In JEE Advanced, Inverse Trigonometric Functions are an important topic in the subject of Mathematics. This topic deals with the reverse of trigonometric functions such as sine, cosine, and tangent, which are necessary to solve many types of problems.
Inverse Trigonometric Functions usually weigh roughly 6-7% out of total marks in the exam, and typically 1-2 questions each year relate to this topic. It is also a significant consideration in other topics, for example, in integration, differentiation, or solving equations. If you know the basics about the topic, the domain, range, etc, you will do well on this topic in JEE Advanced.
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Here are some of the important JEE Advanced Previous Year Questions of the topic Inverse Trigonometric Function (ITF)
Question 1:
Considering only the principal values of the inverse trigonometric functions, the value of \( \tan\left( \sin^{-1}\left( \frac{3}{5} \right) - 2 \cos^{-1}\left( \frac{2}{\sqrt{5}} \right) \right) \) is:
Question 2:
If \( 0 < x < 1 \), then \( \sqrt{1 + x^2 \left[ \left( x \cos\left( \cot^{-1} x \right) + \sin\left( \cot^{-1} x \right) \right)^2 - 1 \right]^{1/2}} \) is equal to:
Question 3:
The value of \( \tan\left[ \cos^{-1}\left( \frac{4}{5} \right) + \tan^{-1}\left( \frac{2}{3} \right) \right] \) is:
Question 4:
The principal value of \( \sin^{-1}\left( \sin \frac{2\pi}{3} \right) \) is:
Question 5:
Let \( \alpha \) and \( \beta \) be real numbers such that \( -\frac{\pi}{4} < \beta < 0 < \alpha < \frac{\pi}{4} \). If \( \sin(\alpha + \beta) = \frac{1}{3} \) and \( \cos(\alpha - \beta) = \frac{2}{3} \), then the greatest integer less than or equal to:
\[ \left( \frac{\sin \alpha}{\cos \beta} + \frac{\cos \beta}{\sin \alpha} + \frac{\cos \alpha}{\sin \beta} + \frac{\sin \beta}{\cos \alpha} \right)^2 \] is ________.
Question 6:
Let \( \alpha \) and \( \beta \) be real numbers such that \( -\frac{\pi}{4} < \beta < 0 < \alpha < \frac{\pi}{4} \). If \( \sin(\alpha + \beta) = \frac{1}{3} \) and \( \cos(\alpha - \beta) = \frac{2}{3} \), then the greatest integer less than or equal to:
\[ \left( \frac{\sin \alpha}{\cos \beta} + \frac{\cos \beta}{\sin \alpha} + \frac{\cos \alpha}{\sin \beta} + \frac{\sin \beta}{\cos \alpha} \right)^2 \] is ________.
Question 7:
In a triangle \( ABC \), let \( AB = \sqrt{23} \), \( BC = 3 \), and \( CA = 4 \). Then the value of \( \frac{\cot A - \cot C}{\cot B} \) is ________.
Question 8:
Let \( x \), \( y \), and \( z \) be positive real numbers. Suppose \( x \), \( y \), and \( z \) are the lengths of the sides of a triangle opposite to its angles \( X \), \( Y \), and \( Z \), respectively. If \( \tan \frac{X}{2} + \tan \frac{Z}{2} = \frac{2y}{x + y + z} \), then which of the following statements is/are TRUE?
Question 9:
The value of \( \lim_{x \to 0} \frac{1}{2} \left( 1 - \cos 2x \right) \) is:
JEE Advanced Inverse Trigonometric function Question Trends (2014-2024)
A detailed year wise analysis with important topics with its difficulty level
Year-wise Question Distribution
| Year | No. of Questions | Key Topics Tested | Difficulty Level |
|---|---|---|---|
| 2024 | 1 | Domain/Range of Inverse Trigonometric Functions , Functional Equations | Moderate |
| 2023 | 1 | Simplification of ITF Expressions | Easy |
| 2022 | 2 | ITF Identities, Trigonometric Substitutions | Moderate-Hard |
| 2021 | 1 | Properties of sin−1x, cos−1x | Moderate |
| 2020 | 1 | Sum of Angles | Easy |
| 2019 | 1 | ITF Equations and Inequalities | Hard |
| 2018 | 2 | Graphs of ITF, Domain Constraints | Moderate |
| 2017 | 1 | ITF Differentiation | Hard |
| 2016 | 1 | Principal Values and Range of ITF | Moderate |
| 2015 | 1 | Simplification using Identities | Easy |
| 2014 | 1 | ITF in Integration (Linked Question) | Hard |
Topic-wise Weightage & Key Concepts
| Topic | Weightage | Key Concepts Tested |
| Identities & Simplification | 40% | - Sum/Difference identities- Complementary angles- Pythagorean conversions |
| Domain & Range | 30% | - Principal values- Composite functions- Inequality constraints |
| Functional Equations | 15% | - Solving ITF equations- Variable substitutions |
| Graphs & Properties | 10% | - Monotonicity- Symmetry- Curve sketching |
| Calculus Applications | 5% | - Differentiation- Integration (rare) |
JEE Advanced Mathematics Paper Analysis (2024 & Trends)
The mathematics syllabus of JEE Advanced assesses students in their analytical thinking, problem-solving abilities, and understanding of concepts. Two papers (Paper 1 and Paper 2), last three hours each, with approximately 15 - 18 questions covering the syllabus collectively. The questions are based/inspired by every part of the syllabus with varying levels of abilities and complexity, ranging from theoretical concepts to challenging numerical problems, requiring a mixture of properties and applications from several disciplines.
Overview (2024)
| Aspect | Details |
|---|---|
| Total Questions | Approximately 15–18 per paper (2 papers) |
| Difficulty | Moderate to Hard; around 30% easy, 40% moderate, and 30% difficult |
| Question Types | Single/Multiple MCQs, Numerical Value, Paragraph-based, Match-the-Column |
| Key Focus Areas | Differential Calculus, Integral Calculus, Matrices & Determinants, Probability, Vector and 3D Geometry |
Chapter-Wise Weightage (2024)
| Topic | Weightage | Difficulty | Remarks |
|---|---|---|---|
| Differential Calculus | 8% | Moderate | Emphasis on application of derivatives and limits |
| Integral Calculus | 19% | Moderate-Hard | Focus on definite integration and differential equations |
| Matrices & Determinants | 8% | Moderate | Application-based questions on matrix operations and properties |
| Probability | 14% | Moderate | Problems involving complex probability distributions |
| Vector and 3D Geometry | 11% | Moderate | Questions on vector algebra and three-dimensional geometry concepts |
Difficulty Trends (2019–2024)
| Year | Difficulty Level | Key Observations |
|---|---|---|
| 2024 | Moderate-Hard | There are an more numerical problems consisting of several concepts |
| 2023 | Moderate | There was a balanced theory and numericals but lengthy problems in calculus |
| 2022 | Hard | Focussed on complex algebraic manipulations and advanced calculus |
| 2021 | Very Hard | It had unusual problems in coordinate geometry and vector algebra |
| 2020 | Moderate | The concept were more NCERT-aligned and straightforward questions in algebra and trigonometry |
JEE Advanced Exam Pattern
The JEE Advanced exam is considered as one of the top engineering entrance examinations in India, which takes place annually for enrollment at IITs and other top institutions to undergraduate programs. JEE Advanced consists of two compulsory papers, Paper 1 and Paper 2, each with three hours of duration. It challenges students’ knowledge of Physics, Chemistry and Mathematics. The exam consists of different question types including Multiple Choice Questions (MCQs), Numerical Value-based Questions and Assertion-Reasoning questions to evaluate the foundational knowledge of topics and ability to solve problems. Both papers have a total of 180 marks, with marks deducted for incorrect answers in a few question types.
| Aspect | Details |
|---|---|
| Mode | Computer-Based Test (CBT) |
| Papers | 2 (Paper 1 & Paper 2) – Both mandatory |
| Duration | 3 hours per paper (with break between papers) |
| Total Subjects | Physics, Chemistry, Mathematics (Equal weightage) |
| Total Questions | ~54–60 (18–20 per subject, split across question types) |
| Language | English & Hindi (option to switch during exam) |
The exam is known for its challenging nature and is conducted in English and Hindi. It is essential to clear the JEE Main exam before appearing for JEE Advanced.
Marking Scheme with Question types in JEE Advanced
Here is a detailed marking scheme for different questions in JEE Advanced exam.
| Question Type | Marks per Q | Negative Marking | Examples |
|---|---|---|---|
| Single Correct MCQ | +3 | -1 | Basic theory/direct formula questions |
| Multiple Correct MCQ | +4 (full) | -2 (if all wrong) / Partial marks* | "Which of the following is/are true?" |
| Numerical Value (NV) | +3 | No negative | Calculations (e.g., wavelength) |
| Paragraph-Based | Varies (2–4) | Depends on sub-type | Linked to a common scenario |
| Match the Following | +2 per match | -1 per wrong match | Column matching (e.g., graphs) |
Partial Marks: For Multiple Correct MCQs, +1 per correct option (if not all selected).
Subject-Wise Distribution
Here is the subject wise questions distribution in the JEE Advanced exam describing number of questions in each subject.
| Subject | MCQs | Numerical | Paragraph/Matching | Total |
|---|---|---|---|---|
| Physics | 6–8 | 4–5 | 2–3 | 12–16 |
| Chemistry | 6–8 | 4–5 | 2–3 | 12–16 |
| Maths | 6–8 | 4–5 | 2–3 | 12–16 |
Subject wise Syllabus: JEE Advanced 2025
Here is a quick syllabus of JEE Advanced 2025 subject wise which helps in preparation of the students. Check JEE Advanced Syllabus for a detailed Syllabus.
| Subject | Key Topics |
|---|---|
| Physics | Mechanics, Electrodynamics, Modern Physics, Optics, Thermodynamics, Error Analysis |
| Chemistry | Physical (Thermo, Kinetics), Organic (Reactions), Inorganic (Coordination, P-Block) |
| Maths | Algebra, Calculus, Coordinate Geometry, Trigonometry, Vectors |
Note:
- Physics: Focus on Modern Physics (20% weightage) & Electrodynamics.
- Chemistry: Organic reactions & Coordination compounds are high-scoring.
- Maths: Calculus (35% weightage) is most critical.
In addition to conceptual knowledge, you should practice multiple mock tests before your exam.
Check:
JEE Advanced Previous Year Question Papers with Detailed Solutions
JEE Advanced 2024 Question Paper Pdf with Solutions
| Paper Name | Question Paper with Solutions |
|---|---|
| JEE Advanced 2024 Paper 1 Question Paper (English) | Check Solutions |
| JEE Advanced 2024 Paper 2 Question Paper (English) | Check Solutions |
| JEE Advanced 2024 Paper 1 Question Paper (Hindi) | Check Solutions |
| JEE Advanced 2024 Paper 2 Question Paper (Hindi) | Check Solutions |
| Architecture Aptitude Test | Check Solutions |
JEE Advanced 2023 Question Paper PDF Download With Solutions
| Paper | Answer Key PDF |
|---|---|
| Question Paper 1 PDF | Check Solutions |
| Question Paper 2 PDF | Check Solutions |
| Question Paper AAT PDF | Check Solutions |
JEE Advanced 2022 Question Paper
| 2022 Question Paper | PDF Download |
|---|---|
| Question Paper 1 PDF | Click here |
| Question Paper 2 PDF | Click here |
JEE Advanced 2021 Question Paper
| Question Paper | PDF Download | ||
|---|---|---|---|
| Question Paper 1 PDF | Physics | Chemistry | Maths |
| Question Paper 2 PDF | Physics | Chemistry | Maths |
JEE Advanced 2020 Question Paper
| Question Paper | PDF Download |
| Question Paper 1 PDF | Click here |
| Question Paper 2 PDF | Click here |
Frequently Asked Questions (FAQs)
Q1. Can I skip trigonometry from my studies for JEE Advanced?
Answer. Absolutely not, it is not recommended to leave out trigonometry from the JEE Advanced syllabus. Trigonometry is a significant subject across both Mathematics and Physics, and is usually present in some of the exam questions in India. Therefore, if you want to do your best in the JEE Advanced exam, understanding trigonometric angles, identities, and equations is essential.
Q2. How many questions come from trigonometry in JEE Advanced?
Answer. Typically, each year there are around 2 to 3 questions each year related to Trigonometry for the JEE Advanced exam. Usually these are problems that ask you to apply various trigonometric identities, and equations, and common questions are inverse trigonometry functions, and applications for problem solving. The number of questions can vary slightly from year to year, but overall Trigonometry is an important topic you will need to be familiar with if you want to be successful on the exam.
Q3. Are Inverse Trigonometric Functions (ITF) an important topic for JEE Advanced?
Answer. Yes, inverse Trigonometric functions (ITF) are very important and a vital topic to study for the JEE Advanced exam. This topic is usually assessed in the exam, and is also a topic that you will find affects your overall score in Trigonometry. ITF topics include applying these concepts to solve equations, study the properties of inverse functions, or apply them in a Differentiation or Integration context. Usually in trigonometry, you should expect around 2-3% of total marks to be weighed on ITF questions, so this topic must be included in your study plan.
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