Download the Class 12th Determinants NCERT Solutions for Class 12 Mathematics Chapter 4 Determinants Exercise 4.3 as a free PDF. Every step in the Class 12th Determinants NCERT Solutions is justified, every formula labelled, and the working is laid out in the format expected on a CBSE board answer script.
- CBSE Weightage: 8-10 marks (full Ch 4)
- JEE Main: 3-4% of paper
- Question Count in Ex 4.3: 5 (minor & cofactor computation + orthogonality identity)
Topics Covered in Class 12 Maths Chapter 4 Exercise 4.3
The Class 12th Determinants NCERT Solutions address this in the same order as the NCERT textbook.
Exercise 4.3 sits between Ex 4.2 (properties) and Ex 4.4 (adjoint and inverse). The 5 questions rehearse the cofactor formula that drives every section after it.
| Sub-concept | Definition / Identity | Questions in Ex 4.3 |
|---|---|---|
| Minor of an element | Mij = determinant left after deleting row i and column j | Q1, Q2 |
| Cofactor of an element | Aij = (-1)i+j Mij | Q1, Q2, Q3 |
| Sign chart for 3×3 | pmatrix+ & - & + - & + & - + & - & +pmatrix | Q2, Q3, Q4 |
| Cofactor expansion (row) | |A| = ai1Ai1 + ai2Ai2 + ai3Ai3 | Q4 |
| Orthogonality identity (MCQ) | j aijAkj = 0 when i ≠ k | Q5 |
Determinants Ex 4 3 Video Walkthrough
Source: Magnet Brains on YouTube
Question-Wise Breakdown of NCERT Class 12 Maths Exercise 4.3
The Class 12th Determinants NCERT Solutions address this in the same order as the NCERT textbook.
The 5 questions move from a 2×2 warm-up to a 3×3 expansion and finish with an orthogonality MCQ. Q4 is the load-bearing board-exam question.
| Q No. | Type | Concept Tested | Difficulty |
|---|---|---|---|
| Q1 | Find minors & cofactors | 2×2 determinants vmatrix2 & -4 0 & 3vmatrix and vmatrixa & c b & dvmatrix | Easy |
| Q2 | Find minors & cofactors | 3×3 determinants: identity I3 and vmatrix1 & 0 & 4 3 & 5 & -1 0 & 1 & 2vmatrix | Medium |
| Q3 | Verify expansion | Expand vmatrix5 & 3 & 8 2 & 0 & 1 1 & 2 & 3vmatrix by 2nd row and 1st column | Medium |
| Q4 | Expansion along row/col | vmatrix1 & x & yz 1 & y & zx 1 & z & xyvmatrix - parametric 3×3 | Hard |
| Q5 | MCQ | Value of a11A21 + a12A22 + a13A23 | Medium |
How will Collegedunia's NCERT Solutions for Class 12 Maths Exercise 4.3 help you?
The Class 12th Determinants NCERT Solutions address this in the same order as the NCERT textbook.
A 3×3 matrix has nine minors and nine cofactors, eighteen sub-results, and one sign error drags every expansion in Ex 4.4 into the wrong answer. Collegedunia's solutions lay out every minor as a separate boxed step so you can audit the chain instead of running it in your head.
The set also teaches orthogonality: the sum a11A21 + a12A22 + a13A23 is zero (not |A| ) because elements come from row 1 and cofactors from row 2. Our solution proves it by showing the sum equals the determinant of a matrix with two identical rows.
Important Formulae & Sign Charts for Exercise 4.3
Every entry in the Class 12th Determinants NCERT Solutions is reproduced in the order it appears in the NCERT textbook.
Every one of the 5 questions reduces to two definitions plus one identity.
F1. Minor: Mij = determinant of the submatrix obtained by deleting the i -th row and j -th column.
F2. Cofactor: Aij = (-1)i+j Mij .
F3. Row- i expansion: |A| = j=1n aij Aij .
F4. Column- j expansion: |A| = i=1n aij Aij .
F5. Orthogonality (mixed sum): j=1n aij Akj = 0 when i ≠ k .
F6. Sign chart 2×2: pmatrix+ & - - & +pmatrix . Sign chart 3×3: pmatrix+ & - & + - & + & - + & - & +pmatrix .
Sample Solved Question from NCERT Class 12 Maths Exercise 4.3
Q1 part (i) solved in the Collegedunia step-format. The sign-attachment line is what CBSE markers look for; do not collapse minor and cofactor into one step.
Question 1(i): Write the minors and cofactors of the elements of vmatrix2 & -4 0 & 3vmatrix .
Step 1 - Compute the four minors. Delete the row and column of each entry; only one entry remains in a 2×2 case.
M11 = 3, M12 = 0, M21 = -4, M22 = 2.
Step 2 - Attach the sign (-1)i+j .
A11 = (-1)2 · 3 = 3, A12 = (-1)3 · 0 = 0,
A21 = (-1)3 · (-4) = 4, A22 = (-1)4 · 2 = 2.
Step 3 - Sanity check via cofactor expansion along row 1. |A| = 2 · A11 + (-4) · A12 = 6 + 0 = 6 , which matches 2 · 3 - (-4) · 0 = 6 . Final answer: M11=3, M12=0, M21=-4, M22=2; A11=3, A12=0, A21=4, A22=2 .
Common Mistakes Students Make in Class 12 Maths Ex 4.3
The Class 12th Determinants NCERT Solutions are written in formal mathematical notation, line by line, in the same convention as the official NCERT print.
These five errors account for the highest mark-loss in the CBSE marking scheme for Chapter 4 Exercise 4.3.
- Skipping the sign factor. Writing the minor as the cofactor. The cofactors of a12 and a21 carry a negative sign; missing this flips the expansion. Costs the full question.
- Deleting the wrong row. For M21 you delete row 2 (not row 1). Students often confuse row index with the value of i .
- Confusing cofactor matrix with adjoint. The adjoint is the transpose of the cofactor matrix. Submitting the cofactor matrix loses 1 to 2 marks in Ex 4.4.
- Misreading the orthogonality identity. In Q5, a11A21 + a12A22 + a13A23 = 0 because elements and cofactors come from different rows.
- Skipping the verification line. Q3 demands you verify row-2 vs column-1 expansion match. Skipping it forfeits 1 mark even when the value is correct.
Class 12 Maths Chapter 4 Determinants Exercise 4.3 Marks Budget
How a typical 5-mark CBSE question on minors and cofactors of a 3×3 matrix is graded, drawn from 2024 and 2023 marking schemes.
| Step | What to write | Marks |
|---|---|---|
| 1 | State the sign chart and the formula Aij = (-1)i+j Mij | 1 |
| 2 | Compute all nine minors (each as a 2×2 determinant) | 2 |
| 3 | Attach the sign factor to each minor (write Aij values) | 1 |
| 4 | Verify by cofactor expansion along any one row, state the determinant value | 1 |
Class 12 Maths Determinants Previous Year Questions Weightage
Cofactor questions appear in roughly half of recent CBSE board papers and every JEE Main session. The orthogonality identity is a recurring JEE Main MCQ.
| Year | CBSE Board | JEE Main | JEE Main |
|---|---|---|---|
| 2025 | Cofactor matrix of 3×3, 5 marks | 2 questions (orthogonality + adjoint) | 1 question (determinant value) |
| 2024 | Minors of given 3×3, 3 marks | 2 questions | - |
| 2023 | Verify expansion along row vs column, 5 marks | 1 question | 1 question |
| 2022 | Cofactor expansion of parametric determinant, 5 marks | 1 question | - |
| 2021 | - | 1 question | - |
| 2026 | Pending (paper not yet held) | 1 question (Jan session) | Pending (exam rescheduled) |
Important Formulae for the Full Chapter 4 Determinants
Ex 4.3 only handles minors and cofactors; here are the five most-asked chapter formulae across CBSE 2021 to 2025.
| # | Formula | Where used |
|---|---|---|
| 1 | |A| = a11A11 + a12A12 + a13A13 | Ex 4.1, Ex 4.3, Ex 4.4 |
| 2 | adj(A) = [Aij]T (transpose of cofactor matrix) | Ex 4.4, Ex 4.5 |
| 3 | A-1 = 1|A| adj(A), |A| ≠ 0 | Ex 4.5 |
| 4 | Area = 12|vmatrixx1 & y1 & 1 x2 & y2 & 1 x3 & y3 & 1vmatrix| | Ex 4.2 |
| 5 | |AB| = |A| · |B| (product rule) | Ex 4.2, Miscellaneous |
Full learn sheet: Class 12 Maths Chapter 4 Determinants Formula Sheet carries all 14 formulae with units, definitions, and solved examples.
Related Resources for Class 12 Maths Chapter 4 Determinants
- NCERT Solutions for Class 12 Maths Chapter 4 Determinants (Full Chapter)
- Class 12 Maths Chapter 4 Determinants Notes
- Class 12 Maths Chapter 4 Determinants Formula Sheet
- Class 12 Maths Chapter 4 Determinants Exemplar Solutions
- Class 12 Maths Chapter 4 Determinants Handwritten Notes
Class 12th Determinants NCERT Solutions: available above as a free PDF download, aligned to the 2026-27 NCERT Class 12 Mathematics syllabus.
NCERT Solutions for Class 12 Mathematics: All Chapters
Chapter-by-chapter NCERT Solutions for the rest of Class 12 Mathematics, each mapped to the 2026-27 print.
| Chapter | NCERT Solutions |
|---|---|
| Chapter 1 | Relations and Functions NCERT Solutions |
| Chapter 2 | Inverse Trigonometric Functions NCERT Solutions |
| Chapter 3 | Matrices NCERT Solutions |
| Chapter 5 | Continuity and Differentiability NCERT Solutions |
| Chapter 6 | Application of Derivatives NCERT Solutions |
| Chapter 7 | Integrals NCERT Solutions |
| Chapter 8 | Application of Integrals NCERT Solutions |
| Chapter 9 | Differential Equations NCERT Solutions |
| Chapter 10 | Vector Algebra NCERT Solutions |
| Chapter 11 | Three Dimensional Geometry NCERT Solutions |
| Chapter 12 | Linear Programming NCERT Solutions |
| Chapter 13 | Probability NCERT Solutions |
Exercise-wise Breakdown of the Determinants Chapter
The Determinants chapter splits into 6 numbered exercises plus a Miscellaneous Exercise. The table below maps every exercise to the specific concept it tests, so students can plan revision per exercise and click straight into the worked solutions.
| Exercise | Topic Tested |
|---|---|
| Exercise 4.1 | Evaluation of 2x2 and 3x3 determinants |
| Exercise 4.2 | Properties of determinants; area of a triangle |
| Exercise 4.3 | Minors and cofactors |
| Exercise 4.4 | Adjoint and inverse of a matrix |
| Exercise 4.5 | Applications: solving systems of linear equations |
| Exercise 4.6 | Consistency of system of linear equations |
| Miscellaneous Exercise | Mixed determinant concepts and applications |
PDF Download Formats and Languages for the Determinants Chapter
The Determinants Class 12 PDF on this page is available in three formats - each suited to a different revision style. The table below summarises what each format is best for:
| Format | Best for | Approx. size |
|---|---|---|
| Normal-resolution PDF | Phone reading, quick revision between classes | 2-3 MB |
| HD PDF | Print-ready, desk study, board hall photocopy | 8-10 MB |
| Handwritten Notes PDF | Mirrors how a topper writes the chapter under Sunday-revision pace | 5-7 MB |
The determinants class 12 ncert pdf and the parallel Hindi-medium edition both follow the same notation and equation numbering as the printed NCERT 2026-27 release. Key points students should know:
- NCERT-faithful: Every definition, theorem and exercise on the determinants class 12 ncert pdf matches the printed textbook line for line.
- Hindi-medium edition: The determinants class 12 pdf is also available in Hindi - same page numbering, same equation labels.
- Formula PDF separate: The determinants class 12 formulas pdf is a one-page A4 reference sheet listing every identity used in the chapter.
- Solutions PDF separate: The determinants class 12 solutions pdf gives every NCERT exercise worked out step by step.
- State-board alignment: Students on the Maharashtra board, HSC, or any state-board syllabus will find the same definitions in this this chapter - only the exercise numbers differ.
Tip: Many toppers keep two parallel copies - a printed formula sheet on A4 for desk revision (the determinants class 12 formulas pdf), and the full these notes on a phone for commute revision. Both files are free and linked above.
Important Questions and Previous Year Trends for the Determinants Chapter
The most repeated question patterns in CBSE Class 12 Maths for the Determinants chapter have settled into a stable cluster across 2019 to 2024 boards. Three question templates account for over 80% of the marks this chapter contributes:
| Template | Typical Marks | What it tests |
|---|---|---|
| Proof / property verification | 3 marks | Students show that a given relation/function/expression satisfies the chapter's definitions. |
| One-step computation | 2 marks | Substitution-based item: plug into a known formula and simplify. |
| Case-study scenario | 4 marks | Real-world setup applying the chapter's definitions, introduced in CBSE 2021+ papers. |
Walking through one example of each template before the exam covers most of the predictable determinants class 12 important questions you will see on board day.
- determinants class 12 previous year questions for 2019-2024 are linked from the PYQ block at the bottom of this page - the exact CBSE phrasings.
- The determinants class 12 important questions with solutions set is reused by toppers in the last fortnight of revision.
- For NCERT Exemplar practice, the matching determinants class 12 extra questions set adds advanced problems suitable for JEE Main and JEE Advanced.
- The MCQ pattern in CBSE has stabilised around 1-2 questions per shift from this chapter - mostly short calculations or assertion-reason items.
Year-wise PYQ Distribution
The table below maps the dominant question type asked from the Determinants chapter across recent CBSE Class 12 Maths boards:
| Year | Dominant Question Type | Approx. Marks |
|---|---|---|
| 2024 | Property verification + case-study item | 5-6 marks |
| 2023 | Computation with proof + assertion-reason MCQ | 5-6 marks |
| 2022 | Long-answer derivation + 2-mark substitution | 5-7 marks |
| 2021 | Definition recall + property check | 4-5 marks |
| 2020 | One-step computation + 3-mark proof | 5 marks |
The full this chapter with solutions set (every year, every paper, every question type) is linked from the PYQ page at the bottom of this article.
How the Determinants Notes Pair with NCERT Solutions and the Formula Sheet
The Determinants Class 12 notes work best when paired with two sister resources from the Class 12 Maths hub. The table below shows how each resource fits into a typical revision week:
| Resource | Use it for | When |
|---|---|---|
| Determinants Notes (this page) | Theory, definitions, exam patterns | First pass, before practice |
| determinants class 12 ncert solutions PDF | Step-by-step solved exercises | Second pass, during NCERT practice |
| determinants class 12 formulas PDF | One-page identity recall | Third pass, alongside mock papers |
| Handwritten Notes PDF | Quick reading in topper's handwriting | Anytime, especially commute revision |
Around 60 percent of the chapter's scoring vocabulary appears on all three pages, so cross-resource use reinforces recall without adding study time.
- The determinants class 12 ncert solutions cover every back-of-chapter exercise plus the miscellaneous exercise.
- The determinants class 12 solutions for each individual exercise are indexed by exercise number on the sister NCERT Solutions page (see the Exercise-wise Breakdown table above for direct links).
- The determinants class 12 formulas reference sheet is the same A4 file students sometimes refer to as determinants class 12 all formulas - it lists every identity used in the chapter.
- State-board references: RD Sharma, ML Aggarwal, Teachoo and the Maharashtra board the resource textbook PDF all share the same core definitions.
- For class-first search phrasings - class 12 determinants solutions, class 12 determinants ncert solutions, ncert class 12 determinants solutions - the same files cover the request.
Reference Books and State-Board Mapping
Students using reference books beyond NCERT, or studying under a state board, can map this chapter cleanly:
| Reference | How it maps to the chapter notes |
|---|---|
| RD Sharma Class 12 Determinants | Question patterns overlap with NCERT at ~70%; an advanced supplement. |
| ML Aggarwal Class 12 Determinants | Solutions style is closer to JEE; good for problem-solving practice. |
| Teachoo the PDF | Free online walkthroughs; useful for video-style learning. |
| Shaalaa determinants class 12 solutions | State-board (Maharashtra HSC) phrasings; same core definitions. |
| Maharashtra board this chapter textbook PDF | Same chapter content under the HSC syllabus; exercise numbers differ. |
| NCERT Exemplar Class 12 Determinants | Advanced problems for JEE Main/JEE Advanced preparation. |
How to Use the Determinants Notes Page Most Effectively
The recommended study plan for these notes chapter splits across three sittings. The table below outlines what to do in each:
| Sitting | Duration | What to do |
|---|---|---|
| Sitting 1: Theory | ~90 minutes | Read the printed NCERT chapter cover to cover. Mark every definition and theorem statement. Then read the formula recall section on this page. |
| Sitting 2: Solved Examples | ~90 minutes | Re-solve every solved example in NCERT without looking at the solution first. Compare your steps against the printed working. Use the determinants class 12 ncert solutions PDF if stuck. |
| Sitting 3: Exercises | ~90 minutes | Attempt back-of-chapter exercises one set per sitting. Track which exercises you finished cleanly and which need a second pass. Click into the linked exercise pages above for verification. |
For students preparing for both CBSE board and JEE Main:
- 60 percent of revision time on NCERT - irreplaceable for board marking-scheme phrasings.
- 40 percent of revision time on JEE-style problem sets - sharpens speed and conceptual depth.
- The these notes set on the previous-year page is the closest free analogue to a JEE-style problem set for this chapter.
- For CUET (UG) Mathematics, focus on definitions and one-step applications - CUET's MCQ pattern rewards reflexive recall.
All NCERT Solutions for Determinants Ex 4.3 with Step-by-Step Working
Every NCERT textbook question for Class 12 Mathematics Chapter 4 Determinants Ex 4.3 is listed below with its full Solution and Expert Solution hidden inside collapsible tabs. Click Check Solution to reveal the step-by-step working; click Expert Solution for the expanded explanation.
Questions
Write Minors and Cofactors of the elements of the following determinants:
(i) vmatrix 2 & -4 0 & 3 vmatrix
(ii) vmatrix a & c b & d vmatrix.
Write Minors and Cofactors of the elements of the following determinants:
(i) vmatrix 1 & 0 & 0 0 & 1 & 0 0 & 0 & 1 vmatrix
(ii) vmatrix 1 & 0 & 4 3 & 5 & -1 0 & 1 & 2 vmatrix.
Using Cofactors of elements of second row, evaluate Δ = vmatrix 5 & 3 & 8 2 & 0 & 1 1 & 2 & 3 vmatrix.
Using Cofactors of elements of third column, evaluate Δ = vmatrix 1 & x & yz 1 & y & zx 1 & z & xy vmatrix.
If Δ = vmatrix a11 & a12 & a13 a21 & a22 & a23 a31 & a32 & a33 vmatrix and Aij is the cofactor of aij, then the value of Δ is given by
(A) a11A31 + a12A32 + a13A33
(B) a11A11 + a12A21 + a13A31
(C) a21A11 + a22A12 + a23A13
(D) a11A11 + a21A21 + a31A31.
Class 12 Mathematics Revision Strategy and Exam Practice Routines
Most CBSE Class 12 students benefit from a three-pass revision rhythm: the first pass is slow and definition-by-definition, the second works through every back-of-chapter problem, and the third uses past board papers at exam pace. JEE and CUET aspirants should add a fourth pass focused on the JEE-specific question bank, because the same chapter content gets tested under different time pressure. Within these passes, a few habits separate students who hit the 85+ band from the rest:
- Read two previous-year marking schemes before the exam — marking-scheme phrasings reward exact wording, which pays off more than another mock paper.
- Write a one-page formula recall sheet per chapter that fits on one side of A4; the night before the exam should be spent only on this sheet and a single full-length mock.
- Solve the CBSE 2026-27 sample paper twice — it is the highest-fidelity guide to question difficulty and lifts mock-paper accuracy by 8 to 12 percent.
- Self-evaluate every two hours by writing the chapter's key results from memory, rather than reading passively.
- Finish back-of-chapter exercises once and revisit the miscellaneous exercise twice — past-board data shows this is worth roughly 2 extra marks.
Common arithmetic slips cost most students at least one mark per paper, and most marks lost in long-answer questions go to incomplete working, not wrong answers. Write every intermediate step in full, even on questions that feel straightforward — method marks are claimed step by step even when the final number is off. The case-study format introduced in recent CBSE boards now appears regularly, framing a real-world scenario that tests definitions plus one-step applications, so practising case studies from the CBSE sample paper translates directly into marks.
Time allocation in the last fortnight matters most. Two thirds of revision time should go to weak chapters, the remaining third to maintaining strong ones; students who revise this chapter twice in the last 10 days score 1.5 to 2 marks higher on past boards. The night before the exam is best spent on:
- The one-page formula recall sheet built earlier in revision.
- A single full-length mock paper at exam timing.
- Avoid learning any new material the night before — sleep matters more.
Mock papers serve two distinct purposes — subject mocks build chapter-level recall while full-paper mocks build time-management discipline. Tracking your own mock-paper scores week by week is the single best predictor of board outcome; a simple spreadsheet with date, paper, score, and one note on a recurring mistake is enough. For students using only one reference, the printed NCERT remains the highest-yield resource — books beyond NCERT add depth but rarely change board outcomes, since the marking scheme rewards NCERT phrasing first. Hindi-medium students can keep the bilingual NCERT edition handy because it follows the same notation, and group study works best when each student picks one sub-topic to explain.
Past CBSE marking schemes from 2020 to 2024 show that average board marks for Class 12 Maths have settled around the 75 to 82 percent band. Students who hit the upper end usually share the same revision rhythm: NCERT first, mock papers second, and previous-year papers third.
Class 12th Determinants NCERT Solutions - Frequently Asked Questions
Ques. How many questions are there in Exercise 4.3 of Class 12 Maths Chapter 4?
Ans. Exercise 4.3 of NCERT Class 12 Maths Chapter 4 Determinants contains 5 questions: Q1 and Q2 are compute-the-minors-and-cofactors questions on 2×2 and 3×3 matrices, Q3 asks you to verify cofactor expansion across rows and columns, Q4 handles a parametric 3×3 determinant, and Q5 is an MCQ testing the orthogonality identity.
Ques. What is the difference between a minor and a cofactor in Class 12 Maths Exercise 4.3?
Ans. The minor Mij is the determinant left after deleting row i and column j of the matrix. The cofactor Aij is the minor multiplied by the sign (-1)i+j .
For elements where i + j is even, the minor and the cofactor are equal; for elements where i + j is odd, the cofactor is the negative of the minor.
Ques. Which question of Exercise 4.3 is most important for CBSE board exams?
Ans. Question 4, the parametric determinant vmatrix1 & x & yz
1 & y & zx
1 & z & xyvmatrix , is the most repeated. Similar 3×3 expansion questions appeared in CBSE 2025, 2023, and 2022 board papers. Q5 (the orthogonality MCQ) is the highest-frequency question in JEE Main, appearing in roughly one of every two recent sessions.
Ques. What is the sign chart used in Exercise 4.3?
Ans. For a 3×3 matrix the sign chart is pmatrix+ & - & + - & + & - + & - & +pmatrix . The (1,1), (1,3), (2,2), (3,1), and (3,3) positions are + ; the rest are - .
For a 2×2 matrix the chart is pmatrix+ & - - & +pmatrix . Memorising both saves time across Ex 4.3 and Ex 4.4.
Ques. Why is the sum a 11 A 21 + a 12 A 22 + a 13 A 23 equal to zero in Q5?
Ans. This is the orthogonality identity for cofactors. The cofactors A21, A22, A23 come from row 2 of the matrix, but the elements a11, a12, a13 come from row 1. The sum equals the determinant of a matrix whose row 1 has been duplicated in place of row 2, which has two identical rows and therefore evaluates to zero.
Ques. Is Exercise 4.3 part of the 2026-27 CBSE syllabus?
Ans. Yes. Determinants is a full chapter in the 2026-27 NCERT Class 12 Maths syllabus, and Exercise 4.3 on minors and cofactors is retained in the new edition. learn of Ex 4.3 is a prerequisite for Ex 4.4 (adjoint and inverse) and Ex 4.5 (solving systems of linear equations using A-1 ).
Ques. Where can I download the free PDF of NCERT Solutions for Class 12 Maths Exercise 4.3?
Ans. the Class 12th Determinants NCERT Solutions is available at the top of the Class 12th Determinants NCERT Solutions. Click the download button to get the step-by-step this resource solutions for all 5 questions of Exercise 4.3, prepared by subject experts as per the 2026-27 NCERT Class 12 Maths textbook.
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