constrains).
NCERT Solutions for Class 12 Maths Chapter 12 Linear Programming Exercises |
Exercise 12.1 |
Exercise 12.2 |
Miscellaneous Exercise |
Also, access the helpful resources for Class 12 Chapter 12 Linear Programming at Physics Wallah
- Important questions for class 12 Maths Chapter 12 Linear Programming
- RD Sharma Solution for Class 12 Maths Chapter 12 Linear Programming
Chapter 13 Probability
This chapter discusses probability. Reviewing the basic facts and formulas studied earlier in Class 11, we reintroduce these formulas in this chapter along with the advanced concepts are mentioned, i.e. conditional probability, properties of conditional probability, multiplication theorem of probability, multiplication rule of probability for more than two events, independent events, Bayesian theorem, sample space distribution, total probability theorem, random variables and their probability distribution, mean of a random variable, probability distribution of a random variable, variance of a random variable, Bernoulli trials and binomial distribution with various examples. Here we have compiled the Solutions to the exercises from this chapter that have thoroughly explained the concept.
Topics Covered in Chapter- 13 Probability of Class 12 Maths:
Multiplication theorem on probability. Conditional probability, independent events, total probability, Baye’s theorem. Random variable and its probability distribution, mean and variance of haphazard variable. Repeated independent (Bernoulli) trials and Binomial distribution.
NCERT Solutions for Class 12 Maths Chapter 13 Probability Exercises |
Exercise 13.1 |
Exercise 13.2 |
Exercise 13.3 |
Exercise 13.4 |
Exercise 13.5 |
Miscellaneous Exercise |
Also, access the helpful resources for Chapter 13 Probability at Physics Wallah
- Important questions for class 12 Maths Chapter 13 Probability
- RD Sharma Solution for Class 12 Maths Chapter 13 Probability
Why NCERT Solutions for Class 12 Maths are important?
The NCERT Solutions for class 12 maths study material provided by the team would help students get top ranks. The complete study material has been divided into various segments. For ease in understanding topics, students can find it divided into chapter wise section.
The very ideology with which the team has been formed is to enhance students' foundations. A stronger foundation of concepts will help students grow in their career. Students can also find important questions in NCERT Solutions for class 12 maths provided by Physics Wallah. NCERT books already known among students for their comprehensive study program. Many a time students cannot complete and solve NCERT textbooks. These NCERT Solutions for class 12 maths are anytime helpful for students at an instant of time.
Maths is among the subject which not only helps students in qualifying their examinations but also helps them in understanding advanced studies in future. The complete study material has been curated by the experts of their field. The duration and efforts put on by this faculties are incomparable. The students learning form NCERT Solutions for class 12 maths attain an extra compatible edge over the rest of the students.
How to Study NCERT Solutions for Class 12 Maths effectively?
Student can enhance their problem-solving skills through, our study material. Many a time students get stuck with difficult numerical. In order to prevent the wastage of time, students can follow NCERT Solutions for class 12 maths prepared byPhysics Wallah.
The entire team of NCERT Solutions for class 12 maths at Physics Wallah have already aced various national level exams. In order to prepare this study material, various previous year question papers have been referred. The complete study material has been prepared flawless and checked various times for any error.
Why Physics Wallah is best for NCERT Solutions for class 12 maths?
Several times students have to travel long distances to get their doubts cleared from expert faculties. Physics Wallah has provided this complete study material right at your home. All the students need to do is study every bit of it and score maximum marks. In order to minimize the efforts students can also find various important questions marked in the study material.
The complete NCERT Solutions for class 12 maths has been provided for free. Students can access the world-class study material with a single click. All that students need to do is, to sign up with us. Study from NCERT Solutions for class 12 maths and climb the ladders of success.
Key Benefit for NCERT Solutions for Class 12 Maths
Physics Wallah academic team consist of expert faculty members across country who prepared NCERT Solutions for class 12 Maths the key highlights of these Solutions are.
- All the questions given in the NCERT exercise are solved.
- All NCERT Solutions for class 12 Maths are errorless and proof-readied several time.
- All answers are explained and written as per the guideline of CBSE to score good marks in class 12 final exam.
- Based on new pattern of CBSE class 12 we have added additional questions which are MCQ based and added several solved examples apart from NCERT Solutions for class 12 Maths.
- Physics Wallah NCERT Solutions for class 12 Maths consist of exercise wise pages to help you find your maths doubts instantly.
- Each question of NCERT Solutions for class 12 Maths is added with all required step which are supposed to write in final exam.
- Step wise Solutions of questions will help you to understand the method of solving the questions and clear your concepts.
- While solving any questions of NCERT text book one must have clarity on the concept for that while reading the theory of maths chapter you must prepare proper notes of the theory and make sure you write and understand all the concept. Write down the important formula use in the chapter in separate notes which helps you in final revision.
- Learning the application of maths and to know how to apply concepts of maths in the numerical required lot of effort to develop such method you must try more than one time and give time to question which are difficult make sure you have tried multiple time to solve the particular question before using NCERT Solutions for class 12 Maths. This way you will develop a skill to attempt the MCQ based question as well as you can handle the new questions.
Frequently Asked Question (FAQs)
Q1. What is Right strategies to score 100 percent marks in CBSE Board of class 12 Maths subject?
Ans. Right approach start with strong planning and it start with preparing right time table and notes. Start your preparation of class 12 Maths subject from the month of April. Do follow NCERT text book for class 12 Maths and read the entire theory mentioned in the text book. While reading and understanding the theory of NCERT text book make sure you have prepared a handout note of the subject.
Try to summarise the entire chapter formula in one page. Solve all questions given in the exercise with the help of Physics Wallah NCERT Solutions for class 12 Maths. Complete your syllabus in the month of October and then start revision with your own notes and do solve previous year papers.
Q2. How to Make balance between Board exam and competitive exam?
Ans. This is most common problem every student face when he was in class 12 and preparing for competitive entrance exam. Right planning is required to make a balance between your competitive entrance exam with board exam. Now a days all most all competitive entrance exam is conducted by NTA and the recommended book of NTA is NCERT.
So start your preparation with NCERT text book. try to solve all question in the exercise and solve questions from NCERT exemplar. Do solve MCQ based question will help you in your entrance exam as well as board exam. Allocate equal time for board as well as competitive exam with effective revision of class 11.
Q3. What are the Important topics of class 12 Maths for CBSE board exam?
Ans. The questions asked in CBSE board exam are from all most all chapters so it is difficult to prepare strategies in important chapters still few topics are important like Differentiability of any functions (This chapter consist of 2 and 4 marks questions)
- Logarithmic Differentiation – (This chapter consist of 4 marks questions)
- Substitution method (In Inverse) – (This chapter consist of 2 and 4 marks questions)
- Higher-order Derivative – (This chapter consist of 4 marks questions)
- Maxima Minima Word Problem – (This chapter consist of 6 marks questions)
- Equation of Tangent and Normal – (This chapter consist of 2 marks questions)
Q4. How sample papers of class 12 Maths can help to score good marks in Board exam?
Ans.If you are preparing for CBSE board exam sample paper for class 12 Maths are must for you. There are few thinks must be keep in mind while solving the sample papers such as solve one sample paper completely at once.
Do solve the sample papers based on time allocated this way you can check your speed of solving the paper and find out your mistake. Physics Wallah uploaded CBSE Sample paper for class 12 Maths with detail Solutions to help you out.
Q5. Should we solve additional MCQ based questions with NCERT text book?
Ans. Yes certainly you must solve MCQ based questions for class 12 Maths. Although NCERT text book consist of objective questions in the exercise but it is highly recommended to solve some addition Maths Questions objective type.
Q6. How to use previous year question papers of CBSE class 12 Board exam?
Ans. To score good marks in any exam need good level of understanding of exam pattern which can be developed through understanding the pattern of entrance exam. You can develop understanding of exam paper by doing deep analysis of previous year question papers. Do solve the questions paper understand the pattern of exam.
Q7. Is NCERT text book and NCERT exercise are enough for board as well as competitive entrance exam?
Ans. Yes NCERT text book is enough for competitive as well as board exam. NCERT text book is well designed and can be completed within the time limit so that you can do perfect revision of entire syllabus which play very important role in entrance as well as board exam.
Recent Concepts
- chapter 1-Relations and Functions Exercise 1.1
- chapter 1-Relations and Functions Exercise 1.2
- chapter 1-Relations and Functions Exercise 1.3
- chapter 1-Relations and Functions Exercise 1.4
- chapter 1-Relations and Functions Miscellaneous Exercise
- chapter 2-Inverse Trigonometric Function Exercise 2.1
- chapter 2-Inverse Trigonometric Function Exercise 2.2
- chapter 2-Inverse Trigonometric Function Miscellaneous Exercise
- chapter 3-Matrices Exercise 3.1
- chapter 3-Matrices Exercise 3.2
- chapter 3-Matrices Exercise 3.3
- chapter 3-Matrices Exercise 3.4
- chapter 3-Matrices Miscellaneous Exercise
- chapter 4-Determinants Exercise 4.1
- chapter 4-Determinants Exercise 4.2
- chapter 4-Determinants Exercise 4.3
- chapter 4-Determinants Exercise 4.4
- chapter 4-Determinants Exercise 4.5
- chapter 4-Determinants Exercise 4.6
- chapter 4-Determinants Miscellaneous Exercise