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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 11/69 Easy Mathematics Inverse Trigonometric Functions Principal value branch Class 12 Level 31

\(\tan^{-1}1\) का मान क्या है?

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

Explanation opens after your attempt

Correct Answer

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

Explanation

Simple Explanation

कारण: \(\tan\frac{\pi}{4}=1\) होता है और आर्कटैन्जेंट का प्रमुख मान (principal value) \(( -\frac{\pi}{2},\frac{\pi}{2})\) के भीतर होता है, इसलिए \(\tan^{-1}1=\frac{\pi}{4}\)। अन्य विकल्प क्यों गलत हैं: \(0\) के लिए \(\tan0=0\) होता है, इसलिए सही नहीं; \(\frac{\pi}{6}\) के लिए \(\tan\frac{\pi}{6}=\frac{1}{\sqrt{3}}\), जो 1 नहीं है; \(\frac{\pi}{2}\) पर \(\tan\) अविभाज्य (undefined) है। परीक्षा टिप: सामान्य त्रिकोणमितीय मान (जैसे \(0,\frac{\pi}{6},\frac{\pi}{4},\frac{\pi}{3},\frac{\pi}{2}\)) याद रखें और arctan का principal range याद रखें। / Reason: \(\tan\frac{\pi}{4}=1\), and the principal value of \(\tan^{-1}x\) lies in \((-\frac{\pi}{2},\frac{\pi}{2})\), so \(\tan^{-1}1=\frac{\pi}{4}\). Why other choices are wrong: \(0\) is wrong because \(\tan0=0\); \(\frac{\pi}{6}\) is wrong because \(\tan\frac{\pi}{6}=1/\sqrt{3}\), not 1; \(\frac{\pi}{2}\) is not valid as \(\tan\) is undefined there. Exam tip: Memorise standard tan values and remember arctan's principal range to pick the correct branch quickly.

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

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

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

Explanation opens after your attempt

Correct Answer

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

Explanation

Simple Explanation

\(\tan\left(-\frac{\pi}{4}\right)=-1\) और \(-\frac{\pi}{4}\) मुख्य परिसर में है। इसलिए उत्तर \(-\frac{\pi}{4}\) है। / Since \(\tan\left(-\frac{\pi}{4}\right)=-1\) and \(-\frac{\pi}{4}\) lies in the principal range, the answer is \(-\frac{\pi}{4}\).

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Question 13/69 Easy Mathematics Inverse Trigonometric Functions Principal value branch Class 12 Level 31

\(\cot^{-1}1\) का प्रमुख मान क्या है?

What is the principal value of \(\cot^{-1}1\)?

Explanation opens after your attempt

Correct Answer

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

Explanation

Simple Explanation

क्योंकि \(\cot\frac{\pi}{4}=1\) और इनवर्स कॉट (प्रमुख मान) सामान्यतः \(0<\cot^{-1}x<\pi\) के अंतराल में परिभाषित किया जाता है, इसलिए \(\cot^{-1}1=\frac{\pi}{4}\)। निकटतम भ्रामक विकल्प \(\frac{\pi}{2}\) गलत है क्योंकि \(\cot\frac{\pi}{2}=0\)। परीक्षा-सुझाव: अरककोट का प्रमुख मान याद रखें — यह हमेशा \((0,\pi)\) के भीतर लिया जाता है। / Since \(\cot(\pi/4)=1\) and the principal value of \(\cot^{-1}x\) is taken in the interval \(0<\cot^{-1}x<\pi\), we get \(\cot^{-1}1=\frac{\pi}{4}\). The closest distractor \(\frac{\pi}{2}\) is incorrect because \(\cot(\pi/2)=0\). Exam tip: remember the principal-value range of arccot is \((0,\pi)\).

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Question 14/69 Easy Mathematics Inverse Trigonometric Functions Principal value branch Class 12 Level 31

\(\sec^{-1}(-1)\) का मान क्या है?

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

Explanation opens after your attempt

Correct Answer

A. \(0\)

Explanation

Simple Explanation

\(\sec\theta=\dfrac{1}{\cos\theta}\) है और \(\sec 0=1\)। सामान्यतः \(\sec^{-1}x\) का principal value लेकर रेंज \([0,\pi]\) (\(\pi/2\) को छोड़कर) ली जाती है, अतः \(\sec^{-1}1=0\)। विकल्प \(\pi\) गलत है क्योंकि \(\sec\pi=-1\)। विकल्प \(\tfrac{\pi}{2}\) और \(-\tfrac{\pi}{2}\) पर \(\sec\) निरूपित नहीं होता (\(\cos\theta=0\) के कारण)। परीक्षा-सूत्र: inverse trig मान लेते समय function की domain और principal range पहले जाँचें। / Since \(\sec\theta=\dfrac{1}{\cos\theta}\) and \(\sec 0=1\), the principal value of \(\sec^{-1}1\) is \(0\). The common principal range for \(\sec^{-1}x\) is \([0,\pi]\) excluding \(\pi/2\), so \(\sec^{-1}1=0\). Option \(\pi\) is incorrect because \(\sec\pi=-1\). Options \(\tfrac{\pi}{2}\) and \(-\tfrac{\pi}{2}\) are invalid since \(\sec\) is undefined where \(\cos\theta=0\). Exam tip: always check the principal range and where the original trig function is defined before selecting the inverse value.

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Question 15/69 Easy Mathematics Inverse Trigonometric Functions Principal value branch Class 12 Level 31

\(cosec^{-1}1\) का मान क्या है?

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

Explanation opens after your attempt

Correct Answer

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

Explanation

Simple Explanation

यदि \(\csc y=1\) तो \(\sin y=1\) होता है। \(\sin y=1\) का सामान्य हल \(y=\frac{\pi}{2}+2k\pi\) (जहाँ \(k\in\mathbb{Z}\)) है। inverse cosec (arccosec) का प्रधान मान प्रायः \([ -\frac{\pi}{2},\frac{\pi}{2}]\setminus\{0\}\) में लिया जाता है, और इस श्रेणी का एकमात्र समाहित हल \(\frac{\pi}{2}\) है। अतः \(cosec^{-1}1=\frac{\pi}{2}\)। निकट की ग़लत विकल्पों के बारे में: \(0\) पर \(\csc\) परिभाषित नहीं है; \(-\frac{\pi}{2}\) पर \(\csc=-1\) होता है; और \(\pi\) पर \(\sin\pi=0\) होने से \(\csc\) परिभाषित नहीं। परीक्षा टिप: arccosec के प्रधान मान और \(\csc y\) को \(\sin y\) से जोडकर सामान्य हल याद रखें। / If \(\csc y=1\) then \(\sin y=1\). The general solution of \(\sin y=1\) is \(y=\frac{\pi}{2}+2k\pi\) (\(k\in\mathbb{Z}\)). The principal value for \(cosec^{-1}x\) is usually taken in \([ -\frac{\pi}{2},\frac{\pi}{2}]\setminus\{0\}\), and the only solution in this interval is \(\frac{\pi}{2}\). Therefore \(cosec^{-1}1=\frac{\pi}{2}\). About distractors: \(0\) is not allowed (\(\csc\) undefined), \(-\frac{\pi}{2}\) gives \(\csc=-1\), and at \(\pi\) \(\sin\pi=0\) so \(\csc\) is undefined. Exam tip: convert \(\csc y=\) value to \(\sin y=\) value and then pick the principal value from the defined range.

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Question 16/69 Easy Mathematics Inverse Trigonometric Functions Principal value branch Class 12 Level 31

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

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

Explanation opens after your attempt

Correct Answer

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

Explanation

Simple Explanation

Inverse cosecant \(\(cosec^{-1}x\)\) का प्रिंसिपल रेंज आम तौर पर \([-\tfrac{\pi}{2},\;\tfrac{\pi}{2}]\) (\(0\) को छोड़कर) लिया जाता है। \(\sin(-\tfrac{\pi}{2})=-1\) होने के कारण \(cosec(-\tfrac{\pi}{2})=-1\)। इसलिए \(cosec^{-1}(-1)=-\tfrac{\pi}{2}\)। विकल्प \(\tfrac{3\pi}{2}\) भी \(cosec= -1\) देता है पर वह प्रिंसिपल रेंज के बाहर है, इसीलिए वह सही उत्तर नहीं है। परीक्षा टिप: किसी inverse trig का मान निकालते समय हमेशा उसकी मानक \(principal\) रेंज याद रखें। / The principal value of inverse cosecant, \(cosec^{-1}x\), is usually taken in \([-\tfrac{\pi}{2},\;\tfrac{\pi}{2}]\) excluding 0. Since \(\sin(-\tfrac{\pi}{2})=-1\), we have \(cosec(-\tfrac{\pi}{2})=-1\); hence \(cosec^{-1}(-1)=-\tfrac{\pi}{2}\). Note that \(\tfrac{3\pi}{2}\) also satisfies \(cosec= -1\) but it lies outside the principal range, so it is not the principal value. Exam tip: always use the principal-range convention for inverse trigonometric functions when answering numerical MCQs.

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Question 17/69 Easy Mathematics Inverse Trigonometric Functions Principal value branch Class 12 Level 31

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

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

Explanation opens after your attempt

Correct Answer

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

Explanation

Simple Explanation

\(\frac{\pi}{6}\) मुख्य परिसर \(\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\) में है। इसलिए मान \(\frac{\pi}{6}\) है। / The angle \(\frac{\pi}{6}\) lies in the principal range \(\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\). Hence the value is \(\frac{\pi}{6}\).

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Question 18/69 Easy Mathematics Inverse Trigonometric Functions Principal value branch Class 12 Level 31

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

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

Explanation opens after your attempt

Correct Answer

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

Explanation

Simple Explanation

\(\frac{2\pi}{3}\) \(\cos^{-1}x\) के मुख्य परिसर \(\left[0,\pi\right]\) में है। इसलिए मान \(\frac{2\pi}{3}\) है। / The angle \(\frac{2\pi}{3}\) lies in the principal range \(\left[0,\pi\right]\) of \(\cos^{-1}x\). Hence the value is \(\frac{2\pi}{3}\).

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Question 19/69 Easy Mathematics Inverse Trigonometric Functions Principal value branch Class 12 Level 31

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

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

Explanation opens after your attempt

Correct Answer

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

Explanation

Simple Explanation

\(\frac{\pi}{6}\) \(\tan^{-1}x\) के मुख्य परिसर में है। इसलिए मान \(\frac{\pi}{6}\) रहेगा। / The angle \(\frac{\pi}{6}\) lies in the principal range of \(\tan^{-1}x\). So the value remains \(\frac{\pi}{6}\).

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Question 20/69 Easy Mathematics Inverse Trigonometric Functions Principal value branch Class 12 Level 31

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

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

Explanation opens after your attempt

Correct Answer

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

Explanation

Simple Explanation

\(\sin\frac{\pi}{6}=\frac{1}{2}\) होता है। इसलिए \(\sin^{-1}\left(\frac{1}{2}\right)=\frac{\pi}{6}\)। / Since \(\sin\frac{\pi}{6}=\frac{1}{2}\), \(\sin^{-1}\left(\frac{1}{2}\right)=\frac{\pi}{6}\).

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Mathematics Principal value branch MCQ Questions for Class 12 General

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Principal value branch Practice Questions

\(\tan^{-1}1\) का मान क्या है?

Simple Explanation

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

Simple Explanation

\(\cot^{-1}1\) का प्रमुख मान क्या है?

Simple Explanation

\(\sec^{-1}(-1)\) का मान क्या है?

Simple Explanation

\(cosec^{-1}1\) का मान क्या है?

Simple Explanation

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

Simple Explanation

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

Simple Explanation

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

Simple Explanation

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

Simple Explanation

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

Simple Explanation

Principal value branch Learning Links

Inverse Trigonometric Functions Chapter

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Mathematics Principal value branch MCQ Questions for Class 12 General

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Principal value branch Practice Questions

\(\tan^{-1}1\) का मान क्या है?

Simple Explanation

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

Simple Explanation

\(\cot^{-1}1\) का प्रमुख मान क्या है?

Simple Explanation

\(\sec^{-1}(-1)\) का मान क्या है?

Simple Explanation

\(cosec^{-1}1\) का मान क्या है?

Simple Explanation

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

Simple Explanation

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

Simple Explanation

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

Simple Explanation

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

Simple Explanation

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

Simple Explanation

Principal value branch Learning Links

Inverse Trigonometric Functions Chapter

Easy Principal value branch Test

Mathematics Subject

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