Important Questions: CBSE Class 12 Mathematics Chapter 4 Determinants

Determinants appear in Section B (Algebra) of the CBSE Class 12 Mathematics board paper. The chapter typically yields one 2-mark question, one 3-mark question, and one 5-mark question, totalling 10 marks. Occasionally, a 1-mark MCQ appears in Section A. The 2025 competency-based pattern also introduces case-based questions where a real-world scenario—such as a network of resistors or a traffic-flow model—requires you to set up and evaluate a determinant or find the inverse of a coefficient matrix. Mastery of properties, minors, cofactors, adjoint, and inverse is non-negotiable. NCERT Exercise 4.1 to 4.6 form the backbone; practise every solved example and miscellaneous exercise problem. The chapter is moderately scoring if you avoid sign errors and choose the right expansion strategy. According to the 2024 CBSE marking scheme, partial marks are awarded for correct method even if the final numerical answer is wrong, so always write intermediate steps clearly.

These questions test quick recall of definitions, properties, and standard results. CBSE often asks about the effect of elementary row or column operations, the value of a determinant after swapping rows, or identifying singular matrices. They appear in Section A of the paper. One mark means no partial credit; accuracy is everything. Always cross-check your answer by substituting back or using a property. Below are five representative MCQs with answers that mirror the difficulty and style of recent board papers. Each question takes roughly 30 seconds to solve if concepts are clear, so use these to warm up before tackling longer problems. Remember that determinant of a skew-symmetric matrix of odd order is always zero, and det(AB) equals det(A) multiplied by det(B)—two facts that recur every year in MCQs.

Two-mark questions usually ask you to evaluate a determinant using properties, find the minor or cofactor of a specific element, or verify a simple property. CBSE marking schemes award one mark for correct method and one for the final answer. Always show at least one intermediate step—for instance, the row operation you applied or the 2×2 determinant you expanded. These questions take roughly two to three minutes in the exam. The key is to choose the simplest expansion: look for zeros or create them using row operations. Below are four standard 2-mark questions taken from NCERT exemplar and previous board papers, complete with step-wise solutions. Practise writing concisely; examiners appreciate clarity over verbosity. If you make an arithmetic slip, partial marks are still yours if the method is sound.

Three-mark questions typically involve evaluating a 3×3 determinant using properties, finding the adjoint or inverse of a 2×2 or simple 3×3 matrix, or proving a determinant identity. The CBSE marking scheme allocates one mark for correct identification of the method, one for intermediate simplification, and one for the final answer. Always label your row operations (e.g. R₂ → R₂ − 2R₁) and write the transformed determinant after each step. Examiners penalize students who jump directly to the answer without showing work. Below are four representative 3-mark questions with complete solutions. Each mirrors the style of recent CBSE papers and should take four to five minutes under exam conditions. Pay special attention to sign conventions when computing cofactors and the order of operations when applying properties.

Five-mark questions are the backbone of your Determinants score. They combine multiple concepts: evaluating a 3×3 determinant using properties, finding the inverse, and applying it to solve a system of three linear equations (Cramer's rule or matrix inversion method). The 2024-25 CBSE sample paper also introduced case-based questions worth four marks, where a short paragraph describes a real scenario—traffic flow at intersections, cost analysis in manufacturing, or a network problem—and you must form a coefficient matrix, compute its determinant or inverse, and interpret the result. Marks are distributed as: one for setup, one for determinant evaluation, one for inverse or adjoint, and two for solving the system or answering sub-questions. Always write the final answer in the required form (x=, y=, z=). Below are three full-length 5-mark questions and one case study with sub-parts, mirroring the latest board pattern.

CBSE follows a predictable pattern when setting Determinants questions. One-mark MCQs test properties (effect of row interchange, scalar multiplication, or transpose on determinant value) and standard results (determinant of a singular matrix, relation between |A| and |adj A|). Two-mark questions ask for evaluation of 2×2 or simple 3×3 determinants, or computation of minors and cofactors of a given element. Three-mark questions demand proof of determinant identities using properties, finding adjoint and verifying the formula A·adj(A)=|A|I, or computing the inverse of a 2×2 or 3×3 matrix. Five-mark questions integrate determinants with systems of linear equations: you may be asked to solve a 3×3 system using Cramer's rule (quoting determinant ratios) or the matrix inversion method (X=A⁻¹B). The 2024 sample paper introduced a four-mark case study where data was presented in matrix form and you had to find the determinant to check consistency, then interpret the result in context. CBSE also loves 'without expansion' proofs—questions that say 'Using properties of determinants, prove…'—which test your ability to apply row and column operations rather than brute-force expansion. Always read the question carefully: if it says 'using properties,' you will lose marks for direct expansion.

According to CBSE examination reports from 2022–2024, determinants questions see a pass rate of around 65%, with the bulk of mark loss due to preventable errors. The most frequent mistake is sign errors in cofactor expansion: students forget that the cofactor Aᵢⱼ carries a factor (−1)^(i+j), leading to wrong signs in the adjoint matrix and hence an incorrect inverse. Expanding along a row or column without zeros is the second-biggest time sink; always apply row or column operations to create zeros before expanding. Writing intermediate steps is crucial—CBSE awards partial marks generously, but only if your method is visible. Another common error is confusing the adjoint (transpose of the cofactor matrix) with the transpose of the original matrix; remember adj(A) ≠ Aᵀ. In questions asking to solve AX=B using A⁻¹, many students compute A⁻¹ correctly but then forget to multiply A⁻¹B, or they multiply in the wrong order (BA⁻¹ is not defined if B is a column vector). When proving identities 'using properties,' do not resort to direct expansion—examiners deduct marks even if your final answer is correct. Finally, arithmetic slips during multiplication of 3×3 matrices or evaluation of 2×2 sub-determinants cost at least one mark per question; double-check your calculations, especially under exam stress.

NCERT Class 12 Mathematics Chapter 4 is organized into clear sub-topics, each carrying distinct question types. Properties of determinants (Section 4.2) include effects of row/column interchanges, scalar multiplication, addition of rows, and the zero determinant for linearly dependent rows. CBSE loves 'prove without expansion' questions here—use operations like Rᵢ→Rᵢ+kRⱼ to simplify before expanding. Minors and cofactors (Section 4.3) are tested through direct computation: given a matrix, find M₂₃ or A₁₂. Remember Mᵢⱼ is the determinant of the (n−1)×(n−1) sub-matrix obtained by deleting row i and column j, while Aᵢⱼ=(−1)^(i+j)Mᵢⱼ. The adjoint (Section 4.4) is the transpose of the cofactor matrix; verify the formula A·adj(A)=|A|I as a 3-mark standard question. The inverse (Section 4.5) is A⁻¹=(1/|A|)adj(A), valid only when |A|≠0. Applications (Section 4.6) tie determinants to solving linear systems: Cramer's rule gives x=Δₓ/Δ, y=Δᵧ/Δ, z=Δᵧ/Δ, where Δ=|A| and Δₓ,Δᵧ,Δᵧ are determinants with columns replaced by the constant vector. Alternatively, use X=A⁻¹B. Both methods yield the same solution; choose based on computational ease. CBSE marks schemes are agnostic to method as long as it is valid and fully shown.

Determinants is a chapter where small conceptual gaps—like mixing up minor and cofactor, or applying the wrong sign—cascade into lost marks. CBSETUTOR.ai offers a 24×7 AI tutor tailored for CBSE Class 12 Mathematics, accessible at a flat ₹999 per month for any class from 6 to 12. Upload a photo of any determinant question—whether it is a tricky 'prove without expansion' problem or a case-based scenario—and receive a step-by-step solution within seconds, complete with annotated row operations and cofactor calculations. The AI tracks which sub-topics you struggle with (properties, inverse, or Cramer's rule) and auto-generates personalized practice sets. Parents in metro cities like Delhi and Bangalore appreciate the 3-day free trial: sign up, let your child solve five determinant questions with instant feedback, and see the improvement before committing. Unlike recorded video lectures that you passively watch, CBSETUTOR.ai is interactive—ask follow-up questions, request alternate methods, or challenge the AI with your own examples. It is like having a patient tutor who never tires, available on your phone during late-night revision sessions. Thousands of Class 12 students used CBSETUTOR.ai in the 2024 board cycle and reported an average score increase of 8-10 marks in the Algebra section, with Determinants and Matrices being the biggest gainers. Try the free trial at cbsetutor.ai and experience how AI-powered learning turns confusion into confidence.

With 10-12 marks at stake, Determinants deserves roughly 8-10 hours of focused preparation spread over two weeks before your board exam. Start by revising NCERT solved examples in Sections 4.2 to 4.6—these are the templates for board questions. Next, solve every problem in Exercises 4.1 to 4.6 and the Miscellaneous Exercise; CBSE often lifts questions verbatim or with minor numerical changes. Allocate three hours to practising 'prove without expansion' problems; master the standard tricks (factor out common terms, apply Rᵢ→Rᵢ+Rⱼ, create zeros). Spend two hours on adjoint and inverse computations—write out the cofactor matrix for at least five different 3×3 matrices until the sign pattern (−1)^(i+j) becomes second nature. Dedicate two hours to solving linear systems using both Cramer's rule and the matrix inversion method; know when each is faster. The final hour should be a timed mock: pick two 5-mark questions from previous year papers, set a 20-minute timer, and solve under exam conditions. During the actual board exam, spend 30 seconds scanning the Determinants questions to pick the easiest first—usually the 2-mark evaluation or minor/cofactor question. For 5-mark questions, if you are stuck on the identity proof, move on and come back; the linear system sub-part often carries three marks and can be attempted independently. Write neatly and box your final answers; CBSE examiners scan hundreds of scripts a day, and a clear presentation earns goodwill marks.