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CBSE Class 12 Mathematics Chapter 4 Determinants Worksheet with Answers

Determinants form a foundation of linear algebra in CBSE Class 12 Mathematics, carrying significant weight in board exams and competitive entrance tests. This worksheet targets Chapter 4 of the NCERT textbook, blending conceptual recall, procedural skill, and higher-order problem-solving across six graded sections. Expect to work through properties, expansions, minors, cofactors, adjoints, inverses, and real-world applications—all mapped to the latest CBSE syllabus.

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Key takeaways

✓Covers all NCERT topics: properties of determinants, minors, cofactors, adjoint, inverse, and applications to linear systems.
✓90-minute mixed-difficulty worksheet with 6 MCQs, 5 fill-in-the-blanks, 6 true/false, 5 short-answer, 3 long-answer HOTS, and 1 case study.
✓Complete answer key with explanations ensures self-assessment and conceptual clarity for board exam preparation.
✓Includes application-based questions on solving systems of equations using Cramer's rule and matrix inversion.
✓Difficulty mix: 40% easy foundational recall, 35% moderate procedural fluency, 25% hard HOTS and case-based reasoning.
✓Suitable for revision during pre-board practice, term-end tests, and JEE Main determinant module drills.
✓Printable PDF format—ideal for timed practice at home or in coaching institutes across India.

Quick Chapter Recap: Determinants

Chapter 4 introduces the determinant as a scalar value derived from a square matrix, essential for testing invertibility and solving linear systems. You begin with 2×2 and 3×3 determinants, expand using Laplace's method along any row or column, and learn properties that simplify computation—such as row/column interchange reversing sign, scalar multiplication, and addition of matrices. Minors and cofactors build the foundation for finding the adjoint of a matrix. The adjoint, when divided by the determinant (if non-zero), yields the inverse. Finally, Cramer's rule and the matrix-inverse method solve systems of linear equations. Mastery here unlocks matrix algebra in higher mathematics and engineering coursework, making this chapter a Board-exam favorite with 8–10 marks typically allocated.

Determinant properties: row/column swap (sign change), scalar multiple, zero row/column yields zero.
Minors Mij obtained by deleting row i, column j; cofactors Cij = (−1)i+j Mij.
Adjoint adj(A) is the transpose of the cofactor matrix; A−1 = adj(A)/|A| when |A| ≠ 0.
Cramer's rule: for AX = B, xi = |Ai|/|A| where Ai replaces column i with B.

Worksheet Metadata: Difficulty & Timing

This worksheet is pitched at moderate-to-hard difficulty, aligning with both term-end board exams and JEE Main's linear-algebra module. Of the 26 total items, roughly 40 per cent test foundational recall (definitions, one-step expansions), 35 per cent demand procedural fluency (property application, adjoint/inverse computation), and 25 per cent probe higher-order thinking through proof-based questions and multi-step case studies. Allocate 90 minutes for a realistic timed sitting—mirror the rhythm of a board exam where Chapter 4 questions might appear in 4-mark or 6-mark slots. Keep a calculator, rough sheets, and the NCERT textbook handy for cross-referencing formulas. Print double-sided to save paper, and attempt all sections in sequence before consulting the answer key at the end.

Difficulty breakdown: Easy 40%, Moderate 35%, Hard/HOTS 25%.
Suggested time: 90 minutes (strict) to simulate exam conditions.
Materials needed: scientific calculator, graph/ruled sheets, NCERT textbook for formula reference.
Attempt all sections sequentially; review answers only after completing Section E and the case study.

Section A: Multiple-Choice Questions (1 mark each)

Each MCQ targets a single concept—determinant properties, cofactor expansion, or quick inverse checks. Choose the correct option and mark your answer clearly. These six questions mirror the objective-style items increasingly common in CBSE sample papers and term exams. They reward speed and conceptual clarity, so aim to finish Section A within 10 minutes. Pay attention to sign conventions when properties involve row swaps or scalar multiples. Remember that a determinant of zero signals a singular (non-invertible) matrix, a frequent trap in distractor options. Use elimination strategies if you are unsure, and always double-check arithmetic for 3×3 expansions written in compact form. Practicing MCQs sharpens pattern recognition, crucial when time is scarce during board exams or JEE Main paper-1 mathematics sections.

Section B: Fill in the Blanks (1 mark each)

Complete each statement with the precise term, formula, or numerical answer. Blank-filling drills recall of definitions, key identities, and standard results from the NCERT Class 12 Mathematics textbook. Write neatly in the space provided (or on your answer sheet if printing single-sided). Target 8 minutes for this section. Common pitfalls include confusing minor with cofactor, or forgetting the transpose step when writing the adjoint. Review the properties table in NCERT Section 4.2 if you hesitate. These five items anchor your understanding of notation and ensure you can state theorems verbally—useful during viva or internal assessments where teachers probe definitions. Mastery here also prevents silly errors in longer derivations found in Sections D and E, where notation slips cascade into wrong final answers.

Section C: True or False (1 mark each)

Decide whether each statement is true or false. Write T or F in the margin. These six assertions test subtle conceptual edges—transpose determinants, adjoint properties, and invertibility criteria. Allocate 7 minutes. A single counterexample suffices to mark a statement false, so think of small 2×2 matrices when in doubt. For instance, |AT| = |A| is always true (transpose does not change determinant value), but |A + B| = |A| + |B| is generally false (determinant is not linear in matrix addition). Carefully read each sentence; words like 'always,' 'never,' and 'if and only if' are clues. Practicing true/false sharpens logical reasoning and prepares you for assertion-reason MCQs in CBSE sample papers. After completing, revisit any uncertain items using the NCERT theorems in Chapter 4 Sections 4.3 and 4.5 to verify your reasoning.

Section D: Short-Answer Questions (2–3 marks each)

Each question demands a brief derivation, property proof, or numerical computation in three to five lines. Show all working—CBSE marking schemes award step marks even if the final answer is incorrect. Reserve 25 minutes for this section. Typical tasks include expanding a 3×3 determinant along a specified row, computing minors and cofactors for given elements, finding the adjoint, verifying A·adj(A) = |A|I, or using properties to simplify before expansion (for example, factoring out common row multiples). Write the property name in parentheses when you apply it; examiners appreciate clarity. These questions mirror 2-mark or 3-mark board items and are high-yield: master the template and you secure easy marks. Practice from NCERT Exercise 4.2 and 4.4 before attempting this section to build confidence in standard patterns.

Always state the property or theorem before applying it (e.g. 'Using C2 → C2 − 2C1').
Show intermediate determinant matrices after each row/column operation for full credit.
Circle or underline your final numeric answer to help the evaluator.
If asked to 'verify,' compute both sides independently and show they match.

Section E: Long-Answer & HOTS Questions (4–6 marks each)

These three problems demand multi-step reasoning, proof synthesis, or application to solve systems of linear equations. Allocate 30 minutes. Expect tasks such as deriving the inverse of a 3×3 matrix using the adjoint method, applying Cramer's rule to a three-variable system, proving determinant identities by row/column manipulations, or showing that certain matrices satisfy A−1 = AT (orthogonal matrices). Structure your solution: state the objective, outline the method (adjoint-inverse or Cramer), perform computations in labeled steps, and box the final answer. Partial credit is generous here—even if you err in arithmetic midway, clear logic and correct method earn 60–70 per cent of marks. Review NCERT Examples 9, 13, and 17 in Chapter 4, which model the extended-response style. These questions separate scoring above 90 from the rest, so practice under timed conditions using past CBSE board papers from 2022, 2023, and 2024.

Outline your method in one sentence before diving into calculations (e.g. 'We compute adj(A) then divide by |A|').
Number each row operation (R1 → R1 − 2R2) to maintain clarity for the evaluator.
If solving a system, write the coefficient matrix A, variable vector X, and constant vector B explicitly.
Double-check determinant signs and cofactor (−1)i+j exponents; sign errors are the most common mistake.

Case-Study Question (4 marks)

A real-world scenario tests your ability to translate narrative context into determinant equations, solve, and interpret results. CBSE introduced case-based items in 2021, and they now appear in every board paper. This question weaves Chapter 4 concepts into an engineering, economics, or physics setting—perhaps network flow, cost optimization, or coordinate geometry. Read the paragraph carefully, extract the coefficient matrix from the description, compute the required determinant or inverse, and answer sub-questions (typically two 1-mark MCQs and one 2-mark short-answer). Allocate 10 minutes. The key skill is mapping word problems to matrix notation; practice similar items from CBSE sample papers 2023–24. Case studies reward analytical reading and real-world application, making them favorite picks for high-difficulty papers. Treat them as bundled mini-questions rather than a single monolith to reduce cognitive load.

Underline key numerical data and variable relationships in the passage first.
Write out the matrix equation AX = B before attempting calculations.
Answer sub-parts in order; often part (i) sets up data needed for part (ii).
Interpret your numerical answer in the context (e.g. 'the cost is ₹450' rather than leaving x = 450 bare).

Answer Key with Explanations

Below are complete solutions for every question, organized by section. Cross-check your responses, award yourself marks, and note recurring mistakes to target revision. Each explanation cites the NCERT theorem or property used, so you can loop back to the textbook if a step is unclear. When practicing at home, resist peeking until you have attempted all sections; self-discipline in timed practice translates directly to exam composure. If your score falls below 70 per cent, revisit the NCERT examples and solved problems in Chapter 4, then reattempt the worksheet after two days. Consistent worksheet drills, combined with personalized doubt resolution—platforms like CBSETUTOR.ai offer 24×7 AI tutor support with photo-upload solving at ₹999/month, one price for Classes 6–12, with a 3-day free trial—ensure mastery well before board exams. Use this answer key as a learning tool, not just a marking rubric.

How CBSETUTOR.ai Supports Determinants Mastery

Determinants can feel mechanical until you hit a tricky property proof or a Cramer's rule question with messy arithmetic—that is when on-demand help makes the difference. CBSETUTOR.ai offers a 24×7 AI tutor accessible from any smartphone: snap a photo of your worksheet question, upload it, and receive a step-by-step solution within seconds. Whether you are stuck on cofactor sign conventions at midnight or need a second explanation of adjoint-transpose relationships before your morning test, the platform delivers instant clarity. Priced at a flat ₹999 per month for all subjects across Classes 6–12—no hidden fees, no per-question charges—it is more affordable than a single private tuition session in metros like Delhi or Mumbai. A 3-day free trial lets your child explore the interface, try photo-solving, and see if the explanations match their learning style. Beyond determinants, the AI covers the entire NCERT syllabus, from Relations & Functions to Differential Equations, ensuring consistent support throughout the academic year. Thousands of CBSE families rely on CBSETUTOR.ai to fill gaps left by large classroom sizes and expensive coaching centres.

24×7 AI tutor with photo-upload solving—ideal for late-night revision and weekend practice.
Flat ₹999/month for Classes 6–12, all subjects; no per-chapter or per-question fees.
3-day free trial with no credit-card lock-in; cancel anytime if it does not suit your needs.
Step-by-step solutions mirror CBSE marking schemes, teaching method alongside answer.
Complements NCERT textbook and worksheets like this one, reinforcing classroom learning at your pace.

Tips for Effective Worksheet Practice

Treat this worksheet as a mini board exam. Print it, set a 90-minute timer, and work in a quiet space with only a calculator and rough paper. Resist the urge to check answers mid-attempt; partial practice fragments focus and reduces exam stamina. After finishing, spend 20 minutes reviewing the answer key, marking wrong responses in red, and writing brief error notes ('forgot cofactor sign,' 'arithmetic slip in row expansion'). Patterns will emerge—perhaps you consistently drop marks in property-based simplifications or rush through adjoint transposition. Target those weak spots with focused NCERT example drills and reattempt similar questions from past board papers. Maintain a dedicated Determinants error log; even five minutes of daily review compounds into significant retention by February pre-boards. Finally, pair worksheet practice with conceptual videos or AI tutor sessions on CBSETUTOR.ai for topics that remain hazy. Worksheets test what you know; guided learning fills what you do not. Together, they form a complete preparation loop.

Simulate exam conditions: timer, no textbook, no interruptions—build mental stamina.
Mark and review immediately after; waiting a day dilutes the learning feedback loop.
Maintain an error log categorized by mistake type (concept gap, arithmetic, notation).
Reattempt incorrect questions after 48 hours to verify that the concept has solidified.
Use NCERT Examples 9–20 in Chapter 4 as worked-model references for long-answer templates.

Frequently asked questions

How many marks does Chapter 4 Determinants typically carry in CBSE Class 12 board exams?+

Determinants usually accounts for 8–10 marks in the board paper, split across one 4-mark or 6-mark long-answer question and two to three short-answer or MCQ items. The chapter often combines with Matrices (Chapter 3) for case-study questions.

What is the difference between a minor and a cofactor?+

The minor Mij of element aij is the determinant of the submatrix obtained by deleting row i and column j. The cofactor Cij equals (−1)i+j times that minor, incorporating the checkerboard sign pattern. Cofactors are used to build the adjoint matrix.

Why does interchanging two rows change the sign of the determinant?+

This is a fundamental property proven by induction on matrix size. Geometrically, swapping rows reverses orientation in the vector space spanned by the matrix. NCERT Chapter 4, Property 2 states it formally; you can verify it quickly with any 2×2 example.

When should I use Cramer's rule versus the matrix-inverse method to solve linear systems?+

Cramer's rule is faster when you need only one or two variables and the system is small (2×2 or 3×3). The inverse method is more efficient if you must find all variables or plan to solve multiple systems with the same coefficient matrix A but different constant vectors B.

How do I remember the formula for a 3×3 determinant expansion?+

Expand along the row or column with the most zeros to minimize arithmetic. Memorize the cofactor sign pattern: + − + in row 1, − + − in row 2, + − + in row 3. Practice daily with one random 3×3 matrix until the pattern becomes automatic; speed matters in board exams.

What happens if the determinant of a matrix is zero?+

A zero determinant means the matrix is singular (non-invertible). The rows or columns are linearly dependent, the system of equations either has no solution or infinitely many solutions, and you cannot compute A−1 using the adjoint method.

Is it necessary to learn all properties of determinants for the board exam?+

Yes. Properties 1–6 in NCERT Section 4.2 are frequently tested in 2-mark 'using properties' questions. Skipping them costs easy marks. Focus especially on row/column operations that create zeros—these simplify expansions and are favorite board-exam shortcuts.

Can I use a calculator to compute determinants during the CBSE board exam?+

CBSE allows basic scientific calculators but not programmable or graphing models. You may use it for arithmetic checks, but you must show manual expansion steps for full credit. Relying solely on calculator determinant functions will earn zero marks, as working must be demonstrated.

How is the adjoint matrix related to the inverse?+

The adjoint adj(A) is the transpose of the cofactor matrix. The inverse A−1 equals adj(A) divided by the determinant |A|, provided |A| ≠ 0. This relationship is central to solving systems via the inverse method and appears in most 6-mark questions.

What are common mistakes students make in determinant questions?+

Sign errors in cofactors (forgetting the (−1)i+j factor), arithmetic slips during 3×3 expansion, confusing minor with cofactor, and neglecting to transpose when writing the adjoint. Double-check every sign and write intermediate matrices clearly to avoid these pitfalls.

How does CBSETUTOR.ai help if I am stuck on a determinant property proof?+

Snap a photo of the question, upload it to the platform, and the AI tutor provides a step-by-step proof with property citations. You can ask follow-up questions in natural language—'Why did you swap rows here?'—and get instant clarifications. It is like having a patient maths teacher available 24×7 at ₹999/month.

Should I memorize the Vandermonde determinant formula?+

The Vandermonde determinant |[1, a, a²], [1, b, b²], [1, c, c²]| = (a−b)(b−c)(c−a) is useful for quick evaluation but not mandatory if you can derive it using column operations. NCERT includes it as an example; knowing the result saves time in objective questions or HOTS proofs.

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