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Important Questions: CBSE Class 10 Mathematics Chapter 9 Some Applications of Trigonometry

Chapter 9 Some Applications of Trigonometry is one of the highest-scoring chapters in CBSE Class 10 Mathematics, bridging pure ratio work from Chapter 8 with real-world problem-solving. Every question boils down to sketching a right triangle, identifying the correct trigonometric ratio, and solving an equation. This page organises 18 carefully chosen board-style questions by marks weightage, complete with model answers, so you can practise efficiently and avoid the common pitfalls that cost marks in the exam hall.

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Key takeaways

✓Chapter 9 Some Applications of Trigonometry typically carries 8-10 marks in the CBSE Class 10 board exam, with most questions worth 2 or 3 marks.
✓Angle of elevation is measured upwards from the horizontal; angle of depression is measured downwards from the horizontal—mixing these is the most common mistake.
✓Every problem requires a clear diagram showing the right triangle, known and unknown sides, and the angle; draw before you calculate.
✓Questions often involve two angles of elevation or depression from different points to find heights or distances using simultaneous equations.
✓Case-based questions in 2024-25 apply heights and distances to real scenarios like lighthouse visibility, kite flying, or building shadows.
✓Competency-focused MCQs test conceptual clarity: recognising which trigonometric ratio applies and checking dimensional consistency in answers.
✓CBSETUTOR.ai offers unlimited doubt-solving for Chapter 9 via photo upload at ₹999/month for all classes, with a 3-day free trial for Class 10 students preparing for boards.

Chapter Overview and Marks Weightage in CBSE Class 10 Board Exam

Some Applications of Trigonometry appears in Unit-III Geometry and Trigonometry, which collectively carries 22 marks in the CBSE Class 10 Mathematics paper. Chapter 9 alone contributes 8-10 marks, typically distributed as one 3-mark question on heights and distances, one or two 2-mark questions on angle of elevation or depression, and sometimes a competency-based MCQ worth 1 mark. From 2023 onwards, CBSE has introduced case-based questions: a 4-5 mark scenario might describe a lighthouse, a building under construction, or a drone survey, followed by three sub-questions testing heights and distances. The chapter has exactly two NCERT exercises—Exercise 9.1 (16 questions) covers straightforward single-angle problems, while the optional questions extend to two angles or indirect measurements. Difficulty-wise, the board paper features at least one direct NCERT adaptation and one twisted version requiring students to set up simultaneous trigonometric equations. Because every mark counts towards your overall percentage, securing full marks in Chapter 9 is both achievable and strategic: the syllabus is narrow, the question types are predictable, and accuracy depends purely on diagram discipline and ratio recall.

Expected weightage: 8-10 marks (one 3-mark + two 2-mark questions, or one case-based 4-5 mark question).
Core topics: heights and distances using angle of elevation and angle of depression.
Difficulty distribution: 60% direct NCERT-style, 40% application or two-step problems.
Time allocation: most students spend 8-12 minutes on this chapter's questions during the 3-hour exam.
High-scoring potential: with proper diagram and correct ratio selection, full marks are routine.

1-Mark Questions: MCQs and Very Short Answers (VSA)

Competency-based MCQs in the CBSE 2024-25 pattern test whether students can identify the correct trigonometric setup without full calculation. Expect questions that ask which ratio is needed, what the angle represents, or which statement about a diagram is true. VSAs may require a one-line formula statement or a direct substitution. These questions reward conceptual clarity: knowing that tan θ = perpendicular/base, recognising that angle of depression from point A equals angle of elevation from point B (alternate angles), and checking units. Even though each carries only 1 mark, an error here signals weak fundamentals that will cascade into 3-mark mistakes. Practise identifying the correct ratio in under 30 seconds by visualising the triangle and labelling sides relative to the given angle. The four MCQs below mirror the style seen in the 2023 and 2024 CBSE sample papers and regional pre-boards across Delhi, Uttar Pradesh, and Maharashtra examination centres.

2-Mark Questions: Short-Answer Type

Two-mark questions typically present a single scenario requiring students to draw a diagram, write the trigonometric ratio, substitute values, and solve. CBSE awards 1 mark for the correct setup (diagram + ratio) and 1 mark for the numerical answer. Common themes include finding the height of a building given angle and distance, finding distance given height and angle, or verifying a given measurement. Students must show the substitution step even if the arithmetic is trivial; skipping straight to the answer risks losing the working mark. Always rationalise denominators unless the answer key specifies otherwise, and include units (metres, etc.) in the final answer. The three questions below are adapted from CBSE 2022, 2023, and 2024 board papers and regional sets from examination centres in Bengaluru, Chennai, and Jaipur. Each model answer demonstrates the minimum detail required for full marks.

3-Mark Questions: Application and Multi-Step Problems

Three-mark questions form the backbone of Chapter 9 scoring. They usually involve two different points of observation, requiring students to form two equations and eliminate one variable. For instance, the angle of elevation from two points on the same straight line, or angles of depression to two boats from a lighthouse. CBSE marks these as: 1 mark for diagram and correct identification of triangles, 1 mark for forming equations, and 1 mark for solving and final answer with units. Common errors include confusing which angle belongs to which triangle, sign mistakes when subtracting distances, and forgetting to rationalise surds. Always label your diagram with variables (h for height, x for unknown distance) and write the two tan or sin equations side by side before attempting elimination. The four questions below represent standard board patterns seen in the 2023 Delhi set, 2024 outside-Delhi set, and compartment exams.

5-Mark Questions and Case-Based Problems

From the 2023-24 session, CBSE introduced case-based or integrated questions worth 4-5 marks. A typical case study provides a real-world context—construction site, airport radar, hill station survey—followed by three sub-questions of varying difficulty. Marks are distributed as 1 + 2 + 2 or 1 + 1 + 3. These questions test reading comprehension, diagram interpretation, and multi-step reasoning. Students must extract numerical data from a paragraph, identify which trigonometric relationship applies, and solve systematically. Partial credit is generous if the method is sound even if arithmetic slips occur. The three case-based examples below reflect the 2024 CBSE sample paper format and the style used in the Delhi, Ajmer, and Patna regional papers. Each sub-question is answered in bullet form to show mark allocation clearly.

How CBSE Frames Questions from Chapter 9

CBSE question setters follow predictable patterns when designing Some Applications of Trigonometry problems. Approximately 40% of questions are direct NCERT adaptations: the same building-and-observer setup with different numbers. Another 40% twist the language—asking for the angle when height and distance are given, or providing the answer and asking students to verify or prove it. The remaining 20% are application or case-based: embedding trigonometry within a real-world narrative like drone surveys, bridge construction, or mountaineering. Since 2020, CBSE has favoured competency-based MCQs that test whether students can identify the correct trigonometric ratio without lengthy calculation, rewarding conceptual understanding over rote drill. In the descriptive section, examiners look for a clear diagram (often worth 0.5 to 1 mark), correct identification of perpendicular and base relative to the angle, proper equation setup, and a final answer with units. Partial marks are awarded if the method is visible even when arithmetic fails, so never skip working steps. The chapter also appears in internal assessments and periodic tests, often as a 10-mark section combining two 2-mark and two 3-mark questions, making consistent practice essential for maintaining term grades that contribute 20% to the board result under the CBSE assessment scheme.

Direct NCERT-style: single angle of elevation or depression, straightforward tan/sin/cos substitution.
Two-angle problems: simultaneous equations from two points of observation, testing algebraic manipulation.
Reverse questions: given height and distance, find the angle; or given the final answer, verify by working backwards.
Proof-based: prove that height equals √(ab) or that two given angles are complementary.
Case-based scenarios: embedding three sub-questions within a 150-word real-world context.
Competency MCQs: identify the correct ratio, recognise alternate angles, check unit consistency.

Common Mistakes Students Make and How to Avoid Them

Even strong students lose 2-3 marks in Chapter 9 due to avoidable errors. The most frequent mistake is confusing angle of elevation with angle of depression: remember that elevation is measured upward from the horizontal, depression downward, and the angle of depression from point A equals the angle of elevation from point B by alternate angles. Second, students often select the wrong trigonometric ratio because they mislabel the sides. Always mark 'perpendicular' and 'base' relative to the given angle before choosing tan, sin, or cos. Third, arithmetic slips occur when rationalising denominators: 1/√3 must be multiplied top and bottom by √3, yielding √3/3, not left as a decimal approximation. Fourth, forgetting units in the final answer costs the last mark; CBSE mark schemes explicitly penalise missing 'm' or 'metres'. Fifth, in two-angle problems, students mix up which distance corresponds to which triangle, leading to incorrect simultaneous equations. Label your diagram with variables (h, x, d) and write both equations before attempting to solve. Sixth, skipping the diagram altogether is risky: even if your algebra is correct, examiners may deduct the diagram mark. Finally, using a calculator set to radians instead of degrees will give nonsensical values; always verify mode before the exam. Practising past-year papers under timed conditions and checking each step against the NCERT solutions will eliminate these mistakes and secure full marks in this high-yield chapter.

Confusing angle of elevation (upward) with angle of depression (downward); draw horizontal reference lines.
Mislabelling perpendicular and base relative to the given angle; annotate the diagram before writing ratios.
Leaving surds un-rationalised or converting to decimals prematurely; work symbolically until the final step.
Omitting units (m, metres) in the answer; CBSE mark schemes deduct 0.5-1 mark for this.
Setting up incorrect simultaneous equations in two-angle problems; always write both equations and check variables.
Skipping the diagram; even a rough sketch can earn 0.5-1 mark and clarify your own thinking.
Calculator errors (radians vs degrees); double-check mode and verify one ratio by hand.

Exam Strategy: Time Management and Answer Presentation

In the 3-hour CBSE Class 10 Mathematics board exam, allocate roughly 10-12 minutes to the Chapter 9 questions. A single 3-mark question should take 4-5 minutes: 1 minute to read and draw the diagram, 2 minutes to set up and solve equations, 1 minute to rationalise and write the final answer with units, and 30 seconds to recheck substitution. Two-mark questions deserve 2.5-3 minutes each, and 1-mark MCQs no more than 45 seconds. For case-based questions, read the entire case first (1 minute), then tackle sub-questions in order, ensuring you extract data correctly. Always start with the diagram: a clear, labelled right triangle not only earns method marks but also prevents ratio confusion. Use a pencil and ruler for the sketch, then write the solution in pen. Show every substitution step; even if the arithmetic is trivial, examiners award partial credit for method when the final answer is wrong. If you are stuck on a 3-mark question, attempt the diagram and write the ratio setup to secure 1-1.5 marks, then move on and return if time permits. Rationalising surds is mandatory unless the question specifies otherwise; leaving 1/√3 un-rationalised will lose 0.5 marks. Finally, underline or box your final answer so it stands out during marking. Practising 15-20 timed questions in the week before the exam will build speed and accuracy, ensuring you complete Chapter 9 with full marks and enough time left for the tougher Coordinate Geometry or Probability questions.

Allocate 10-12 minutes total for Chapter 9 across all questions on the paper.
Spend 1 minute on the diagram; it earns marks and clarifies your approach.
Show every substitution step; partial credit requires visible working.
Rationalise surds unless told otherwise; 1/√3 = √3/3, not 0.577.
Box or underline the final answer with units to make it easy to spot.
If stuck on a 3-mark question, secure method marks (diagram + ratio) and move on.

CBSETUTOR.ai: 24×7 Doubt-Solving for Heights and Distances

Chapter 9 questions often hinge on a single conceptual clarity—identifying the correct triangle or choosing the right ratio—that a quick expert hint can unlock. CBSETUTOR.ai provides Class 10 students with round-the-clock access to an AI tutor trained on NCERT and CBSE board patterns. Simply photograph your Some Applications of Trigonometry problem—whether it is an NCERT exercise, a sample paper question, or a tricky two-angle proof—and receive a step-by-step solution within seconds, complete with annotated diagrams and ratio explanations in natural Indian English. The platform covers all of Class 10 Mathematics (Standard and Basic), so you can clarify Trigonometric Identities from Chapter 8, Surface Areas from Chapter 13, or Probability from Chapter 15 in the same session. Pricing is straightforward: ₹999 per month for all subjects and classes 6-12, with no hidden fees or per-question charges. A 3-day free trial lets students test the photo-upload solver and practice modules before committing. Especially valuable during board revision, when doubt-clearing slots with school teachers are scarce, CBSETUTOR.ai ensures that no conceptual gap remains unresolved, helping students convert every Chapter 9 question into full marks on exam day.

Instant step-by-step solutions for NCERT Exercise 9.1 and board-style questions via photo upload.
Annotated diagrams showing perpendicular, base, and angle labels to prevent ratio confusion.
Covers all Class 10 Mathematics chapters, so you can revise Trigonometry, Geometry, and Algebra in one place.
Flat ₹999/month for classes 6-12, all subjects; one price, unlimited questions.
3-day free trial with no credit-card requirement; test the AI tutor on your toughest Chapter 9 problems.
Especially useful for self-study during board exam preparation when teacher availability is limited.

Frequently asked questions

How many marks does Chapter 9 Some Applications of Trigonometry carry in the CBSE Class 10 board exam?+

Chapter 9 contributes 8-10 marks out of the total 80 in the CBSE Class 10 Mathematics theory paper. Typically one 3-mark question on heights and distances, one or two 2-mark questions, and sometimes a 1-mark MCQ or a 4-5 mark case-based question appear.

What is the difference between angle of elevation and angle of depression?+

Angle of elevation is measured upwards from the horizontal line of sight to an object above the observer. Angle of depression is measured downwards from the horizontal to an object below. By alternate angles, the angle of depression from point A equals the angle of elevation from point B when both lie on a vertical plane.

Which trigonometric ratios are most commonly used in Chapter 9 questions?+

Tangent (tan θ = perpendicular/base) is used in about 70% of problems because most questions involve finding height given distance or vice versa. Sine and cosine appear when the hypotenuse (ladder length, string length) is known or required. Always check which sides are given before choosing the ratio.

How should I approach two-angle problems where observations are made from two different points?+

Draw one diagram showing both observation points and label unknowns (h for height, x for one distance). Write two separate equations using tan or sin from each point. Eliminate one variable by substitution or addition, solve for the remaining variable, then back-substitute to find the other. Always verify units and rationalise surds in the final answer.

Do I need to rationalise denominators in Chapter 9 answers?+

Yes, unless the question specifies a decimal approximation. Leaving 1/√3 un-rationalised will cost 0.5 marks. Multiply numerator and denominator by the surd: 1/√3 becomes √3/3. CBSE mark schemes require exact rationalised forms in board exams.

Is drawing a diagram compulsory for every Chapter 9 question?+

While not always explicitly mentioned, CBSE mark schemes allocate 0.5-1 mark for a correct, labelled diagram in 2-mark and 3-mark questions. Even if your algebraic method is sound, omitting the diagram can cost you. A clear sketch also prevents ratio and angle confusion, so always draw before you calculate.

What are the most common mistakes in Some Applications of Trigonometry questions?+

Top errors: confusing elevation with depression, mislabelling perpendicular and base, forgetting to rationalise surds, omitting units in the final answer, mixing up distances in two-angle problems, and calculator mode errors (radians vs degrees). Each of these can lose 0.5-1 marks per question.

How do case-based questions in Chapter 9 differ from traditional word problems?+

Case-based questions present a 150-200 word real-world scenario (construction site, lighthouse, drone survey) followed by three sub-questions worth 1+2+2 or 1+1+3 marks. You must extract numerical data from the paragraph, draw a diagram, and solve each part step by step. Partial marks are awarded for method even if arithmetic slips occur.

Can I use a calculator for trigonometric values in the board exam?+

CBSE allows simple calculators (non-programmable) in the Class 10 Mathematics exam. However, most Chapter 9 questions use standard angles (30°, 45°, 60°) whose exact ratios (1/√3, 1, √3, etc.) you must know by heart. Using a calculator for these risks rounding errors and wastes time; memorise the ratios and work symbolically.

How much time should I spend on Chapter 9 questions during the 3-hour board exam?+

Allocate 10-12 minutes total: about 4-5 minutes for a 3-mark question, 2.5-3 minutes per 2-mark question, and 45 seconds per MCQ. If you are stuck, secure method marks (diagram + ratio setup) and move on, returning only if time permits after completing the rest of the paper.

Are NCERT Exercise 9.1 questions enough to score full marks in the board exam?+

NCERT Exercise 9.1 (16 questions) covers all fundamental concepts and roughly 60% of board questions are direct adaptations. However, to handle twisted two-angle problems and case-based questions, practise the optional NCERT examples, CBSE sample papers, and at least three previous years' board sets. This ensures you see every question type and build speed.

How can CBSETUTOR.ai help with Chapter 9 preparation?+

CBSETUTOR.ai offers instant step-by-step solutions for any Some Applications of Trigonometry problem via photo upload. The AI tutor provides annotated diagrams, ratio selection logic, and worked solutions in Indian English. At ₹999/month for all classes and subjects, with a 3-day free trial, it is an affordable way to resolve doubts 24×7 during board exam revision when teacher availability is limited.

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