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Edited By Ramraj Saini | Updated on Dec 04, 2023 12:16 PM IST | #CBSE Class 12th

**NCERT Solutions for miscellaneous exercise chapter 3 class 12 Matrices **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 3 Miscellaneous Exercise. These **miscellaneous exercise chapter 3** Class 12 consist of questions from all the different topics of this chapter. If you have done with all the exercises problems, you should be able to these miscellaneous exercise chapter 3 Class 12 problems. Miscellaneous exercises are considered to be difficult as compared to other exercises but students who have solved NCERT exercises on their own can solve these exercises easily.

You may find difficulties while these problems, you can take help from NCERT Solutions for Class 12 Maths Chapter 3 Miscellaneous Exercise. Miscellaneous exercises are not very important for the board exams as over 90% of the questions in the board exams are not asked from Class 12 Maths chapter 3 miscellaneous exercise solutions. **Miscellaneous exercise class 12 chapter 3**** **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 enumerated in NCERT Book together using the link provided below.

**Also, see**

**Question:1 **Let , show that , where I is the identity matrix of order 2 and .

**Answer:**

Given :

To prove :

For n=1,

The result is true for n=1.

Let result be true for n=k,

Now, we prove that the result is true for n=k+1,

Put the value of in above equation,

Hence, the result is true for n=k+1.

Thus, we have where ,.

**Question 2. **If then show that , .

**Answer:**

Given :

To prove:

For n=1, we have

Thus, the result is true for n=1.

Now, take n=k,

For, n=k+1,

Thus, the result is true for n=k+1.

Hence, we have , where .

**Question 3. **If , then prove that , where *n* is any positive integer.

**Answer:**

Given :

To prove:

For n=1, we have

Thus, result is true for n=1.

Now, take result is true for n=k,

For, n=k+1,

Thus, the result is true for n=k+1.

Hence, we have , where .

**Question 4. **If A and B are symmetric matrices, prove that is a skew symmetric matrix.

**Answer:**

If A, B are symmetric matrices then

and

we have,

Hence, we have

Thus,( AB-BA)' is skew symmetric.

**Question 5. **Show that the matrix B′AB is symmetric or skew symmetric according as A is symmetric or skew symmetric.

**Answer:**

Let be a A is symmetric matrix , then

Consider,

Replace by

i.e.

Thus, if A is a symmetric matrix than is a symmetric matrix.

Now, let A be a skew-symmetric matrix, then .

Replace by -,

i.e. .

Thus, if A is a skew-symmetric matrix then is a skew-symmetric matrix.

Hence, the matrix B′AB is symmetric or skew-symmetric according to as A is symmetric or skew-symmetric.

**Question 6. **Find the values of *x*, *y*, *z* if the matrix satisfy the equation

**Answer:**

Thus equating the terms element wise

**Question 10(a) **A manufacturer produces three products x, y, z which he sells in two markets.

Annual sales are indicated below:

** Market Products**

I 10,000 2,000 18,000

II 6,000 20,000 8,000

If unit sale prices of x, y and z are ` 2.50, ` 1.50 and ` 1.00, respectively, find the total revenue in each market with the help of matrix algebra.

**Answer:**

The unit sale prices of x, y and z are ` 2.50, ` 1.50 and ` 1.00, respectively.

The total revenue in the market I with the help of matrix algebra can be represented as :

The total revenue in market II with the help of matrix algebra can be represented as :

Hence, total revenue in the market I is 46000 and total revenue in market II is 53000.

**Question 10(b). **A manufacturer produces three products x, y, z which he sells in two markets.

Annual sales are indicated below:

** Market Products**

I 10,000 2,000 18,000

II 6,000 20,000 8,000

If the unit costs of the above three commodities are ` 2.00, ` 1.00 and 50 paise respectively. Find the gross profit.

**Answer:**

The unit costs of the above three commodities are ` 2.00, ` 1.00 and 50 paise respectively.

The total cost price in market I with the help of matrix algebra can be represented as :

Total revenue in the market I is 46000 , gross profit in the market is

The total cost price in market II with the help of matrix algebra can be represented as :

Total revenue in market II is 53000, gross profit in the market is

**Question 11. **Find the matrix *X* so that

**Answer:**

The matrix given on R.H.S is matrix and on LH.S is matrix.Therefore, X has to be matrix.

Let X be

Taking,

Hence, we have

Matrix X is .

**Answer:**

A and B are square matrices of the same order such that ,

To prove : ,

For n=1, we have

Thus, the result is true for n=1.

Let the result be true for n=k,then we have

Now, taking n=k+1 , we have

Thus, the result is true for n=k+1.

Hence, we have , .

To prove:

For n=1, we have

Thus, the result is true for n=1.

Let the result be true for n=k,then we have

Now, taking n=k+1 , we have

Thus, the result is true for n=k+1.

Hence, we have and for all .

**Question 13 **Choose the correct answer in the following questions:

** **If is such that

(A)

(B)

(C)

(D)

**Answer:**

Thus we obtained that

Option C is correct.

**Question 14. **If the matrix A is both symmetric and skew-symmetric, then

(A) A is a diagonal matrix

(B) A is a zero matrix

(C) A is a square matrix

(D) None of these

**Answer:**

** **If the matrix A is both symmetric and skew-symmetric, then

and

Hence, A is a zero matrix.

Option B is correct.

**Question 15. **If A is square matrix such that , then is equal to

(A) *A*

(B) *I – A*

(C) *I*

(D) *3A*

**Answer:**

A is a square matrix such that

(Replace by )

Hence, we have

Option C is correct.

**More about ****NCERT Solutions for Class 12 Maths Chapter 3 Miscellaneous Exercise:-**

There are 3 miscellaneous solved examples are given in the NCERT textbook before themiscellaneous exercise which could be solved if you have conceptual clarity. You can solve the 12 long answer type questions and 3 multiple choice types questions given in the miscellaneous exercise of NCERT syllabus chapter 3 Class 12. Solving these problems will check your understanding of this chapter.

**Also Read| ****Matrices Class 12 Maths Chapter Notes**

**Benefits of NCERT for Class 12 Maths Chapter 3 ****Miscellaneous Exercise:-**

- Class 12 Maths chapter 3 miscellaneous exercise solutions are available at one place which are helpful for the students who are facing problems while solving them.
- First, try to solve previous exercises then you will be able to solve the miscellaneous exercise.
- Class 12 Maths chapter 3 miscellaneous exercise solutions are also helpful for the students who are preparing for the engineering entrance exams as some questions in these exams are directly asked from miscellaneous exercises.
- You can use Class 12 Maths chapter 3 miscellaneous solutions for quick revision before the exam.

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Download EBook**Comprehensive Coverage:**The solutions encompass all the topics covered in miscellaneous exercise class 12 chapter 3, ensuring a thorough understanding of the concepts.**Step-by-Step Solutions:**In this class 12 chapter 3 maths miscellaneous solutions, each problem is solved systematically, providing a stepwise approach to aid in better comprehension for students.**Accuracy and Clarity:**Solutions for class 12 maths miscellaneous exercise chapter 3 are presented accurately and concisely, using simple language to help students grasp the concepts easily.**Conceptual Clarity:**In this class 12 maths ch 3 miscellaneous exercise solutions, emphasis is placed on conceptual clarity, providing explanations that assist students in understanding the underlying principles behind each problem.**Inclusive Approach:**Solutions for class 12 chapter 3 miscellaneous exercise cater to different learning styles and abilities, ensuring that students of various levels can grasp the concepts effectively.**Relevance to Curriculum:**The solutions for miscellaneous exercise class 12 chapter 3 align closely with the NCERT curriculum, ensuring that students are prepared in line with the prescribed syllabus.

As per latest 2024 syllabus. Maths formulas, equations, & theorems of class 11 & 12th chapters

**Also see-**

- 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

**Happy learning!!!**

1. If matrix B is the inverse of matrix A, then what is inverse of B ?

If matrix B is the inverse of matrix A, then matrix A is also the inverse of matrix B.

2. Which is the Best Book for NCERT Class 12 Maths ?

NCERT textbook is the most important book for the students who are preparing for board exams. CBSE usually use syllabus similar to the NCERT syllabus. For more questions from the chapter Matrices NCERT exemplar for Class 12 Maths can be used.

3. Where can I get the NCERT exemplar for Class 12 Maths chapter 3?

Here you will get NCERT exemplar for class 12 maths chapter 3.

4. Number of chapters in the NCERT class 12 maths ?

There are 13 chapters in the NCERT Class 12 maths.

5. what is the weightage of calculus in CBSE class 12 maths ?

Calculus carries 35 marks weighatge in the CBSE final board exam.

6. what is the weightage of Algebra in CBSE class 12 maths ?

Algebra carries 10 marks weighatge in the CBSE final board exam.

7. What are miscellaneous exercises ?

As the name suggests miscellaneous exercises consist of a mixture of questions from all the exercises of the chapter.

8. Do miscellaneous exercises are important for competitive exams ?

Yes, miscellaneous exercises are very important for competitive exams like JEE, SRMJEE, etc.

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6 minHave a question related to CBSE Class 12th ?

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.
**

Hello student,

**
If you are planning to appear again for class 12th board exam with PCMB as a private candidate here is the right information you need:
**

- No school admission needed! Register directly with CBSE. (But if you want to attend the school then you can take admission in any private school of your choice but it will be waste of money)
- You have to appear for the 2025 12th board exams.
- Registration for class 12th board exam starts around September 2024 (check CBSE website for exact dates).
- Aim to register before late October to avoid extra fees.
- Schools might not offer classes for private students, so focus on self-study or coaching.

**
Remember
**
, these are tentative dates based on last year. Keep an eye on the CBSE website ( https://www.cbse.gov.in/ ) for the accurate and official announcement.

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 help you.

**
Good luck with your studies!
**

Hello Aspirant , Hope your doing great . As per your query , your eligible for JEE mains in the year of 2025 , Every candidate can appear for the JEE Main exam
**
6 times over three consecutive years
**
. The JEE Main exam is held two times every year, in January and April.

Hi there,

Hope you are doing fine

Yes you are certainly eligible for giving the jee exam in the year 2025. You must pass the maths exam with at least 75% criteria as required by jee and provide the marksheet and the passing certificate while registering for the exam.

Pursuing maths as an additional subject while taking biology as your main subject does not offer any hindrance in you appearing for the jee examination. It is indeed an privilege to pursue both maths and biology as the subjects and prepare for the same.

There will be no issue in filling the form while registering for the exam as it will only require your basic details and marksheet which you can provide by attaching the marksheet of maths also. Also, a detailed roadmap is also available on the official websites on how to fill the registration form. So you can fill the form easily.

Hope this resolves your query.

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