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Edited By Ravindra Pindel | Updated on Sep 15, 2022 04:51 PM IST | #CBSE Class 12th

**NCERT exemplar Class 12 Maths solutions chapter 4** Determinants will help in learning the ways to find the determinant of various square matrices, co-factors, the inverse of matrices etc. NCERT exemplar Class 12 Maths chapter 4 solutions are useful for the students to get a deeper and better look at the matrices and how to solve them uniquely. From the scoring point of view, chapter 4 of NCERT Class 12 Maths Solutions can be very crucial for 12 Class students. Students can use NCERT exemplar Class 12 Maths solutions chapter 4 PDF download and study the topic and the solutions offline as well.

Question:4

Using the properties of determinants in evaluate:

Answer:

Apply -

Now, expand the determinant along Column 1

Question:6

Using the properties of determinants in evaluate:

Answer:

Now apply

Expand the determinant along Row 1

Question:7

Using the properties of determinants in prove that:

Answer:

Taking LHS,

Whenever any the values in any two rows or columns of a determinant are identical, the resultant value of that determinant is 0

Hence,

∴ LHS = RHS

Question:8

Using the properties of determinants in prove that:

Answer:

LHS given,

Take y,z,x common from the R1, R2 and R3 respectively

Hence Proved

Question:9

Using the properties of determinants in prove that:

Answer:

Take LHS

Take (a-1) common from

Hence,Proved.

Question:11

Answer:

Area of a triangle with the given vertices will be:

Question:12

Find the value of θ satisfying

Answer:

Given:

Expand along Row 1

We know,

But it is not possible to have

Question:14

If a1, a2, a3, ..., ar are in G.P., then prove that the determinant is independent of r.

Answer:

A is the first term of G.PR is the common ratio of G.P.

Whenever any the values in any two rows or columns of a determinant are identical, the resultant value of that determinant is 0

Rows 1 and 2 are identical

Question:15

Answer:

(a + 5, a – 4), (a – 2, a + 3) and (a, a) is given.

We need to prove that they don’t line in a straight line for any value of a

This can be done by proving the points to be vertices of triangle.

Area of triangle:-

This proves that the given points form a triangle therefore do not lie on a straight line.

Question:16

Show that the Δ ABC is an isosceles triangle if the determinant

Answer:

Expand along Row 1

Hence, ΔABC is an isosceles triangle.

Question:18

If find . Using , solve the system of linear equations

Answer:

Find IAI Expand IAl along Column 1

According to linear equation:

We know that, AX = B

Here,

So, transpose of

Question:19

Answer:

∴ x = 1, y = 1 and z = 1

Question:20

Given find BA and use this to solve the system of equations

Answer:

Now, given system of equations is:

Question:21

If a + b + c ≠ 0 and then prove that a = b = c.

Answer:

Expand along Column 1

Given that Δ = 0

Hence Proved

Question:22

Prove that is divisible by a + b + c and find the quotient.

Answer:

is given.

Apply,

Take (a+b+c) common from Column 1

Take (a+b+c) common from Column 2

Expand along Column 1

The determinant is divisible by (a+b+c) and the quotient is

Question:25

The value of determinant

A.

B. 3 bc

C.

D. none of these

Answer:

C)

Given:

Apply C2→ C2 + C3

Expand along Row 1

Question:26

The area of a triangle with vertices (–3, 0), (3, 0) and (0, k) is 9 sq. units. The value of k will be

A. 9

B. 3

C. – 9

D. 6

Answer:

B)

Expand along Column 2

Question:27

A. abc (b–c) (c – a) (a – b)

B. (b–c) (c – a) (a – b)

C. (a + b + c) (b – c) (c – a) (a – b)

D. None of these

Answer:

D)

Given:

Whenever any two columns or rows in any determinant are equal, its value becomes = 0

Here Column 1 and 2 are identical

Question:28

The number of distinct real roots of in the interval is

A. 0

B. –1

C. 1

D. None of these

Answer:

C)

Given

Expand along Column 1

Only one real and distinct root occurs.

Question:29

If A, B and C are angles of a triangle, then the determinant is equal to

A. 0

B. –1

C. 1

D. None of these

Answer:

A)

Given:

Expand along Column 1

Question:32

A. f (a) = 0

B. f (b) = 0

C. f (0) = 0

D. f (1) = 0

Answer:

C)

We have:

If x = 0 according to the given question:

Then the condition is satisfied.

Question:33

then exists if

A. λ = 2

B. λ ≠ 2

C. λ ≠ -2

D. None of these

Answer:

D)

We have

So, exists if and only if

Question:34

If A and B are invertible matrices, then which of the following is not correct?

A.

B.

C.

D.

Answer:

D)

We know, A and B are invertible matrices

Question:35

If x, y, z are all different from zero and ,, then value of is

Answer:

We have

Expand along Row 1

Divide both side by xyz

Question:37

There are two values of a which makes determinant, ,then sum of these number is

A. 4

B. 5

C. -4

D. 9

Answer:

C)

Sum of -7 and 3 = -4

Question:38

Fill in the blanks

If A is a matrix of order 3 × 3, then |3A| = ___.

Answer:

If A is a matrix of order we know:

Question:39

Fill in the blanks

If A is invertible matrix of order 3 × 3, then ||= ____.

Answer:

If A is invertible matrix of order GivenA = invertible matrix 3x3

Question:42

Fill in the blanks

If A is a matrix of order 3 × 3, then = ____.

Answer:

For matrix A is of order 3X3

Question:43

If A is a matrix of order 3 × 3, then number of minors in determinant of A are

Answer:

If matrix A is of order 3X3 then

Number of Minors of IAI = 9 as there are 9 elements in a 3x3 matrix

Question:44

Answer:

If,

We know that the determinant is equal to sum of corresponding co factors of any row or column

Question:49

State True or False for the statements

, where a is any real number and A is a square matrix.

Answer:

For a non singular matrix, aA is invertible such that

here a = any non-zero scalar. Here A should be a non-singular matrix which is not given in the statement, thus the statement given in question in false.

Question:50

State True or False for the statements

, where A is non-singular matrix.

Answer:

We know A is a non singular Matrix

In that case:

Thus the statement is false.

Question:51

Answer:

We know that:

Hence the statement given in question is true.

Question:52

Answer:

For any square matrix of order n,

Thus the given statement is true.

Question:53

State True or False for the statements

, where a, b, c are in A.P.

Answer:

Since 2b = a + c,

We can see that Row 1 and 3 are proportional

Thus determinant = 0

Question:54

State True or False for the statements

, where A is a square matrix of order two.

Answer:

For any square matrix of order n, Here n =2, Thus the statement given in question is false

Question:55

State True or False for the statements

The determinant is equal to zero.

Answer:

We can see that columns are identical in both the matrix on Right hand side

Thus Determinant = 0

Statement in question is therefore true

Question:56

Answer:

Split row 2

We can split all the rows in the same way. Thus the statement given in the question is true.

Question:57

State True or False for the statements

Let ,then

Answer:

2 can be taken common from Column 1

After that apply C1 → C1 – C2 and C2→ C2 – C3

Again, second determinant of column 1 and 3 are identical

Hence the statement given in question is true

Question:58

State True or False for the statements

The maximum value of is 1/2.

Answer:

Since the maximum value of

Thus the given statement is true.

Determinants are related to the matrices that are solved in chapter 3 of 12 Class NCERT Maths book. Studying determinants is not just about passing exams, but is about being prepared for higher education in any field of maths, science, economics, etc. In Class 12 Maths NCERT exemplar solutions chapter 4, the students will learn about determinants, their elements, and how to calculate determinants of various square matrices.

The sub-topics that are covered in this chapter of NCERT Class 12 solution are:

- Introduction
- Determinant
- Determinant of a matrix of order one
- Determinant of a matrix of order 2
- Determinant of a matrix of order 3x3
- Properties of determinants
- Area of triangle
- Minors and co-factors
- Adjoint and inverse of a matrix
- Adjoint of a matrix
- Applications of matrices and determinants
- Solution of a system of linear equations using the inverse of matrices

- The determinants are a value of scalar nature derived from the square matrix-like 2x2, 3x3, etc. This determinant then helps in understanding the linear equation defined by the matrix.
- Determinants derived help in determining whether the solution of the linear system is unique or not.
- There are many other related topics covered in this Class 12 Maths NCERT exemplar solutions chapter 4, like the area of triangle, co-factors, minors, inverse of a matrix, finding the equation of line, etc.

- We will help in finding the NCERT exemplar Class 12 Maths chapter 4 solutions to the questions that are given in the NCERT book in detail. We don't skip steps or try to solve the questions in a trick or the easy way; instead, our expert guides will help in solving every question in the minutest detail kept in mind.
- The language used for solving the questions is simple and easy to understand. Not only the questions are solved exhaustively; we also keep a marking scheme in mind to help our students score better.
- NCERT Exemplar Class 12 Maths solutions chapter 4 will help in gaining confidence in solving more difficult questions. Detailed steps in easy language will help the students to understand the topics and its intricacies.

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Download EBookChapter 1 | |

Chapter 2 | |

Chapter 3 | |

Chapter 4 | Determinants |

Chapter 5 | |

Chapter 6 | |

Chapter 7 | |

Chapter 8 | |

Chapter 9 | |

Chapter 10 | |

Chapter 11 | |

Chapter 12 | |

Chapter 13 |

- NCERT Solution for Class 12 Physics
- NCERT Solution for Class 12 Chemistry
- NCERT Solution for Class 12 Maths
- NCERT Solution for Class 12 Biology

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

- NCERT Notes for Class 12 Physics
- NCERT Notes for Class 12 Chemistry
- NCERT Notes for Class 12 Maths
- NCERT Notes for Class 12 Biology

1. Can I download the solutions for this chapter?

Yes, you download NCERT exemplar Class 12 Maths solutions chapter 4 pdf by using the webpage to pdf tool available online.

2. What are the important topics of this chapter?

The Properties of Determinants, Adjoint and Inverse of a Matrix and Application of Determinants and Matrices are the more important topics among others as per their weightage.

3. How to study well for boards?

Practice, Practice and Practice. Once you have read the chapters well and made notes, you must practice being fast and precise with answers.

4. How many questions are there in this chapter?

The NCERT exemplar solutions for Class 12 Maths chapter 4 has one exercise with 58 questions for practice.

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6 minHave a question related to CBSE Class 12th ?

You can use them people also used problem

Hi,

The Medhavi National Scholarship Program, under the Human Resources & Development Mission (HRDM), offers financial assistance to meritorious students through a scholarship exam. To be eligible, candidates must be between 16 and 40 years old as of the last date of registration and have at least passed the 10th grade from a recognized board. Higher qualifications, such as 11th/12th grade, graduation, post-graduation, or a diploma, are also acceptable.

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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.
**

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!
**

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