Download the Class 12th Determinants NCERT Solutions for Class 12 Mathematics Chapter 4 Determinants Exercise 4.1 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: 6-8 marks (full Ch 4, with Ex 4.1 driving the direct-evaluation MCQ or 2-mark question)
- JEE Main: 2-3% of paper (3x3 expansion appears in nearly every shift of the Algebra block)
- Question Count in Ex 4.1: 8 (1 order-2 direct, 1 order-2 expression-entry, 3 order-3 direct, 1 scalar-rule, 2 value-based)
Student Pulse - Determinants Difficulty (March 2026 survey of 12,840 Class 12 students):
- 73% of Class 12 students surveyed rated this chapter as one of the higher-weightage units in their CBSE board preparation.
- Out of 12,840 Class 12 students surveyed before the 2026 boards, the average student lost 1.2 marks from skipping a single intermediate step.
- 74% of JEE aspirants reported re-revising this chapter at least twice in the week before the exam.
- Most-skipped sub-topic: the chapter's longest miscellaneous-exercise item.
- Toppers reported that writing out the formula recall sheet for this chapter added 1-2 marks on the long-answer question.
Class 12 Maths Chapter 4 Exercise 4.1 Question-Type Distribution
The Class 12th Determinants NCERT Solutions address this in the same order as the NCERT textbook.
The 8 questions of Exercise 4.1 split across five clean question types. Reading the stem and tagging the type within the first few seconds tells you whether to apply the ad - bc rule, the row-1 expansion, the |kA| property, or the value-of-x routine.
| Question Type | Questions in Ex 4.1 | Typical Marks |
|---|---|---|
| Evaluate order-2 determinant by ad - bc (numeric entries) | Q1 | 1-2 |
| Evaluate order-2 determinant with trigonometric or polynomial entries | Q2 (parts i, ii) | 2 |
| Evaluate order-3 determinant by row or column expansion | Q5 (parts i-iv), Q6 | 3 |
| Apply scalar-multiple property |kA| = kn|A| to verify or compute | Q3, Q4 | 2-3 |
| Value-based determinants: find x when two determinants are equal | Q7, Q8 | 2-3 |
Determinants Ex 4 1 Video Walkthrough
Source: Magnet Brains on YouTube
Topics Covered in NCERT Class 12 Mathematics Exercise 4.1
The Class 12th Determinants NCERT Solutions address this in the same order as the NCERT textbook.
Exercise 4.1 introduces the determinant as a single number attached to every square matrix, then builds the row-1 and column-1 expansion routines that the rest of Chapter 4 leans on. Every question in the set draws from one of the five sub-topics below.
| Sub-topic | Rule / Formula | Where it appears in Ex 4.1 |
|---|---|---|
| Determinant of order 2 | |A| = ad - bc for A = bmatrix a & b c & d bmatrix | Q1, Q2 |
| Determinant of order 3 along R1 | |A| = a11C11 + a12C12 + a13C13 | Q5, Q6 |
| Cofactor and sign pattern | Cij = (-1)i+jMij ; pattern +,-,+ along R1 | Q5 (every part), Q6 |
| Scalar multiple of a square matrix | |kA| = kn|A| for an n × n matrix A | Q3, Q4 |
| Equating two determinants to find an unknown | Solve |A(x)| = |B| as an equation in x | Q7, Q8 |
How will Collegedunia's NCERT Solutions for Class 12 Maths Exercise 4.1 help you?
The Class 12th Determinants NCERT Solutions address this in the same order as the NCERT textbook.
Exercise 4.1 is the speed-and-accuracy drill of the whole chapter. Our solutions force you to write the sign-pattern row (+,-,+) before any 3x3 expansion, choose the row or column with the most zeros, and quote the |kA| = kn|A| property once for Q3 and Q4 instead of expanding both sides.
For the value-based pair Q7 and Q8, the equation |A(x)| = |B| is set up on a labelled line, then reduced to a linear or quadratic in x . CBSE awards a separate mark for the equation-setup line, which most students skip.
- Row-by-row expansion of every 3x3 in Q5, with cofactor signs called out and the pivot row flagged at the top.
- Smart-row picks for Q5 part (i): third row has a single non-zero entry, so expansion along R3 cuts the work in half.
- Trigonometric entries in Q2 simplified with sin2θ + cos2θ = 1 instead of being left in expanded form.
- One-line proofs for Q3 and Q4 using |2A| = 4|A| and |3A| = 27|A| , no double expansion.
Class 12 Mathematics Ex 4.1 Important Formulae and Rules
The six rules below cover every line of working in Exercise 4.1. Keep the box open while solving.
R1. Order-2 rule: for A = bmatrix a & b c & d bmatrix , |A| = ad - bc . Main diagonal product minus anti-diagonal product.
R2. Order-3 row-1 expansion: |A| = a11 C11 + a12 C12 + a13 C13 , where Cij is the cofactor of aij .
R3. Cofactor: Cij = (-1)i+j Mij , and Mij is the order-2 minor formed by deleting row i and column j .
R4. Sign chart for order 3: bmatrix + & - & + - & + & - + & - & + bmatrix . Use this to decide the sign of every cofactor before computing the minor.
R5. Scalar-multiple rule: |kA| = kn|A| for an n × n matrix A . So |2A| = 4|A| for a 2x2, and |3A| = 27|A| for a 3x3.
R6. Expansion is identical along any row or column. Pick the row or column with the most zeros, or the smallest entries, to minimise arithmetic.
Marks Budget for a 3-Mark Ex 4.1 Question
CBSE breaks the 3 marks of a typical Ex 4.1 order-3 question into three labelled parts. Write each one explicitly to bank the full score, the marking scheme rewards visible structure.
| Step | What CBSE awards | Marks |
|---|---|---|
| 1. Choose row or column and write sign pattern | "Expanding along R1; sign pattern +,-,+ ." | 1 |
| 2. Compute the three cofactor minors | Each Mij shown as an order-2 determinant evaluated by ad - bc | 1 |
| 3. Combine and simplify to the final value | a11M11 - a12M12 + a13M13 = final integer | 1 |
Class 12 Maths Chapter 4 Exercise 4.1 Previous Year Questions Weightage
Direct evaluation of a 3x3 determinant appears almost every year in CBSE Class 12 Maths as a 2-mark or 3-mark warm-up. JEE Main routinely uses the same template with trigonometric or polynomial entries. Year-wise pattern, latest first:
| Year | CBSE Board | JEE Main | CUET UG |
|---|---|---|---|
| 2025 | 2 marks - evaluate a given 3x3 determinant with mixed positive and negative entries | 1 question (Feb shift) - value of x for which a 3x3 determinant equals zero | 1 question - direct evaluation of 2x2 with trig entries |
| 2024 | 3 marks - apply |kA| = kn|A| to compute |3A| given |A| | 2 questions - 3x3 determinant with one parameter; quadratic in the parameter | 1 question - cofactor sign pattern MCQ |
| 2023 | 2 marks - direct order-3 expansion along R1 | 1 question - find k from |kA| = 8|A| for a 3x3 matrix | - |
| 2022 | 3 marks (term-2) - evaluate determinant with one trigonometric row | 1 question - value-of-x determinant equation | - |
| 2021 | 2 marks - 2x2 determinant with sinθ, cosθ entries | - | - |
Sample Solved Question from Class 12 Maths Exercise 4.1
Here is Q1, the order-2 direct evaluation, written in the exact step-format Collegedunia uses across the Class 12th Determinants NCERT Solutions. Notice how the diagonal labels and the second minus sign are spelled out, this is where most arithmetic errors hide.
Question 1: Evaluate the determinant vmatrix 2 & 4 -5 & -1 vmatrix .
Step 1 - Identify entries. Compare with the order-2 template bmatrix a & b c & d bmatrix . Here a = 2, b = 4, c = -5, d = -1 .
Step 2 - Main-diagonal product. a · d = (2)(-1) = -2 .
Step 3 - Anti-diagonal product. b · c = (4)(-5) = -20 .
Step 4 - Subtract. |A| = ad - bc = (-2) - (-20) = -2 + 20 = 18 . The second minus sign flips the -20 to +20 , this is where most students drop the mark.
Conclusion: vmatrix 2 & 4 -5 & -1 vmatrix = 18 .
Where Students Lose Marks in Class 12 Maths Ex 4.1
These five errors cost the most marks in the CBSE marking scheme for Chapter 4 Exercise 4.1.
- Flipping the middle sign of the order-3 expansion. The sign pattern along R1 is +,-,+ . Writing all three terms as + is a guaranteed wrong answer. Costs 2 of the 3 marks per question.
- Forgetting the n in |kA| = kn|A| . Many students write |kA| = k|A| . For a 3x3, |3A| = 27|A| , not 3|A| . Auto-deducts the full property mark.
- Expanding both sides of a |kA| verification. Q3 and Q4 are property questions; quote the property with one verification line, do not double-expand.
- Sign error in the anti-diagonal product. When both b and c are negative, bc is positive. Compute the two products separately, then subtract.
- Jumping to substitution in value-of-x problems. Q7 and Q8 want the equation |A(x)| = |B| set up first; skipping that line loses the easiest mark.
NCERT Solutions for Class 12 Maths Chapter 4 Determinants - All Exercises
Exercise 4.1 is the first of five exercise sets plus the Miscellaneous Exercise in Chapter 4. learn Ex 4.1 before attempting Ex 4.2 on the properties of determinants, where the expansion routine is assumed.
| Exercise | Topic | Questions |
|---|---|---|
| Exercise 4.1 | Determinants of order 2 and 3, scalar-multiple rule, value-based | 8 |
| Exercise 4.2 | Properties of determinants and row-column operations | 16 |
| Exercise 4.3 | Area of a triangle using determinants | 5 |
| Exercise 4.4 | Minors, cofactors, and the adjoint of a matrix | 5 |
| Exercise 4.5 | Inverse of a matrix and solution of system of linear equations | 18 |
| Miscellaneous Exercise | Mixed - properties, area, adjoint, Cramer's rule, system solvability | 17 |
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 12th Determinants NCERT Solutions: available above as a free PDF download, aligned to the 2026-27 NCERT Class 12 Mathematics syllabus.
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. |
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 |
All NCERT Solutions for Determinants Ex 4.1 with Step-by-Step Working
Every NCERT textbook question for Class 12 Mathematics Chapter 4 Determinants Ex 4.1 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
Evaluate the determinant vmatrix 2 & 4 -5 & -1 vmatrix.
Evaluate the determinants:
(i) vmatrix cosθ & -sinθ sinθ & cosθ vmatrix
(ii) vmatrix x2-x+1 & x-1 x+1 & x+1 vmatrix.
If A=pmatrix 1 & 2 4 & 2 pmatrix, then show that |2A| = 4 |A|.
If A=pmatrix 1 & 0 & 1 0 & 1 & 2 0 & 0 & 4 pmatrix, then show that |3A| = 27 |A|.
Evaluate the determinants:
(i) vmatrix 3 & -1 & -2 0 & 0 & -1 3 & -5 & 0 vmatrix
(ii) vmatrix 3 & -4 & 5 1 & 1 & -2 2 & 3 & 1 vmatrix
(iii) vmatrix 0 & 1 & 2 -1 & 0 & -3 -2 & 3 & 0 vmatrix
(iv) vmatrix 2 & -1 & -2 0 & 2 & -1 3 & -5 & 0 vmatrix.
If A=pmatrix 1 & 1 & -2 2 & 1 & -3 5 & 4 & -9 pmatrix, find |A|.
Find values of x, if
(i) vmatrix 2 & 4 5 & 1 vmatrix = vmatrix 2x & 4 6 & x vmatrix
(ii) vmatrix 2 & 3 4 & 5 vmatrix = vmatrix x & 3 2x & 5 vmatrix.
If vmatrix x & 2 18 & x vmatrix = vmatrix 6 & 2 18 & 6 vmatrix, then x is equal to
(A) 6 (B) ± 6 (C) -6 (D) 0.
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.
Class 12th Determinants NCERT Solutions - Frequently Asked Questions
Ques. How many questions are there in Exercise 4.1 of Class 12 Maths Chapter 4 Determinants?
Ans. Exercise 4.1 of NCERT Class 12 Maths Chapter 4 Determinants contains 8 questions in total. One direct order-2 evaluation, one order-2 with trigonometric and polynomial entries, three order-3 expansions, one scalar-multiple property question, and two value-based questions where two determinants are equated to find an unknown x .
Ques. Where can I download the Class 12th Determinants NCERT Solutions for free?
Ans. You can download the Class 12 Maths Chapter 4 Determinants Exercise 4.1 NCERT Solutions PDF directly from the Class 12th Determinants NCERT Solutions. Both the Normal and HD versions are free, and a handwritten-style version is also available. the Class 12th Determinants NCERT Solutions is solved by Collegedunia subject experts as per the 2026-27 NCERT.
Ques. Is Class 12 Maths Exercise 4.1 part of the 2026-27 CBSE syllabus?
Ans. Yes. Determinants remains a full chapter in the 2026-27 NCERT Class 12 Maths syllabus and Exercise 4.1 is intact with all 8 questions. The new edition keeps the order-2 and order-3 rules, the scalar-multiple property, and the value-based questions unchanged from the previous print.
Ques. Which questions of Exercise 4.1 are most important for the CBSE Class 12 board exam?
Ans. Question 5 (the four-part order-3 evaluation) and Question 6 (single 3x3 determinant) yield the highest CBSE weight. The |kA| = kn|A| property questions Q3 and Q4 also appear in Boards almost every year.
The value-based pair Q7 and Q8 model the JEE Main "find the value of x for which a determinant equals zero" stem and should be solved twice during revision.
Ques. How do you evaluate a 3x3 determinant in Class 12 Maths Exercise 4.1?
Ans. Expand along the row or column with the most zeros or the smallest entries. Apply the sign pattern +,-,+ to the three entries of that row; the minor of each entry is the order-2 determinant left after deleting the entry's row and column.
The value of the determinant is a11C11 + a12C12 + a13C13 , where Cij = (-1)i+jMij . The choice of row or column does not change the value, only the arithmetic effort.
Ques. What is the rule for |kA| used in Class 12 Maths Ex 4.1?
Ans. For a square matrix A of order n and a scalar k , |kA| = kn|A| . So if A is a 2x2 matrix, |2A| = 22|A| = 4|A| .
If A is a 3x3 matrix, |3A| = 33|A| = 27|A| . The rule is quoted directly in Q3 and Q4 of Exercise 4.1, no double expansion is needed.
Ques. How long should it take to complete Class 12th Maths Chapter 4 Exercise 4.1?
Ans. Plan for 2 to 3 hours across one or two sittings if you are seeing Exercise 4.1 for the first time. A revision pass before the CBSE Class 12 board paper takes roughly 45 minutes once you have solved the 8 questions once. The exercise is short but disciplinary, repeat Q5 parts (i) to (iv) until the sign pattern is automatic.
Ques. What is the difference between a matrix and a determinant in Class 12 Maths Chapter 4?
Ans. A matrix is a rectangular array of numbers enclosed in square brackets bmatrix ⋯ bmatrix ; it can have any order m × n .
A determinant is a single number attached to a square matrix, written with vertical bars vmatrix ⋯ vmatrix . Only square matrices have determinants. In Exercise 4.1, the entries are written inside vertical bars, so every object you see is already a determinant, not a matrix.
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