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Mathematics Principal value branch 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 Properties of inverse trigonometric functions 48 MCQs

Principal value branch Practice Questions

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Question 41/69 Easy Mathematics Inverse Trigonometric Functions Principal value branch Class 12 Level 32

यदि \(\theta=\cos^{-1}\left(\frac{12}{13}\right)\), तो \(\sin\theta\) का मान क्या है?

If \(\theta=\cos^{-1}\left(\frac{12}{13}\right)\), what is the value of \(\sin\theta\)?

Explanation opens after your attempt
Correct Answer

C. \(\frac{5}{13}\)

Explanation

Simple Explanation

\(\cos\theta=\frac{12}{13}\) और \(\theta\in\left[0,\pi\right]\) है। इस स्थिति में \(\sin\theta=\frac{5}{13}\) धनात्मक होगा। / Here \(\cos\theta=\frac{12}{13}\) and \(\theta\in\left[0,\pi\right]\). In this case, \(\sin\theta=\frac{5}{13}\) is positive.

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Question 42/69 Easy Mathematics Inverse Trigonometric Functions Principal value branch Class 12 Level 32

\(\sin^{-1}\left(\sin\left(-\frac{\pi}{4}\right)\right)\) का मान क्या है?

What is the value of \(\sin^{-1}\left(\sin\left(-\frac{\pi}{4}\right)\right)\)?

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Correct Answer

A. -\(\frac{\pi}{4}\)

Explanation

Simple Explanation

कोण \((-\frac{\pi}{4}) (\sin^{-1}x)\) के मुख्य परिसर में है। इसलिए मान \(-\frac{\pi}{4}\) है। / The angle \(-\frac{\pi}{4}\) lies in the principal range of \(\sin^{-1}x\). Hence the value is \(-\frac{\pi}{4}\).

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Question 43/69 Easy Mathematics Inverse Trigonometric Functions Principal value branch Class 12 Level 32

\(\cos^{-1}\left(\cos 0\right)\) का मान क्या है?

What is the value of \(\cos^{-1}\left(\cos 0\right)\)?

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Correct Answer

A. 0

Explanation

Simple Explanation

\(\cos0=1\)। \(\cos^{-1}1\) वह कोण है जिसका कोसाइन 1 हो और जो इन्फर्स कोसाइन की प्रमुख सीमा \([0,\pi]\) में आता हो। इसलिए \(\cos^{-1}1=0\)। अन्य विकल्प गलत हैं क्योंकि \(\cos\frac{\pi}{2}=0\), \(\cos\pi=-1\) और \(-\pi\) प्रमुख सीमा \([0,\pi]\) में नहीं आता। परीक्षा टिप: सामान्य रूप से \(\cos^{-1}(\cos\theta)\) निकालते समय \(\theta\) को प्रमुख सीमा \([0,\pi]\) में समकक्ष कोण पर लाएँ। / Since \(\cos0=1\) and \(\cos^{-1}1\) is the angle in the principal range \([0,\pi]\) whose cosine equals 1, we get \(\cos^{-1}1=0\). The other choices are incorrect: \(\cos(\pi/2)=0\), \(\cos\pi=-1\), and \(-\pi\) is outside the principal range of arccos. Exam tip: When evaluating \(\cos^{-1}(\cos\theta)\), reduce or map \(\theta\) into the principal range \([0,\pi]\) and use that equivalent angle.

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Question 44/69 Medium Mathematics Inverse Trigonometric Functions Principal value branch Class 12 Level 31

\(\tan^{-1}(\tan\frac{3\pi}{4})\) का मुख्य मान क्या है?

What is the principal value of \(\tan^{-1}(\tan\frac{3\pi}{4})\)?

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Correct Answer

C. \(-\frac{\pi}{4}\)

Explanation

Simple Explanation

\(\tan\frac{3\pi}{4}=-1\) और \(\tan^{-1}(-1)=-\frac{\pi}{4}\)। उत्तर \(\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\) में होना चाहिए। / Since \(\tan\frac{3\pi}{4}=-1), (\tan^{-1}(-1)=-\frac{\pi}{4}\). The answer must lie in \(\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\).

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Question 45/69 Medium Mathematics Inverse Trigonometric Functions Principal value branch Class 12 Level 31

यदि \(\alpha=\sin^{-1}\left(-\frac{\sqrt{3}}{2}\right)\), तो \(\alpha\) किसके बराबर है?

If \(\alpha=\sin^{-1}\left(-\frac{\sqrt{3}}{2}\right)\), what is \(\alpha\) equal to?

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Correct Answer

A. \(-\frac{\pi}{3}\)

Explanation

Simple Explanation

\(\sin\left(-\frac{\pi}{3}\right)=-\frac{\sqrt{3}}{2}\) और यह मुख्य परिसर में है। इसलिए \(-\frac{\pi}{3}\) सही है। / Since \(\sin\left(-\frac{\pi}{3}\right)=-\frac{\sqrt{3}}{2}\) and it lies in the principal range. Therefore \(-\frac{\pi}{3}\) is correct.

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Question 46/69 Medium Mathematics Inverse Trigonometric Functions Principal value branch Class 12 Level 31

\(\cos^{-1}\left(\frac{\sqrt{2}}{2}\right)\) का मान क्या है?

What is the value of \(\cos^{-1}\left(\frac{\sqrt{2}}{2}\right)\)?

Explanation opens after your attempt
Correct Answer

B. \(\frac{\pi}{4}\)

Explanation

Simple Explanation

\(\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}\) और \(\frac{\pi}{4}\in\left[0,\pi\right]\)। विशेष मानों की तालिका याद रखें। / Since \(\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}\) and \(\frac{\pi}{4}\in\left[0,\pi\right]\). Remember the table of special values.

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Question 47/69 Medium Mathematics Inverse Trigonometric Functions Principal value branch Class 12 Level 31

\(\sin^{-1}\left(\sin\left(-\frac{5\pi}{6}\right)\right)\) का मुख्य मान क्या है?

What is the principal value of \(\sin^{-1}\left(\sin\left(-\frac{5\pi}{6}\right)\right)\)?

Explanation opens after your attempt
Correct Answer

B. \(-\frac{\pi}{6}\)

Explanation

Simple Explanation

\(\sin\left(-\frac{5\pi}{6}\right)=-\frac{1}{2}\), इसलिए \(\sin^{-1}\left(-\frac{1}{2}\right)=-\frac{\pi}{6}\)। उत्तर मुख्य परिसर में होना चाहिए। / Since \(\sin\left(-\frac{5\pi}{6}\right)=-\frac{1}{2}\), \(\sin^{-1}\left(-\frac{1}{2}\right)=-\frac{\pi}{6}\). The answer must lie in the principal range.

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Question 48/69 Medium Mathematics Inverse Trigonometric Functions Principal value branch Class 12 Level 31

वास्तविक संख्या \(x\) के लिए \(\tan^{-1}x\) का मुख्य मान किस अंतराल में होता है?

For a real number \(x\), in which interval does the principal value of \(\tan^{-1}x\) lie?

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Correct Answer

B. \(\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\)

Explanation

Simple Explanation

\(\tan^{-1}x\) का मुख्य मान हमेशा \(\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\) में लिया जाता है। इस अंतराल में \(\tan \theta\) एक-एक फलन है और प्रत्येक वास्तविक \(x\) के लिए एक ही कोण देता है। विकल्प C सही नहीं है, क्योंकि \(\theta=\pm\frac{\pi}{2}\) पर \(\tan\theta\) अपरिभाषित होता है। परीक्षा टिप: \(\tan^{-1}x\) के लिए सिरों वाले कोण कभी शामिल नहीं होते। / The principal value of \(\tan^{-1}x\) is always chosen from \(\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\). In this interval, \(\tan\theta\) is one-to-one, so every real \(x\) corresponds to exactly one angle. Option C is incorrect because \(\tan\theta\) is undefined at \(\theta=\pm\frac{\pi}{2}\). Exam tip: the endpoint angles are never included in the principal-value range of \(\tan^{-1}x\).

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Question 49/69 Medium Mathematics Inverse Trigonometric Functions Principal value branch Class 12 Level 31

\(\sin^{-1}\left(\sin\frac{7\pi}{6}\right)\) का मुख्य मान क्या है?

What is the principal value of \(\sin^{-1}\left(\sin\frac{7\pi}{6}\right)\)?

Explanation opens after your attempt
Correct Answer

C. \(-\frac{\pi}{6}\)

Explanation

Simple Explanation

\(\sin\frac{7\pi}{6}=-\frac{1}{2}\), इसलिए \(\sin^{-1}\left(-\frac{1}{2}\right)=-\frac{\pi}{6}\)। उत्तर हमेशा \(\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\) में रखें। / Since \(\sin\frac{7\pi}{6}=-\frac{1}{2}\), \(\sin^{-1}\left(-\frac{1}{2}\right)=-\frac{\pi}{6}\). Keep the answer in \(\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\).

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Question 50/69 Medium Mathematics Inverse Trigonometric Functions Principal value branch Class 12 Level 31

\(\cos^{-1}\left(\cos\left(-\frac{\pi}{4}\right)\right)\) का मुख्य मान क्या है?

What is the principal value of \(\cos^{-1}\left(\cos\left(-\frac{\pi}{4}\right)\right)\)?

Explanation opens after your attempt
Correct Answer

B. \(\frac{\pi}{4}\)

Explanation

Simple Explanation

\(\cos\left(-\frac{\pi}{4}\right)=\frac{1}{\sqrt{2}}\), इसलिए \(\cos^{-1}\left(\frac{1}{\sqrt{2}}\right)=\frac{\pi}{4})। (\cos^{-1}\) का उत्तर ऋणात्मक नहीं होता। / Since \(\cos\left(-\frac{\pi}{4}\right)=\frac{1}{\sqrt{2}}\), \(\cos^{-1}\left(\frac{1}{\sqrt{2}}\right)=\frac{\pi}{4}). The answer of (\cos^{-1}\) is not negative.

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