Toppers Study Toppers Study Toppers StudyDetailed NCERT Solutions for 12 Mathematics I 4. Determinants to simplify learning. Understand chapters clearly and practice with free solutions for better results.
Detailed NCERT Solutions for 12 Mathematics I 4. Determinants to simplify learning. Understand chapters clearly and practice with free solutions for better results.
Preparing for exams becomes easier with Miscellaneous Exercise on Chapter - 4. Whether you are studying for board exams or mid-term exams, 12 Mathematics I Chapter 4. Determinants solutions provide quick revising points, well-structured answers, and additional practice material to help you score better.
ncert_solutions
is independent of 
Ans. Let 
Expanding along R1,
= 
= 
which is independent of
Ans. L.H.S. = 
Multiplying R1 by 
R2 by 
and R3 by 
, 
= 
= 
= 
[Interchanging C1 and C3]
= 
= 
[Interchanging C2 and C3]
Proved.
Ans. Let 
Expanding along R1,
=
= 
= 
= 
= 1
and are real numbers and 
Show that either 
or 
Ans. Given: 
Here, Either 
……….(i)
Or 
[Expanding along first row]
and 
and 
and 
and 
……….(ii)
hance, from eq. (i) and (ii), either or 
Ans. Given: 
Either 
……….(i)
Or 
But this is contrary as given that 
.
hance, from eq. (i), is only the solution.
Ans. L.H.S. = 
= 
= 
= 
= 
= 
= 
= 
= R.H.S. Proved.
and B = 
find 
Ans. Given: 
and B = 
Since, 
[Reversal law] ……….(i)
Now 
= 
= 
hance, 
exists.
and 
and 
adj. B = 
= 
From eq. (i), 
= 
verify that:(i) 
(ii) 
Ans. Given: Matrix A = 
= 
Therefore, 
exists.
and 
and 
adj. A = 
= B (say)
= 
………(i)
= 
= 
hance, exists.
and 
and 
adj. B = 
= 
= 
= 
….(ii)
Now to find 
(say), where
C = 
= 
C = 
C = 
= 
=
hance, 
exists.
and 
and 
adj. A = 
= 
……….(iii)
Again 
= 
= 
= A (given)
(i)
=
[From eq. (ii) and (iii)]
(ii) 
=
Ans. Let 
= 
= 
= 
= 
= 
= 
= 
= 
= 
Ans. Let 
= 
= 
= 
= 
Ques.11. 
Ans. L.H.S. = 
= 
=
= 
= 
= 
= 
= 
= 
= 
= R.H.S.
Ans. L.H.S. = 
= 
= 
(say) ……….(i)
Now 
= 
= 
= 
From eq. (i), L.H.S. = 
……….(ii)
Now 
= 
Expanding along C1, 
= 
= 
= 
= 
= 
From eq. (i), L.H.S. = 
= 
= R.H.S.
Ques.13. 
Ans. L.H.S. = 
= 
= 
= 
= 
= 
= 
= 
= 
= R.H.S.
Ques.14. 
Ans. L.H.S. = 
= 
= 
= 
= 
= 1 = R.H.S.
= 0Ans. L.H.S. = 
= 
= 
= 
= 
= 
[
C2 and C3 have become identical]
= 0 = R.H.S
Ques.16. Solve the system of the following equations: (Using matrices):
Ans. Putting 
and 
in the given equations,
the matrix form of given equations is 
[AX= B]
Here, A = 
X = 
and B = 
= 
= 
exists and unique solution is 
……….(i)
Now 
and 
and 
adj. A = 
= 
And 
Miscellaneous Exercise on Chapter - 4 are created by experts to give step-by-step explanations. Around 60–70% of exam questions are based on NCERT concepts.
Our 12 Mathematics I Chapter 4. Determinants solutions help you understand the core concepts and practice effectively.
Revision is the key to exam success. Our notes for 12 Mathematics I highlight important formulas, key definitions,
and exam-ready points from Chapter 4. Determinants. These quick revision notes make last-minute preparation easy.
Every NCERT chapter ends with exercises, and solving them is crucial. Our Miscellaneous Exercise on Chapter - 4 include complete solutions for
12 Mathematics I Chapter 4. Determinants exercises. With step-by-step answers, you gain clarity and
confidence to attempt similar exam questions.
To boost your preparation, we also provide additional important questions with answers. These are prepared from previous year board papers,
sample papers, and important concepts of Chapter 4. Determinants. Practicing these ensures you are well-prepared for both board and mid-term exams.
Our Miscellaneous Exercise on Chapter - 4 are useful for both board exams and mid-term exams.
For 12 Mathematics I, we provide notes, exercises, and important Q&A so that you can
revise smartly and write perfect answers in exams.
In short, Miscellaneous Exercise on Chapter - 4 for 12 Mathematics I Chapter 4. Determinants
are a complete study package. With quick revising points, NCERT exercises,
and additional important questions, you can prepare effectively for exams.
Make these solutions your study companion and excel in your academic journey.
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📘 Why Miscellaneous Exercise on Chapter - 4 are Important?
✍️ Quick Revising Points as Notes in Page-1
📚 NCERT Exercise Solutions
📝 Additional Important Questions & Answers
🎯 Useful for Board & Mid-Term Exams
🌟 Final Words
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