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

यदि \(f:\mathbb{R}\setminus{2}\to\mathbb{R}\), (f(x)=\frac{x^-2-4}{x-2}), तो (f) की एकैकीता पर सही निष्कर्ष क्या है?

If \(f:\mathbb{R}\setminus{2}\to\mathbb{R}\), (f(x)=\frac{x^-2-4}{x-2}), what is the correct conclusion about the one-one nature of (f)?

Explanation opens after your attempt

Correct Answer

A. एकैकी हैIt is one-one

Step 1

Concept

For \(x\neq 2\), \(\frac{x^2-4}{x-2}=x+2\).

Step 2

Why this answer is correct

In (x+2), the coefficient of (x) is non-zero, so equal images give equal inputs.

Step 3

Exam Tip

While simplifying, keep the removed point outside the domain. चरण 1: \(x\neq 2\) पर \(\frac{x^2-4}{x-2}=x+2\) हो जाता है। चरण 2: (x+2) में (x) का गुणांक शून्य नहीं है, इसलिए समान छवि से समान आगत मिलेगा। चरण 3: सरलीकरण करते समय हटाए गए बिंदु को क्षेत्र से बाहर ही रखें।

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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:
यदि (f:\mathbb{R}\setminus{2}\to\mathbb{R}), (f(x)=\frac{x^2-4}{x-2}), तो (f) की एकैकीता पर सही निष्कर्ष क्या है?

Question English:
If (f:\mathbb{R}\setminus{2}\to\mathbb{R}), (f(x)=\frac{x^2-4}{x-2}), what is the correct conclusion about the one-one nature of (f)?

Options:
A. एकैकी है / It is one-one
B. एकैकी नहीं है / It is not one-one
C. परिभाषित नहीं है क्योंकि हर जगह हर शून्य है / It is undefined because denominator is zero everywhere
D. स्थिर है / It is constant

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

Explanation Hindi:
चरण 1: (x\neq 2) पर (\frac{x^2-4}{x-2}=x+2) हो जाता है। चरण 2: (x+2) में (x) का गुणांक शून्य नहीं है, इसलिए समान छवि से समान आगत मिलेगा। चरण 3: सरलीकरण करते समय हटाए गए बिंदु को क्षेत्र से बाहर ही रखें।

Explanation English:
Step 1: For (x\neq 2), (\frac{x^2-4}{x-2}=x+2). Step 2: In (x+2), the coefficient of (x) is non-zero, so equal images give equal inputs. Step 3: While simplifying, keep the removed point outside the domain.

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यदि \(f:\mathbb{R}\setminus{2}\to\mathbb{R}\), (f(x)=\frac{x-2-4}{x-2}), तो (f) की एकैकीता पर सही निष्कर्ष क्या है?

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यदि \(f:\mathbb{R}\setminus{2}\to\mathbb{R}\), (f(x)=\frac{x^-2-4}{x-2}), तो (f) की एकैकीता पर सही निष्कर्ष क्या है?