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NCERT Solutions for Exercise 4.2 Class 12 Maths Chapter 4 Determinants are discussed here. These NCERT solutions are created by subject matter expert at Careers360 considering the latest syllabus and pattern of CBSE 2023-24. In this article, you will get NCERT solutions for Class 12 Maths chapter 4 exercise 4.2. These Exercise 4.2 Class 12 Maths solutions are consist of questions related to properties of determinants. Properties of determinants make it easy for us to finding determinants without complicated calculations. There are 6 properties of determinants related to operation on the determinants given in the NCERT textbook before the Class 12 Maths ch 4 ex 4.2. You are advised to go through the proof of these properties given in the textbook to get a better understanding. There are some examples given after each property which will also help you to get conceptual clarity.
12th class Maths exercise 4.2 answers are designed as per the students demand covering comprehensive, step by step solutions of every problem. Practice these questions and answers to command the concepts, boost confidence and in depth understanding of concepts. Students can find all exercise together using the link provided below.
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Question:1 Using the property of determinants and without expanding, prove that
Answer:
We can split it in manner like;
So, we know the identity that If any two rows (or columns) of a determinant are identical (all corresponding elements are same), then the value of the determinant is zero.
Clearly, expanded determinants have identical columns.
Hence the sum is zero.
Question:2 Using the property of determinants and without expanding, prove that
Answer:
Given determinant
Applying the rows addition then we have;
So, we have two rows and identical hence we can say that the value of determinant = 0
Therefore .
Question:3 Using the property of determinants and without expanding, prove that
Answer:
Given determinant
So, we can split it in two addition determinants:
[ Here two columns are identical ]
and [ Here two columns are identical ]
Therefore we have the value of determinant = 0.
Question:4 Using the property of determinants and without expanding, prove that
Answer:
We have determinant:
Applying we have then;
So, here column 3 and column 1 are proportional.
Therefore, .
Question:5 Using the property of determinants and without expanding, prove that
Answer:
Given determinant :
Splitting the third row; we get,
.
Then we have,
On Applying row transformation and then ;
we get,
Applying Rows exchange transformation and , we have:
also
On applying rows transformation, and then
and then
Then applying rows exchange transformation;
and then . we have then;
So, we now calculate the sum =
Hence proved.
Question:6 Using the property of determinants and without expanding, prove that
Answer:
We have given determinant
Applying transformation, we have then,
We can make the first row identical to the third row so,
Taking another row transformation: we have,
So, determinant has two rows identical.
Hence .
Question:7 Using the property of determinants and without expanding, prove that
Answer:
Given determinant :
As we can easily take out the common factors a,b,c from rows respectively.
So, get then:
Now, taking common factors a,b,c from the columns respectively.
Now, applying rows transformations and then we have;
Expanding to get R.H.S.
Question:8(i) By using properties of determinants, show that:
We have the determinant
Applying the row transformations and then we have:
Now, applying we have:
or
Hence proved.
Question:8(ii) By using properties of determinants, show that:
Answer:
Given determinant :
,
Applying column transformation and then
We get,
Now, applying column transformation , we have:
Hence proved.
Question:9 By using properties of determinants, show that:
Answer:
We have the determinant:
Applying the row transformations and then , we have;
Now, applying ; we have
Now, expanding the remaining determinant;
Hence proved.
Question:10(i) By using properties of determinants, show that:
Answer:
Given determinant:
Applying row transformation: then we have;
Taking a common factor: 5x+4
Now, applying column transformations and
Question:10(ii) By using properties of determinants, show that:
Answer:
Given determinant:
Applying row transformation we get;
[taking common (3y + k) factor]
Now, applying column transformation and
Hence proved.
Question:11(i) By using properties of determinants, show that:
Answer:
Given determinant:
We apply row transformation: we have;
Taking common factor (a+b+c) out.
Now, applying column tranformation and then
We have;
Hence Proved.
Question:11(ii) By using properties of determinants, show that:
Answer:
Given determinant
Applying we get;
Taking 2(x+y+z) factor out, we get;
Now, applying row transformations, and then .
we get;
Hence proved.
Question:12 By using properties of determinants, show that:
Answer:
Give determinant
Applying column transformation we get;
[after taking the (1+x+x^{2 }) factor common out.]
Now, applying row transformations, and then .
we have now,
As we know
Hence proved.
Question:13 By using properties of determinants, show that:
Answer:
We have determinant:
Applying row transformations, and then we have;
taking common factor out of the determinant;
Now expanding the remaining determinant we get;
Hence proved.
Question:14 By using properties of determinants, show that:
Answer:
Given determinant:
Let
Then we can clearly see that each column can be reduced by taking common factors like a,b, and c respectively from C_{1,}C_{2,}and C_{3.}
We then get;
Now, applying column transformations: and
then we have;
Now, expanding the remaining determinant:
.
Hence proved.
Question:15 Choose the correct answer. Let A be a square matrix of order , then is equal to
(A) (B) (C) (D)
Answer:
Assume a square matrix A of order of .
Then we have;
(Taking the common factors k from each row.)
Therefore correct option is (C).
Question:16 Choose the correct answer.
Which of the following is correct
(A) Determinant is a square matrix.
(B) Determinant is a number associated to a matrix.
(C) Determinant is a number associated to a square matrix.
(D) None of these
Answer:
The answer is (C) Determinant is a number associated to a square matrix.
As we know that To every square matrix of order n, we can associate a number (real or complex) called determinant of the square matrix A, where element of A.
This article NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.2 is consists of questions related to properties of determinants. In Class 12th Maths chapter 4 exercise 4.2 there are 16 questions including 2 multiple choice type questions. There are 11 examples given in NCERT book before the exercise 4.2 Class 12 Maths. First, try to solve these examples given in the textbook. It will help you to get conceptual clarity and solving NCERT problems. NCERT syllabus Class 12th Maths chapter 4 exercise 4.2 questions are very important for the board exam as well as for the engineering competitive exams.
Also Read| Determinants Class 12 Chapter 4 Notes
Also see-
Subject Wise NCERT Exampler Solutions
Happy learning!!!
The value of the determinant remains unchanged when the rows and columns of determinants are interchanged.
If any two rows of a determinant are interchanged then the sign of the determinant change.
The sign of the determinant change when any two columns of a determinant are interchanged.
The value of the determinant is zero if any two rows of a determinant are identical.
The value of the determinant is zero if any two columns of a determinant are identical.
If the order of the square matrix is 2 then |kA| = k^2|A|.
There are 16 questions including two multiple choice questions are given in this exercise. All questions are useful to get a conceptual clarity. Following NCERT syllabus is beneficial for the CBSE board exam
By clicking on the link you will get NCERT solutions. NCERT Solutions for Mathematics and Science are given chapter wise.
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Hello there! Thanks for reaching out to us at Careers360.
Ah, you're looking for CBSE quarterly question papers for mathematics, right? Those can be super helpful for exam prep.
Unfortunately, CBSE doesn't officially release quarterly papers - they mainly put out sample papers and previous years' board exam papers. But don't worry, there are still some good options to help you practice!
Have you checked out the CBSE sample papers on their official website? Those are usually pretty close to the actual exam format. You could also look into previous years' board exam papers - they're great for getting a feel for the types of questions that might come up.
If you're after more practice material, some textbook publishers release their own mock papers which can be useful too.
Let me know if you need any other tips for your math prep. Good luck with your studies!
It's understandable to feel disheartened after facing a compartment exam, especially when you've invested significant effort. However, it's important to remember that setbacks are a part of life, and they can be opportunities for growth.
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I hope this information helps you.
Hi,
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hello mahima,
If you have uploaded screenshot of your 12th board result taken from CBSE official website,there won,t be a problem with that.If the screenshot that you have uploaded is clear and legible. It should display your name, roll number, marks obtained, and any other relevant details in a readable forma.ALSO, the screenshot clearly show it is from the official CBSE results portal.
hope this helps.
Hello Akash,
If you are looking for important questions of class 12th then I would like to suggest you to go with previous year questions of that particular board. You can go with last 5-10 years of PYQs so and after going through all the questions you will have a clear idea about the type and level of questions that are being asked and it will help you to boost your class 12th board preparation.
You can get the Previous Year Questions (PYQs) on the official website of the respective board.
I hope this answer helps you. If you have more queries then feel free to share your questions with us we will be happy to assist you.
Thank you and wishing you all the best for your bright future.
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