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Mathematics Principal value branch MCQ Questions for Class 12 General

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Principal value branch Practice Questions

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Question 21/69 Easy Mathematics Inverse Trigonometric Functions Principal value branch Class 12 Level 31

\(\sin^{-1}\left(\frac{\sqrt{3}}{2}\right)\) का मान क्या है?

What is the value of \(\sin^{-1}\left(\frac{\sqrt{3}}{2}\right)\)?

Explanation opens after your attempt

Correct Answer

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

Explanation

Simple Explanation

क्योंकि \(\sin\frac{\pi}{3}=\frac{\sqrt{3}}{2}\) और \(\sin^{-1}x\) का principal मान क्षेत्र \([-\frac{\pi}{2},\,\frac{\pi}{2}]\) है, अतः \(\sin^{-1}\!\left(\frac{\sqrt{3}}{2}\right)=\frac{\pi}{3}\)। विकल्प -\(\frac{\pi}{3}\) गलत है क्योंकि \(\sin(-\frac{\pi}{3})=-\frac{\sqrt{3}}{2}\)। \(\frac{\pi}{6}\) और \(\frac{\pi}{4}\) के साइन क्रमशः \(\frac{1}{2}\) और \(\frac{\sqrt{2}}{2}\) होते हैं, इसलिए वे उपयुक्त नहीं हैं। परीक्षा सुझाव: सामान्य कोणों के \(\sin,\cos\) मान याद रखें। / Since \(\sin\frac{\pi}{3}=\frac{\sqrt{3}}{2}\) and the principal value of \(\sin^{-1}x\) lies in \([-\frac{\pi}{2},\frac{\pi}{2}]\), we have \(\sin^{-1}\!\left(\frac{\sqrt{3}}{2}\right)=\frac{\pi}{3}\). The option \(-\frac{\pi}{3}\) is incorrect because \(\sin(-\frac{\pi}{3})=-\frac{\sqrt{3}}{2}\). \(\frac{\pi}{6}\) and \(\frac{\pi}{4}\) are wrong as their sines are \(\frac{1}{2}\) and \(\frac{\sqrt{2}}{2}\) respectively. Exam tip: memorize standard sine/cosine values for common angles \(0, \(\frac{\pi}{6}\), \(\frac{\pi}{4}\), \(\frac{\pi}{3}\), \(\frac{\pi}{2}\)\).

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Question 22/69 Easy Mathematics Inverse Trigonometric Functions Principal value branch Class 12 Level 31

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

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

Explanation opens after your attempt

Correct Answer

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

Explanation

Simple Explanation

\(\cos\frac{\pi}{3}=\frac{1}{2}\) होता है। इसलिए \(\cos^{-1}\left(\frac{1}{2}\right)=\frac{\pi}{3}\)। / Since \(\cos\frac{\pi}{3}=\frac{1}{2}\), \(\cos^{-1}\left(\frac{1}{2}\right)=\frac{\pi}{3}\).

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Question 23/69 Easy Mathematics Inverse Trigonometric Functions Principal value branch Class 12 Level 31

\(\cos^{-1}\left(\frac{\sqrt{3}}{2}\right)\) का मान क्या है?

What is the value of \(\cos^{-1}\left(\frac{\sqrt{3}}{2}\right)\)?

Explanation opens after your attempt

Correct Answer

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

Explanation

Simple Explanation

आर्ककोस (principal value) का मान हमेशा \([0,\pi]\) पर लिया जाता है। ज्ञात है कि \(\cos\frac{\pi}{6}=\frac{\sqrt{3}}{2}\)। अतः \(\cos^{-1}\!\left(\frac{\sqrt{3}}{2}\right)=\frac{\pi}{6}\)। अन्य विकल्प गलत हैं क्योंकि \(\cos\frac{\pi}{3}=\tfrac{1}{2},\;\cos\frac{\pi}{4}=\tfrac{\sqrt{2}}{2}\) और \(\cos\frac{5\pi}{6}=-\tfrac{\sqrt{3}}{2}\)। परीक्षा टिप: आर्ककोस की रेंज \([0,\pi]\) और मानक त्रिकोणमितीय मान याद रखें। / The principal value of arccos is taken in the interval \([0,\pi]\). Since \(\cos\frac{\pi}{6}=\frac{\sqrt{3}}{2}\), we have \(\cos^{-1}\!\left(\frac{\sqrt{3}}{2}\right)=\frac{\pi}{6}\). The other choices are incorrect: \(\cos\frac{\pi}{3}=\tfrac{1}{2},\;\cos\frac{\pi}{4}=\tfrac{\sqrt{2}}{2}\), and \(\cos\frac{5\pi}{6}=-\tfrac{\sqrt{3}}{2}\). Exam tip: memorize standard trig values and remember arccos range \([0,\pi]\).

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Question 24/69 Easy Mathematics Inverse Trigonometric Functions Principal value branch Class 12 Level 31

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

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

Explanation opens after your attempt

Correct Answer

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

Explanation

Simple Explanation

कारण: \(\tan\frac{\pi}{6}=\frac{1}{\sqrt{3}}\) और आर्कटैन (principal value) का मान क्षेत्र \((-\frac{\pi}{2},\frac{\pi}{2})\) में लिया जाता है। इसलिए \(\tan^{-1}\left(\frac{1}{\sqrt{3}}\right)=\frac{\pi}{6}\). निकटतम भ्रामक विकल्प की जाँच: \(\tan\frac{\pi}{4}=1\), इसलिए \(\frac{\pi}{4}\) गलत है; \(\tan\frac{\pi}{3}=\sqrt{3}\) और \(\tan\frac{\pi}{2}\) पर परिभाषित नहीं है। परीक्षात्मक सुझाव: प्रमुख कोणों के मान (\(0,\frac{\pi}{6},\frac{\pi}{4},\frac{\pi}{3}\) आदि) याद रखें और आर्कफंक्शन का principal-range ध्यान में रखें। / Reason: \(\tan\frac{\pi}{6}=\frac{1}{\sqrt{3}}\), and the principal value of arctan lies in \((-\frac{\pi}{2},\frac{\pi}{2})\). Hence \(\tan^{-1}\left(\frac{1}{\sqrt{3}}\right)=\frac{\pi}{6}\). Checking close distractors: \(\tan\frac{\pi}{4}=1\), so \(\frac{\pi}{4}\) is incorrect; \(\tan\frac{\pi}{3}=\sqrt{3}\) and \(\tan\frac{\pi}{2}\) is undefined. Exam tip: Memorise standard angle values and remember the principal range for inverse trig functions when answering quickly.

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Question 25/69 Easy Mathematics Inverse Trigonometric Functions Principal value branch Class 12 Level 31

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

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

Explanation opens after your attempt

Correct Answer

B. \(\frac{2\pi}{3}\)

Explanation

Simple Explanation

\(\cos\frac{2\pi}{3}=-\frac{1}{2}\) और \(\frac{2\pi}{3}\) मुख्य परिसर में है। इसलिए उत्तर \(\frac{2\pi}{3}\) है। / Since \(\cos\frac{2\pi}{3}=-\frac{1}{2}\) and \(\frac{2\pi}{3}\) is in the principal range, the answer is \(\frac{2\pi}{3}\).

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Question 26/69 Easy Mathematics Inverse Trigonometric Functions Principal value branch Class 12 Level 31

\(sin^{-1}\left(\sin\frac{2\pi}{3}\right)\) का मुख्य मान क्या है?

What is the principal value of \(sin^{-1}\left(\sin\frac{2\pi}{3}\right)\)?

Explanation opens after your attempt

Correct Answer

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

Explanation

Simple Explanation

\(sin\frac{2\pi}{3}=\frac{\sqrt{3}}{2}\) और \(sin^{-1}\left(\frac{\sqrt{3}}{2}\right)=\frac{\pi}{3}\)। मुख्य मान हमेशा निर्धारित परिसर में लेना चाहिए। / Since \(sin\frac{2\pi}{3}=\frac{\sqrt{3}}{2}\) and \(sin^{-1}\left(\frac{\sqrt{3}}{2}\right)=\frac{\pi}{3}\). Always choose the principal range value.

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Question 27/69 Easy Mathematics Inverse Trigonometric Functions Principal value branch Class 12 Level 31

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

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

Explanation opens after your attempt

Correct Answer

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

Explanation

Simple Explanation

\(tan\frac{3\pi}{4}=-1\) है और \(tan^{-1}\left(-1\right)=-\frac{\pi}{4}\)। मुख्य परिसर \(\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\) याद रखें। / We have \(tan\frac{3\pi}{4}=-1\) and \(tan^{-1}\left(-1\right)=-\frac{\pi}{4}\). Remember the principal range \(\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\).

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Question 28/69 Easy Mathematics Inverse Trigonometric Functions Principal value branch Class 12 Level 31

\(\sec^{-1}2\) का मान क्या है?

What is the value of \(\sec^{-1}2\)?

Explanation opens after your attempt

Correct Answer

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

Explanation

Simple Explanation

यदि \(\sec\theta=2\) हो तो \(\cos\theta=\dfrac{1}{2}\) होता है क्योंकि \(\sec\theta=\dfrac{1}{\cos\theta}\)। मुख्य मान (principal value) के लिए \(\sec^{-1}x\) का परिमाण आमतौर पर \([0,\pi]\) (\(\theta\neq\tfrac{\pi}{2}\)) लिया जाता है।\n\(\cos\theta=\tfrac{1}{2}\) से \(\theta=\tfrac{\pi}{3}\) आता है, जो इस रेंज में है। इसलिए \(\sec^{-1}2=\tfrac{\pi}{3}\)।\nगलत विकल्पों पर टिप्पणी: \(\tfrac{\pi}{6}\) पर \(\sec=2/\sqrt{3}\), \(\tfrac{\pi}{4}\) पर \(\sec=\sqrt{2}\), और \(\tfrac{2\pi}{3}\) पर \(\sec=-2\) होता है — इसलिए वे सही नहीं हैं।\nपरीक्षा सुझाव: पहले \(\sec^{-1}x\) को \(\cos\theta=1/x\) में बदलें और मुख्य मान की सीमा याद रखें। / If \(\sec\theta=2\), then \(\cos\theta=\dfrac{1}{2}\) because \(\sec\theta=\dfrac{1}{\cos\theta}\). For the principal value, \(\sec^{-1}x\) is taken in \([0,\pi]\) with \(\theta\neq\tfrac{\pi}{2}\).\nSolving \(\cos\theta=\tfrac{1}{2}\) gives \(\theta=\tfrac{\pi}{3}\), which lies in the principal range. Hence \(\sec^{-1}2=\tfrac{\pi}{3}\).\nWhy distractors fail: at \(\tfrac{\pi}{6}\), \(\sec=2/\sqrt{3}\); at \(\tfrac{\pi}{4}\), \(\sec=\sqrt{2}\); at \(\tfrac{2\pi}{3}\), \(\sec=-2\).\nExam tip: convert \(\sec^{-1}x\) to \(\cos\theta=1/x\) and then choose the solution inside the principal-value interval.

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Question 29/69 Easy Mathematics Inverse Trigonometric Functions Principal value branch Class 12 Level 31

\(cosec^{-1}2\) का मान क्या है?

What is the value of \(cosec^{-1}2\)?

Explanation opens after your attempt

Correct Answer

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

Explanation

Simple Explanation

क्योंकि \(cosec\,\frac{\pi}{6}=\dfrac{1}{\sin(\pi/6)}=\dfrac{1}{1/2}=2\)। इसलिए प्रमुख मान (principal value) के रूप में \(cosec^{-1}2=\frac{\pi}{6}\) होगा। अन्य विकल्प गलत हैं क्योंकि \(cosec\,\frac{\pi}{3}=\dfrac{1}{\sin(\pi/3)}=\dfrac{2}{\sqrt{3}}\), \(cosec\,\frac{\pi}{4}=\sqrt{2}\) और \(cosec\,\frac{\pi}{2}=1\) — ये कोई भी 2 के बराबर नहीं हैं। परीक्षा टिप: \(cosec\theta=1/\sin\theta\) याद रखें और सर्वमान्य कोणों के sin मानों का उपयोग करें। / Since \(cosec\,\frac{\pi}{6}=\dfrac{1}{\sin(\pi/6)}=\dfrac{1}{1/2}=2\), the principal value is \(cosec^{-1}2=\frac{\pi}{6}\). The other options are incorrect because \(cosec\,\frac{\pi}{3}=\dfrac{2}{\sqrt{3}}\), \(cosec\,\frac{\pi}{4}=\sqrt{2}\) and \(cosec\,\frac{\pi}{2}=1\), none of which equals 2. Exam tip: use \(cosec\theta=1/\sin\theta\) and remember standard sine values for common angles.

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Question 30/69 Easy Mathematics Inverse Trigonometric Functions Principal value branch Class 12 Level 32

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

What is \(cos^{-1}\left(-x\right)\) equal to when \(x\in\left[-1,1\right]\)?

Explanation opens after your attempt

Correct Answer

B. \(\pi-\cos^{-1}x\)

Explanation

Simple Explanation

\(cos^{-1}\left(-x\right)=\pi-\cos^{-1}x\) मुख्य मानों के कारण सही है। इसे विषम फलन की तरह न मानें। / The relation \(cos^{-1}\left(-x\right)=\pi-\cos^{-1}x\) is correct because of principal values. Do not treat it as an odd function.

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Mathematics Principal value branch MCQ Questions for Class 12 General

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Principal value branch Practice Questions

\(\sin^{-1}\left(\frac{\sqrt{3}}{2}\right)\) का मान क्या है?

Simple Explanation

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

Simple Explanation

\(\cos^{-1}\left(\frac{\sqrt{3}}{2}\right)\) का मान क्या है?

Simple Explanation

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

Simple Explanation

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

Simple Explanation

\(sin^{-1}\left(\sin\frac{2\pi}{3}\right)\) का मुख्य मान क्या है?

Simple Explanation

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

Simple Explanation

\(\sec^{-1}2\) का मान क्या है?

Simple Explanation

\(cosec^{-1}2\) का मान क्या है?

Simple Explanation

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

Simple Explanation

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Mathematics Principal value branch MCQ Questions for Class 12 General

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Principal value branch Practice Questions

\(\sin^{-1}\left(\frac{\sqrt{3}}{2}\right)\) का मान क्या है?

Simple Explanation

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

Simple Explanation

\(\cos^{-1}\left(\frac{\sqrt{3}}{2}\right)\) का मान क्या है?

Simple Explanation

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

Simple Explanation

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

Simple Explanation

\(sin^{-1}\left(\sin\frac{2\pi}{3}\right)\) का मुख्य मान क्या है?

Simple Explanation

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

Simple Explanation

\(\sec^{-1}2\) का मान क्या है?

Simple Explanation

\(cosec^{-1}2\) का मान क्या है?

Simple Explanation

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

Simple Explanation

Principal value branch Learning Links

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