Download the Class 12th Determinants NCERT Solutions for Class 12 Mathematics Chapter 4 Determinants Exercise 4.2 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: 2-4 marks (the area / collinearity question is almost annual)
- JEE Main: 1 MCQ every 2-3 shifts on the area-by-determinant form
- Question Count in Ex 4.2: 5 (area, value of k , collinearity, equation of a line)
Solutions curated by Collegedunia subject experts, mapped to the 2026-27 NCERT, and verified against the last five years of CBSE board, JEE Main, and CUET UG papers.
The walkthrough below maps each line of working to the CBSE step-mark scheme.
Also Check:
- Determinants Class 12 Maths Chapter 4 NCERT Solutions (Full Chapter)
- Determinants Class 12 Maths Notes
- the PDF Maths Formula Sheet
Why Exercise 4.2 is the Highest-Yield Sub-Topic of Chapter 4 Determinants
The Class 12th Determinants NCERT Solutions address this in the same order as the NCERT textbook.
The area-of-triangle determinant is the densest mark-per-line topic in Chapter 4. A correctly written 4-mark Ex 4.2 answer takes under 4 minutes, the highest mark-per-minute return in the Maths paper.
Determinants Ex 4 2 Video Walkthrough
Source: Magnet Brains on YouTube
How the Class 12th Determinants NCERT Solutions 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.
Every solution follows a fixed five-step format: state the formula, substitute coordinates, expand, take modulus, state the verdict.
- Sign-aware computation so the modulus is never missed.
- Expansion along row 3 by default (the row of 1s gives the simplest cofactors).
- Two-line collinearity test for Q2 and Q5.
- Equation-of-line via determinants with the variable point (x, y) in row 1 and the final form boxed in standard ax + by + c = 0 shape.
NCERT Class 12 Maths Exercise 4.2 Question-Type Distribution
The Class 12th Determinants NCERT Solutions address this in the same order as the NCERT textbook.
The 5 questions split into four distinct task types. Knowing the type lets you pick the right formula variant in the first 10 seconds of reading.
| Question Type | Questions in Ex 4.2 | Typical Marks |
|---|---|---|
| Area of a triangle from three given vertices | Q1 (i, ii, iii) | 2-3 |
| Collinearity test ( det = 0 ) | Q2, Q5 | 2-4 |
| Find k so that the area equals a stated value | Q3 (i, ii) | 3-4 |
| Equation of line through two given points | Q4 (i, ii) | 4 |
Topics Covered in Class 12 Mathematics Chapter 4 Exercise 4.2
Every question in the set draws from this short toolkit of one primary formula and three application patterns.
| Sub-topic | Formula / Test | Appears in |
|---|---|---|
| Area of a triangle in determinant form | Δ = 12|detbmatrix x1 & y1 & 1 x2 & y2 & 1 x3 & y3 & 1bmatrix| | Q1, Q3 |
| Collinearity of three points | Collinear det = 0 | Q2, Q5 |
| Value-of-parameter from given area | 12|det| = given value, two cases from modulus | Q3 |
| Equation of a line through two points | |matrix x & y & 1 x1 & y1 & 1 x2 & y2 & 1 matrix| = 0 | Q4 |
Marks Budget for a 4-Mark Class 12 Ex 4.2 Question
CBSE splits the 4 marks of a typical Ex 4.2 question into four labelled steps. Writing each one explicitly is the most reliable way to bank the full score.
| Step | What CBSE awards | Marks |
|---|---|---|
| 1. State the formula | Write Δ = 12|det(…)| with the 3-by-3 matrix | 1 |
| 2. Substitute coordinates | Each (xi, yi) placed in the correct row | 1 |
| 3. Expand the determinant | Cofactor expansion shown with intermediate value | 1 |
| 4. Apply modulus and conclude | "Hence Δ = … sq units" OR "Hence collinear" | 1 |
Sample Solved Question from Class 12 Maths Exercise 4.2
Below is Q4 (i), the equation-of-a-line type, in the exact step-format used across the Class 12th Determinants NCERT Solutions.
Question 4 (i): Find the equation of the line joining (1, 2) and (3, 6) using determinants.
Step 1. Locus of (x, y) on the line: |matrix x & y & 1 1 & 2 & 1 3 & 6 & 1 matrix| = 0 .
Step 2. Expand along row 1: x(2 - 6) - y(1 - 3) + (6 - 6) = 0 ⇒ -4x + 2y = 0 .
Step 3. Divide by -2 : 2x - y = 0 . Final answer: y = 2x .
Class 12 Maths Chapter 4 Exercise 4.2 Previous Year Questions Weightage (2021-2025)
Area and collinearity questions sourced from Exercise 4.2 appear in CBSE almost every year and in JEE Main once every two to three shifts. Year-wise pattern, latest first:
| Year | CBSE Board | JEE Main | CUET UG |
|---|---|---|---|
| 2025 | 3 marks - area of triangle from given vertices | 1 question (Jan) - find k for collinearity | 1 question - collinearity by determinant |
| 2024 | 2 marks - check collinearity of three points | 2 questions - area = given value, find unknown | 1 question - area of triangle |
| 2023 | 4 marks - equation of line by determinant | 1 question - collinearity condition | - |
| 2022 | 3 marks - area with variable vertex (k, 0) | - | - |
| 2021 | 2 marks - show three points collinear | 1 question - area formula application | - |
Full year-wise PYQ map: the PDF Maths Chapter 4 NCERT Solutions
Common Question Stems CBSE Uses in Class 12 Exercise 4.2
These four phrasings cover roughly 95 per cent of CBSE, JEE Main, and CUET questions sourced from Ex 4.2.
- "Find the area of the triangle whose vertices are … ." Apply the area formula, expand by row 3, take modulus.
- "Show that the points … are collinear." Build the determinant; if it is zero, points are collinear.
- "If the area is k sq units, find a ." Set 12|det| = k and solve both + and - cases.
- "Equation of the line joining two points using determinants." Put (x, y, 1) in row 1, simplify to ax + by + c = 0 .
Where Students Lose Marks in Class 12 Maths Ex 4.2
The five errors below cost the most marks in the CBSE scheme for Ex 4.2.
- Forgetting the 12 factor. Reporting the determinant value as the area doubles it. Loses 1 to 2 marks.
- Skipping the modulus. A determinant of -10 gives area 5 sq units, not -5 .
- Returning only one value of k . |D| = c splits into D = ± c ; both roots are valid. Auto-deducts 2 marks.
- Wrong cofactor sign. Without the alternating + - + on row 1 the expansion is off by a sign.
- No conclusion line. CBSE awards a separate mark for the closing verdict.
How to Study Class 12th Maths Exercise 4.2 in 3 Hours
The set is short and self-contained. Plan 3 hours for the first pass; the revision sweep before the board paper takes under 30 minutes.
- Hour 1. Q1 (three sub-parts on direct area computation), until the formula is automatic.
- Hour 2. Q2 and Q5 (collinearity tests). Both reduce to "determinant equals zero".
- Hour 3. Q3 (find k , modulus splits into two cases) and Q4 (equation of line, mechanical once row order is fixed).
NCERT Solutions for Class 12 Maths Chapter 4 Determinants - All Exercises
Exercise 4.2 is the second of five exercise sets plus the Miscellaneous Exercise. Wrap Ex 4.2 before attempting Ex 4.3 on properties of determinants.
| Exercise | Topic | Questions |
|---|---|---|
| Exercise 4.1 | Determinant of order 1, 2, 3; cofactor expansion | 8 |
| Exercise 4.2 | Area, collinearity, equation of line | 5 |
| Exercise 4.3 | Minors, cofactors, properties of determinants | 5 |
| Exercise 4.4 | Adjoint and inverse of a matrix | 18 |
| Exercise 4.5 | System of linear equations: inverse and Cramer's rule | 16 |
| Miscellaneous Exercise | Mixed - properties, adjoint, inverse, systems | 18 |
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 4.2 with Step-by-Step Working
Every NCERT textbook question for Class 12 Mathematics Chapter 4 Determinants Ex 4.2 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
Find area of the triangle with vertices at the points given in each of the following:
(i) (1,0), (6,0), (4,3)
(ii) (2,7), (1,1), (10,8)
(iii) (-2,-3), (3,2), (-1,-8).
Show that points A(a,b+c), B(b,c+a), C(c,a+b) are collinear.
Find values of k if area of triangle is 4 sq units and vertices are
(i) (k,0), (4,0), (0,2)
(ii) (-2,0), (0,4), (0,k).
(i) Find equation of line joining (1,2) and (3,6) using determinants.
(ii) Find equation of line joining (3,1) and (9,3) using determinants.
If area of triangle is 35 sq units with vertices (2,-6),(5,4) and (k,4), then k is
(A) 12 (B) -2 (C) -12,-2 (D) 12,-2.
More Determinants Class 12 Maths Resources
Related Links:
- Matrices Class 12 Maths Chapter 3 NCERT Solutions
- Continuity and Differentiability Class 12 Maths Chapter 5 NCERT Solutions
- NCERT Solutions for All Classes
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.
- this chapter 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
These notes 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 this Class 12 page 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 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 4.2 of Class 12 Maths Chapter 4 Determinants?
Ans. Exercise 4.2 of NCERT Class 12 Maths Chapter 4 Determinants contains 5 questions. The set covers four task types: finding the area of a triangle from three given vertices, testing collinearity of three points, finding the value of a parameter k when the area is specified, and writing the equation of a line through two given points using determinants.
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 4.2 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. Every solution is solved by Collegedunia subject experts as per the 2026-27 NCERT syllabus.
Ques. What is the formula for area of a triangle using determinants in Class 12 Maths Ex 4.2?
Ans. The area Δ of a triangle with vertices (x1, y1) , (x2, y2) , (x3, y3) is Δ = 12|detbmatrix x1 & y1 & 1 x2 & y2 & 1 x3 & y3 & 1bmatrix| .
The outer vertical bars denote the absolute value, since area cannot be negative, and the 12 factor is mandatory in every CBSE answer.
Ques. How do you check if three points are collinear using determinants in Exercise 4.2?
Ans. Three points are collinear if and only if the determinant |matrix x1 & y1 & 1 x2 & y2 & 1 x3 & y3 & 1 matrix| evaluates to zero.
The reason is that collinear points form a degenerate triangle whose area is zero, and the area formula contains this exact determinant. Questions Q2 and Q5 of Exercise 4.2 are direct applications.
Ques. Is Class 12 Maths Exercise 4.2 part of the 2026-27 CBSE syllabus?
Ans. Yes. Chapter 4 Determinants remains a full chapter in the 2026-27 NCERT Class 12 Maths syllabus, and Exercise 4.2 is intact with all 5 questions. The new edition keeps the area-of-triangle determinant, the collinearity test, and the equation-of-line determinant unchanged from the previous print.
Ques. Which questions of Exercise 4.2 are most important for the CBSE Class 12 board exam?
Ans. Q3 (find the value of k for a given area) and Q4 (equation of a line through two points) carry the highest CBSE weight from Exercise 4.2. The same question stems appear almost every year in CBSE Class 12 or in CUET UG with only the numerical coordinates changed. Q2 and Q5 on collinearity are popular 2-mark questions.
Ques. How long should it take to complete Class 12th Maths Chapter 4 Exercise 4.2?
Ans. Plan for 2 to 3 hours across one or two sittings if you are seeing Exercise 4.2 for the first time. A revision pass before the CBSE Class 12 board paper takes under 30 minutes once you have already solved the 5 questions once, since the formula and method are uniform across all four task types.
Ques. Why does the area formula in Class 12 Maths Ex 4.2 contain a modulus?
Ans. The determinant of a 3 × 3 matrix can be positive, negative, or zero depending on the order in which the vertices are listed. Since area is a physical quantity and must be non-negative, the modulus (absolute value) discards the sign. The CBSE marking scheme awards a separate step mark for applying the modulus and reporting a positive area in square units.
Comments