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Important Questions: CBSE Class 12 Mathematics Chapter 6 Application of Derivatives

Q: How many marks does Application of Derivatives carry in CBSE Class 12 board exam?

Application of Derivatives typically carries 10 to 12 marks in the CBSE Class 12 Mathematics board paper. This includes one or two 1-mark MCQs, one 2-mark VSA, one or two 3-mark short-answer questions, and one 5-mark long-answer or case-based question.

Q: What are the most important topics in Chapter 6 for board exams?

The four pillars are rate of change, increasing and decreasing functions using the first derivative, tangents and normals, and maxima-minima optimization problems. CBSE especially loves 5-mark real-world optimization questions on area, volume, cost or profit.

Q: How do I find intervals where a function is increasing or decreasing?

Differentiate the function to get f'(x). Solve f'(x) = 0 to find critical points. Draw a sign chart to test the sign of f'(x) in each interval. Where f'(x) > 0, the function is strictly increasing; where f'(x) < 0, it is strictly decreasing.

Q: What is the second derivative test for maxima and minima?

If f'(c) = 0, check f''(c). If f''(c) 0, then x = c is a local minimum. If f''(c) = 0, the test is inconclusive; use the first derivative test or check values around c.

Application of Derivatives is one of the most scoring yet formula-intensive chapters in CBSE Class 12 Mathematics. The 2024-25 board paper allocates 10-12 marks to this chapter, blending direct theory with real-world optimization and geometrical applications. This page presents 18 board-standard important questions sorted by marks weightage, complete with concise solutions and working steps, so you can practice efficiently and learn the examiner's language.

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Key takeaways

✓Application of Derivatives typically carries 10-12 marks in the CBSE Class 12 Mathematics board exam under Calculus, making it a high-scoring unit.
✓Questions are distributed as 1-mark MCQ, 2-mark numerical, 3-mark problem-solving, and 5-mark case-based or optimization problems.
✓Rate of change, increasing/decreasing intervals, tangents and normals, and maxima-minima are the four pillars tested every year.
✓CBSE loves real-world optimization problems (volume, cost, profit) in 5-mark questions; model answers require clear method and final statement.
✓Common mistakes include sign errors in the first derivative test and forgetting to check endpoints in closed-interval maxima-minima problems.
✓CBSETUTOR.ai offers instant photo-upload doubt solving and step-by-step walkthroughs for every Application of Derivatives problem at ₹999/month for all classes.

Chapter Overview and Marks Weightage in CBSE Board Exam

Application of Derivatives is Chapter 6 in the NCERT Class 12 Mathematics textbook and falls under the Calculus unit, which together with Vectors and Three-Dimensional Geometry forms the backbone of Term-2 board exams. The chapter builds on Limits and Continuity (Chapter 5) to solve real-life problems involving rates, slopes, and optimization. Historically, CBSE allocates 10 to 12 marks to Application of Derivatives across four question types: one or two 1-mark MCQs, at least one 2-mark numerical, one or two 3-mark problems, and a 5-mark case-based or long-answer question. The 2024 sample paper and the February 2024 SET-1 board paper both featured a 5-mark case study on maximizing profit and a 3-mark tangent-normal problem. Understanding question patterns helps you prioritize revision topics and manage exam time effectively.

Typical distribution: 1 MCQ (1 mark), 1 VSA (2 marks), 1-2 short answer (3 marks each), 1 long or case-based (5 marks).
Core topics: rate of change, increasing/decreasing functions via first derivative, tangents and normals, maxima and minima via second derivative test.
Highest scoring section: 5-mark optimization problems on area, volume, cost or distance.
Interlinkage: Questions often combine algebra, trigonometry and geometry; keep formulae for areas and volumes handy.

1-Mark Questions (MCQ and Assertion-Reason)

The CBSE Class 12 board paper typically includes one or two 1-mark multiple-choice questions from Application of Derivatives. These test direct definitions, derivative rules for increasing/decreasing functions, or the sign of slopes. Assertion-Reason questions appeared in the 2024 paper. The key is speed and formula recall. Practice these 1-mark questions in under one minute each. Always eliminate obviously wrong options first, and remember that CBSE loves testing sign conventions and interval notation. Below are four representative 1-mark questions with answers to sharpen your MCQ skills and build confidence before the board exam starts.

2-Mark Questions (Very Short Answer)

Two-mark questions in Application of Derivatives demand a clean statement of method, one or two lines of differentiation, and a final answer with units or an interval. CBSE examiners award one mark for correct method and one mark for the final answer. Common 2-mark topics include finding intervals of increase/decrease, proving that a function is always increasing or decreasing, computing the slope of a tangent at a given point, or solving a simple rate-of-change problem. Write your working clearly and box the final answer. Below are three typical 2-mark questions with step-by-step solutions that mirror the concise style the board expects in a two-mark answer.

3-Mark Questions (Short Answer)

Three-mark questions require a structured answer with clear steps: stating the formula, differentiating, solving an equation, and interpreting the result. CBSE typically gives one mark for method, one for computation, and one for the final answer or verification. Popular 3-mark topics include finding points where tangent or normal satisfies a given condition, proving a given value is a local maximum or minimum using the second derivative test, solving word problems on related rates, and finding absolute maxima or minima on a closed interval. Always write the second derivative or critical-point check explicitly; partial credit depends on clear working. Below are four exam-standard 3-mark questions with full solutions.

5-Mark Questions (Long Answer and Case-Based)

Five-mark questions are the crown jewels of Application of Derivatives. CBSE awards one mark for understanding the problem, one for setting up the function, one for differentiation, one for solving the equation (critical points or maxima/minima), and one for the final answer with proper statement and units. Optimization problems—maximize area, minimize cost, find dimensions—are almost guaranteed in every board paper. Case-based questions present a real-world scenario (construction, packaging, profit analysis) followed by four sub-questions worth 1+1+1+2 or 1+1+1.5+1.5 marks. Always draw a diagram if geometry is involved, label variables clearly, and verify endpoints if the domain is a closed interval. Below are three full 5-mark questions with detailed step-by-step solutions that reflect the depth expected in board answers.

How CBSE Frames Questions from Application of Derivatives

Understanding the examiner's blueprint helps you anticipate question patterns and allocate revision time wisely. CBSE does not randomly pick problems; every year the question paper follows a consistent structure reflecting the NCERT exercise difficulty and the four cognitive levels—Knowledge, Understanding, Application, and HOTS (Higher Order Thinking Skills). Application of Derivatives questions are weighted heavily toward Application and HOTS, especially in 3-mark and 5-mark problems. The 2024 board paper included a 5-mark case study on profit optimization and a 3-mark tangent-normal problem, both requiring multi-step reasoning. CBSE loves real-world contexts: packaging, construction, motion, economics. The board also tests your ability to interpret the sign of the first derivative (increasing/decreasing) and the second derivative (concavity, maxima/minima). Assertion-Reason MCQs test conceptual clarity without lengthy computation. To excel, solve all NCERT exercises, especially the miscellaneous exercise, and practice previous years' board papers from 2020 onward. CBSETUTOR.ai provides chapter-wise question banks and instant photo-upload doubt solving, so you can clarify tricky optimization problems at ₹999/month for all classes 6-12, with a 3-day free trial.

Distribution: ~40% direct application (tangent, normal, rate of change), ~60% problem-solving (optimization, maxima-minima with constraints).
Favourite contexts: Geometry (box, cone, cylinder), Economics (cost, revenue, profit), Motion (related rates, shadow problems).
Language: CBSE uses phrases like 'strictly increasing', 'absolute maximum', 'rate of change', 'justify your answer'—mirror this in your solution.
Case-based pattern: 1 passage + 4 sub-questions totaling 4-5 marks; sub-questions test formula recall, differentiation, and final interpretation.
Marking scheme: Partial credit for method even if arithmetic is wrong; always write f'(x), solve f'(x) = 0, and state the conclusion.

Common Mistakes Students Make in Application of Derivatives

Year after year, CBSE examiners report the same handful of errors that cost students 2-3 marks per question. The most frequent mistake is sign errors when applying the first or second derivative test: forgetting that f'(x) > 0 means increasing and f'(x) < 0 means decreasing, or confusing f''(x) > 0 (concave up, local minimum) with f''(x) < 0 (concave down, local maximum). Many students also neglect to check boundary points when finding absolute maxima or minima on a closed interval; NCERT Miscellaneous Exercise question 10 explicitly requires checking f(a), f(b), and f(critical points). Another common pitfall is incorrect implicit differentiation for tangent-normal problems on circles or ellipses—students forget to apply the chain rule to y terms. In related-rates problems, students often mix up which variable is constant and which is changing with time. Finally, poor final statements lose marks: writing 'x = 3' instead of 'The side of the square to be cut is 3 cm' or forgetting units. To avoid these, practice writing full solutions under timed conditions, double-check signs before solving equations, and always re-read the question to ensure your answer matches what is asked. CBSETUTOR.ai offers a mistake-tracker feature that logs your frequent errors and suggests targeted practice, all for a flat ₹999/month across classes 6-12.

Sign confusion: f'(x) > 0 → increasing; f'(x) < 0 → decreasing. Write a sign chart for clarity.
Ignoring endpoints: Always evaluate f(a), f(b), and f(critical points) for absolute extrema on [a, b].
Implicit differentiation slip: When differentiating x² + y² = r², remember d(y²)/dx = 2y(dy/dx).
Units and statements: '3 cm' not '3'; 'Maximum profit is ₹1300' not 'P = 1300'.
Second derivative test: f''(c) > 0 → local min; f''(c) < 0 → local max. If f''(c) = 0, test is inconclusive.

Topic-Wise Breakdown: Rate of Change and Increasing/Decreasing Functions

Rate of change is the simplest application of derivatives and usually appears as a 1-mark or 2-mark question. If y = f(x), then dy/dx represents the instantaneous rate of change of y with respect to x. NCERT Exercise 6.1 covers direct problems: find the rate of change of area of a circle with respect to its radius, or the rate at which volume of a sphere changes. In related-rates problems, you have two or more quantities changing with time; differentiate both sides of an equation with respect to t using the chain rule. Increasing and decreasing functions form the backbone of 3-mark questions. A function f is strictly increasing on an interval I if f'(x) > 0 for all x in I, and strictly decreasing if f'(x) < 0. To find intervals, solve f'(x) = 0 to get critical points, then test the sign of f'(x) in each interval. CBSE loves asking you to prove that a given function is always increasing or always decreasing by showing f'(x) has constant sign. Practice NCERT Examples 7, 8, 9 and Exercise 6.2 Q1-Q16 thoroughly. These form 15-20% of the chapter's board weightage.

Key formula: dy/dx = rate of change of y w.r.t. x; dy/dt = (dy/dx)(dx/dt) for related rates.
Strictly increasing: f'(x) > 0 on interval I; strictly decreasing: f'(x) < 0 on I.
Method: Find f'(x), solve f'(x) = 0, draw sign chart, state intervals.
Board tip: Write 'f is strictly increasing on (a, b)' not just '(a, b)'—clarity earns marks.

Topic-Wise Breakdown: Tangents and Normals

Tangent and normal problems are a staple of 2-mark and 3-mark questions. The slope of the tangent to the curve y = f(x) at the point (x₀, y₀) is f'(x₀). The equation of the tangent is y − y₀ = f'(x₀)(x − x₀). The normal is perpendicular to the tangent, so its slope is −1/f'(x₀) (provided f'(x₀) ≠ 0). NCERT Exercise 6.3 covers finding tangent and normal equations, finding points where tangent is parallel or perpendicular to a given line, and proving that tangent at a point on a curve satisfies a certain condition. CBSE often gives you a curve in implicit form—like x² + y² = 25 or x²/a² + y²/b² = 1—so you must use implicit differentiation to find dy/dx. A 3-mark question might ask for the tangent at a given point and then verify that it passes through another point. Always write the equation in standard form ax + by + c = 0 at the end. NCERT Miscellaneous Exercise Q15 and Q17 are excellent practice for board-level tangent-normal questions.

Tangent slope at (x₀, y₀): m = f'(x₀). Equation: y − y₀ = m(x − x₀).
Normal slope: −1/m. Equation: y − y₀ = (−1/m)(x − x₀).
Implicit differentiation: For F(x, y) = 0, dy/dx = −(∂F/∂x)/(∂F/∂y).
Parallel/perpendicular condition: Tangent parallel to y = mx + c means f'(x₀) = m; perpendicular means f'(x₀) = −1/m.

Topic-Wise Breakdown: Maxima and Minima

Maxima and minima questions account for nearly half the chapter's board weightage and almost always appear as 3-mark or 5-mark problems. A function f has a local maximum at x = c if f'(c) = 0 and f''(c) < 0; it has a local minimum if f'(c) = 0 and f''(c) > 0. This is the second derivative test. If f''(c) = 0, the test is inconclusive and you must check the sign of f'(x) on either side of c (first derivative test). For absolute (global) maxima and minima on a closed interval [a, b], evaluate f at all critical points inside (a, b) and at the endpoints a and b; the largest value is the absolute maximum and the smallest is the absolute minimum. NCERT Exercise 6.5 is packed with optimization problems: find two numbers whose sum is constant and product is maximum, find dimensions of a box with maximum volume, minimize cost, maximize area. The board loves word problems, so practice translating English sentences into equations. CBSE marking schemes reward clear labeling of variables, the objective function, the derivative, solving f'(x) = 0, verifying the second derivative, and a final statement with units. CBSETUTOR.ai walks through every NCERT optimization problem with photo-solve and step-by-step hints at ₹999/month for all classes, with a 3-day trial to test the platform.

Local maximum: f'(c) = 0 and f''(c) < 0. Local minimum: f'(c) = 0 and f''(c) > 0.
Absolute extrema on [a, b]: Evaluate f(a), f(b), f(all critical points); compare values.
Optimization steps: (1) Define variables and constraints. (2) Write objective function. (3) Differentiate. (4) Solve f'(x) = 0. (5) Verify with f''(x) or endpoints. (6) State answer with units.
Board tip: Always check physical constraints (e.g. x > 0, length < total perimeter) before finalizing answer.

Recommended Practice Strategy and Resources

To score full marks in Application of Derivatives, follow a three-phase practice plan. Phase one (Weeks 1-2): Solve all NCERT Examples and Exercises 6.1 to 6.5 twice—first with the textbook open, then closed. Focus on understanding the method rather than memorizing solutions. Phase two (Weeks 3-4): Attempt NCERT Miscellaneous Exercise questions and previous years' CBSE board papers (2019-2024). Time yourself: 1-mark in 1 minute, 2-mark in 3-4 minutes, 3-mark in 6 minutes, 5-mark in 10 minutes. Check your solutions against marking schemes available on cbse.gov.in. Phase three (Week 5-exam): Revise formulae, practice case-based questions from CBSE sample papers, and drill common-mistake scenarios—sign errors, endpoint checks, implicit differentiation. Keep a formula sheet: rate of change, tangent/normal slopes, second derivative test conditions, and standard volumes (cube, cylinder, cone, sphere). CBSETUTOR.ai offers a personalized question bank that adapts to your performance, unlimited photo-upload doubt solving, and chapter tests with instant feedback, all for ₹999/month across classes 6-12 with a 3-day free trial—a single subscription that grows with your child from middle school through boards.

Daily drill: 2 MCQs, 1 VSA (2-mark), 1 short answer (3-mark) for speed and accuracy.
Weekly mock: One full 5-mark optimization problem under timed conditions; self-assess with marking scheme.
Error log: Maintain a notebook of mistakes; revisit weekly to ensure you do not repeat sign or endpoint errors.
Resources: NCERT textbook and Exemplar, CBSE question banks (2020-2024), class notes, and CBSETUOR.ai for 24×7 AI tutor support.

Frequently asked questions

How many marks does Application of Derivatives carry in CBSE Class 12 board exam?+

Application of Derivatives typically carries 10 to 12 marks in the CBSE Class 12 Mathematics board paper. This includes one or two 1-mark MCQs, one 2-mark VSA, one or two 3-mark short-answer questions, and one 5-mark long-answer or case-based question.

What are the most important topics in Chapter 6 for board exams?+

The four pillars are rate of change, increasing and decreasing functions using the first derivative, tangents and normals, and maxima-minima optimization problems. CBSE especially loves 5-mark real-world optimization questions on area, volume, cost or profit.

How do I find intervals where a function is increasing or decreasing?+

Differentiate the function to get f'(x). Solve f'(x) = 0 to find critical points. Draw a sign chart to test the sign of f'(x) in each interval. Where f'(x) > 0, the function is strictly increasing; where f'(x) < 0, it is strictly decreasing.

What is the second derivative test for maxima and minima?+

If f'(c) = 0, check f''(c). If f''(c) < 0, then x = c is a local maximum. If f''(c) > 0, then x = c is a local minimum. If f''(c) = 0, the test is inconclusive; use the first derivative test or check values around c.

How do I write the equation of a tangent to a curve given in implicit form?+

Use implicit differentiation to find dy/dx. Substitute the given point coordinates to get the slope m. Then use the point-slope form y − y₀ = m(x − x₀) and simplify to standard form ax + by + c = 0.

What are common mistakes in optimization problems?+

Forgetting to check endpoints when the domain is a closed interval, sign errors in the derivative, missing physical constraints (e.g. dimensions must be positive), and not stating the final answer with proper units or a clear sentence.

How should I approach a 5-mark case-based question?+

Read the passage carefully, identify the variables and constraints, answer each sub-question in order, show all working, and ensure each sub-answer is complete. CBSE awards marks for method even if the final arithmetic is wrong, so write every step clearly.

Do I need to memorize formulas for volumes and areas?+

Yes. Keep a formula sheet for volumes of cube, cuboid, cylinder, cone, sphere, and areas of circle, rectangle, triangle. CBSE optimization problems often require these. Write them on the first page of your answer sheet for quick reference during the exam.

Can CBSETUTOR.ai help with Application of Derivatives practice?+

Absolutely. CBSETUTOR.ai offers chapter-wise question banks, instant photo-upload solutions for tricky optimization problems, step-by-step walkthroughs of NCERT exercises, and adaptive practice tests. All of this for a flat ₹999/month for classes 6-12, with a 3-day free trial so you can try before committing.

How much time should I spend on this chapter during revision?+

Given its 10-12 mark weightage and high scoring potential, allocate at least 8-10 hours over two weeks. Split time equally between concept clarity (NCERT examples), numerical practice (exercises), and timed mock tests (previous board papers). Daily practice of 2-3 questions ensures retention and speed.

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