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Question Hard Mathematics Relations and Functions One-one function Class 12 Level 22

फलन \(f:\mathbb{R}\to\mathbb{R}\) के लिए (f(x)=\begin{cases}x+3,&x<0\x^-2+3,&x\geq0\end{cases})। सही विकल्प चुनिए।

For the function \(f:\mathbb{R}\to\mathbb{R}\), (f(x)=\begin{cases}x+3,&x<0\x^-2+3,&x\geq0\end{cases}). Choose the correct option.

Explanation opens after your attempt

Correct Answer

A. एकैकी हैIt is one-one

Step 1

Concept

For (x<0), the range of (x+3) is less than (3).

Step 2

Why this answer is correct

For \(x\geq0\), the range of \(x^2+3\) is at least (3).

Step 3

Exam Tip

The two ranges do not overlap and both pieces are one-one on their domains, so the whole function is one-one. चरण 1: (x<0) पर (x+3) का परास (3) से कम है। चरण 2: \(x\geq0\) पर \(x^2+3\) का परास (3) या उससे अधिक है। चरण 3: दोनों परास नहीं टकराते और दोनों भाग अपने क्षेत्र में एकैकी हैं, इसलिए पूरा फलन एकैकी 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: Hard
Level: 22

Question Hindi:
फलन (f:\mathbb{R}\to\mathbb{R}) के लिए (f(x)=\begin{cases}x+3,&x<0\x^2+3,&x\geq0\end{cases})। सही विकल्प चुनिए।

Question English:
For the function (f:\mathbb{R}\to\mathbb{R}), (f(x)=\begin{cases}x+3,&x<0\x^2+3,&x\geq0\end{cases}). Choose the correct option.

Options:
A. एकैकी है / It is one-one
B. एकैकी नहीं है / It is not one-one
C. स्थिर है / It is constant
D. केवल प्राकृतिक संख्याओं पर एकैकी है / It is one-one only on natural numbers

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

Explanation Hindi:
चरण 1: (x<0) पर (x+3) का परास (3) से कम है। चरण 2: (x\geq0) पर (x^2+3) का परास (3) या उससे अधिक है। चरण 3: दोनों परास नहीं टकराते और दोनों भाग अपने क्षेत्र में एकैकी हैं, इसलिए पूरा फलन एकैकी है।

Explanation English:
Step 1: For (x<0), the range of (x+3) is less than (3). Step 2: For (x\geq0), the range of (x^2+3) is at least (3). Step 3: The two ranges do not overlap and both pieces are one-one on their domains, so the whole function is one-one.

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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- 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.
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- Text overlays for 16:9 and 9:16.
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- Hashtags: #MuftShiksha #FreeEducation #MCQ #Quiz #Class12 #Study

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Question Hard Mathematics Relations and Functions One-one function Class 12 Level 22

फलन \(f:\mathbb{R}\to\mathbb{R}\) को (f(x)=\begin{cases}x+2,&x<0\2x+2,&x\geq0\end{cases}) से परिभाषित किया गया है। सही विकल्प चुनिए।

The function \(f:\mathbb{R}\to\mathbb{R}\) is defined by (f(x)=\begin{cases}x+2,&x<0\2x+2,&x\geq0\end{cases}). Choose the correct option.

Explanation opens after your attempt

Correct Answer

B. एकैकी नहीं हैIt is not one-one

Step 1

Concept

In a piecewise function, the images from different pieces must be compared.

Step 2

Why this answer is correct

The left part gives values less than (2), while the right part gives values at least (2).

Step 3

Exam Tip

Since each part is one-one and their image ranges do not overlap, the function is one-one. चरण 1: अलग-अलग भागों से समान छवि मिल सकती है। चरण 2: (f(-1)=1) और (f\left\(-\frac{1}{2}\right\)=\frac{3}{2}) केवल बाएँ भाग में अलग हैं, पर (f(-1)=1) और (f\left\(-\frac{1}{2}\right\)\neq1); अब (f(-1)=1) और दाएँ भाग में (2x+2=1) देने वाला \(x=-\frac{1}{2}\) क्षेत्र में नहीं है। इस जाँच से टकराव नहीं मिलता। चरण 3: फिर भी सीमा पर (f(0)=2) और बाएँ भाग में (x+2=2) देने वाला (x=0) क्षेत्र में नहीं है, इसलिए दोनों भाग अलग छवि-क्षेत्र रखते हैं और फलन एकैकी है।

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AI Video Prompt 16:9 + 9:16

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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.
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Question Page:
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MCQ Data:
Class: Class 12
Subject: Mathematics
Chapter: Relations and Functions
Topic: One-one function
Difficulty: Hard
Level: 22

Question Hindi:
फलन (f:\mathbb{R}\to\mathbb{R}) को (f(x)=\begin{cases}x+2,&x<0\2x+2,&x\geq0\end{cases}) से परिभाषित किया गया है। सही विकल्प चुनिए।

Question English:
The function (f:\mathbb{R}\to\mathbb{R}) is defined by (f(x)=\begin{cases}x+2,&x<0\2x+2,&x\geq0\end{cases}). Choose the correct option.

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:
B. एकैकी नहीं है / It is not one-one

Explanation Hindi:
चरण 1: अलग-अलग भागों से समान छवि मिल सकती है। चरण 2: (f(-1)=1) और (f\left(-\frac{1}{2}\right)=\frac{3}{2}) केवल बाएँ भाग में अलग हैं, पर (f(-1)=1) और (f\left(-\frac{1}{2}\right)\neq1); अब (f(-1)=1) और दाएँ भाग में (2x+2=1) देने वाला (x=-\frac{1}{2}) क्षेत्र में नहीं है। इस जाँच से टकराव नहीं मिलता। चरण 3: फिर भी सीमा पर (f(0)=2) और बाएँ भाग में (x+2=2) देने वाला (x=0) क्षेत्र में नहीं है, इसलिए दोनों भाग अलग छवि-क्षेत्र रखते हैं और फलन एकैकी है।

Explanation English:
Step 1: In a piecewise function, the images from different pieces must be compared. Step 2: The left part gives values less than (2), while the right part gives values at least (2). Step 3: Since each part is one-one and their image ranges do not overlap, the function is one-one.

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

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फलन \(f:\mathbb{R}\to\mathbb{R}\) के लिए (f(x)=\begin{cases}x+3,&x<0\x-2+3,&x\geq0\end{cases})। सही विकल्प चुनिए।

Concept

Why this answer is correct

Exam Tip

फलन \(f:\mathbb{R}\to\mathbb{R}\) को (f(x)=\begin{cases}x+2,&x<0\2x+2,&x\geq0\end{cases}) से परिभाषित किया गया है। सही विकल्प चुनिए।

Concept

Why this answer is correct

Exam Tip

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फलन \(f:\mathbb{R}\to\mathbb{R}\) के लिए (f(x)=\begin{cases}x+3,&x<0\x^-2+3,&x\geq0\end{cases})। सही विकल्प चुनिए।