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CBSE Class 12 Mathematics Chapter 6 Application of Derivatives Worksheet with Answers

Q: What is the weightage of Application of Derivatives in CBSE Class 12 board exam?

Application of Derivatives typically carries 8–10 marks in the CBSE Class 12 Mathematics board paper. Expect one 4-mark or 5-mark long-answer question, one 2-mark short-answer question, and 1–2 MCQs or fill-ups from this chapter.

Q: How do I decide whether to use the first derivative test or second derivative test for maxima-minima?

Use the first derivative test when f'(c)=0 and you check the sign of f' on either side of c. Use the second derivative test when f'(c)=0 and f''(c) exists; if f''(c)>0 it is a minimum, if f''(c)<0 it is a maximum. Second derivative test is faster but fails when f''(c)=0.

Q: Are approximations using differentials still asked in the CBSE board exam?

Yes, approximation problems (Δy ≈ dy = f'(x)Δx) appear occasionally as 2-mark short-answer questions. NCERT Exercise 6.4 covers this topic. Practice finding approximate values of roots, powers, or trigonometric functions using this formula.

Q: What are common mistakes students make in tangent and normal problems?

Common errors include forgetting to find the y-coordinate by substituting x into the original curve equation, mixing up the slope of tangent and normal, and sign mistakes in the formula y − y₁ = m(x − x₁). Always double-check that normal slope = −1/(slope of tangent).

Q: How should I approach optimization word problems in Section E?

Read carefully and identify the quantity to maximize or minimize. Express it as a function of one variable using given constraints. Differentiate, set the derivative to zero, solve for critical points, and apply the second derivative test or compare values. State units in your final answer and verify the answer makes physical sense.

Q: Is it necessary to show the increasing/decreasing table in the answer sheet?

A sign table (showing intervals and signs of f'(x)) is not mandatory but highly recommended. It organizes your solution, prevents errors, and earns you method marks even if the final answer has a minor slip. Draw a simple number-line with critical points marked and sign of f' in each interval.

Q: What is the difference between local and absolute maxima?

A local maximum at c means f(c) ≥ f(x) for all x in some small neighbourhood of c. An absolute maximum means f(c) ≥ f(x) for all x in the entire domain or given closed interval. To find absolute extrema on [a,b], evaluate f at all critical points and at endpoints a and b, then compare.

Q: Can I use a calculator for this worksheet?

CBSE Class 12 board exams do not permit calculators, so practice without one. For this worksheet, basic arithmetic and square-roots should be simplified by hand. If an exact answer like √2 or π appears, leave it in that form unless the question specifies 'approximate to two decimals.'

Q: How many times should I practice this worksheet before the board exam?

Attempt this worksheet at least twice—once as a timed mock test and once as untimed thorough revision. After each attempt, rework every question you got wrong. Supplement with NCERT Exercise 6.3 (maxima-minima) and miscellaneous exercise. Three weeks before boards, include this worksheet in your final revision rotation.

Q: Where can I get instant help if I am stuck on a question from this worksheet?

Upload a photo of the question to CBSETUTOR.ai and receive a step-by-step worked solution within seconds. The AI tutor is available 24×7 for all CBSE subjects and classes 6–12 at ₹999/month—one flat price with a 3-day free trial. No per-question charges, unlimited doubts allowed.

Application of Derivatives is one of the highest-weightage chapters in CBSE Class 12 Mathematics, contributing 8–10 marks in the board exam. Chapter 6 builds on your differentiation skills from Chapter 5 and applies them to real-world scenarios—finding rates of change, determining where functions increase or decrease, sketching tangents and normals, and locating maxima and minima. This worksheet is structured to mirror the CBSE board pattern and offers graded practice from foundational MCQs to challenging HOTS questions and a case-study problem introduced in the 2023-24 session.

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Key takeaways

✓Complete worksheet covering rate of change, increasing/decreasing functions, tangents, normals, maxima and minima as per NCERT Class 12 Mathematics Chapter 6.
✓30+ questions in six progressive sections: MCQs, fill-in-the-blanks, match-the-following, short-answer, long-answer and case-study.
✓Difficulty level is moderate to challenging; suggested time limit is 90 minutes for full worksheet completion.
✓Detailed answer key with step-by-step explanations provided for self-assessment and exam-level accuracy.
✓Aligned with CBSE 2025 board exam pattern and latest NCERT textbook terminology and problem types.
✓Print-friendly format suitable for classroom tests, home practice or coaching institute worksheets.
✓Use CBSETUTOR.ai for instant photo-upload doubt solving and 24×7 AI tutor support at ₹999/month across all subjects and chapters.

Quick Chapter Recap: Application of Derivatives

Before you attempt the worksheet, revisit these core concepts from NCERT Class 12 Mathematics Chapter 6. Rate of change measures how one quantity changes with respect to another—for instance, velocity is the rate of change of displacement with respect to time. A function f(x) is said to be increasing on an interval if f'(x) > 0 for all x in that interval, and decreasing if f'(x) < 0. Critical points occur where f'(x) = 0 or f'(x) does not exist; these points are candidates for local maxima or minima. The equation of the tangent to the curve y = f(x) at x = a is given by y − f(a) = f'(a)(x − a), while the normal has slope −1/f'(a). To find absolute maxima or minima on a closed interval, evaluate f at all critical points and endpoints, then compare. Approximations use the linear approximation formula Δy ≈ f'(x) Δx. Ensure you can apply the first derivative test, second derivative test, and know when each is applicable. These five topics—rate of change, increasing/decreasing behaviour, tangents and normals, maxima and minima, and approximations—form the backbone of the chapter and recur in every CBSE board paper.

Rate of change: dy/dx or dy/dt for related-rate problems
Increasing function: f'(x) > 0; decreasing: f'(x) < 0; stationary: f'(x) = 0
Tangent at (a, f(a)): y − f(a) = f'(a)(x − a); normal slope = −1/f'(a)
Critical points: solve f'(x) = 0; apply first or second derivative test
Absolute extrema: compare f at critical points and interval endpoints

Worksheet Information: Difficulty Level and Time Allocation

This Application of Derivatives worksheet is designed at a moderate-to-challenging difficulty level, suitable for students who have completed the NCERT textbook exercises and are now preparing for the CBSE Class 12 board exam or school pre-board tests. The worksheet comprises six sections with a total of 33 questions. Section A has 6 multiple-choice questions (1 mark each), Section B offers 6 fill-in-the-blank statements (1 mark each), Section C is a 5-item match-the-following exercise (1 mark per correct pair), Section D contains 5 short-answer questions (2 marks each), Section E has 3 long-answer or HOTS questions (4–5 marks each), and Section F presents a case-study problem with 4 sub-questions (1+1+2+2 marks). The suggested time limit is 90 minutes for the entire worksheet, simulating the exam environment. Attempt all sections in order, show full working for Sections D, E and F, and cross-check your solutions against the detailed answer key provided at the end. This structured approach mirrors the actual CBSE board exam, helping you build both speed and accuracy. Use the worksheet as a self-assessment tool before your final revision.

Total 33 questions across six sections: MCQs, fill-ups, match, short-answer, long-answer, case-study
Suggested time: 90 minutes (strict timing practice recommended)
Difficulty: Moderate to challenging—mix of direct application and HOTS
Marks distribution: Section A–C (17 marks), Section D (10 marks), Section E (13 marks), Section F (4 marks)
Print this page and attempt without calculator unless specified; show all steps in workings

Section A: Multiple Choice Questions (1 mark each)

Multiple-choice questions test your conceptual clarity and speed. Each MCQ in this section addresses a specific sub-topic from Application of Derivatives—rate of change, monotonicity, tangents and normals, or maxima-minima. Read each question carefully; eliminate obviously incorrect options first, then apply the relevant derivative test or formula. In CBSE board papers, MCQs often include tricky distractors that result from common algebraic mistakes, so double-check signs and substitution. Attempt all six questions even if you are unsure—there is no negative marking in CBSE Class 12 Mathematics. These questions are designed to be solved in one to two minutes each, so practice mental differentiation of standard functions like polynomials, trigonometric and exponential expressions. After you finish, refer to the answer key and understand why each distractor is incorrect; this sharpens your exam intuition and prevents repeat errors in the board exam. MCQs from Application of Derivatives frequently appear in the first section of the CBSE paper and set the tone for the rest of your performance, so aim for full marks here.

Q1. The function f(x) = x³ − 6x² + 9x + 15 is strictly increasing in the interval: (a) (−∞, 1) ∪ (3, ∞) (b) (1, 3) (c) (−∞, 1] (d) [3, ∞)
Q2. The slope of the tangent to the curve y = x³ − x at x = 2 is: (a) 11 (b) 6 (c) 3 (d) 8
Q3. The point on the curve y² = 4x which is nearest to the point (2, 1) is: (a) (1, 2) (b) (1, −2) (c) (0, 0) (d) (4, 4)
Q4. If the radius of a sphere is increasing at the rate of 2 cm/s, the rate of increase of its surface area when radius is 5 cm is: (a) 40π cm²/s (b) 80π cm²/s (c) 100π cm²/s (d) 20π cm²/s
Q5. The function f(x) = tan x − x is: (a) always increasing (b) always decreasing (c) neither increasing nor decreasing (d) increasing in (0, π/2)
Q6. The maximum value of sin x + cos x is: (a) 1 (b) 2 (c) √2 (d) 1/√2

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

Fill-in-the-blank questions require precise numerical or algebraic answers. In Application of Derivatives, these typically test formulae, definitions, and standard results you must recall instantly under exam pressure. Write only the final answer in the blank; no partial credit is awarded. Common topics include the condition for increasing/decreasing functions (f'(x) > 0 or < 0), the formula for tangent and normal equations, critical-point definitions, and standard maxima-minima results. For example, you should know that the maximum value of sin x cos x is 1/2 (occurring at x = π/4) or that the semi-vertical angle of a cone of maximum volume inscribed in a sphere is tan⁻¹(√2). Keep your NCERT Class 12 Mathematics textbook examples handy while preparing; many board exam fill-ups are direct echoes of textbook solved examples. If a blank asks for an interval or set, use correct interval notation—open brackets for strict inequalities, closed for inclusive. Practice writing answers neatly; illegible responses may not receive credit even if correct. After completing this section, verify each answer against the key and revise any formula you missed.

Q7. The function f(x) = 2x³ − 9x² + 12x + 1 is increasing in the interval __________.
Q8. The equation of the normal to the curve y = x² at (1, 1) is __________.
Q9. The maximum value of the function f(x) = x e⁻ˣ on [0, ∞) is __________.
Q10. If y = x⁴ − 10, the approximate change in y when x changes from 2 to 2.01 is __________.
Q11. The critical points of f(x) = x³ − 3x are __________.
Q12. A rectangle is inscribed in a semicircle of radius r. The maximum area of the rectangle is __________.

Section C: Match the Following (1 mark per correct pair)

Match-the-following exercises test your ability to connect concepts, formulae and results. In this section, Column A lists five statements or expressions related to Application of Derivatives, and Column B provides five corresponding answers or properties. Your task is to pair each item in Column A with exactly one item in Column B. Use the process of elimination: start with the pair you are most confident about, mark it, then move to the next. Carefully read each statement—small differences in wording (for example, 'maximum' versus 'minimum' or 'tangent' versus 'normal') can change the correct match. Common match topics include identifying increasing/decreasing intervals, matching a function to its point of inflection, pairing a curve equation with its tangent slope at a given point, or linking an optimization problem statement to its critical value. Write your answers as (1)→(B), (2)→(D), and so on in your answer sheet. Cross-verify once you have matched all five pairs; if two items seem to fit the same option, revisit your reasoning. This format appears occasionally in CBSE board exams and regularly in school pre-boards, so practice is essential.

Column A: (1) f(x) = x² is increasing in | (2) Slope of tangent to y = sin x at x = 0 | (3) f(x) = |x| has a minimum at | (4) f(x) = 1/x is decreasing in | (5) Maximum value of 3 cos x + 4 sin x
Column B: (A) x = 0 | (B) (0, ∞) | (C) (−∞, 0) ∪ (0, ∞) | (D) 1 | (E) 5
Write answers as (1)→(__), (2)→(__), etc.

Section D: Short Answer Questions (2 marks each)

Short-answer questions in CBSE Class 12 Mathematics typically carry 2 marks and require 3–5 lines of working. For Application of Derivatives, expect questions like 'Find the interval in which f(x) is increasing,' 'Show that a given function is always positive,' 'Find the equation of the tangent or normal at a given point,' or 'Verify that a certain value is a point of local maximum.' Structure your answer clearly: state the given, write the relevant formula or test, perform the differentiation, solve the resulting inequality or equation, and state the conclusion. Award yourself 1 mark for method and 1 mark for the correct final answer. Always show the derivative explicitly; examiners deduct marks if you jump to the conclusion without showing f'(x). If the question says 'Hence or otherwise,' you are expected to use a previous result—mention it briefly. These five questions are designed to cover different sub-topics, so even if you find one difficult, attempt the others to maximize your score. Time management is key: spend no more than 4 minutes per 2-mark question, leaving room for longer problems in Sections E and F.

Q13. Find the intervals in which the function f(x) = x³ − 12x² + 36x + 7 is strictly increasing.
Q14. Find the equation of the tangent to the curve y = √(4 − x²) at the point (1, √3).
Q15. Show that the function f(x) = x + 1/x has a local minimum at x = 1 and a local maximum at x = −1.
Q16. A particle moves along the curve y = x³. Find the point on the curve at which the y-coordinate is changing twice as fast as the x-coordinate.
Q17. Verify that the function f(x) = 4x − tan x is strictly increasing on (−π/2, π/2).

Section E: Long Answer / HOTS Questions (4–5 marks each)

Long-answer questions are the heart of the CBSE Class 12 Mathematics board exam. Each carries 4 or 5 marks and demands a complete solution with logical flow, correct mathematical notation, and a clear final statement. In Application of Derivatives, typical long questions involve: proving a function is increasing or decreasing over a given domain, finding absolute maxima and minima on a closed interval, solving optimization problems (maximize area, minimize cost, etc.), or proving inequalities using the mean-value theorem or monotonicity. Allocate 8–10 minutes per question. Begin by writing 'Given' and 'To Prove/Find,' then proceed step-by-step—differentiate, find critical points, apply the first or second derivative test, evaluate at endpoints if required, and state your conclusion. For optimization, always define your variable clearly, express the quantity to be optimized as a function of one variable, and justify why your answer is a maximum or minimum (using the second derivative test or comparing values). The three questions below span different application areas; attempt all three even if one seems unfamiliar. Partial marks are generous in board exams, so show all working even if you cannot reach the final answer. Practice writing neatly and underlining or boxing the final answer for easy identification by the examiner.

Q18. (5 marks) Prove that the function f(x) = log(1 + x) − 2x/(2 + x) is strictly increasing on (−1, ∞).
Q19. (5 marks) A window is in the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is 10 m, find the dimensions that will allow the maximum amount of light to pass through.
Q20. (4 marks) Find the absolute maximum and minimum values of f(x) = x³ − 12x on the interval [−1, 3]. Justify your answer using the closed-interval method.

Section F: Case Study Question (1+1+2+2 marks)

Case-study questions were introduced in the CBSE Class 12 board pattern from 2020-21 and are now a permanent feature. A case study presents a short real-world or theoretical scenario—typically 80–100 words—followed by 4 sub-questions of varying marks (usually 1, 1, 2 and 2). For Application of Derivatives, expect scenarios involving rates of change in engineering contexts (cooling, motion, economics), geometric optimization (packaging, construction), or biological models (population growth, spread of disease). Read the paragraph carefully and underline key numerical data and relationships. The first two sub-questions are often direct MCQs or one-step calculations (1 mark each), while the last two require short derivations or applications of maxima-minima techniques (2 marks each). Even if the context is unfamiliar, the underlying mathematics is always from the NCERT syllabus—so translate the word problem into a function, differentiate, and solve. Attempt all four sub-questions; they are usually independent, so you can score even if you skip one. This section rewards careful reading and the ability to model real situations mathematically, skills that will serve you well beyond the board exam.

Complete Answer Key with Explanations

Below is the full answer key for all 33 questions in this worksheet. Each answer includes a brief explanation or the key step, enabling you to self-correct and understand where you went wrong. For MCQs, the correct option is indicated along with why the distractors are incorrect. For fill-in-the-blanks and match-the-following, the exact answer is given with the supporting reasoning. Short-answer and long-answer solutions are presented step-by-step, showing differentiation, critical-point analysis, and final conclusions. Use this key actively: after attempting the worksheet under timed conditions, mark your own paper, award partial marks honestly, and identify recurring mistakes—sign errors, algebraic slips, or conceptual gaps. If you consistently struggle with a particular question type (for example, optimization or tangent-normal equations), revisit the corresponding NCERT Class 12 Mathematics textbook sections and solved examples. For instant doubt-clearing and worked solutions with photo upload, consider joining CBSETUTOR.ai at ₹999/month—one flat fee for 24×7 AI tutor support across all subjects and classes 6–12, plus a 3-day free trial to explore the platform risk-free.

Section A Answers: Q1(a), Q2(a), Q3(a), Q4(b), Q5(a), Q6(c)
Section B Answers: Q7: (−∞,1)∪(2,∞) | Q8: x+2y=3 | Q9: 1/e | Q10: 0.32 | Q11: −1, 1 | Q12: r²
Section C Answers: (1)→(B), (2)→(D), (3)→(A), (4)→(C), (5)→(E)
Section D, E, F: Detailed step-by-step solutions provided below

How to Use This Worksheet for Maximum Benefit

To extract full value from this CBSE Class 12 Mathematics Chapter 6 Application of Derivatives worksheet, follow these best practices. First, print the page or copy the questions into a notebook—screen-based practice reduces retention. Set a timer for 90 minutes, find a quiet space, and attempt the worksheet in one sitting without referring to notes or solutions; this simulates real board-exam conditions and reveals your true speed and accuracy. Write all steps clearly, especially for Sections D, E and F, as if an examiner will grade your work. After time is up, stop immediately and mark your attempt using the answer key. Award marks honestly, including partial credit where steps are correct even if the final answer is wrong. Analyze your errors: were they conceptual (misunderstanding increasing/decreasing conditions), computational (sign errors, wrong derivative), or time-management issues (ran out of time on long questions)? Focus your revision on weak areas—if you lost marks in maxima-minima, redo NCERT Examples 17–25 from Chapter 6. Repeat this worksheet after a week to track improvement. For continuous support, CBSETUTOR.ai offers unlimited doubt-solving via photo upload, chapter-wise mock tests, and an AI tutor available 24×7—all at ₹999/month for every class from 6 to 12, with a 3-day free trial so you can test the platform before committing. Structured, timed worksheet practice combined with instant doubt resolution is the most effective formula for board exam success in Class 12 Mathematics.

Print or write out all questions; avoid on-screen attempts for serious practice
Set a strict 90-minute timer; stop when time is up to build exam discipline
Show full working in Sections D, E, F; partial marks are awarded in board exams
Self-mark using the answer key; analyse error patterns (concept vs. calculation vs. speed)
Revisit weak topics in NCERT textbook and reattempt the worksheet after one week
Use CBSETUTOR.ai for instant AI tutor support and photo-upload doubt solving at ₹999/month; 3-day free trial available

Frequently asked questions

What is the weightage of Application of Derivatives in CBSE Class 12 board exam?+

Application of Derivatives typically carries 8–10 marks in the CBSE Class 12 Mathematics board paper. Expect one 4-mark or 5-mark long-answer question, one 2-mark short-answer question, and 1–2 MCQs or fill-ups from this chapter.

How do I decide whether to use the first derivative test or second derivative test for maxima-minima?+

Use the first derivative test when f'(c)=0 and you check the sign of f' on either side of c. Use the second derivative test when f'(c)=0 and f''(c) exists; if f''(c)>0 it is a minimum, if f''(c)<0 it is a maximum. Second derivative test is faster but fails when f''(c)=0.

Are approximations using differentials still asked in the CBSE board exam?+

Yes, approximation problems (Δy ≈ dy = f'(x)Δx) appear occasionally as 2-mark short-answer questions. NCERT Exercise 6.4 covers this topic. Practice finding approximate values of roots, powers, or trigonometric functions using this formula.

What are common mistakes students make in tangent and normal problems?+

Common errors include forgetting to find the y-coordinate by substituting x into the original curve equation, mixing up the slope of tangent and normal, and sign mistakes in the formula y − y₁ = m(x − x₁). Always double-check that normal slope = −1/(slope of tangent).

How should I approach optimization word problems in Section E?+

Read carefully and identify the quantity to maximize or minimize. Express it as a function of one variable using given constraints. Differentiate, set the derivative to zero, solve for critical points, and apply the second derivative test or compare values. State units in your final answer and verify the answer makes physical sense.

Is it necessary to show the increasing/decreasing table in the answer sheet?+

A sign table (showing intervals and signs of f'(x)) is not mandatory but highly recommended. It organizes your solution, prevents errors, and earns you method marks even if the final answer has a minor slip. Draw a simple number-line with critical points marked and sign of f' in each interval.

What is the difference between local and absolute maxima?+

A local maximum at c means f(c) ≥ f(x) for all x in some small neighbourhood of c. An absolute maximum means f(c) ≥ f(x) for all x in the entire domain or given closed interval. To find absolute extrema on [a,b], evaluate f at all critical points and at endpoints a and b, then compare.

Can I use a calculator for this worksheet?+

CBSE Class 12 board exams do not permit calculators, so practice without one. For this worksheet, basic arithmetic and square-roots should be simplified by hand. If an exact answer like √2 or π appears, leave it in that form unless the question specifies 'approximate to two decimals.'

How many times should I practice this worksheet before the board exam?+

Attempt this worksheet at least twice—once as a timed mock test and once as untimed thorough revision. After each attempt, rework every question you got wrong. Supplement with NCERT Exercise 6.3 (maxima-minima) and miscellaneous exercise. Three weeks before boards, include this worksheet in your final revision rotation.

Where can I get instant help if I am stuck on a question from this worksheet?+

Upload a photo of the question to CBSETUTOR.ai and receive a step-by-step worked solution within seconds. The AI tutor is available 24×7 for all CBSE subjects and classes 6–12 at ₹999/month—one flat price with a 3-day free trial. No per-question charges, unlimited doubts allowed.

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