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NCERT Solutions Class 12 Maths Chapter 4 Exercise 4.1 - Determinants

NCERT Solutions Class 12 Maths Chapter 4 Exercise 4.1 - Determinants

Edited By Komal Miglani | Updated on May 09, 2025 11:48 AM IST | #CBSE Class 12th

Suppose your friend told you that the following 2×2 matrix is given:[2345] and asked you to find its determinant. How would you approach this problem? This is where the concept of determinants comes in. Determinants provide a systematic way to find a scalar value from a square matrix, which is crucial in solving systems of equations, finding the area, and understanding matrix properties. The first exercise of this chapter of the NCERT book is based on the basics of determinants and calculating determinants of different orders of square matrices. NCERT Class 12 Maths Chapter 4 - Determinants, Exercise 4.1 introduces us to the basic idea of determinants and how to evaluate them. This article on the NCERT Solutions for Exercise 4.1 Class 12 Maths Chapter 4 offers clear and step-by-step solutions for the exercise problems to help the students understand the method and logic behind it. For syllabus, notes, and PDF, refer to this link: NCERT.

LiveCBSE Board Result 2025 LIVE: Class 10th, 12th results soon on cbse.gov.in; marksheets, DigiLocker codeMay 9, 2025 | 11:27 AM IST

The Central Board of Secondary Education (CBSE) has not announced any schedule to declare the CBSE Class 10, 12 board results 2025. However, as per previous years trends, the board is expected to declare the results in the mid-May. 


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Class 12 Maths Chapter 4 Exercise 4.1 Solutions: Download PDF

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Determinants Exercise:4.1

Question:1 Evaluate the following determinant- |24−5−1|

Answer:

The determinant is evaluated as follows

|24−5−1|=2(−1)−4(−5)=−2+20=18

Question:2(i) Evaluate the following determinant- |cos⁡θ−sin⁡θsin⁡θcos⁡θ|

Answer:

The given two by two determinant is calculated as follows

|cos⁡θ−sin⁡θsin⁡θcos⁡θ|=cos⁡θ(cos⁡θ)−(−sin⁡θ)sin⁡θ=cos2⁡θ+sin2⁡θ=1

Question:2(ii) Evaluate the following determinant- |x2−x+1x−1x+1x+1|

Answer:

We have determinant |x2−x+1x−1x+1x+1|

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

=(x+1)(x2−x+1−x+1)=(x+1)(x2−2x+2)

=x3−2x2+2x+x2−2x+2

=x3−x2+2

Question:3 If A=[1242] , then show that |2A|=4|A|

Answer:

Given determinant A=[1242] then we have to show that |2A|=4|A|,

So, A=[1242] then, 2A=2[1242]=[2484]

Hence we have |2A|=|2484|=2(4)−4(8)=−24

So, L.H.S. = |2A| = -24

then calculating R.H.S. 4|A|

We have,

|A|=|1242|=1(2)−2(4)=−6

hence R.H.S becomes 4|A|=4×(−6)=−24

Therefore L.H.S. =R.H.S.

Hence proved.

Question:4 If A=[101012004] then show that |3A|=27|A|

Answer:

Given MatrixA=[101012004]

Calculating 3A=3[101012004]=[3030360012]

So, |3A|=3(3(12)−6(0))−0(0(12)−0(6))+3(0−0)=3(36)=108

calculating 27|A|,

|A|=|101012004|=1|1204|−0|0204|+1|0100|=4−0+0=4

So, 27|A|=27(4)=108

Therefore |3A|=27|A|.

Hence proved.

Question:5(i) Evaluate the determinants.

|3−1−200−13−50|

Answer:

Given the determinant |3−1−200−13−50|;

now, calculating its determinant value,

|3−1−200−13−50|=3|0−1−50|−(−1)|0−130|+(−2)|003−5|

=3(0−5)+1(0+3)−2(0−0)=−15+3−0=−12.

Question:5(ii) Evaluate the determinants.

|3−4511−2231|

Answer:

Given determinant |3−4511−2231|;

Now calculating the determinant value;

|3−4511−2231|=3|1−231|−(−4)|1−221|+5|1123|

=3(1+6)+4(1+4)+5(3−2)=21+20+5=46.

Question:5(iii) Evaluate the determinants.

|012−10−3−230|

Answer:

Given determinant |012−10−3−230|;

Now calculating the determinant value;

|012−10−3−230|=0|0−130|−1|−1−3−20|+2|−10−23|

=0−1(0−6)+2(−3−0)=6−6=0

Question:5(iv) Evaluate the determinants.

|2−1202−13−50|

Answer:

Given determinant: |2−1−202−13−50|,

We now calculate determinant value:

|2−1−202−13−50|=2|2−1−50|−(−1)|0−130|+(−2)|023−5|

=2(0−5)+1(0+3)−2(0−6)=−10+3+12=5

Question:6 If A=[11−221−354−9] , then find |A|.

Answer:

Given the matrix A=[11−221−354−9] then,

Finding the determinant value of A;

|A|=1|1−34−9|−1|2−35−9|−2|2154|

=1(−9+12)−1(−18+15)−2(8−5)=3+3−6=0

Question:7(i) Find values of x, if

|2451|=|2x46x|

Answer:

Given that |2451|=|2x46x|

First, we solve the determinant value of L.H.S. and equate it to the determinant value of R.H.S.,

|2451|=2(1)−4(5)=2−20=−18 and |2x46x|=2x(x)−4(6)=2x2−24

So, we have then,

−18=2x2−24 or 3=x2 or x=±3

Question:7(ii) Find values of x, if

|2345|=|x32x5|

Answer:

Given |2345|=|x32x5|;

So, we here equate both sides after calculating each side's determinant values.

L.H.S. determinant value;

|2345|=2(5)−3(4)=10−12=−2

Similarly R.H.S. determinant value;

|x32x5|=5(x)−3(2x)=5x−6x=−x

So, we have then;

−2=−x or x=2.

Question:8 If |x218x|=|62186| , then x is equal to

(A) 6 (B) ±6 (C) −6 (D) 0

Answer:

Solving the L.H.S. determinant ;

|x218x|=x(x)−2(18)=x2−36

and solving R.H.S determinant;

|62186|=36−36=0

So equating both sides;

x2−36=0 or x2=36 or x=±6

Hence answer is (B).


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Background wave

Topics covered in Chapter 4, Determinants: Exercise 4.1

Here are the main topics covered in NCERT Class 12 Chapter 4, Determinants: Exercise 4.1.

1. Definition:

We know that multiplication and addition are basic operations in matrices. In determinants, we evaluate a scalar value from a square matrix using specific rules. For example, for a 2×2 matrix

[abcd], the determinant is given by ad−bc

2. Determinant of a 2×2 Matrix: Let the matrix A be defined as:

A=[a11a12a21a22]

To evaluate the determinant of a 2×2 matrix, use the formula:

det(A)=a11a22−a12a21

3. Determinant of a 3×3 Matrix:

For 3×3 matrices, the determinant is calculated using expansion by minors.

|abcdefghi|=a(ei−fh)−b(di−fg)+c(dh−eg)

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Given below are some useful links for subject-wise NCERT solutions of class 12.

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Frequently Asked Questions (FAQs)

1. what is determinant ?

The determinant is a scalar value of a square matrix which characterize some properties of the matrix.

2. If square matrix A is order of 2 then | 2A | = ?

|2A| = 2^2 = 4 | A |

3. How many exercises are there in Class 12 Maths chapter 4 ?

There are 6 exercises and one miscellaneous exercise given in the NCERT textbook Class 12 Maths chapter 4.

4. Does CBSE provides NCERT solutions for Class 12 Maths chapter 4?

No, CBSE doesn't provide NCERT solutions, you can get NCERT solutions for chapter 4 Class 12 Maths.

5. What is the determinant of a singular matrix ?

The determinant of a singular matrix is always zero.

6. What is non-singular matrix?

A matrix that is not singular is called a non-singular matrix. The determinant of a non-singular matrix is non-zero.  

7. What is syllabus for CBSE Class 12 Maths ?
8. how many chapters are there in NCERT Class 12 Maths syllabus?

There are 13 chapters in the NCERT Class 12 Maths book.

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Changing from the CBSE board to the Odisha CHSE in Class 12 is generally difficult and often not ideal due to differences in syllabi and examination structures. Most boards, including Odisha CHSE , do not recommend switching in the final year of schooling. It is crucial to consult both CBSE and Odisha CHSE authorities for specific policies, but making such a change earlier is advisable to prevent academic complications.

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Manasvini Gupta 30 January,2025
quartely question paper for mathematics cbse

Hello there! Thanks for reaching out to us at Careers360.

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Jefferson 25 September,2024
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Kanishka kaushikii 17 September,2024
i scored 84 percent in cbse 2 years ago.Am i eligible for medhavi scholarship if give 12th again after 2 drop years and will got 75+ in mpboard

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Yuvan 30 August,2024
i upload 12th board result screenshot in ncweb form from the cbse site so it is valid

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Ashvadha 12 July,2024
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