NCERT Solutions for Class 12 Maths Chapter 4 Determinants

Whether working on Exercise 4.3 or solving complex problems in Exercise 4.6, students can rely on NCERT Simple Solutions for a structured approach. The solutions are available in PDF format, making them accessible for quick revisions before exams. NCERT Revised Solutions for class 12 Maths chapter 4 Determinants all exercise with miscellaneous in Hindi and English Medium new session 2024-25. The questions of each exercise is changed as per the new syllabus for CBSE 2024-25.

• Exercise 4.1 Solutions
• Exercise 4.2 Solutions
• Exercise 4.3 Solutions
• Exercise 4.4 Solutions
• Exercise 4.5 Solutions
• Miscellaneous Exercise 4 Solutions

Key Features of Class 12 Mathematics Determinants NCERT Textbook Solutions

Class 12th Mathematics Determinants NCERT Detailed Solutions are designed to help students develop problem-solving skills with clear explanations. NCERT Exercise 4.4 introduces the application of determinants in real-life scenarios, such as calculating areas of triangles and verifying collinearity. Similarly, NCERT Book Exercise 4.5 focuses on determinants of matrices and their role in finding the inverse of a matrix. Class 12 NCERT Mathematics Book solutions emphasize the properties of determinants, ensuring students grasp key concepts easily.

NCERT Class 12th Math Textbook Chapter 4 solutions are comprehensive and include detailed methods for solving problems, making them ideal for both beginners and advanced learners. The Miscellaneous Exercise Chapter 4 offers challenging problems to test overall understanding. Downloading the PDF solutions for Class 12 Mathematics Chapter 4 allows students to study offline at their convenience. Practicing these solutions regularly builds confidence in tackling determinant-based questions in exams.

Important Points for Class 12 Maths Chapter 4 Determinants

Key points for exams include the properties of determinants, calculation of minors and cofactors, adjoint and inverse of matrices and solving linear equations using determinants. Focus on NCERT exercises like 4.3, 4.5 and the Miscellaneous Exercise for advanced problems. Practice applications in geometry, such as collinearity and area of triangles.

Benefits of Solving NCERT Class 12 Mathematics Chapter 4 Exercises

Solving NCERT Textbook Class 12 Maths Chapter 4 exercises is essential for students aiming to excel in the topic of determinants. The step-by-step NCERT Mathematics solutions for exercises such as Exercise 4.1, Exercise 4.2 and NCERT Book Exercise 4.3 simplify complex mathematical operations, like evaluating determinants and finding minors and co-factors. NCERT Textbook Math solutions for Class 12 Determinants Miscellaneous Exercise ensure students get ample practice for advanced problems.

Class 12 Mathematics Book solutions are well-structured to align with board exam requirements and help students understand the application of determinants in solving real-world problems, such as system equations. NCERT Solutions for Class 12 Maths Chapter 4 PDF downloads provide easy access to study materials anywhere. By distributing their study time across all exercises, including Exercise 4.6, students can thoroughly cover the topic. These solutions make learning determinants simple and effective, boosting confidence for exams.

Class XII Maths Chapter 4 solutions is given here. UP board solutions for class 12 Maths Chapter 4 are same as the NCERT Sols for 12th Maths Chapter 4. UP Board intermediate students are also using following NCERT Textbooks for new session, so they also can use these solutions for their help in problem solving. Other boards students like MP Board, Uttarakhand, and Gujrat Board can use these textbook solutions to prepare their exams. The solutions of questions are helpful for all the students who are using 12th NCERT Books. To understand the 12th Maths Chapter 4 properly, we must know everything about matrices. So, first learn the concepts of the chapter 3 Matrices to do Chapter 4 Determinants.

In 12th Mathematics chapter 4 we shall study determinants up to order three only with real entries. The history about determinant is given below to know more about this fact. Download NCERT solutions for class 12 Maths chapter 4 determinants all six exercise with miscellaneous in PDF format. UP Board Solutions for Class 12 Maths Chapter 4 is same as NCERT Sols for 12th Maths Chapter 4 Determinants. So, UP Board students can also take the benefits of these solutions.

If all the rows of a determinant are converted into the corresponding columns, the value of the determinant remains same.

If two rows (columns) of a determinant are interchanged, the value of the new determinant is the additive inverse of the value of the given determinant.

The value of a determinant gets multiplied by k, if every entry in any of its row (column) is multiplied by k.

If some or all elements of a row or column of a determinant are expressed as sum of two (or more) terms, then the determinant can be express as sum of two (or more) determinants.

If the corresponding entries in any two rows ( or columns) are identical, the value of the determinant is zero.

The value of a determinant does not changes if any of its rows (columns) is multiplied by non-zero real number k and added to another row (column).

Minor – Removing entries of the column and the row containing a given element of a determinant and keeping the surviving entries as they are, yields a determinant called the minor of the given element.
Cofactor – If we multiply the minor of an element by (-1)^(i+j), where i is the number of the row and j is the number of the column containing the element, then we get the cofactor of that element.

1. The Chinese early developed the idea of subtracting columns and rows as in simplification of a determinant using rods. Seki Kowa, the greatest of the Japanese Mathematicians of seventeenth century in his work ‘Kai Fukudai no Ho’ in 1683 showed that he had the idea of determinants as well as their expansion.
2. T. Hayashi, “The Fakudoi and Determinants in Japanese Mathematics,” in the proc. of the Tokyo Math. Soc., V. Vendermonde was the first to recognise determinants as independent functions. He may be called the formal founder.
3. Laplace (1772), gave general method of expanding a determinant in terms of its complementary minors.
4. Lagrange, in 1773, treated determinants of the second and third orders and used them for purpose other than the solution of equations.
5. Gauss, in 1801, used determinants in his theory of numbers.
6. Jacques – Philippe – Marie Binet, in 1812, stated the theorem relating to the product of two matrices of m-columns and n-rows, which for the special case of m = n reduces to the multiplication theorem.
7. Cauchy, in 1812, presented one on the same subject. He used the word ‘determinant’ in its present sense. He gave the proof of multiplication theorem more satisfactory than Binet’s.
8. The greatest contributor to the theory was Carl Gustav Jacob Jacobi, after this the word determinant received its final acceptance.