CBSE Class 12 Determinants PYQs Analysis (2021-2025)

Based on detailed analysis of last 5 years' papers. Perfect for 2026 Boards prep!

Key Trends

Weightage Breakdown: 1-2 MCQs/assertion-reason (1 mark), 1-2 short answers (2-3 marks), 1 long answer on properties/evaluation/inverse using determinants (4-5 marks).
Repeated Topics: Properties of determinants (row/column operations), evaluation of 3×3 determinants, adjoint & inverse of matrix using det, area of triangle using coordinates, solving linear equations using determinants (Cramer's rule), singular/non-singular matrices.
Difficulty: Medium; calculation-intensive but property-based shortcuts save time. More objective & application questions in recent years.
Changes: Increased focus on properties to simplify determinants and competency-based questions (2023–2025).

Most Important and Repeated Questions

Question Example	Type/Marks	Years Repeated	Notes
Using properties of determinants, prove that det begins with row operations like R1 → R1 + kR2, etc., or evaluate \|a b c; b c a; c a b\| = (a+b+c)(ab+bc+ca) or similar cyclic.	Proof/Evaluation (4-5 marks)	2021, 2022, 2023, 2024, 2025	Repeated 5x; Highest repeat; master row/column properties to reduce to zero rows or factorize.
Find the area of triangle with vertices (x1,y1), (x2,y2), (x3,y3) using determinant formula. Show it's zero if collinear.	Short Answer/Proof (2-3 marks)	2021 Term 1, 2022, 2023, 2024	Repeated 4x; Formula: (1/2)\| x1(y2-y3) + x2(y3-y1) + x3(y1-y2) \|; collinear if det=0.
If A is a square matrix, show that adj(A) = \|A\| A^{-1} if invertible, or find adj of given matrix.	Short Answer (2-3 marks)	2022, 2023, 2024, 2025	Repeated 4x; Key property; often asked with 2×2 or 3×3 matrix.
Assertion: If det(A) = 0, then A is singular. Reason: Non-invertible matrix has zero determinant.	Assertion-Reason (1 mark)	2023, 2024, 2025	Repeated 3x; Both true, reason explains assertion.
Evaluate the determinant \|1 2 3; 4 5 6; 7 8 λ\| and find λ if det=0.	Short Answer (2 marks)	2021 Term 2, 2022, 2023, 2025	Repeated 4x; Expand or use properties; λ= something specific (often makes rows dependent).
Solve the system of equations using determinants (Cramer's rule): ax+by+cz=p, etc. (3 equations).	Long Answer (4 marks)	2022, 2024, 2025	Repeated 3x; x = Δx/Δ, etc.; show if unique/infinite/no solution based on Δ=0.
MCQ: The value of determinant remains unchanged if two rows are interchanged? (a) Yes (b) No, sign changes (c) Becomes zero	MCQ (1 mark)	2021 Term 1, 2023, 2024	Repeated 3x; Answer (b) sign changes (property).
Prove that det(AB) = det(A) det(B) for square matrices.	Proof (2-3 marks)	2021 Term 2, 2023	Repeated 2x; Multiplicative property; often asked theoretically.
Case-based: Given determinant expression or matrix, apply properties to simplify and find value or condition for zero.	Case-Based (4 marks)	2023, 2025	Repeated 2x; Focus on quick property application.
If A is 3×3 matrix with det(A)=5, find det(3A), det(A'), det(adj A).	Short Answer (2 marks)	2021 Term 1, 2024	Repeated 2x; det(3A)=27×5, det(A')=5, det(adj A)=25.

Best Format for Preparation and Answering

Study Format: Memorize all 10+ properties of determinants (row/column ops). Practice evaluation of 3×3 using properties (avoid full expansion). Solve 10 PYQs on area & adjoint daily.
Answer Format: Show row/column operations step-by-step (R1 → R1 - kR2). For proofs: State property → apply → conclude. Calculations: Expand along row with max zeros if possible.
Tips: Properties are game-changer (save time, high marks). Time: 8–10 mins for 4–5 mark questions. Pair with Matrices chapter for better understanding of adj & inverse.

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