NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.1 (Ex 4.1)

Mathematics is a subject which is used by scientists to find quantitative solutions for experimental laws. Mathematics is applied in many fields including Computer Science, Social Sciences, Finance, Medicine, Natural Sciences, Engineering etc. So, it is a subject that is applied in the curriculum of all the subjects for further studies. The course of Mathematics for Class 12 introduces many essential concepts of Mathematics which form the basic concept for higher-level studies including engineering, architecture, etc. For students to score their best in Class 12, they need to have a strong conceptual base so that they would not only score good grades but be able to apply those fundamentals in the future.

Chapter- 4 of Class 12 Mathematics NCERT is about Determinants. To study the concept of Determinants, students must know the basic concept of Matrices, which is the previous chapter of the NCERT curriculum. In Mathematics, a determinant is said to be a scalar value that is a function of the entries of a square matrix. NCERT is like the building block for any student who has to appear for the board examinations. Extramarks provides students with the NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.1 so that they can get detailed and easy solutions.

A determinant can also be explained as an associated number to every square matrix. This may be thought of as a function which associates each square matrix with a unique number (real or complex). To learn more about determinants, students can download the NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.1 provided by Extramarks.

Click on the given link to download the NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.1.

NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.1 (Ex 4.1)

Students can download the NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.1 here. The solutions provided by Extramarks are accurate and reliable and are curated by the experts of Extramarks.

Access NCERT Solutions for Class 12 Maths Chapter 4 – Determinants

NCERT Solutions for Class 12 Maths Chapter 4 – Determinants

Class 12, Chapter- 4 Determinants, includes an Introduction of Determinants, About Determinants, Properties of Determinants, Area of a Triangle, Minors and Cofactors, Adjoint and Inverse of a Matrix and Applications of Determinants and Matrices. The chapter has a wide range of topics covered. Extramarks provides the NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.1 so that students don’t waste their time searching for answers and get authentic solutions without having to look anywhere else. The Extramarks’ website provides a wide range of tools such as K12 study material, live doubt-solving sessions and so much more, to clear all the queries and doubts of students. This helps students focus on their goals, excel in their studies and score well in their board examinations.

NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.1 includes the following topics- Introduction, Determinants, Determinant of a Matrix of Order one x one, Determinant of a Matrix of Order two x two, Determinant of a Matrix of Order three x three. Exercise 4.1 Class 12th Maths is composed of eight questions. In the beginning, students might find these solutions tough, but the NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.1 provided by Extramarks help the students understand the concept better so that they can solve all complicated questions. Board examinations often leave students with a lot of questions, including how to prepare, revise, manage time and stay focused. The most basic step to preparing for the board exam is to practice from the NCERT textbooks. Extramarks is a platform that provides students with all the necessary help to score well in board examinations. Along with the syllabus for the NCERT Solutions Class 12 Mathematics. Extramarks also provides students with NCERT Solutions for all the respective subjects of various other academic years. Students can refer to Extramarks’ website for the following:

NCERT Solutions Class 11

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NCERT Solutions Class 6

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Click on the given link to download the NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.1 in PDF Format.

Class 12 Maths Exercise 4.1 Solutions

Students must practice the necessary exercises from the NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.1 to strengthen their preparation. Class 12 Maths Exercise 4.1 Solutions are listed below.

Exercise 4.1 Class 12 Maths – Question 1

The first question of the NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.1 is about 2×2 matrices and students have to find the determinant for it. For questions like these, students can refer to the NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.1 provided by Extramarks. The components of the matrix are Integers in this question, but in other cases, the components can vary. They can be fractions, decimals or natural numbers, but in all these cases the process of finding the determinant will be the same. The first step to find a determinant is to identify rows and columns. To further know how to solve these questions, students can download the NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.1 and subscribe to the Extramarks website.

Exercise 4.1 Maths Class 12 – Question 2

The second question of the NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.1 is also about 2×2 matrices and students have to find the determinant for it. The components of the matrix are trigonometric identities and algebraic expressions. The Fundamentals and the Identities of Trigonometry should also be kept in mind while solving this problem. The solution method involves expanding the matrix and finding the value, similarly as it is done in the case of natural numbers. With regular practice, students can easily understand the fundamentals of the chapter and solve any complicated problem that comes in front of them. For a detailed explanation of the solution, students can subscribe to Extramarks, and download the NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.1.

Exercise 4.1 Class 12 – Question 3

The third question of the NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.1 asks students to prove the given statement. Before attempting this question, students should have prior knowledge about the operation of multiplication in matrices. Students should also know the multiplication of a determinant with a natural number. Students should keep in mind that both these methods are different from each other. For more detailed explanations, students can download the NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.1.

In such questions where statements are to be proved, students need to check whether the idea of the question seems correct. Thus, students can easily verify whether their solution is accurate or if it contains any errors. It is very easy for them to identify errors if there are any errors on both sides of the steps. For further assistance, students are advised to download the NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.1, provided by Extramarks.

 Exercise 4.1 Class 12 Maths – Question 4

The fourth question of the NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.1 is about a 3×3 matrix. Students are asked to prove that multiplying its determinant with 27 is equivalent to multiplying the matrix with 3. For solving this question, they should have an understanding of multiplying a 3×3 matrix with a natural number. Students often confuse the multiplication of rows and columns of a matrix. For a better understanding of these concepts, Extramarks provides students with the NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.1.

Exercise 4.1 Class 12 Maths – Question 5

The fifth question of the NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.1 asks students to find the determinants of four matrices. All four matrices are 3×3 matrices. But, this question is a bit different from other questions. If students find a particular row to have more than one element as zero, then they can reduce the determinant equation to a single term. The other two terms are multiplied by zero, so they get cancelled out. For a step-by-step solution, students can download the NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.1.

These solutions help students understand the fundamentals of Exercise- 4.1. They are easy to understand and provide students with accurate solutions to their questions. Extramarks also offers students tools like K12 learning resources for board examinations, assessment centres, smart class solutions etc. These tools make studies easy and enjoyable for students so that they can confidently sit for their board examinations. For further explanation, students can download the NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.1.

Exercise 4.1 Class 12 Maths – Question 6

The sixth question of the NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.1 has a very straightforward solution. Students are asked to find out the value of the determinant of a 3×3 matrix. They should be mindful while expanding the rows for calculation. With every expansion, one of the components in the first row gets changed, and you have to consider the other elements.

Students should concentrate well while performing the determinant calculations, as any small mistake can make the whole solution incorrect. For further assistance, students can download the NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.1 provided by Extramarks.

Exercise 4.1 Class 12 Maths – Question 7

The seventh question of the NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.1 has two matrices given, where students have to find the value of ‘x’ from one of the matrices by equating them. To compare them, they must first frame an equation. Students should be careful while dealing with some terms, as they are likely to obtain two values for the same. One of the values will be negative, and the other one will be positive; therefore, they need to consider both values while writing their final answer.

For a better understanding of the solution, download the NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.1, provided by Extramarks.

Exercise 4.1 Class 12 Maths – Question 8

The eighth question of the NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.1 is a bit similar to the previous one. Students are required to find the value of ‘x’ from the two given matrices, which should be equal.

Therefore, at first, they have to calculate the determinant of the matrices in terms of ‘x’ and after that equate them to solve for the value of ‘x.

Solving these types of problems for the first time often seems a little difficult, but with regular practice, students can get a better hold of the topic.

In this question, students are given four options from which they can choose an answer. The answer is in the form of square roots, which determines that there are two values. For a detailed solution, download the NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.1 provided by Extramarks.

Click here to download the NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.1.

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Extramarks is an online platform which has always focused on helping students achieve their best. It has provided them with many tools which have made learning easy and enjoyable. Extramarks equips students with in-depth performance reports to track their progress, live classes from top faculty, doubt-solving sessions, K12 live class programs and much more. Extramarks is a one-stop solution to all the problems of the students.

Extramarks provides step-by-step detailed answers to all the questions of the students, like the Class 12th Maths Exercise 4.1. If students practice the NCERT questions provided by Extramarks daily, they will eventually get a better understanding of the subject and score well in the board examinations.

Along with the NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.1, Extramarks also provides students with sample papers and the past years’ question papers. This helps students get a better understanding of the blueprint of the question paper for the board examinations and helps them practice time management, which is essential for the examinations.

Tools like ‘In-depth Performance Reports to Track Progress’ provided by the Extramarks website help students keep a track of their performance. It also provides students with proper study material so that they don’t waste their time looking for answers. Extramarks makes students revise their core fundamentals very efficiently so that they can solve any question in their examination effortlessly.

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NCERT Solution Class 12 Maths of Chapter 4 Other Exercises

Chapter 4 – Determinants of  Other Exercises
Exercise 4.2
10 Questions & Solutions (4 Short Answers, 10 Long Answers)
Exercise 4.3
5 Questions & Solutions (2 Short Answers, 3 Long Answers)
Exercise 4.4
5 Questions & Solutions (2 Short Answers, 3 Long Answers)
Exercise 4.5
18 Questions & Solutions (4 Short Answers, 14 Long Answers)
Exercise 4.6
16 Questions & Solutions (3 Short Answers, 13 Long Answers)

 

Q.1 Evaluate the determinants

   2    4−5−1

Ans

| 2    4−5−1|=2×(−1)−4(−5)

          =−2+20          =18

Q.2 Evaluate the determinants

|(x2−x+1) (x−1)(x+1) (x+1)|

Ans

|x2−x+1 x−1x+1 x+1|=(x+1)(x2−x+1)−(x−1)(x+1)  =(x3−1)−(x2−1)=x3−1−x2+1  =x3−x2

Q.3

If A=1 24 2, then show that |2A| = 4 |A|.

Ans

Given:[A]=[1 24 2]   |A|=|1 24 2|          =2−8          =−6  2A=[2 48 4]L.H.S.=|2A|=|2 48 4|          =8−32          =−24          =4×(−6)          =4|A|=R.H.S.

Q.4 

If A=[1 0 10 1 20 0 4], then show that |3A| =27|A|.

Ans

Given  A=[1 0 10 1 20 0 4] 3A=[3 0 30 3 60 0 12]  |A|=|3 0 30 3 60 0 12|          =3(36−0)−0(0−0)+3(0−0)          =108+0+0          =108|3A|=|9 0 90 9 180 0 36|          =9(324−0)−0(0−0)+9(0−0)          =2916+0+0          =27×108          =27×|A||3A|=27|A|Hence Proved.

Q.5


Evaluate the determinants(i)     |3−1−20    0−13−5  0|     (ii)   |3−4   51    1−22    3    1|(iii)   |   0 1   2   1 0−3−2 3   0|    (iv) |2−1−20    2−13−5    0|

Ans

 (i)     |3−1−20    0−13−5   0|=3|   0−1−5   0|−(−1)|0−13   0|+(−2)|0    03−5|           [Expanding​​​ along R1]        =3(0−5)+1(0+3)−2(0−0)        =−15+3−0        =−12(ii)|3−4   51    1−22    3   1|=3|1−23   1|−(−4)|1−22   1|+5|1 12 3|     =3(1+6)+4(1+4)+5(3−2)=21+20+5=46

 ( iii )     |     0 1 2 −1   0    −3 −2 3 0 |=0| 0 −3 3    0 |−1| −1 −3 −2    0 |+2| −1 0 −2 3 |            [ Expanding​​​ along R 1 ]         =0( 0+9 )−1( 0−6 )+2( −3−0 )         =0+6−6         =0 ( iv )| 2 −1 −2 0     2 −1 3 −5    0 |=2|    2 −1 −5    0 |−( −1 )| 0 −1 3    0 |−2| 0     2 3 −5 |                                            [ Expanding​​​ along R 1 ]      =2( 0−5 )+1( 0+3 )−2( 0−6 ) =−10+3+12 =5 MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYBg9LrFfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=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@CB47@

Q.6 

If A=[1 1−22 1−35 4−9],   find  |A|.

Ans

 A=[1 1−22 1−35 4−9],  |A|=|1 1−22 1−35 4−9|[Expending​ along R1]       =1|1−34−9|−1|2−35−9|+(−2)|2 15 4 |      =(−9+12)−1(−18+15)−2(8−5)      =3+3−6      =0

Q.7

Find value so fx, if(i)|2 45 1|=|2x 4 6 x| (ii) |2 34 5|=|x 32x 5|

Ans

(i) |2 45 1|=|2x 4 6 x|⇒       2−20=2x2−24⇒         −18=2x2−24⇒  24−18=2x2⇒      x2=62⇒        x=±3(ii) |2 34 5|=|x 32x 5|⇒10−12=5x−6x⇒           −2=−x⇒       x=2.

Q.8

If  |x 218 x|=|6 218 6|, then x is equal to(A) 6(B) ±6(C )−6(D) 0.

Ans

∵ |x 218 x|=|6 2 18 6|∴x2−36=36−36⇒   x2=36⇒            x=±6Therefore, option (B) is correct.

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FAQs (Frequently Asked Questions)

1. Are the NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.1 difficult?

With regular practice and proper guidance, these questions can be easily solved, and concepts can be mastered so that students can score well in the board examinations.

2. Is it important to do all the questions of the NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.1?

Yes, students should practice all the questions of Chapter 12 Maths Exercise 4.1, as it is crucial to have strong fundamentals and fast calculation speed to succeed in the board exams.

3. How can students clear their doubts about the NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.1?

Students can refer to the solutions provided by Extramarks to clear their doubts and further, subscribe to Extramarks’ website to have access to live doubt-solving sessions and get expert guidance for pursuing their further studies. 

4. Is it necessary to know matrices before learning the NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.1?

Yes, it is important to know Chapter 3 Matrices before starting to learn Chapter 4 Determinants, as the determinant is defined as the real or a complex number that can be associated with a square matrix. 

5. Is it significant to practice the solved questions of Class 12 Maths Exercise 4.1?

In Mathematics, practice is the key to scoring good marks, so students should practice every question they can, especially questions from the NCERT textbook since the NCERT is the first step for board exams.

6. Is the NCERT Exemplar book needed for the preparation of Exercise 4.1 Class 12th Maths?

Mathematics is a subject which requires a lot of practice, practising the examples given in the NCERT book and the NCERT Exemplar will help in building the concepts. The more students will practice, the better they will get.

7. Will the NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.1 help students in any other competitive examination?

Yes, according to the changes in the admission pattern of Delhi University, the university is conducting an entrance examination which is entirely based on the content of NCERT. Also, there are other universities which follow the same pattern. So yes, the NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.1 will help students in the preparation for certain entrance examinations.

8. Give a preview of NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.1.

NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.1 include the following topics- Introduction, Determinants, Determinant of a Matrix of Order one 

Determinant of a Matrix of Order two etc. Exercise 4.1 Class 12 Maths consists of eight questions. In the beginning, students might find these calculations difficult and confusing at some steps, but with proper practice and guidance, students will be able to solve these questions very efficiently. Extramarks provides students with an easy and step-by-step approach to the NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.1.