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Mathematics Properties of inverse trigonometric functions MCQ Questions for Class 12 General

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01 Practice Topic Domain and range 24 MCQs 02 Practice Topic Graphs of inverse trigonometric functions 3 MCQs 03 Practice Topic Introduction to inverse trigonometric functions 106 MCQs 04 Practice Topic Principal value branch 69 MCQs

Properties of inverse trigonometric functions Practice Questions

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Question 11/48 Easy Mathematics Inverse Trigonometric Functions Properties of inverse trigonometric functions Class 12 Level 32

यदि \(\tan^{-1}(-x)\) दिया हो, तो इसका सही रूप क्या होगा?

What is the correct form of \(\tan^{-1}(-x)\)?

Explanation opens after your attempt
Correct Answer

A. \(-\tan^{-1}x\)

Explanation

Simple Explanation

\(\tan^{-1}x\) एक विषम फलन है और इसका प्रमुख मान क्षेत्र \((-\tfrac{\pi}{2},\tfrac{\pi}{2})\) है। इसलिए किसी भी वास्तविक \(x\) के लिए \(\tan^{-1}(-x)= -\tan^{-1}x\)। विकल्प B गलत है क्योंकि वह चिह्न बदलने की बात नहीं करता; विकल्प C और D सामान्य रूप से समान नहीं होते क्योंकि वे प्रमुख मान के बाहर के रूप हैं। परीक्षा-टिप: किसी भी inverse-trig के लिए पहले उसकी parity (odd/even) और प्रमुख मान सीमा याद रखें। / The arctan function is odd and has principal value range \((-\tfrac{\pi}{2},\tfrac{\pi}{2})\). Hence for any real \(x\), \(\tan^{-1}(-x)= -\tan^{-1}x\). Option B is incorrect because it does not change the sign; options C and D do not generally equal \(\tan^{-1}(-x)\) within the principal range. Exam tip: check whether the inverse trig function is odd or even and remember its principal value interval.

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Question 12/48 Easy Mathematics Inverse Trigonometric Functions Properties of inverse trigonometric functions Class 12 Level 32

यदि \(\theta=\tan^{-1}\left(\frac{3}{4}\right)\), तो \(\sin\theta\) का मान क्या है?

If \(\theta=\tan^{-1}\left(\frac{3}{4}\right)\), what is the value of \(\sin\theta\)?

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Correct Answer

A. \(\frac{3}{5}\)

Explanation

Simple Explanation

\(\tan\theta=\frac{3}{4}\) से लंब (3) और आधार (4) लें। कर्ण (5) होगा, इसलिए \(\sin\theta=\frac{3}{5}\)। / From \(\tan\theta=\frac{3}{4}\), take opposite side (3) and adjacent side (4). The hypotenuse is (5), so \(\sin\theta=\frac{3}{5}\).

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Question 13/48 Easy Mathematics Inverse Trigonometric Functions Properties of inverse trigonometric functions Class 12 Level 32

\(\sin\left(\cos^{-1}x\right)\) का मान क्या है जब \(x\in\left[-1,1\right]\)?

What is the value of \(\sin\left(\cos^{-1}x\right)\) when \(x\in\left[-1,1\right]\)?

Explanation opens after your attempt
Correct Answer

A. \(\sqrt{1-x^2}\)

Explanation

Simple Explanation

यदि \(\theta=\cos^{-1}x\), तो \(\cos\theta=x\) और \(\theta\in\left[0,\pi\right]\)। इस परिसर में \(\sin\theta\ge0\), इसलिए मान \(\sqrt{1-x^2}\) है। / If \(\theta=\cos^{-1}x\), then \(\cos\theta=x\) and \(\theta\in\left[0,\pi\right]\). In this range \(\sin\theta\ge0\), so the value is \(\sqrt{1-x^2}\).

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Question 14/48 Easy Mathematics Inverse Trigonometric Functions Properties of inverse trigonometric functions Class 12 Level 32

\(\cos\left(\sin^{-1}x\right)\) का मान क्या है जब \(x\in\left[-1,1\right]\)?

What is the value of \(\cos\left(\sin^{-1}x\right)\) when \(x\in\left[-1,1\right]\)?

Explanation opens after your attempt
Correct Answer

A. \(\sqrt{1-x^2}\)

Explanation

Simple Explanation

यदि \(\theta=\sin^{-1}x\), तो \(sin\theta=x\) और \(\theta\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\)। यहाँ \(cos\theta\ge0\), इसलिए \(\sqrt{1-x^2}\) सही है। / If \(\theta=\sin^{-1}x\), then \(\sin\theta=x\) and \(\theta\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\). Here \(cos\theta\ge0\), so \(\sqrt{1-x^2}\) is correct.

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Question 15/48 Easy Mathematics Inverse Trigonometric Functions Properties of inverse trigonometric functions Class 12 Level 32

\(\tan\left(\sin^{-1}\left(\frac{3}{5}\right)\right)\) का मान क्या है?

What is the value of \(\tan\left(\sin^{-1}\left(\frac{3}{5}\right)\right)\)?

Explanation opens after your attempt
Correct Answer

A. \(\frac{3}{4}\)

Explanation

Simple Explanation

यदि \(\theta=\sin^{-1}\left(\frac{3}{5}\right)\), तो \(\sin\theta=\frac{3}{5}\) और \(\cos\theta=\frac{4}{5}\)। इसलिए \(\tan\theta=\frac{3}{4}\)। / If \(\theta=\sin^{-1}\left(\frac{3}{5}\right)\), then \(\sin\theta=\frac{3}{5}\) and \(\cos\theta=\frac{4}{5}\). Hence \(\tan\theta=\frac{3}{4}\).

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Question 16/48 Easy Mathematics Inverse Trigonometric Functions Properties of inverse trigonometric functions Class 12 Level 32

\(\cot\left(\cos^{-1}\left(\frac{5}{13}\right)\right)\) का मान क्या है?

What is the value of \(\cot\left(\cos^{-1}\left(\frac{5}{13}\right)\right)\)?

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Correct Answer

A. \(\frac{5}{12}\)

Explanation

Simple Explanation

यदि \(\theta=\cos^{-1}\left(\frac{5}{13}\right)\), तो \(\cos\theta=\frac{5}{13}\) और \(\sin\theta=\frac{12}{13}\)। इसलिए \(\cot\theta=\frac{5}{12}\)। / If \(\theta=\cos^{-1}\left(\frac{5}{13}\right)\), then \(\cos\theta=\frac{5}{13}\) and \(\sin\theta=\frac{12}{13}\). Hence \(\cot\theta=\frac{5}{12}\).

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Question 17/48 Easy Mathematics Inverse Trigonometric Functions Properties of inverse trigonometric functions Class 12 Level 32

यदि \(\theta=\tan^{-1}\left(\frac{7}{24}\right)\) हो, तो \(\sec\left(\tan^{-1}\left(\frac{7}{24}\right)\right)\) का मान क्या है?

If \(\theta=\tan^{-1}\left(\frac{7}{24}\right)\), what is the value of \(\sec\left(\tan^{-1}\left(\frac{7}{24}\right)\right)\)?

Explanation opens after your attempt
Correct Answer

A. \(\frac{25}{24}\)

Explanation

Simple Explanation

लेते हैं \(\theta=\tan^{-1}\left(\frac{7}{24}\right)\)। तब \(\tan\theta=\frac{7}{24}=\frac{\text{opposite}}{\text{adjacent}}\)। एक समकोण त्रिभुज में opposite = 7, adjacent = 24 लिया जाए तो पाइथागोरस से hypotenuse = \(\sqrt{7^2+24^2}=25\)। \(\sec\theta=\frac{1}{\cos\theta}=\frac{\text{hypotenuse}}{\text{adjacent}}=\frac{25}{24}\) इसलिए सही उत्तर \(\frac{25}{24}\) है। निकटतम भ्रमित विकल्प \(\frac{24}{25}\) (विकल्प B) विकर्ण का अनुपात नहीं बल्कि कॉसाइन का मान है (या सही उत्तर का倒数 नहीं)। विकल्प C \(\frac{7}{25}\) वास्तव में \(\sin\theta\) है और विकल्प D \(\frac{25}{7}\) \(\csc\theta\) के बराबर है। परीक्षा‑टिप: arctan के लिए तुरंत 7‑24‑25 त्रिभुज याद रखें और त्रिभुज बनाकर opposite/adjacent/hypotenuse सम्बन्ध लागू करें। / Let \(\theta=\tan^{-1}(7/24)\), so \(\tan\theta=7/24=\) opposite/adjacent. Using a right triangle with opposite = 7 and adjacent = 24 gives hypotenuse = \(\sqrt{7^2+24^2}=25\). Then \(\sec\theta=1/\cos\theta=\) hypotenuse/adjacent = \(25/24\), so option A is correct. Closest distractor \(24/25\) (B) is the reciprocal and corresponds to \(\cos\theta\), not \(\sec\theta\). Option C (\(7/25\)) equals \(\sin\theta\), and D (\(25/7\)) equals \(\csc\theta\). Exam tip: draw the 7‑24‑25 right triangle for arctan problems and read off opposite, adjacent, hypotenuse to compute sin, cos, sec quickly.

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Question 18/48 Easy Mathematics Inverse Trigonometric Functions Properties of inverse trigonometric functions Class 12 Level 32

\(cosec\left(\cos^{-1}\left(\frac{8}{17}\right)\right)\) का मान क्या है?

What is the value of \(cosec\left(\cos^{-1}\left(\frac{8}{17}\right)\right)\)?

Explanation opens after your attempt
Correct Answer

A. \(\frac{17}{15}\)

Explanation

Simple Explanation

यदि \(cos\theta=\frac{8}{17}\), तो \(sin\theta=\frac{15}{17}\)। इसलिए \(cosec\theta=\frac{17}{15}\)। / If \(cos\theta=\frac{8}{17}\), then \(sin\theta=\frac{15}{17}\). Hence \(cosec\theta=\frac{17}{15}\).

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Question 19/48 Easy Mathematics Inverse Trigonometric Functions Properties of inverse trigonometric functions Class 12 Level 32

\(\sin^{-1}\left(\cos\frac{\pi}{3}\right)\) का मान क्या है?

What is the value of \(\sin^{-1}\left(\cos\frac{\pi}{3}\right)\)?

Explanation opens after your attempt
Correct Answer

A. \(\frac{\pi}{6}\)

Explanation

Simple Explanation

\(\cos\frac{\pi}{3}=\frac{1}{2}\) और \(\sin^{-1}\left(\frac{1}{2}\right)=\frac{\pi}{6}\)। पहले अंदर का मान निकालें। / Since \(\cos\frac{\pi}{3}=\frac{1}{2}\) and \(\sin^{-1}\left(\frac{1}{2}\right)=\frac{\pi}{6}\). First find the inner value.

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Question 20/48 Easy Mathematics Inverse Trigonometric Functions Properties of inverse trigonometric functions Class 12 Level 32

\(\cos^{-1}\left(\sin\frac{\pi}{6}\right)\) का मान क्या है?

What is the value of \(\cos^{-1}\left(\sin\frac{\pi}{6}\right)\)?

Explanation opens after your attempt
Correct Answer

B. \(\frac{\pi}{3}\)

Explanation

Simple Explanation

\(\sin\frac{\pi}{6}=\frac{1}{2}\) और \(\cos^{-1}\left(\frac{1}{2}\right)=\frac{\pi}{3}\)। मानक कोण याद रखें। / Since \(\sin\frac{\pi}{6}=\frac{1}{2}\) and \(\cos^{-1}\left(\frac{1}{2}\right)=\frac{\pi}{3}\). Remember standard angles.

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