Chapter 4: Determinants Class 12 NCERT Solutions – Questions and Answers PDF Free Download (Class 12th Maths) in Hindi

Determinants Class 12 NCERT Solutions PDF Free Download: Chapter 4 of Class 12 Math’s, titled Determinants, plays a crucial role in understanding advanced algebra and is heavily used in vectors, matrices, and calculus. It builds on the knowledge of matrices and introduces the concept of determinants, their properties, applications in solving systems of linear equations using Cramer’s Rule, and calculating area of triangles.

This chapter is foundational for competitive exams like JEE, CUET, and board exams. Through this chapter, students learn how to find the determinant of a matrix, use cofactor expansion, apply properties of determinants, and work with adjoint and inverse of a matrix using determinants. The clarity in concepts and formulas is vital to excel. To support your learning, we’ve provided NCERT solutions along with free downloadable handwritten notes that are concise, board-exam friendly, and easy to revise.

Key Features of Chapter 4: Determinants – Class 12 Math’s NCERT Solutions & Handwritten Notes PDF

• Subject: Maths (Determinants Class 12 NCERT Solutions – Class 12 Math’s)
• Language : Hindi
• Total pages : 54
• File size: 22.4 MB
• Format : PDF
• Well structured and easy to understand
• Includes importance formulas and definitions
• Covers all NCERT syllabus topics
• Useful for quick revision before exam

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Images of Chapter 4: Determinants Class 12 NCERT Solutions & Handwritten Notes PDF

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📘 Definition of Determinants (Class 12 Math’s – Chapter 4)

In mathematics, a determinant is a numerical value that is calculated from a square matrix (a matrix having the same number of rows and columns). It provides important information about the matrix and the system of equations it may represent. The determinant is denoted by symbols like det(A) or |A|, where A is a square matrix. For a 2×2 matrix:
If A=[abcd]A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}A=[ac​bd​], then its determinant is given by: ∣A∣=ad−bc|A| = ad – bc∣A∣=ad−bc. For a 3×3 matrix or higher-order matrices, the determinant is calculated using cofactors and minors, often by expanding along a row or column.

Determinants are crucial in linear algebra. A non-zero determinant means the matrix is invertible (it has an inverse), while a zero determinant indicates the matrix is singular (non-invertible). They are used in solving systems of linear equations using Cramer’s Rule, and also in calculating the area of a triangle from its vertices.

Geometrically, the determinant of a 2×2 or 3×3 matrix can be interpreted as a scaling factor or area/volume factor for linear transformations. Thus, determinants not only have algebraic importance but also a strong geometric meaning in vector spaces and transformations.

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📌 Why Use Our Class 12 Math’s Handwritten Notes?

✅ 1. Easy-to-Understand Language
Our notes are written in simple and clear language, making complex mathematical concepts easier to grasp—even for average students.

✅ 2. Neatly Organized & Exam-Focused
All chapters are presented in a structured manner, following the NCERT sequence. Important formulas, theorems, and solved examples are highlighted for quick revision.

✅ 3. Saves Time
Instead of spending hours making your own notes, you can start practicing and revising immediately. These notes are concise yet complete.

✅ 4. Handwritten for Better Memory Retention
Studies show that handwritten material improves memory recall. Our notes are designed to reinforce concepts visually and mentally.

✅ 5. Includes Important Board Questions & Short Tricks
We’ve added previous year board questions, tips, and shortcut methods to help you solve problems faster in exams.

✅ 6. Ideal for Last-Minute Revision
During pre-board or final exams, you need crisp summaries. Our notes provide exactly that without any fluff.

✅ 7. Printable and Shareable PDFs
Download once, and use it on your phone or take a printout – study anytime, anywhere.

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📘 Class 12 Math’s Handwritten Notes – Chapters List

📚 Chapter No.	📝 Chapter Name
1️⃣	Relations and Functions
2️⃣	Inverse Trigonometric Functions
3️⃣	Matrices
4️⃣	Determinants
5️⃣	Continuity and Differentiability
6️⃣	Application of Derivatives
7️⃣	Integrals
8️⃣	Application of Integrals
9️⃣	Differential Equations
🔟	Vector Algebra
1️⃣1️⃣	Three Dimensional Geometry
1️⃣2️⃣	Linear Programming
1️⃣3️⃣	Probability

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📚 How to Prepare for Chapter 4: Determinants – Class 12 Math’s NCERT Solutions & Handwritten Notes PDF

✅ 1. Understand the Basics First

• Begin by revising the definition and concept of determinants.
• Understand how to compute determinants for 2×2 and 3×3 matrices.
• Learn the difference between minor and cofactor—a frequent source of confusion.

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✅ 2. Focus on Properties of Determinants

• Memorize the key properties of determinants (like row/column operations).
• Solve questions that test these properties directly.
• This section helps in solving problems efficiently without full expansion.

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✅ 3. Practice Adjoint & Inverse of Matrices

• Learn how to calculate the adjoint of a matrix using cofactors.
• Use the formula A−1=1∣A∣⋅adj(A)A^{-1} = \frac{1}{|A|} \cdot \text{adj}(A)A−1=∣A∣1​⋅adj(A), and ensure you only apply it when ∣A∣≠0|A| ≠ 0∣A∣=0.

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✅ 4. Master Cramer’s Rule

• Understand how to solve linear equations using Cramer’s Rule.
• Practice problems where you compute multiple determinants D,Dx,Dy,DzD, D_x, D_y, D_zD,Dx​,Dy​,Dz​.

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✅ 5. Use Handwritten Notes Wisely

• Refer to handwritten notes for revision—especially important formulas and solved examples.
• Keep a one-page summary of all key properties and formulas.

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✅ 6. Solve NCERT & Exemplars

• Practice all NCERT exercise problems and miscellaneous exercises.
• Refer to NCERT solutions PDF to cross-check your answers.

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✅ 7. Attempt Previous Year Questions (PYQs)

• Identify frequently asked patterns.
• Practice questions involving determinants in application-based contexts (like area of triangle).

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✅ 8. Revise Regularly

• Keep revising determinant properties.
• Use flashcards or revision sheets for last-minute brushing up.

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🌟 Why Are Determinants Class 12 NCERT Solutions Handwritten Notes Special for You?

✅ Tailored for Board Exams
These notes are designed as per the latest CBSE NCERT 2025-26 syllabus with a sharp focus on what’s important for board exams—no extra or confusing content.

✅ Made by Toppers, For Toppers
Created by previous year toppers and subject experts, these notes follow a smart approach to explain even tough topics like Determinants in the simplest way possible.

✅ Handwritten = High Memory Retention
Research proves that handwritten material improves recall. You’ll find it easier to remember formulas, steps, and tricks when you revise from these clean, visual notes.

✅ Concept + Formula + PYQs = All-in-One
You get all key concepts, formulas, solved NCERT questions, and previous year questions in one place—no need to refer to multiple books.

✅ Time-Saving & Revision-Friendly
Perfect for quick revision before exams. Each page is neat, well-structured, and helps you scan through topics in minutes.

✅ Offline & Portable
Download once, study anytime—even without the internet.

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🎯 Benefits of Using Handwritten Notes for Class 12 Math’s

✅ 1. Improves Memory & Understanding
Writing and reading from handwritten notes helps in better retention of concepts and formulas compared to printed textbooks.

✅ 2. Well-Organized & Chapter-Wise
The notes are neatly structured chapter-wise, making it easy to find and revise topics like Determinants, Matrices, and more.

✅ 3. Covers Important Points Only
They highlight only the most important concepts, key formulas, and board exam questions—saving you from unnecessary overload.

✅ 4. Best for Last-Minute Revision
During pre-boards or finals, handwritten notes act as a quick-revision tool. One glance and you’re ready to recall everything.

✅ 5. Time-Saving
No need to prepare your own notes from scratch—just download and start studying instantly.

✅ 6. Includes Tricks & Tips
Smart shortcuts and solving techniques are often written in the margins—giving you an edge in exams.

✅ 7. Designed by Toppers
These notes are prepared by top-performing students and experienced teachers, ensuring clarity and quality.

✅ 8. Boosts Confidence
When you revise using handwritten notes regularly, your confidence builds up, helping reduce exam fear.

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📘 Chapter 4 – Determinants: Important Formulae Table

🔢 Concept	🧮 Formula / Explanation
Determinant of 2×2 Matrix	(
Determinant of 3×3 Matrix	Expansion using cofactors along any row/column
Minor of an element aija_{ij}aij​	Determinant of the matrix obtained by deleting ith row and jth column
Cofactor of aija_{ij}aij​	Aij=(−1)i+j⋅MijA_{ij} = (-1)^{i+j} \cdot M_{ij}Aij​=(−1)i+j⋅Mij​
Properties of Determinants	– Interchanging rows/columns changes sign – If two rows/columns are equal → det = 0
Row/Column Multiplication	If one row/column is multiplied by scalar kkk, then (
Determinant of Identity Matrix	(
Multiplicative Property	(
Adjoint of a Matrix AAA	Transpose of cofactor matrix: adj(A)=[Cij]T\text{adj}(A) = [C_{ij}]^Tadj(A)=[Cij​]T
Inverse of a Matrix (using Determinant)	( A^{-1} = \frac{1}{
Cramer’s Rule	x=DxD, y=DyD, z=DzDx = \frac{D_x}{D},\ y = \frac{D_y}{D},\ z = \frac{D_z}{D}x=DDx​​, y=DDy​​, z=DDz​​, if D≠0D \neq 0D=0
Area of Triangle using Determinant	( \text{Area} = \frac{1}{2} \left

How to Download Determinants Class 12 NCERT Solutions PDF?

Step-by-step instructions:

1. Click on the “Download PDF” link beside each chapter.
2. The file will open in Google Drive or a direct link.
3. Press the download button () and save to your device.

You can also print them if you prefer reading on paper.

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❓ FAQs on Class 12 Math’s Handwritten Notes PDF Download

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❌ Top 8 Common Mistakes Students Make in Class 12 Math’s

1️⃣ Skipping NCERT Questions
Many students rely only on reference books and ignore NCERT. In reality, most board exam questions come directly from NCERT or slightly modified versions.

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2️⃣ Memorizing Without Understanding
Blindly memorizing formulas or steps without understanding the concepts leads to mistakes in application-based questions.

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3️⃣ Not Practicing Graph-Based Questions
In chapters like Linear Programming or Application of Derivatives, students often skip graph questions which carry easy but crucial marks.

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4️⃣ Ignoring Units & Final Answers
Forgetting to write units (like cm², probability in fraction/decimal form) or skipping the final boxed answer results in loss of marks.

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5️⃣ Calculation Errors in Determinants & Matrices
Signs, cofactor expansion, and row/column operations often lead to small calculation mistakes, especially in Determinants.

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6️⃣ Not Revising Formulas Regularly
Math is formula-based. Forgetting key formulas in Probability, Integrals, Derivatives, etc., causes silly mistakes.

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7️⃣ Weak Time Management During Exam
Spending too much time on 1 or 2 tough questions leads to rushing or leaving easy questions toward the end.

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8️⃣ Skipping Step-Wise Presentation
Marks in board exams are awarded step-wise. Writing only the final answer or skipping reasoning reduces partial marks.

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🎯 Class 12 Maths Preparation Tips (Short & Effective)

✅ 1. Start with NCERT First
Begin your preparation by thoroughly understanding every concept and example from the NCERT textbook. It is the most important book for the CBSE Board exam.

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✅ 2. Make a Formula Sheet
Prepare a chapter-wise formula chart and revise it regularly—especially for chapters like Integrals, Probability, and Determinants.

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✅ 3. Practice Daily
Solve at least 10–15 questions daily, mixing simple, moderate, and HOTS-level problems to improve speed and accuracy.

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✅ 4. Solve Previous Year Papers (PYQs)
Go through last 5–10 years of board papers. This helps in understanding question patterns and important topics.

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✅ 5. Focus on Presentation
Always write step-by-step solutions, underline final answers, and use proper mathematical notation to fetch full marks.

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✅ 6. Revise from Handwritten Notes
Use handwritten notes to revise formulas, methods, and important shortcuts quickly before tests and exams.

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✅ 7. Clear Your Doubts Early
Don’t let small doubts pile up. Take help from your teacher, tutor, or online tools immediately.

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✅ 8. Take Mock Tests Weekly
Attempt full-length timed mock tests to improve time management and reduce exam anxiety.

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🔚 Conclusion – Chapter 4: Determinants (Class 12 Math’s)

Chapter 4: Determinants is a foundational topic in Class 12 Maths that strengthens your understanding of linear algebra and its real-world applications. It introduces you to the concept of calculating a scalar value from a square matrix, which is crucial for determining the invertibility of matrices, solving systems of linear equations using Cramer’s Rule, and calculating areas using coordinate geometry.

The chapter also emphasizes the importance of cofactors, minors, adjoint, and inverse of a matrix, all of which are vital in both academic and competitive examinations like JEE, CUET, NDA, and beyond. Mastering the properties of determinants helps simplify calculations and save time during exams.

To succeed in this chapter, it’s important to practice regularly, understand the logic behind each property, and revise formulas frequently. Using handwritten notes and NCERT solutions makes learning easier and more effective by presenting the content in a simplified and exam-focused format.

In conclusion, a strong grasp of Determinants will not only help you score full marks in this chapter but also build a strong base for higher-level mathematics. Stay consistent, solve plenty of examples, and revise smartly for best results.

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