Download the Class 12th Determinants NCERT Solutions for Class 12 Mathematics Chapter 4 Determinants Miscellaneous Exercise 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 (Matrices and Determinants cluster, includes one 5-marker)
- JEE Main Weightage: 4-6% (2 questions per shift on properties and inverse)
- Exercise Profile: 18 questions (14 subjective + 4 MCQs Q15-Q18) following 6 solved examples
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.
These NCERT Solutions are prepared by Collegedunia's senior Mathematics faculty against the 2026-27 NCERT print, every property invocation named, every cofactor written in full, every CBSE marking-scheme step preserved.
The walk-through below opens with why this exercise carries disproportionate weight, then maps the questions concept by concept.
Also Check:
- Class 12 Maths Chapter 4 Determinants Notes
- Class 12 Maths Chapter 4 Determinants Formula Sheet
- Class 12 Maths Chapter 4 Determinants Exemplar Solutions
Why the Miscellaneous Exercise of Determinants Matters for CBSE Class 12th and JEE Main
The Class 12th Determinants NCERT Solutions address this in the same order as the NCERT textbook.
Of the 18 problems, roughly 11 are direct CBSE long-answer templates and 4 map straight to JEE Main MCQs. The remaining three (Q5, Q7, Q14) test conceptual nuance, the trap cases NCERT has retained in the current 2026-27 syllabus: when an inverse fails, when a system is inconsistent, and when the area-of-triangle determinant returns a signed value.
For JEE aspirants, the property chain |kA| = kn |A| , |adj A| = |A|n-1 , |A-1| = 1/|A| used in Q5, Q6, Q15 recurs across every shift. This is the single highest-ROI hour in the Class 12th Determinants NCERT Solutions.
Determinants Misc Video Walkthrough
Source: Magnet Brains on YouTube
How the NCERT Solutions Class 12 Maths on the Class 12th Determinants NCERT Solutions Help You
The Class 12th Determinants NCERT Solutions address this in the same order as the NCERT textbook.
Each of the 18 step-by-step solutions in the NCERT Solutions Class 12 Maths follows a four-line structure that CBSE examiners reward:
- State the property or theorem being applied (Property 1, Property 5, Theorem 4) before any algebra. The marking scheme carries an explicit method mark for naming it.
- Show the row or column operation on a separate line with notation R2 → R2 - 2R1 above the arrow.
- Compute the cofactor matrix in full, with each Aij = (-1)i+j Mij shown, then transpose to get the adjoint.
- Conclude with a one-line statement connecting the result to the question (matrix is singular, system is consistent, area equals 4 square units).
This is the same step grammar Collegedunia faculty use when marking pre-board mock answer sheets, training the habit of writing the property-name line, the single most under-claimed mark on the Determinants long-answer slot.
Top 5 Properties You Need Before Attempting the Miscellaneous Exercise
The Class 12th Determinants NCERT Solutions address this in the same order as the NCERT textbook.
The 18 questions assume these five tools are ready to deploy. Lock them in; the NCERT Solutions Class 12 Maths become 60% shorter.
| # | Property | Form |
|---|---|---|
| 1 | Scalar multiple | |kA| = kn |A| for n × n |
| 2 | Det. of adjoint | |adj A| = |A|n-1 |
| 3 | Inverse via adjoint | A-1 = 1|A| adj(A) , |A| ≠ 0 |
| 4 | Product rule | |AB| = |A||B| ; |AT| = |A| |
| 5 | Area of triangle | Area = 12| |xi, yi, 1| | |
Full learn formula sheet: Class 12 Maths Chapter 4 Determinants Formula Sheet, all 14 properties with derivations.
Concepts Tested in the Class 12 Maths Chapter 4 Determinants Miscellaneous Exercise
The 18 problems cluster into five concept families, mapped below to the NCERT section and the CBSE marks range.
| Cluster | Underlying Concept | NCERT Section | Marks |
|---|---|---|---|
| Q1, Q2 | Evaluate determinant via row or column operations | 4.2 Properties | 3-4 |
| Q3, Q4 | Prove an identity via property-based reduction | 4.2 Properties | 4-5 |
| Q5, Q6, Q7 | Adjoint and inverse; |adj A| , |A-1| | 4.4 Adjoint, Inverse | 3-5 |
| Q8-Q14 | 3-variable linear system via matrix-inverse method | 4.5 Applications | 5-6 |
| Q15-Q18 | MCQs on properties, cofactor signs, singular matrices | 4.2 to 4.4 | 1-2 |
Notice the heavy concentration on the system-of-equations cluster (Q8 to Q14). That is the revision priority, the matrix inverse method accounts for seven of the 18 questions and is the format CBSE uses for its 5-mark slot in every recent paper.
Question-by-Question Solution Index for Determinants Miscellaneous Exercise
Below is the index of all 18 Miscellaneous Exercise problems with the technique applied. Open the NCERT Solutions Class 12 Maths for the full step-by-step solution to each.
| Q No. | Question Type | Technique Used |
|---|---|---|
| 1 | Prove determinant is independent of θ | Expand along R1 ; sinθ and cosθ terms cancel |
| 2 | Evaluate 3 × 3 trigonometric determinant | Expand along R2 ; apply sin2 + cos2 = 1 ; answer is 1 |
| 3 | Verify A · adj(A) = |A| · I for a 3 × 3 | Compute every cofactor; transpose; multiply and compare with |A| I |
| 4 | Prove |1, x, x2; x2, 1, x; x, x2, 1| = (1 - x3)2 | C1 → C1 + C2 + C3 ; factor; expand the residual |
| 5 | Evaluate symmetric determinant in a, b, c | R1 → R1 + R2 + R3 ; factor (a+b+c) ; answer (a+b+c)3 |
| 6 | Prove a cyclic-row determinant equals zero implies a+b+c=0 | Column-sum operation; factor; expand the residual |
| 7 | Solve 2x-y=-2 , 3x+4y=3 by matrix inverse | X = A-1B with |A|=11 ; x=-5/11, y=12/11 |
| 8 | Evaluate |1+a, 1, 1; 1, 1+b, 1; 1, 1, 1+c| | Subtract R1 from R2, R3 ; expand; answer abc + ab + bc + ca |
| 9 | From |1+x, 1, 1; ...| = 0 , find 1/x + 1/y + 1/z | Use Q8 result; divide by xyz ; answer -1 |
| 10 | Find A-1 of a 3 × 3 , solve a 3-variable system | Cofactor, transpose, divide by |A| ; answer x=0, y=5, z=3 |
| 11 | Solve reciprocal system in 1/x, 1/y, 1/z | Substitute u, v, w ; matrix-inverse; answer x=2, y=3, z=5 |
| 12 | Prove Vandermonde-style identity in α, β, γ | R3 → R3 + R1 ; factor differences; expand |
| 13 | Prove |x2+1, xy, xz; ...| = 1 + x2 + y2 + z2 | Scale rows by x, y, z ; column operations; factor |
| 14 | Verify Q11 answer by direct substitution | Substitute computed values back into all three equations |
| 15 | MCQ: |AT B A| when B is order-3 skew-symmetric | |AT B A| = |A|2 |B| ; |B| = 0 ; answer (A) 0 |
| 16 | MCQ: |adj(A)| when |A| = 4 , order 3 | |adj A| = |A|n-1 = 16 ; answer (B) |
| 17 | MCQ: |2 A B-1| when |A| = 2, |B| = 3 | 23 |A|/|B| = 16/3 |
| 18 | MCQ: Form of (AB)-1 | Reversal rule; answer (C) B-1 A-1 |
Roughly 70% of these techniques recur verbatim in CBSE 2021-2025 board papers and JEE Main 2024-25 shifts, which is why this set has the highest mark-per-hour return in the Class 12th Determinants NCERT Solutions.
Previous Year Question Trends from the Determinants Chapter
The chapter has been a steady 5 to 8 mark contributor in CBSE Class 12 Maths board papers, with the long-answer slot almost always sourced from the Miscellaneous Exercise pool.
| Year | CBSE Marks | Question Type | Source Pattern |
|---|---|---|---|
| 2025 | 5 | Solve a 3-variable system via matrix-inverse method | Modelled on Misc Q10, Q11 |
| 2024 | 5 | Prove a determinant identity via row-sum reduction | Modelled on Misc Q5, Q12 |
| 2023 | 6 | Find A-1 , use it to solve a system | Modelled on Misc Q10 |
| 2022 | 5 | Property-based determinant evaluation | Modelled on Misc Q1, Q2 |
| 2021 | 4 | Prove |adj A| = |A|n-1 for given matrix | Modelled on Misc Q3, Q16 |
Full year-wise PYQ map: Class 12 Maths Chapter 4 Determinants Notes, includes the full chapter-level PYQ breakdown by topic.
Common Mistakes Students Make in the Determinants Miscellaneous Exercise
The Class 12th Determinants NCERT Solutions are written in formal mathematical notation, line by line, in the same convention as the official NCERT print.
How to Study Determinants Effectively and the Time Required for Chapter 4
Budget roughly 12 to 14 study hours across Chapter 4, split as:
Sample Fully-Solved Question Walk-Through, Miscellaneous Exercise Q10
Q10 is the canonical CBSE 5-marker template. Below is the five-step working that earns every method mark.
Question. If A = pmatrix 1 & -1 & 2 0 & 2 & -3 3 & -2 & 4 pmatrix , find A-1 and solve x - y + 2z = 1 , 2y - 3z = 1 , 3x - 2y + 4z = 2 .
Step 1, |A| via R1 expansion: 1(8-6) - (-1)(0+9) + 2(0-6) = 2 + 9 - 12 = -1 ≠ 0 , so inverse exists.
Step 2, cofactors: A11=2, A12=-9, A13=-6; A21=0, A22=-2, A23=-1; A31=-1, A32=3, A33=2 .
Step 3, adjoint (transpose): adj(A) = pmatrix 2 & 0 & -1 -9 & -2 & 3 -6 & -1 & 2 pmatrix .
Step 4, inverse: A-1 = (1/|A|) adj(A) = pmatrix -2 & 0 & 1 9 & 2 & -3 6 & 1 & -2 pmatrix .
Step 5, solve X = A-1B with B = (1,1,2)T : X = (0, 5, 3)T . Verify: 0-5+6=1 , 10-9=1 , 0-10+12=2 . All three check.
Score breakdown is typically 1 mark for |A| , 2 marks for the cofactor matrix, 1 mark for the adjoint, and 1 mark for the final answer with verification.
All NCERT Solutions for Determinants Misc with Step-by-Step Working
Every NCERT textbook question for Class 12 Mathematics Chapter 4 Determinants Misc 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
Prove that the determinant vmatrix x & sinθ & cosθ -sinθ & -x & 1 cosθ & 1 & x vmatrix is independent of θ.
Evaluate vmatrix cosβ & cosβ & -sinα -sinβ & cosβ & 0 sinβ & sinβ & cosα vmatrix.
If A-1 = pmatrix 3 & -1 & 1 -15 & 6 & -5 5 & -2 & 2 pmatrix and B = pmatrix 1 & 2 & -2 -1 & 3 & 0 0 & -2 & 1 pmatrix, find (AB)-1.
Let A = pmatrix 1 & -2 & 1 -2 & 3 & 1 1 & 1 & 5 pmatrix. Verify that
(i) [adjA]-1 = adj(A-1)
(ii) (A-1)-1 = A.
Evaluate vmatrix x & y & x+y y & x+y & x x+y & x & y vmatrix.
Evaluate vmatrix 1 & x & y 1 & x+y & y 1 & x & x+y vmatrix.
Solve the system of equations:
2x + 3y + 10z = 4, 4x - 6y + 5z = 1, 6x + 9y - 20z = 2.
If x,y,z are nonzero real numbers, then the inverse of matrix A = pmatrix x & 0 & 0 0 & y & 0 0 & 0 & z pmatrix is
(A) pmatrix x-1 & 0 & 0 0 & y-1 & 0 0 & 0 & z-1 pmatrix
(B) xyzpmatrix x-1 & 0 & 0 0 & y-1 & 0 0 & 0 & z-1 pmatrix
(C) 1xyzpmatrix x & 0 & 0 0 & y & 0 0 & 0 & z pmatrix
(D) 1xyzpmatrix 1 & 0 & 0 0 & 1 & 0 0 & 0 & 1 pmatrix.
Let A = pmatrix 1 & sinθ & 1 -sinθ & 1 & sinθ -1 & -sinθ & 1 pmatrix, where 0 ≤ θ ≤ 2π. Then
(A) det(A) = 0
(B) det(A) ∈ (2,∞)
(C) det(A) ∈ (2,4)
(D) det(A) ∈ [2,4].
More Class 12 Maths Chapter 4 Determinants Resources
- Class 12 Maths Chapter 4 Determinants Notes
- Class 12 Maths Chapter 4 Determinants Formula Sheet
- Class 12 Maths Chapter 4 Determinants NCERT Book PDF
- Class 12 Maths Chapter 4 Determinants NCERT Exemplar Book PDF
- Class 12 Maths Chapter 4 Determinants Exemplar Solutions
- Class 12 Maths Chapter 4 Determinants Handwritten Notes
NCERT Solutions for Class 12 Maths: All Chapters
The table below links the NCERT Solutions pages for every other chapter of Class 12 Maths in the 2026-27 syllabus.
| Chapter | Resource |
|---|---|
| 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 |
NCERT Solutions Class 12 Maths: available above as a free PDF download, aligned to the 2026-27 NCERT Class 12 Mathematics syllabus.
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. |
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 the Miscellaneous Exercise of Class 12 Maths Chapter 4 Determinants?
Ans. The Miscellaneous Exercise of Class 12 Maths Chapter 4 Determinants contains 18 questions. Q1 to Q14 are subjective short and long-answer problems, and Q15 to Q18 are objective MCQ-type questions. The set follows six solved examples (Examples 22 to 27) at the end of the chapter.
Ques. Is the Miscellaneous Exercise of Determinants important for the CBSE Class 12 board exam?
Ans. Yes. The CBSE 5-mark long-answer slot on Determinants in the Class 12 board paper is almost always modelled on a Miscellaneous Exercise problem, particularly the matrix-inverse method for solving a 3-variable system. Skipping this exercise routinely costs students the entire 5-mark slot.
Ques. Which Miscellaneous Exercise questions are most important for Class 12th Maths Determinants?
Ans. Q5, Q8, Q9, Q10, Q11, Q12, Q13, and the four MCQs Q15 to Q18 are the highest-yield. They cover property-based reduction, cofactor and adjoint computation, the inverse formula, and the matrix method for solving systems, which recur in every recent CBSE board paper and JEE Main shift.
Ques. Is this Class 12th Determinants NCERT Solutions aligned with the 2026-27 NCERT syllabus?
Ans. Yes. the Class 12th Determinants NCERT Solutions is fully aligned with the current 2026-27 NCERT print of Class 12 Maths. Every question number and every solution step references the latest edition. Collegedunia revises this Solutions PDF every academic year.
Ques. Can the same Miscellaneous Exercise questions help with JEE Main?
Ans. Yes. JEE Main consistently uses two MCQs per shift drawn from the determinant-property and adjoint-inverse patterns of this exercise. The property |adj A| = |A|n-1 in Q16 and the reversal rule (AB)-1 = B-1 A-1 in Q18 appear in JEE Main 2024 and 2025 shifts almost verbatim.
Ques. What is the difference between Exercise 4.5 and the Miscellaneous Exercise on Determinants?
Ans. Exercise 4.5 tests one technique per question (cofactor, adjoint, inverse, or a single system to solve). The Miscellaneous Exercise mixes two or three techniques per question, demands the property-name justification, and includes the four MCQs that test the meta-properties (singular, skew-symmetric, scalar multiple, reversal rule). It is the application-level version of Exercise 4.5.
Ques. How many pages is the Class 12 Maths Chapter 4 Determinants Miscellaneous Exercise Solutions PDF?
Ans. The Solutions PDF for the Miscellaneous Exercise of Class 12 Maths Chapter 4 Determinants runs approximately 28 to 32 pages and covers all 18 questions with full step-by-step step-by-step solutions, including the cofactor matrices and adjoint computations expanded line by line.
Ques. Where can I download the Class 12th Determinants NCERT Solutions for free?
Ans. The free PDF of all 18 Miscellaneous Exercise solutions is available at the top of the Class 12th Determinants NCERT Solutions. Click the Download PDF button to save the complete step-by-step step-by-step solutions to your device. Both Normal and HD versions are available, and both are free.
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