CBSE Class 12 Application of Derivatives 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), 2-3 short answers (2-3 marks), 1-2 long answers/case-based (4-6 marks) on maxima-minima or approximations.
Repeated Topics: Increasing/decreasing functions, local maxima/minima using first & second derivative test, absolute max/min in closed interval, rate of change problems, tangent/normal equations, approximation using differentials, optimization word problems (area, volume, cost).
Difficulty: Medium to high; word problems & sign chart for increasing/decreasing are key. Application-based questions increased in 2023–2025.
Changes: More real-life optimization and case-based questions in recent papers.

Most Important and Repeated Questions

Question Example	Type/Marks	Years Repeated	Notes
Find intervals where f(x) = x³ - 3x² + 3x + 2 (or similar cubic) is increasing or decreasing. Use first derivative test.	Short Answer (3 marks)	2021, 2022, 2023, 2024, 2025	Repeated 5x; f'(x) = 3(x-1)² ≥ 0 → increasing everywhere; critical point x=1 (point of inflection).
Find local maxima and minima of f(x) = x³ - 3x + 2 (or -x³ + 3x - 2). Use second derivative test. Also find absolute max/min in [a,b].	Long Answer (4-5 marks)	2021 Term 2, 2022, 2023, 2024	Repeated 4x; Critical points x=±1; f''(x)=6x → max at x=-1, min at x=1.
Find the approximate value of √26 or (1.01)^5 or cos(59°) using differentials (linear approximation).	Short Answer (2-3 marks)	2022, 2023, 2024, 2025	Repeated 4x; dy ≈ f'(x)dx; e.g., √26 ≈ 5.099 using x=25, dx=1.
Assertion: If f'(x) > 0 in an interval, then f is strictly increasing. Reason: By definition of derivative and mean value theorem.	Assertion-Reason (1 mark)	2023, 2024, 2025	Repeated 3x; Both true, reason explains.
Find equation of tangent and normal to y = x³ - 3x + 1 at x=1 (or at point where slope is given).	Short Answer (3 marks)	2021 Term 1, 2022, 2023, 2025	Repeated 4x; Slope = dy/dx at point; tangent: y - y1 = m(x - x1); normal perpendicular.
An open box is to be made from a square sheet of side 12 cm by cutting equal squares from corners. Find side of square cut so volume is maximum.	Long Answer (4-6 marks)	2022, 2023, 2024, 2025	Repeated 4x; V = x(12-2x)²; max at x=2 cm (second derivative test or sign chart).
MCQ: The function f(x) = x² - 4x + 5 has minimum value at x= ? (a) 1 (b) 2 (c) 3 (d) 4	MCQ (1 mark)	2021 Term 1, 2023, 2024	Repeated 3x; Answer (b) 2 (vertex form or f'=0).
A spherical balloon is being inflated at 10 cm³/s. Find rate of increase of radius when r=5 cm.	Short Answer (3 marks)	2023, 2024	Repeated 2x; V=4/3 π r³ → dr/dt = (dV/dt)/(4π r²) = 10/(4π×25).
Case-based: Given cost/revenue function or geometric figure, find maximum profit/area or minimum cost using derivative.	Case-Based (4-6 marks)	2023, 2025	Repeated 2x; Set derivative=0, check second derivative or sign change.
Show that the rectangle of maximum area inscribed in a circle of radius r has diagonal equal to diameter (or sides √2 r).	Proof/Short Answer (3 marks)	2021 Term 2, 2024	Repeated 2x; A = x √(4r² - x²); max when x = r√2.

Best Format for Preparation and Answering

Study Format: Make sign charts for f'(x) to find increasing/decreasing & critical points. Practice 10–12 optimization word problems. Memorize second derivative test & approximation formula.
Answer Format: Increasing/Decreasing: Solve f'(x)>0 or <0 → intervals. Max/Min: Find critical points (f'=0), test f'' or first derivative sign change. Word problems: Define variables → function → derivative=0 → verify max/min.
Tips: Optimization & increasing/decreasing are highest repeat & marks. Always show second derivative or sign change for nature of extrema. Time: 12–15 mins for 5–6 mark questions. Very scoring chapter if practiced well.

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