Welcome to StudyMats. This comprehensive educational guide for Class 12 Inverse Trigonometric Functions Important Questions is prepared to assist students in preparing effectively for their school, board, and competitive examinations. All questions, exercise solutions, and concept notes strictly adhere to the latest NCERT curriculum and CBSE guidelines.
Class 12 Maths Chapter 2: Inverse Trigonometric Functions Important Questions
Mastering Inverse Trigonometric Functions (प्रतिलोम त्रिकोणमितीय फलन) is essential for scoring high in CBSE Class 12 Board Examinations and competitive tests like JEE Main. In this chapter, questions on Principal Value Branches, domain & range restrictions, and algebraic simplification of inverse trigonometric expressions are frequently asked in 1-mark (MCQ), 2-mark, and 3-mark sections.
Below is a curated set of 25 bilingual (English & Hindi) important practice questions carefully selected from NCERT textbooks, previous years' board papers (PYQs), and standard exam patterns. Work through each problem step-by-step to solidify your conceptual understanding.
- \(\sin^{-1} x\): Domain \([-1, 1]\), Principal Value Branch \([-\pi/2, \pi/2]\)
- \(\cos^{-1} x\): Domain \([-1, 1]\), Principal Value Branch \([0, \pi]\)
- \(\tan^{-1} x\): Domain \(\mathbb{R}\), Principal Value Branch \((-\pi/2, \pi/2)\)
25 Bilingual Important Questions (Practice & Board Exam Based)
Q1. Principal value of \(\sin^{-1}(-1/2)\) is:
\(\sin^{-1}(-1/2)\) का मुख्य मान ज्ञात कीजिए।
Q2. Find the value of \(\tan^{-1}(\sqrt{3}) - \sec^{-1}(-2)\)
\(\tan^{-1}(\sqrt{3}) - \sec^{-1}(-2)\) का मान ज्ञात कीजिए।
Q3. Find the domain of \(\cos^{-1}(2x-1)\)
\(\cos^{-1}(2x-1)\) का प्रांत (domain) ज्ञात कीजिए।
Q4. Write the principal value of \(\cos^{-1}(\cos(7\pi/6))\)
\(\cos^{-1}(\cos(7\pi/6))\) का मुख्य मान लिखिए।
Q5. Find the value of \(\cos^{-1}(\cos(13\pi/6))\)
\(\cos^{-1}(\cos(13\pi/6))\) का मान ज्ञात कीजिए।
Q6. Find \(\tan^{-1}(1) + \cos^{-1}(-1/2) + \sin^{-1}(-1/2)\)
\(\tan^{-1}(1) + \cos^{-1}(-1/2) + \sin^{-1}(-1/2)\) का मान ज्ञात कीजिए।
Q7. Find the value of \(\sin(\sec^{-1}x + \csc^{-1}x)\), \(|x| \geq 1\)
\(\sin(\sec^{-1}x + \csc^{-1}x)\) का मान ज्ञात कीजिए, जहाँ \(|x| \geq 1\)
Q8. Find the value of \(\tan^{-1}(\sqrt{3}) - \cot^{-1}(-\sqrt{3})\)
\(\tan^{-1}(\sqrt{3}) - \cot^{-1}(-\sqrt{3})\) का मान ज्ञात कीजिए।
Q9. Find the value of \(\sin^{-1}(\sin(3\pi/5))\)
\(\sin^{-1}(\sin(3\pi/5))\) का मान ज्ञात कीजिए।
Q10. Find the principal value of \(\text{cosec}^{-1}(2)\)
\(\text{cosec}^{-1}(2)\) का मुख्य मान ज्ञात कीजिए।
Q11. Find the value of \(\sin(\tan^{-1} x)\), \(|x| < 1\)
\(\sin(\tan^{-1} x)\) का मान ज्ञात कीजिए, जहाँ \(|x| < 1\)
Q12. Prove that \(2\sin^{-1}(3/5) = \tan^{-1}(24/7)\)
सिद्ध कीजिए: \(2\sin^{-1}(3/5) = \tan^{-1}(24/7)\)
Q13. Solve for \(x\): \(\sin^{-1}(1-x) - 2\sin^{-1}x = \pi/2\)
\(x\) के लिए हल कीजिए: \(\sin^{-1}(1-x) - 2\sin^{-1}x = \pi/2\)
Q14. Prove that \(\tan^{-1}(1/2) + \tan^{-1}(1/3) = \pi/4\)
सिद्ध कीजिए: \(\tan^{-1}(1/2) + \tan^{-1}(1/3) = \pi/4\)
Q15. Find the value of \(\tan(1/2 \cos^{-1}(\sqrt{5}/3))\)
\(\tan(1/2 \cos^{-1}(\sqrt{5}/3))\) का मान ज्ञात कीजिए।
Q16. Simplify: \(\tan^{-1}(\sqrt{\frac{1-\cos x}{1+\cos x}})\), \(0 < x < \pi\)
सरल कीजिए: \(\tan^{-1}(\sqrt{\frac{1-\cos x}{1+\cos x}})\), जहाँ \(0 < x < \pi\)
Q17. Prove that \(\sin^{-1}(8/17) + \sin^{-1}(3/5) = \tan^{-1}(77/36)\)
सिद्ध कीजिए: \(\sin^{-1}(8/17) + \sin^{-1}(3/5) = \tan^{-1}(77/36)\)
Q18. Find the value of \(\cos(\sin^{-1}(1/3) + \cos^{-1}(1/3))\)
\(\cos(\sin^{-1}(1/3) + \cos^{-1}(1/3))\) का मान ज्ञात कीजिए।
Q19. Prove that \(\tan^{-1}(1/4) + \tan^{-1}(2/9) = \frac{1}{2} \cos^{-1}(3/5)\)
सिद्ध कीजिए: \(\tan^{-1}(1/4) + \tan^{-1}(2/9) = \frac{1}{2} \cos^{-1}(3/5)\)
Q20. Solve for \(x\): \(\tan^{-1}(x+1) + \tan^{-1}(x-1) = \tan^{-1}(8/31)\)
\(x\) के लिए हल कीजिए: \(\tan^{-1}(x+1) + \tan^{-1}(x-1) = \tan^{-1}(8/31)\)
Q21. Prove that \(\tan^{-1}x + \tan^{-1}(\frac{2x}{1-x^2}) = \tan^{-1}(\frac{3x-x^3}{1-3x^2})\)
सिद्ध कीजिए: \(\tan^{-1}x + \tan^{-1}(\frac{2x}{1-x^2}) = \tan^{-1}(\frac{3x-x^3}{1-3x^2})\)
Q22. Solve for \(x\): \(\tan^{-1}(\frac{x-1}{x-2}) + \tan^{-1}(\frac{x+1}{x+2}) = \frac{\pi}{4}\)
\(x\) के लिए हल कीजिए: \(\tan^{-1}(\frac{x-1}{x-2}) + \tan^{-1}(\frac{x+1}{x+2}) = \frac{\pi}{4}\)
Q23. Prove: \(9\pi/8 - 9/4 \sin^{-1}(1/3) = 9/4 \sin^{-1}(2\sqrt{2}/3)\)
सिद्ध कीजिए: \(9\pi/8 - 9/4 \sin^{-1}(1/3) = 9/4 \sin^{-1}(2\sqrt{2}/3)\)
Q24. Prove that \(\cot^{-1}(\frac{\sqrt{1+\sin x} + \sqrt{1-\sin x}}{\sqrt{1+\sin x} - \sqrt{1-\sin x}}) = x/2\)
सिद्ध कीजिए: \(\cot^{-1}(\frac{\sqrt{1+\sin x} + \sqrt{1-\sin x}}{\sqrt{1+\sin x} - \sqrt{1-\sin x}}) = x/2\)
Q25. Prove that \(\tan^{-1}(\frac{\cos x}{1-\sin x}) - \cot^{-1}(\frac{\sqrt{1+\cos x}}{\sqrt{1-\cos x}}) = \pi/4\)
सिद्ध कीजिए: \(\tan^{-1}(\frac{\cos x}{1-\sin x}) - \cot^{-1}(\frac{\sqrt{1+\cos x}}{\sqrt{1-\cos x}}) = \pi/4\)
💡 Board Exam Preparation Tips for Inverse Trigonometry
When solving Inverse Trigonometric problems in your board exam, keep these golden tips in mind:
- Always check the Principal Value Branch: The most common mistake students make is evaluating expressions outside the standard range (for instance, \(\cos^{-1}(\cos(7\pi/6)) \neq 7\pi/6\) because \(7\pi/6 \notin [0, \pi]\)). Always reduce angles into the principal domain.
- Substitution Method: For algebraic proofs, use standard substitutions such as \(x = \tan \theta\) or \(x = \sin \theta\) to simplify trigonometric identities smoothly.
- Verify Extreme Domain Values: Ensure that solutions obtained in equations actually satisfy the original domain of each inverse function.
📚 Related Class 12 Study Materials
Strengthen your Class 12 preparation with these recommended study resources on StudyMats:
- CBSE Class 12 Complete Study Notes & Question Banks
- Class 12 Mathematics Chapter-wise Important Questions
- NCERT Solutions & Bilingual Practice Sets
Frequently Asked Questions (FAQs)
Q1: What is the weightage of Inverse Trigonometric Functions in CBSE Class 12?
Typically, Unit 1 (Relations and Functions + Inverse Trigonometry) carries around 8 to 10 marks in the CBSE Class 12 Maths theory paper.
Q2: Are these questions bilingual?
Yes, all questions are provided in both English and Hindi with exact mathematical notations (LaTeX) so students from English and Hindi medium can prepare seamlessly.
💡 Important Exam & Study Tips
- Master NCERT Fundamentals: Ensure thorough familiarity with standard textbook definitions, formulas, and in-text examples before solving advanced problems.
- Step-by-Step Presentation: In board examinations, neat presentation, correct formula application, and labeled steps earn maximum step-marking.
- Practice Under Timed Conditions: Regular practice with previous years' question papers (PYQs) builds exam speed and accuracy.
📚 Related Study Materials on StudyMats
Access more chapter-wise notes and practice papers:
- StudyMats Home — Complete Educational Study Material
- 2026 Study Notes & Question Papers
- Bilingual Study Notes & Question Papers
- Board Exams Study Notes & Question Papers
- CBSE Study Notes & Question Papers
❓ Frequently Asked Questions (FAQs)
Q1: Are these solutions aligned with the latest rationalized syllabus?
Yes, all study notes and practice questions on StudyMats strictly follow the updated NCERT textbook pattern and CBSE assessment guidelines.
Q2: How can I prepare this chapter for highest marks?
Review the theory thoroughly, solve all textbook exercise questions, and repeatedly practice the high-frequency questions provided in this guide.