Class 12 positive domain MCQ Questions

Question 1/7 Expert Mathematics Relations and Functions Onto function Class 12 Level 25

फलन (f:\(0,\infty\)\to[2,\infty)), (f(x)=x+\frac{1}{x}), के लिए सही कथन क्या है?

For (f:\(0,\infty\)\to[2,\infty)), (f(x)=x+\frac{1}{x}), what is the correct statement?

Explanation opens after your attempt

Correct Answer

A. सर्वाच्छादक हैIt is onto

Step 1

Concept

\(x+\frac{1}{x}\ge2\), and (2) is obtained at (x=1).

Step 2

Why this answer is correct

For every (y>2), the equation \(x+\frac{1}{x}=y\) has a positive solution.

Step 3

Exam Tip

Check separately whether the minimum value is actually attained or only approached. चरण 1: \(x+\frac{1}{x}\ge2\) और (x=1) पर (2) मिलता है। चरण 2: (2) से बड़े हर (y) के लिए समीकरण \(x+\frac{1}{x}=y\) का धनात्मक हल मिल जाता है। चरण 3: न्यूनतम मान मिलता है या केवल सीमा है, यह अलग से जाँचें।

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MCQ Data:
Class: Class 12
Subject: Mathematics
Chapter: Relations and Functions
Topic: Onto function
Difficulty: Expert
Level: 25

Question Hindi:
फलन (f:(0,\infty)\to[2,\infty)), (f(x)=x+\frac{1}{x}), के लिए सही कथन क्या है?

Question English:
For (f:(0,\infty)\to[2,\infty)), (f(x)=x+\frac{1}{x}), what is the correct statement?

Options:
A. सर्वाच्छादक है / It is onto
B. सर्वाच्छादक नहीं क्योंकि (2) नहीं मिलता / It is not onto because (2) is not obtained
C. सर्वाच्छादक नहीं क्योंकि (3) नहीं मिलता / It is not onto because (3) is not obtained
D. परास केवल ((2,\infty)) है / The range is only ((2,\infty))

Correct Answer:
A. सर्वाच्छादक है / It is onto

Explanation Hindi:
चरण 1: (x+\frac{1}{x}\ge2) और (x=1) पर (2) मिलता है। चरण 2: (2) से बड़े हर (y) के लिए समीकरण (x+\frac{1}{x}=y) का धनात्मक हल मिल जाता है। चरण 3: न्यूनतम मान मिलता है या केवल सीमा है, यह अलग से जाँचें।

Explanation English:
Step 1: (x+\frac{1}{x}\ge2), and (2) is obtained at (x=1). Step 2: For every (y>2), the equation (x+\frac{1}{x}=y) has a positive solution. Step 3: Check separately whether the minimum value is actually attained or only approached.

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Question 2/7 Expert Mathematics Relations and Functions Onto function Class 12 Level 25

यदि (f:\(0,\infty\)\to\(0,\infty\)), (f(x)=x+\frac{1}{x}), तो (f) सर्वाच्छादक क्यों नहीं है?

If (f:\(0,\infty\)\to\(0,\infty\)), (f(x)=x+\frac{1}{x}), why is (f) not onto?

Explanation opens after your attempt

Correct Answer

A. क्योंकि (1) छवि नहीं बनताBecause (1) is not an image

Step 1

Concept

\(x+\frac{1}{x}\ge2\) for (x>0).

Step 2

Why this answer is correct

The codomain (\(0,\infty\)) contains (1), but it is not an image.

Step 3

Exam Tip

The standard inequality \(x+\frac{1}{x}\ge2\) is useful in such problems. चरण 1: \(x+\frac{1}{x}\ge2\) जब (x>0)। चरण 2: सहप्रांत (\(0,\infty\)) में (1) है, पर वह छवि नहीं बनता। चरण 3: प्रसिद्ध असमता \(x+\frac{1}{x}\ge2\) ऐसे प्रश्नों में बहुत काम आती है।

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MCQ Data:
Class: Class 12
Subject: Mathematics
Chapter: Relations and Functions
Topic: Onto function
Difficulty: Expert
Level: 25

Question Hindi:
यदि (f:(0,\infty)\to(0,\infty)), (f(x)=x+\frac{1}{x}), तो (f) सर्वाच्छादक क्यों नहीं है?

Question English:
If (f:(0,\infty)\to(0,\infty)), (f(x)=x+\frac{1}{x}), why is (f) not onto?

Options:
A. क्योंकि (1) छवि नहीं बनता / Because (1) is not an image
B. क्योंकि (2) छवि नहीं बनता / Because (2) is not an image
C. क्योंकि (3) छवि नहीं बनता / Because (3) is not an image
D. क्योंकि सभी धनात्मक मान मिलते हैं / Because all positive values are obtained

Correct Answer:
A. क्योंकि (1) छवि नहीं बनता / Because (1) is not an image

Explanation Hindi:
चरण 1: (x+\frac{1}{x}\ge2) जब (x>0)। चरण 2: सहप्रांत ((0,\infty)) में (1) है, पर वह छवि नहीं बनता। चरण 3: प्रसिद्ध असमता (x+\frac{1}{x}\ge2) ऐसे प्रश्नों में बहुत काम आती है।

Explanation English:
Step 1: (x+\frac{1}{x}\ge2) for (x>0). Step 2: The codomain ((0,\infty)) contains (1), but it is not an image. Step 3: The standard inequality (x+\frac{1}{x}\ge2) is useful in such problems.

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Question 3/7 Hard Mathematics Relations and Functions Onto function Class 12 Level 25

यदि (f:\(0,\infty\)\to [2,\infty)), (f(x)=x+\frac{1}{x}), तो (f) कैसा है?

If (f:\(0,\infty\)\to [2,\infty)), (f(x)=x+\frac{1}{x}), what type is (f)?

Explanation opens after your attempt

Correct Answer

A. आच्छादक हैOnto

Step 1

Concept

The minimum value of \(x+\frac{1}{x}\) is (2), attained at (x=1).

Step 2

Why this answer is correct

As \(x\to0^+\) or \(x\to\infty\), the value goes to \(\infty\), so all of \([2,\infty\)) is obtained.

Step 3

Exam Tip

After finding the minimum, compare it with the codomain. चरण 1: \(x+\frac{1}{x}\) का न्यूनतम मान (2) है जो (x=1) पर मिलता है। चरण 2: \(x\to0^+\) या \(x\to\infty\) पर मान \(\infty\) की ओर जाता है इसलिए \([2,\infty\)) पूरा मिलता है। चरण 3: न्यूनतम मान मिलने के बाद सहक्षेत्र से तुलना करें।

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AI Video Prompt 16:9 + 9:16

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MCQ Data:
Class: Class 12
Subject: Mathematics
Chapter: Relations and Functions
Topic: Onto function
Difficulty: Hard
Level: 25

Question Hindi:
यदि (f:(0,\infty)\to [2,\infty)), (f(x)=x+\frac{1}{x}), तो (f) कैसा है?

Question English:
If (f:(0,\infty)\to [2,\infty)), (f(x)=x+\frac{1}{x}), what type is (f)?

Options:
A. आच्छादक है / Onto
B. आच्छादक नहीं है / Not onto
C. परास ((-\infty,2]) है / Range is ((-\infty,2])
D. केवल (2) मिलता है / Only (2) is obtained

Correct Answer:
A. आच्छादक है / Onto

Explanation Hindi:
चरण 1: (x+\frac{1}{x}) का न्यूनतम मान (2) है जो (x=1) पर मिलता है। चरण 2: (x\to0^+) या (x\to\infty) पर मान (\infty) की ओर जाता है इसलिए ([2,\infty)) पूरा मिलता है। चरण 3: न्यूनतम मान मिलने के बाद सहक्षेत्र से तुलना करें।

Explanation English:
Step 1: The minimum value of (x+\frac{1}{x}) is (2), attained at (x=1). Step 2: As (x\to0^+) or (x\to\infty), the value goes to (\infty), so all of ([2,\infty)) is obtained. Step 3: After finding the minimum, compare it with the codomain.

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Question 4/7 Medium Mathematics Relations and Functions One-one function Class 12 Level 24

यदि (f:\(0,\infty\)\to\mathbb{R}) और (f(x)=\frac{1}{x}) है, तो (f) के बारे में सही कथन क्या है?

If (f:\(0,\infty\)\to\mathbb{R}) and (f(x)=\frac{1}{x}), what is the correct statement about (f)?

Explanation opens after your attempt

Correct Answer

A. एकैकी हैOne-one

Step 1

Concept

Assume (f(a)=f(b)), where (a,b>0).

Step 2

Why this answer is correct

From \(\frac{1}{a}=\frac{1}{b}\), we get (a=b).

Step 3

Exam Tip

In the reciprocal function, equal images occur only for equal inputs. चरण 1: मान लें (f(a)=f(b)), जहां (a,b>0)। चरण 2: \(\frac{1}{a}=\frac{1}{b}\) से (a=b) मिलता है। चरण 3: व्युत्क्रम फलन में समान प्रतिबिंब तभी होगा जब आगत समान हों।

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AI Video Prompt 16:9 + 9:16

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MCQ Data:
Class: Class 12
Subject: Mathematics
Chapter: Relations and Functions
Topic: One-one function
Difficulty: Medium
Level: 24

Question Hindi:
यदि (f:(0,\infty)\to\mathbb{R}) और (f(x)=\frac{1}{x}) है, तो (f) के बारे में सही कथन क्या है?

Question English:
If (f:(0,\infty)\to\mathbb{R}) and (f(x)=\frac{1}{x}), what is the correct statement about (f)?

Options:
A. एकैकी है / One-one
B. एकैकी नहीं है / Not one-one
C. नियत फलन है / Constant function
D. डोमेन में कोई संख्या नहीं है / Domain has no number

Correct Answer:
A. एकैकी है / One-one

Explanation Hindi:
चरण 1: मान लें (f(a)=f(b)), जहां (a,b>0)। चरण 2: (\frac{1}{a}=\frac{1}{b}) से (a=b) मिलता है। चरण 3: व्युत्क्रम फलन में समान प्रतिबिंब तभी होगा जब आगत समान हों।

Explanation English:
Step 1: Assume (f(a)=f(b)), where (a,b>0). Step 2: From (\frac{1}{a}=\frac{1}{b}), we get (a=b). Step 3: In the reciprocal function, equal images occur only for equal inputs.

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Question 5/7 Medium Mathematics Relations and Functions One-one function Class 12 Level 23

फलन (f:\(0,\infty\)\to\mathbb{R}), (f(x)=\frac{1}{x}) के लिए सही विकल्प चुनिए।

Choose the correct option for (f:\(0,\infty\)\to\mathbb{R}), (f(x)=\frac{1}{x}).

Explanation opens after your attempt

Correct Answer

A. एकैकी हैIt is one-one

Step 1

Concept

Let (f(a)=f(b)), where (a,b>0).

Step 2

Why this answer is correct

From \(\frac{1}{a}=\frac{1}{b}\), we get (a=b).

Step 3

Exam Tip

In this reciprocal form, equal images lead to equal inputs. चरण 1: मान लें (f(a)=f(b)) जहाँ (a,b>0)। चरण 2: \(\frac{1}{a}=\frac{1}{b}\) से (a=b) मिलता है। चरण 3: प्रतिलोम रूप वाले ऐसे फलन में बराबर छवि बराबर आगत देती है।

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AI Video Prompt 16:9 + 9:16

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MCQ Data:
Class: Class 12
Subject: Mathematics
Chapter: Relations and Functions
Topic: One-one function
Difficulty: Medium
Level: 23

Question Hindi:
फलन (f:(0,\infty)\to\mathbb{R}), (f(x)=\frac{1}{x}) के लिए सही विकल्प चुनिए।

Question English:
Choose the correct option for (f:(0,\infty)\to\mathbb{R}), (f(x)=\frac{1}{x}).

Options:
A. एकैकी है / It is one-one
B. एकैकी नहीं है / It is not one-one
C. स्थिर है / It is constant
D. फलन नहीं है / It is not a function

Correct Answer:
A. एकैकी है / It is one-one

Explanation Hindi:
चरण 1: मान लें (f(a)=f(b)) जहाँ (a,b>0)। चरण 2: (\frac{1}{a}=\frac{1}{b}) से (a=b) मिलता है। चरण 3: प्रतिलोम रूप वाले ऐसे फलन में बराबर छवि बराबर आगत देती है।

Explanation English:
Step 1: Let (f(a)=f(b)), where (a,b>0). Step 2: From (\frac{1}{a}=\frac{1}{b}), we get (a=b). Step 3: In this reciprocal form, equal images lead to equal inputs.

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4. 15-18 sec: Add countdown/timer effect: "Socho... correct answer kya hoga?"
5. 18-23 sec: Reveal correct answer with green glow and subtle success sound.
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Question 6/7 Easy Mathematics Relations and Functions One-one function Class 12 Level 24

यदि (f:\(0,\infty\)\to\mathbb{R}) और (f(x)=\frac{2}{x}) हो तो (f) कैसा है?

If (f:\(0,\infty\)\to\mathbb{R}) and (f(x)=\frac{2}{x}), what type of function is (f)?

Explanation opens after your attempt

Correct Answer

A. एकैकीOne-one

Step 1

Concept

From (f(a)=f(b)), we get \(\frac{2}{a}=\frac{2}{b}\).

Step 2

Why this answer is correct

Since (a,b>0), this gives (a=b).

Step 3

Exam Tip

A reciprocal-type rule is one-one on its proper domain. चरण 1: (f(a)=f(b)) से \(\frac{2}{a}=\frac{2}{b}\) मिलता है। चरण 2: (a,b>0) होने से (a=b) निष्कर्ष निकलता है। चरण 3: व्युत्क्रम जैसा नियम अपने उचित प्रांत पर एकैकी होता है।

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AI Video Prompt 16:9 + 9:16

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MCQ Data:
Class: Class 12
Subject: Mathematics
Chapter: Relations and Functions
Topic: One-one function
Difficulty: Easy
Level: 24

Question Hindi:
यदि (f:(0,\infty)\to\mathbb{R}) और (f(x)=\frac{2}{x}) हो तो (f) कैसा है?

Question English:
If (f:(0,\infty)\to\mathbb{R}) and (f(x)=\frac{2}{x}), what type of function is (f)?

Options:
A. एकैकी / One-one
B. एकैकी नहीं / Not one-one
C. स्थिर / Constant
D. अपरिभाषित / Undefined

Correct Answer:
A. एकैकी / One-one

Explanation Hindi:
चरण 1: (f(a)=f(b)) से (\frac{2}{a}=\frac{2}{b}) मिलता है। चरण 2: (a,b>0) होने से (a=b) निष्कर्ष निकलता है। चरण 3: व्युत्क्रम जैसा नियम अपने उचित प्रांत पर एकैकी होता है।

Explanation English:
Step 1: From (f(a)=f(b)), we get (\frac{2}{a}=\frac{2}{b}). Step 2: Since (a,b>0), this gives (a=b). Step 3: A reciprocal-type rule is one-one on its proper domain.

Video Structure:
1. 0-2 sec: Premium Muft Shiksha intro with logo glow and title: "MCQ Challenge".
2. 2-8 sec: Show question in large readable text. Keep Hindi and English clean, no overlap.
3. 8-15 sec: Animate A/B/C/D option cards one by one.
4. 15-18 sec: Add countdown/timer effect: "Socho... correct answer kya hoga?"
5. 18-23 sec: Reveal correct answer with green glow and subtle success sound.
6. 23-30 sec: Show short explanation with one exam tip.
7. 30-35 sec: CTA: "More free MCQs ke liye visit karein MuftShiksha.com".

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- Make 16:9 version for YouTube/Facebook.
- Make 9:16 version for Shorts/Reels.
- Add Hindi-English natural voice-over.
- Add readable subtitles.
- Add light background music, low volume.
- Keep all text inside safe margins.
- Use smooth transitions, no clutter, no watermark except Muft Shiksha branding.
- Formula/Math/Chemistry text must stay exactly readable; use LaTeX-style clean rendering where needed.

Output Needed:
- Final narration script.
- Scene-by-scene visual direction.
- Text overlays for 16:9 and 9:16.
- Thumbnail idea.
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Question 7/7 Expert Mathematics Relations and Functions Functions Class 12 Level 19

यदि (f:\(0,\infty\)\to\mathbb{R}) को (f(x)=x+\frac{1}{x}) से दिया गया है, तो (f) का परास क्या है?

If (f:\(0,\infty\)\to\mathbb{R}) is given by (f(x)=x+\frac{1}{x}), what is the range of (f)?

Explanation opens after your attempt

Correct Answer

A. \([2,\infty\))

Step 1

Concept

For positive (x), \(x+\frac{1}{x}\geq2\).

Step 2

Why this answer is correct

At (x=1), the value (2) is attained.

Step 3

Exam Tip

The AM-GM idea is very useful for such range questions. चरण 1: धनात्मक (x) के लिए \(x+\frac{1}{x}\geq2\) होता है। चरण 2: (x=1) पर मान (2) मिल जाता है। चरण 3: ऐसे प्रश्न में गुणोत्तर और अंकगणितीय माध्य का विचार बहुत उपयोगी है।

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AI Video Prompt 16:9 + 9:16

Is question ka premium MCQ video banane ke liye ready prompt. Copy karke Sora, Runway, Canva AI, CapCut AI, ChatGPT video workflow ya editor me use karein.

Create a premium AI MCQ quiz video for Muft Shiksha in BOTH formats: 16:9 landscape and 9:16 vertical short.

Brand:
- Name: Muft Shiksha
- Website: https://muftshiksha.com
- Style: dark navy futuristic learning game UI, neon cyan/blue glow, gold reward highlights, clean Devanagari text, premium educational look.
- Do not add wrong facts. Do not change the question, options or answer.

Question Page:
https://muftshiksha.com/question/if-f-0-infty-to-mathbb-r-is-given-by-f-x-x-frac-1-x-what-is-the-range-of-f

MCQ Data:
Class: Class 12
Subject: Mathematics
Chapter: Relations and Functions
Topic: Functions
Difficulty: Expert
Level: 19

Question Hindi:
यदि (f:(0,\infty)\to\mathbb{R}) को (f(x)=x+\frac{1}{x}) से दिया गया है, तो (f) का परास क्या है?

Question English:
If (f:(0,\infty)\to\mathbb{R}) is given by (f(x)=x+\frac{1}{x}), what is the range of (f)?

Options:
A. ([2,\infty))
B. ((2,\infty))
C. (\mathbb{R})
D. ((-\infty,2])

Correct Answer:
A. ([2,\infty))

Explanation Hindi:
चरण 1: धनात्मक (x) के लिए (x+\frac{1}{x}\geq2) होता है। चरण 2: (x=1) पर मान (2) मिल जाता है। चरण 3: ऐसे प्रश्न में गुणोत्तर और अंकगणितीय माध्य का विचार बहुत उपयोगी है।

Explanation English:
Step 1: For positive (x), (x+\frac{1}{x}\geq2). Step 2: At (x=1), the value (2) is attained. Step 3: The AM-GM idea is very useful for such range questions.

Video Structure:
1. 0-2 sec: Premium Muft Shiksha intro with logo glow and title: "MCQ Challenge".
2. 2-8 sec: Show question in large readable text. Keep Hindi and English clean, no overlap.
3. 8-15 sec: Animate A/B/C/D option cards one by one.
4. 15-18 sec: Add countdown/timer effect: "Socho... correct answer kya hoga?"
5. 18-23 sec: Reveal correct answer with green glow and subtle success sound.
6. 23-30 sec: Show short explanation with one exam tip.
7. 30-35 sec: CTA: "More free MCQs ke liye visit karein MuftShiksha.com".

Custom Support:
- Make 16:9 version for YouTube/Facebook.
- Make 9:16 version for Shorts/Reels.
- Add Hindi-English natural voice-over.
- Add readable subtitles.
- Add light background music, low volume.
- Keep all text inside safe margins.
- Use smooth transitions, no clutter, no watermark except Muft Shiksha branding.
- Formula/Math/Chemistry text must stay exactly readable; use LaTeX-style clean rendering where needed.

Output Needed:
- Final narration script.
- Scene-by-scene visual direction.
- Text overlays for 16:9 and 9:16.
- Thumbnail idea.
- Hashtags: #MuftShiksha #FreeEducation #MCQ #Quiz #Class12 #Study

Open Question

positive domain MCQ Questions for Class 12

Practice Questions

फलन (f:\(0,\infty\)\to[2,\infty)), (f(x)=x+\frac{1}{x}), के लिए सही कथन क्या है?

Concept

Why this answer is correct

Exam Tip

यदि (f:\(0,\infty\)\to\(0,\infty\)), (f(x)=x+\frac{1}{x}), तो (f) सर्वाच्छादक क्यों नहीं है?

Concept

Why this answer is correct

Exam Tip

यदि (f:\(0,\infty\)\to [2,\infty)), (f(x)=x+\frac{1}{x}), तो (f) कैसा है?

Concept

Why this answer is correct

Exam Tip

यदि (f:\(0,\infty\)\to\mathbb{R}) और (f(x)=\frac{1}{x}) है, तो (f) के बारे में सही कथन क्या है?

Concept

Why this answer is correct

Exam Tip

फलन (f:\(0,\infty\)\to\mathbb{R}), (f(x)=\frac{1}{x}) के लिए सही विकल्प चुनिए।

Concept

Why this answer is correct

Exam Tip

यदि (f:\(0,\infty\)\to\mathbb{R}) और (f(x)=\frac{2}{x}) हो तो (f) कैसा है?

Concept

Why this answer is correct

Exam Tip

यदि (f:\(0,\infty\)\to\mathbb{R}) को (f(x)=x+\frac{1}{x}) से दिया गया है, तो (f) का परास क्या है?

Concept

Why this answer is correct

Exam Tip

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positive domain MCQ Questions for Class 12

Practice Questions

फलन (f:\(0,\infty\)\to[2,\infty)), (f(x)=x+\frac{1}{x}), के लिए सही कथन क्या है?

Concept

Why this answer is correct

Exam Tip

यदि (f:\(0,\infty\)\to\(0,\infty\)), (f(x)=x+\frac{1}{x}), तो (f) सर्वाच्छादक क्यों नहीं है?

Concept

Why this answer is correct

Exam Tip

यदि (f:\(0,\infty\)\to [2,\infty)), (f(x)=x+\frac{1}{x}), तो (f) कैसा है?

Concept

Why this answer is correct

Exam Tip

यदि (f:\(0,\infty\)\to\mathbb{R}) और (f(x)=\frac{1}{x}) है, तो (f) के बारे में सही कथन क्या है?

Concept

Why this answer is correct

Exam Tip

फलन (f:\(0,\infty\)\to\mathbb{R}), (f(x)=\frac{1}{x}) के लिए सही विकल्प चुनिए।

Concept

Why this answer is correct

Exam Tip

यदि (f:\(0,\infty\)\to\mathbb{R}) और (f(x)=\frac{2}{x}) हो तो (f) कैसा है?

Concept

Why this answer is correct

Exam Tip

यदि (f:\(0,\infty\)\to\mathbb{R}) को (f(x)=x+\frac{1}{x}) से दिया गया है, तो (f) का परास क्या है?

Concept

Why this answer is correct

Exam Tip

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