# NCERT Solutions for Exercise 4.2 Class 12 Maths Chapter 4 - Determinants

## Determinants Exercise:4.2

Question:1 Using the property of determinants and without expanding, prove that

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

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

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

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

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

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

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:

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:

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:

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:

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:

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:

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:

Give determinant

Applying column transformation we get;

[after taking the (1+x+x2 ) 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:

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:

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 C1,C2,and C3.

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)

Assume a square matrix A of order of .

Then we have;

(Taking the common factors k from each row.)

Therefore correct option is (C).

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

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.

## More About NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.2

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

## Benefits of NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.2

• Class 12 Maths chapter 4 exercise 4.2 solutions are prepared by the subject matter experts who know how best to answer in order to perform well in the board exams.
• Class 12th Maths chapter 4 exercise 4.2 questions are prepared in a very descriptive manner which you will get easily.
• NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.2 are important in competitive exams like JEE, SRMJEE, etc.
• As most of the time, one question from this exercise is asked in the board exam, so you are advised to be thorough with them.
• You can use these NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.2 for reference.

## Key Features Of NCERT Solutions for Exercise 4.2 Class 12 Maths Chapter 4

• Comprehensive Coverage: The solutions encompass all the topics covered in ex 4.2 class 12, ensuring a thorough understanding of the concepts.
• Step-by-Step Solutions: In this class 12 maths ex 4.2, each problem is solved systematically, providing a stepwise approach to aid in better comprehension for students.
• Accuracy and Clarity: Solutions for class 12 ex 4.2 are presented accurately and concisely, using simple language to help students grasp the concepts easily.
• Conceptual Clarity: In this 12th class maths exercise 4.2 answers, emphasis is placed on conceptual clarity, providing explanations that assist students in understanding the underlying principles behind each problem.
• Inclusive Approach: Solutions for ex 4.2 class 12 cater to different learning styles and abilities, ensuring that students of various levels can grasp the concepts effectively.
• Relevance to Curriculum: The solutions for class 12 maths ex 4.2 align closely with the NCERT curriculum, ensuring that students are prepared in line with the prescribed syllabus.

Also see-

• NCERT Solutions for Class 12 Maths Chapter 4
• NCERT Exemplar Solutions Class 12 Maths Chapter 4

## NCERT Solutions of Class 12 Subject Wise

• NCERT Solutions for Class 12 Maths
• NCERT Solutions for Class 12 Physics
• NCERT Solutions for Class 12 Chemistry
• NCERT Solutions for Class 12 Biology

Subject Wise NCERT Exampler Solutions

• NCERT Exemplar Solutions for Class 12th Maths
• NCERT Exemplar Solutions for Class 12th Physics
• NCERT Exemplar Solutions for Class 12th Chemistry
• NCERT Exemplar Solutions for Class 12th Biology

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