Search: injectivity

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

यदि (f:\(0,\infty\)\to\mathbb{R}) तथा (f(x)=\log x) है, तो (f) के बारे में सही निष्कर्ष क्या है?

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

Explanation opens after your attempt

Correct Answer

A. (f) एक-एक है(f) is one-one

Step 1

Concept

The logarithmic function increases on its proper domain (\(0,\infty\)).

Step 2

Why this answer is correct

If \(\log x_1=\log x_2\), then \(x_1=x_2\).

Step 3

Exam Tip

While checking injectivity of a logarithmic function, always check its domain first. चरण 1: लघुगणकीय फलन अपने सही प्रांत (\(0,\infty\)) पर बढ़ता है। चरण 2: यदि \(\log x_1=\log x_2\), तो \(x_1=x_2\) ही होगा। चरण 3: लघुगणकीय फलन की एक-एकता जांचते समय पहले उसका प्रांत अवश्य देखें।

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MCQ Data:
Class: Class 12
Subject: Mathematics
Chapter: Relations and Functions
Topic: One-one function
Difficulty: Expert
Level: 22

Question Hindi:
यदि (f:(0,\infty)\to\mathbb{R}) तथा (f(x)=\log x) है, तो (f) के बारे में सही निष्कर्ष क्या है?

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

Options:
A. (f) एक-एक है / (f) is one-one
B. (f) एक-एक नहीं है / (f) is not one-one
C. (f) हर जगह शून्य है / (f) is zero everywhere
D. (f) ऋणात्मक आगतों पर भी परिभाषित है / (f) is also defined for negative inputs

Correct Answer:
A. (f) एक-एक है / (f) is one-one

Explanation Hindi:
चरण 1: लघुगणकीय फलन अपने सही प्रांत ((0,\infty)) पर बढ़ता है। चरण 2: यदि (\log x_1=\log x_2), तो (x_1=x_2) ही होगा। चरण 3: लघुगणकीय फलन की एक-एकता जांचते समय पहले उसका प्रांत अवश्य देखें।

Explanation English:
Step 1: The logarithmic function increases on its proper domain ((0,\infty)). Step 2: If (\log x_1=\log x_2), then (x_1=x_2). Step 3: While checking injectivity of a logarithmic function, always check its domain first.

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

यदि \(f:\mathbb{R}\to\mathbb{R}\) तथा (f(x)=x^-3-6x^-2+9x) है, तो कौन सा विकल्प (f) के एक-एक न होने को सिद्ध करता है?

If \(f:\mathbb{R}\to\mathbb{R}\) and (f(x)=x^-3-6x^-2+9x), which option proves that (f) is not one-one?

Explanation opens after your attempt

Correct Answer

A. (f(0)=f(3))

Step 1

Concept

To disprove injectivity it is enough to show equal output for two different inputs.

Step 2

Why this answer is correct

(f(0)=0) and (f(3)=27-54+27=0), while \(0\neq3\).

Step 3

Exam Tip

In polynomial questions use simple values to quickly find repeated images. चरण 1: एक-एकता को गलत सिद्ध करने के लिए दो अलग आगतों का समान मान दिखाना पर्याप्त है। चरण 2: (f(0)=0) और (f(3)=27-54+27=0), जबकि \(0\neq3\)। चरण 3: बहुपद में आसान मान रखकर समान प्रतिबिंब जल्दी खोजें।

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MCQ Data:
Class: Class 12
Subject: Mathematics
Chapter: Relations and Functions
Topic: One-one function
Difficulty: Expert
Level: 22

Question Hindi:
यदि (f:\mathbb{R}\to\mathbb{R}) तथा (f(x)=x^3-6x^2+9x) है, तो कौन सा विकल्प (f) के एक-एक न होने को सिद्ध करता है?

Question English:
If (f:\mathbb{R}\to\mathbb{R}) and (f(x)=x^3-6x^2+9x), which option proves that (f) is not one-one?

Options:
A. (f(0)=f(3))
B. (f(1)=f(2))
C. (f(-1)=f(1))
D. (f(2)=f(4))

Correct Answer:
A. (f(0)=f(3))

Explanation Hindi:
चरण 1: एक-एकता को गलत सिद्ध करने के लिए दो अलग आगतों का समान मान दिखाना पर्याप्त है। चरण 2: (f(0)=0) और (f(3)=27-54+27=0), जबकि (0\neq3)। चरण 3: बहुपद में आसान मान रखकर समान प्रतिबिंब जल्दी खोजें।

Explanation English:
Step 1: To disprove injectivity it is enough to show equal output for two different inputs. Step 2: (f(0)=0) and (f(3)=27-54+27=0), while (0\neq3). Step 3: In polynomial questions use simple values to quickly find repeated images.

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

यदि \(f:\mathbb{R}\to\mathbb{R}\) तथा (f(x)=2x^-3-5) है, तो (f) के लिए सही कथन चुनिए।

If \(f:\mathbb{R}\to\mathbb{R}\) and (f(x)=2x^-3-5), choose the correct statement about (f).

Explanation opens after your attempt

Correct Answer

B. (f) एक-एक है(f) is one-one

Step 1

Concept

\(2x^3-5\) is a transformation of the cube function.

Step 2

Why this answer is correct

From \(2x_1^3-5=2x_2^3-5\), we get \(x_1^3=x_2^3\), so \(x_1=x_2\).

Step 3

Exam Tip

Stretching and shifting the cube function does not destroy injectivity. चरण 1: \(2x^3-5\) घन फलन का रूपांतरण है। चरण 2: \(2x_1^3-5=2x_2^3-5\) से \(x_1^3=x_2^3\), इसलिए \(x_1=x_2\)। चरण 3: घन फलन पर खिंचाव और स्थानांतरण करने से एक-एकता नहीं बदलती।

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MCQ Data:
Class: Class 12
Subject: Mathematics
Chapter: Relations and Functions
Topic: One-one function
Difficulty: Expert
Level: 22

Question Hindi:
यदि (f:\mathbb{R}\to\mathbb{R}) तथा (f(x)=2x^3-5) है, तो (f) के लिए सही कथन चुनिए।

Question English:
If (f:\mathbb{R}\to\mathbb{R}) and (f(x)=2x^3-5), choose the correct statement about (f).

Options:
A. (f) एक-एक नहीं है / (f) is not one-one
B. (f) एक-एक है / (f) is one-one
C. (f) स्थिर है / (f) is constant
D. (f) (x=0) पर परिभाषित नहीं है / (f) is not defined at (x=0)

Correct Answer:
B. (f) एक-एक है / (f) is one-one

Explanation Hindi:
चरण 1: (2x^3-5) घन फलन का रूपांतरण है। चरण 2: (2x_1^3-5=2x_2^3-5) से (x_1^3=x_2^3), इसलिए (x_1=x_2)। चरण 3: घन फलन पर खिंचाव और स्थानांतरण करने से एक-एकता नहीं बदलती।

Explanation English:
Step 1: (2x^3-5) is a transformation of the cube function. Step 2: From (2x_1^3-5=2x_2^3-5), we get (x_1^3=x_2^3), so (x_1=x_2). Step 3: Stretching and shifting the cube function does not destroy injectivity.

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

यदि \(g\circ f\) एक-एक है, तो (g) के बारे में कौन सा कथन सबसे सटीक है?

If \(g\circ f\) is one-one, which statement about (g) is the most accurate?

Explanation opens after your attempt

Correct Answer

D. (g) को कम से कम (f(A)) पर एक-एक व्यवहार करना होगा(g) must behave one-one at least on (f(A))

Step 1

Concept

In \(g\circ f\), (g) is used only on the range of (f).

Step 2

Why this answer is correct

(g) may not be one-one on its whole domain but can behave distinctly on (f(A)).

Step 3

Exam Tip

In composition, distinguish the used part from the whole domain. चरण 1: संयोजन \(g\circ f\) में (g) केवल (f) के परास पर काम करता है। चरण 2: (g) पूरे प्रांत पर एक-एक न भी हो, फिर भी (f(A)) पर अलग मान दे सकता है। चरण 3: संयोजन में प्रयुक्त भाग और पूरे प्रांत में अंतर समझना जरूरी है।

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MCQ Data:
Class: Class 12
Subject: Mathematics
Chapter: Relations and Functions
Topic: One-one function
Difficulty: Expert
Level: 22

Question Hindi:
यदि (g\circ f) एक-एक है, तो (g) के बारे में कौन सा कथन सबसे सटीक है?

Question English:
If (g\circ f) is one-one, which statement about (g) is the most accurate?

Options:
A. (g) पूरे प्रांत पर अवश्य एक-एक है / (g) must be one-one on its whole domain
B. (g) अवश्य स्थिर है / (g) must be constant
C. (g) फलन नहीं हो सकता / (g) cannot be a function
D. (g) को कम से कम (f(A)) पर एक-एक व्यवहार करना होगा / (g) must behave one-one at least on (f(A))

Correct Answer:
D. (g) को कम से कम (f(A)) पर एक-एक व्यवहार करना होगा / (g) must behave one-one at least on (f(A))

Explanation Hindi:
चरण 1: संयोजन (g\circ f) में (g) केवल (f) के परास पर काम करता है। चरण 2: (g) पूरे प्रांत पर एक-एक न भी हो, फिर भी (f(A)) पर अलग मान दे सकता है। चरण 3: संयोजन में प्रयुक्त भाग और पूरे प्रांत में अंतर समझना जरूरी है।

Explanation English:
Step 1: In (g\circ f), (g) is used only on the range of (f). Step 2: (g) may not be one-one on its whole domain but can behave distinctly on (f(A)). Step 3: In composition, distinguish the used part from the whole domain.

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

यदि \(f:A\to B\) एक-एक है, तो निम्न में से कौन सा कथन इसके बराबर अर्थ रखता है?

If \(f:A\to B\) is one-one, which of the following statement has equivalent meaning?

Explanation opens after your attempt

Correct Answer

D. यदि \(a\neq b\), तो (f(a)\neq f(b))If \(a\neq b\), then (f(a)\neq f(b))

Step 1

Concept

Injectivity means different inputs have different images.

Step 2

Why this answer is correct

Therefore if \(a\neq b\), then (f(a)\neq f(b)).

Step 3

Exam Tip

Remember both forms of the definition to handle proofs easily. चरण 1: एक-एकता का अर्थ है अलग आगतों के प्रतिबिंब अलग हों। चरण 2: इसलिए \(a\neq b\) होने पर (f(a)\neq f(b)) होना चाहिए। चरण 3: परिभाषा को दोनों रूपों में याद रखें, इससे प्रमाण आसान होते हैं।

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MCQ Data:
Class: Class 12
Subject: Mathematics
Chapter: Relations and Functions
Topic: One-one function
Difficulty: Expert
Level: 22

Question Hindi:
यदि (f:A\to B) एक-एक है, तो निम्न में से कौन सा कथन इसके बराबर अर्थ रखता है?

Question English:
If (f:A\to B) is one-one, which of the following statement has equivalent meaning?

Options:
A. यदि (a=b), तो (f(a)\neq f(b)) / If (a=b), then (f(a)\neq f(b))
B. यदि (f(a)=f(b)), तो (a\neq b) / If (f(a)=f(b)), then (a\neq b)
C. हर (a) के लिए (f(a)) समान है / (f(a)) is same for every (a)
D. यदि (a\neq b), तो (f(a)\neq f(b)) / If (a\neq b), then (f(a)\neq f(b))

Correct Answer:
D. यदि (a\neq b), तो (f(a)\neq f(b)) / If (a\neq b), then (f(a)\neq f(b))

Explanation Hindi:
चरण 1: एक-एकता का अर्थ है अलग आगतों के प्रतिबिंब अलग हों। चरण 2: इसलिए (a\neq b) होने पर (f(a)\neq f(b)) होना चाहिए। चरण 3: परिभाषा को दोनों रूपों में याद रखें, इससे प्रमाण आसान होते हैं।

Explanation English:
Step 1: Injectivity means different inputs have different images. Step 2: Therefore if (a\neq b), then (f(a)\neq f(b)). Step 3: Remember both forms of the definition to handle proofs easily.

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

यदि \(f:\mathbb{R}\to\mathbb{R}\) तथा (f(x)=x^-5-x) है, तो कौन सा विकल्प (f) के एक-एक न होने को सिद्ध करता है?

If \(f:\mathbb{R}\to\mathbb{R}\) and (f(x)=x^-5-x), which option proves that (f) is not one-one?

Explanation opens after your attempt

Correct Answer

D. (f(-1)=f(0)=f(1))

Step 1

Concept

To disprove injectivity, show the same value at different inputs.

Step 2

Why this answer is correct

(f(-1)=0), (f(0)=0), and (f(1)=0).

Step 3

Exam Tip

Multiple roots giving the same output quickly break injectivity. चरण 1: एक-एकता को गलत करने के लिए अलग आगतों पर समान मान दिखाएं। चरण 2: (f(-1)=0), (f(0)=0), और (f(1)=0) है। चरण 3: बहुपद में शून्य देने वाले अनेक मान एक-एकता को तुरंत तोड़ देते हैं।

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MCQ Data:
Class: Class 12
Subject: Mathematics
Chapter: Relations and Functions
Topic: One-one function
Difficulty: Expert
Level: 22

Question Hindi:
यदि (f:\mathbb{R}\to\mathbb{R}) तथा (f(x)=x^5-x) है, तो कौन सा विकल्प (f) के एक-एक न होने को सिद्ध करता है?

Question English:
If (f:\mathbb{R}\to\mathbb{R}) and (f(x)=x^5-x), which option proves that (f) is not one-one?

Options:
A. (f(2)=f(3))
B. (f(-2)=f(2))
C. (f(1)=f(2))
D. (f(-1)=f(0)=f(1))

Correct Answer:
D. (f(-1)=f(0)=f(1))

Explanation Hindi:
चरण 1: एक-एकता को गलत करने के लिए अलग आगतों पर समान मान दिखाएं। चरण 2: (f(-1)=0), (f(0)=0), और (f(1)=0) है। चरण 3: बहुपद में शून्य देने वाले अनेक मान एक-एकता को तुरंत तोड़ देते हैं।

Explanation English:
Step 1: To disprove injectivity, show the same value at different inputs. Step 2: (f(-1)=0), (f(0)=0), and (f(1)=0). Step 3: Multiple roots giving the same output quickly break injectivity.

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

यदि \(f:\left[0,\frac{\pi}{2}\right\)\to\mathbb{R}) तथा (f(x)=\tan x) है, तो सही कथन चुनिए।

If \(f:\left[0,\frac{\pi}{2}\right\)\to\mathbb{R}) and (f(x)=\tan x), choose the correct statement.

Explanation opens after your attempt

Correct Answer

D. (f) एक-एक है(f) is one-one

Step 1

Concept

On the given interval \(\tan x\) is strictly increasing.

Step 2

Why this answer is correct

Therefore two different (x)-values cannot have the same tangent value.

Step 3

Exam Tip

Taking \(\tan x\) on a suitable restricted interval gives injectivity. चरण 1: दिए गए अंतराल पर \(\tan x\) लगातार बढ़ता है। चरण 2: इसलिए दो अलग (x) के लिए समान \(\tan x\) नहीं मिलेगा। चरण 3: \(\tan x\) को सही सीमित अंतराल पर लेने से एक-एकता मिलती है।

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MCQ Data:
Class: Class 12
Subject: Mathematics
Chapter: Relations and Functions
Topic: One-one function
Difficulty: Expert
Level: 22

Question Hindi:
यदि (f:\left[0,\frac{\pi}{2}\right)\to\mathbb{R}) तथा (f(x)=\tan x) है, तो सही कथन चुनिए।

Question English:
If (f:\left[0,\frac{\pi}{2}\right)\to\mathbb{R}) and (f(x)=\tan x), choose the correct statement.

Options:
A. (f) एक-एक नहीं है / (f) is not one-one
B. (f) स्थिर है / (f) is constant
C. (f) ऋणात्मक मान देता है / (f) gives negative values
D. (f) एक-एक है / (f) is one-one

Correct Answer:
D. (f) एक-एक है / (f) is one-one

Explanation Hindi:
चरण 1: दिए गए अंतराल पर (\tan x) लगातार बढ़ता है। चरण 2: इसलिए दो अलग (x) के लिए समान (\tan x) नहीं मिलेगा। चरण 3: (\tan x) को सही सीमित अंतराल पर लेने से एक-एकता मिलती है।

Explanation English:
Step 1: On the given interval (\tan x) is strictly increasing. Step 2: Therefore two different (x)-values cannot have the same tangent value. Step 3: Taking (\tan x) on a suitable restricted interval gives injectivity.

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5. 18-23 sec: Reveal correct answer with green glow and subtle success sound.
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Question Expert Mathematics Relations and Functions One-one function Class 12 Level 22

फलन \(f:\mathbb{R}\to\mathbb{R}\) को (f(x)=x+1) यदि (x<0) और (f(x)=x^-2) यदि \(x\ge0\) से परिभाषित किया गया है। सही कथन क्या है?

The function \(f:\mathbb{R}\to\mathbb{R}\) is defined by (f(x)=x+1) if (x<0) and (f(x)=x^-2) if \(x\ge0\). What is the correct statement?

Explanation opens after your attempt

Correct Answer

D. (f) एक-एक नहीं है क्योंकि दो भागों के परास मिलते हैं(f) is not one-one because the ranges of the two pieces overlap

Step 1

Concept

In a piecewise function check the ranges of different pieces.

Step 2

Why this answer is correct

(f\left\(-\frac{1}{2}\right\)=\frac{1}{2}) and (f\left\(\frac{1}{\sqrt{2}}\right\)=\frac{1}{2}), while the inputs are different.

Step 3

Exam Tip

Same value from different pieces breaks injectivity. चरण 1: टुकड़ों में दिए फलन में अलग-अलग भागों के परास देखें। चरण 2: (f\left\(-\frac{1}{2}\right\)=\frac{1}{2}) और (f\left\(\frac{1}{\sqrt{2}}\right\)=\frac{1}{2}), जबकि दोनों आगत अलग हैं। चरण 3: अलग भागों से समान मान मिलना एक-एकता को तोड़ देता है।

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MCQ Data:
Class: Class 12
Subject: Mathematics
Chapter: Relations and Functions
Topic: One-one function
Difficulty: Expert
Level: 22

Question Hindi:
फलन (f:\mathbb{R}\to\mathbb{R}) को (f(x)=x+1) यदि (x<0) और (f(x)=x^2) यदि (x\ge0) से परिभाषित किया गया है। सही कथन क्या है?

Question English:
The function (f:\mathbb{R}\to\mathbb{R}) is defined by (f(x)=x+1) if (x<0) and (f(x)=x^2) if (x\ge0). What is the correct statement?

Options:
A. (f) एक-एक है / (f) is one-one
B. (f) फलन नहीं है / (f) is not a function
C. (f) हर जगह स्थिर है / (f) is constant everywhere
D. (f) एक-एक नहीं है क्योंकि दो भागों के परास मिलते हैं / (f) is not one-one because the ranges of the two pieces overlap

Correct Answer:
D. (f) एक-एक नहीं है क्योंकि दो भागों के परास मिलते हैं / (f) is not one-one because the ranges of the two pieces overlap

Explanation Hindi:
चरण 1: टुकड़ों में दिए फलन में अलग-अलग भागों के परास देखें। चरण 2: (f\left(-\frac{1}{2}\right)=\frac{1}{2}) और (f\left(\frac{1}{\sqrt{2}}\right)=\frac{1}{2}), जबकि दोनों आगत अलग हैं। चरण 3: अलग भागों से समान मान मिलना एक-एकता को तोड़ देता है।

Explanation English:
Step 1: In a piecewise function check the ranges of different pieces. Step 2: (f\left(-\frac{1}{2}\right)=\frac{1}{2}) and (f\left(\frac{1}{\sqrt{2}}\right)=\frac{1}{2}), while the inputs are different. Step 3: Same value from different pieces breaks injectivity.

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5. 18-23 sec: Reveal correct answer with green glow and subtle success sound.
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Question Expert Mathematics Relations and Functions One-one function Class 12 Level 22

यदि \(f:\mathbb{R}\to\mathbb{R}\) तथा (f(x)=\frac{x}{1+x^-2}) है, तो (f) के बारे में सही कथन क्या है?

If \(f:\mathbb{R}\to\mathbb{R}\) and (f(x)=\frac{x}{1+x^-2}), what is the correct statement about (f)?

Explanation opens after your attempt

Correct Answer

C. (f) एक-एक नहीं है क्योंकि (f(2)=f\left\(\frac{1}{2}\right\))(f) is not one-one because (f(2)=f\left\(\frac{1}{2}\right\))

Step 1

Concept

For a rational function check repeated values carefully.

Step 2

Why this answer is correct

(f(2)=\frac{2}{5}) and (f\left\(\frac{1}{2}\right\)=\frac{2}{5}), while \(2\neq\frac{1}{2}\).

Step 3

Exam Tip

Same value at two different inputs destroys injectivity. चरण 1: भिन्न वाले फलन में समान मानों की जांच करें। चरण 2: (f(2)=\frac{2}{5}) और (f\left\(\frac{1}{2}\right\)=\frac{2}{5}), जबकि \(2\neq\frac{1}{2}\)। चरण 3: दो अलग आगतों पर समान मान मिलने से एक-एकता नहीं रहती।

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MCQ Data:
Class: Class 12
Subject: Mathematics
Chapter: Relations and Functions
Topic: One-one function
Difficulty: Expert
Level: 22

Question Hindi:
यदि (f:\mathbb{R}\to\mathbb{R}) तथा (f(x)=\frac{x}{1+x^2}) है, तो (f) के बारे में सही कथन क्या है?

Question English:
If (f:\mathbb{R}\to\mathbb{R}) and (f(x)=\frac{x}{1+x^2}), what is the correct statement about (f)?

Options:
A. (f) एक-एक है / (f) is one-one
B. (f) परिभाषित नहीं है / (f) is not defined
C. (f) एक-एक नहीं है क्योंकि (f(2)=f\left(\frac{1}{2}\right)) / (f) is not one-one because (f(2)=f\left(\frac{1}{2}\right))
D. (f) स्थिर है / (f) is constant

Correct Answer:
C. (f) एक-एक नहीं है क्योंकि (f(2)=f\left(\frac{1}{2}\right)) / (f) is not one-one because (f(2)=f\left(\frac{1}{2}\right))

Explanation Hindi:
चरण 1: भिन्न वाले फलन में समान मानों की जांच करें। चरण 2: (f(2)=\frac{2}{5}) और (f\left(\frac{1}{2}\right)=\frac{2}{5}), जबकि (2\neq\frac{1}{2})। चरण 3: दो अलग आगतों पर समान मान मिलने से एक-एकता नहीं रहती।

Explanation English:
Step 1: For a rational function check repeated values carefully. Step 2: (f(2)=\frac{2}{5}) and (f\left(\frac{1}{2}\right)=\frac{2}{5}), while (2\neq\frac{1}{2}). Step 3: Same value at two different inputs destroys injectivity.

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1. 0-2 sec: Premium Muft Shiksha intro with logo glow and title: "MCQ Challenge".
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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.
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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Question Expert Mathematics Relations and Functions One-one function Class 12 Level 22

एक-एक फलन की रेखाचित्र द्वारा जांच के लिए कौन सा नियम सही है?

Which rule is correct for checking a one-one function from its graph?

Explanation opens after your attempt

Correct Answer

B. हर क्षैतिज रेखा रेखाचित्र को अधिकतम एक बार काटेEvery horizontal line cuts the graph at most once

Step 1

Concept

Injectivity means the same (y)-value should not come from two different (x)-values.

Step 2

Why this answer is correct

A horizontal line represents a fixed (y)-value.

Step 3

Exam Tip

If any horizontal line cuts the graph twice, the function is not one-one. चरण 1: एक-एकता का अर्थ है कि एक ही (y)-मान दो अलग (x)-मानों से न मिले। चरण 2: क्षैतिज रेखा एक निश्चित (y)-मान को दिखाती है। चरण 3: यदि कोई क्षैतिज रेखा दो बार काटती है, तो फलन एक-एक नहीं होगा।

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MCQ Data:
Class: Class 12
Subject: Mathematics
Chapter: Relations and Functions
Topic: One-one function
Difficulty: Expert
Level: 22

Question Hindi:
एक-एक फलन की रेखाचित्र द्वारा जांच के लिए कौन सा नियम सही है?

Question English:
Which rule is correct for checking a one-one function from its graph?

Options:
A. हर ऊर्ध्वाधर रेखा रेखाचित्र को दो बार काटे / Every vertical line cuts the graph twice
B. हर क्षैतिज रेखा रेखाचित्र को अधिकतम एक बार काटे / Every horizontal line cuts the graph at most once
C. हर क्षैतिज रेखा रेखाचित्र को कम से कम दो बार काटे / Every horizontal line cuts the graph at least twice
D. रेखाचित्र का वक्र होना जरूरी है / The graph must be curved

Correct Answer:
B. हर क्षैतिज रेखा रेखाचित्र को अधिकतम एक बार काटे / Every horizontal line cuts the graph at most once

Explanation Hindi:
चरण 1: एक-एकता का अर्थ है कि एक ही (y)-मान दो अलग (x)-मानों से न मिले। चरण 2: क्षैतिज रेखा एक निश्चित (y)-मान को दिखाती है। चरण 3: यदि कोई क्षैतिज रेखा दो बार काटती है, तो फलन एक-एक नहीं होगा।

Explanation English:
Step 1: Injectivity means the same (y)-value should not come from two different (x)-values. Step 2: A horizontal line represents a fixed (y)-value. Step 3: If any horizontal line cuts the graph twice, the function is not one-one.

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

किस स्थिति में \(f:A\to B\) का प्रतिलोम संबंध (f(A)\to A) एक फलन बनेगा?

In which case will the inverse relation of \(f:A\to B\) become a function from (f(A)) to (A)?

Explanation opens after your attempt

Correct Answer

A. जब (f) एक-एक होWhen (f) is one-one

Step 1

Concept

In the inverse relation each image is linked back to its original input.

Step 2

Why this answer is correct

If (f) is one-one, each image has only one original input.

Step 3

Exam Tip

Injectivity is the key condition for the inverse to be a function. चरण 1: प्रतिलोम में हर प्रतिबिंब को उसके मूल आगत से जोड़ा जाता है। चरण 2: यदि (f) एक-एक है, तो हर प्रतिबिंब का केवल एक मूल आगत होगा। चरण 3: प्रतिलोम को फलन बनाने के लिए एक-एकता जरूरी शर्त है।

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MCQ Data:
Class: Class 12
Subject: Mathematics
Chapter: Relations and Functions
Topic: One-one function
Difficulty: Expert
Level: 22

Question Hindi:
किस स्थिति में (f:A\to B) का प्रतिलोम संबंध (f(A)\to A) एक फलन बनेगा?

Question English:
In which case will the inverse relation of (f:A\to B) become a function from (f(A)) to (A)?

Options:
A. जब (f) एक-एक हो / When (f) is one-one
B. जब (f) स्थिर हो और (A) में कई तत्व हों / When (f) is constant and (A) has many elements
C. जब (B) रिक्त हो / When (B) is empty
D. जब (f) परिभाषित न हो / When (f) is not defined

Correct Answer:
A. जब (f) एक-एक हो / When (f) is one-one

Explanation Hindi:
चरण 1: प्रतिलोम में हर प्रतिबिंब को उसके मूल आगत से जोड़ा जाता है। चरण 2: यदि (f) एक-एक है, तो हर प्रतिबिंब का केवल एक मूल आगत होगा। चरण 3: प्रतिलोम को फलन बनाने के लिए एक-एकता जरूरी शर्त है।

Explanation English:
Step 1: In the inverse relation each image is linked back to its original input. Step 2: If (f) is one-one, each image has only one original input. Step 3: Injectivity is the key condition for the inverse to be a function.

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

यदि \(f:\mathbb{Z}\to\mathbb{Z}\) तथा (f(n)=|n|) है, तो (f) एक-एक क्यों नहीं है?

If \(f:\mathbb{Z}\to\mathbb{Z}\) and (f(n)=|n|), why is (f) not one-one?

Explanation opens after your attempt

Correct Answer

C. क्योंकि (f(2)=f(-2))Because (f(2)=f(-2))

Step 1

Concept

Absolute value gives the same value for opposite integers.

Step 2

Why this answer is correct

\(2\neq-2\), but (|2|=2) and (|-2|=2).

Step 3

Exam Tip

One repeated image is enough to break injectivity. चरण 1: निरपेक्ष मान विपरीत पूर्णांकों को समान मान देता है। चरण 2: \(2\neq-2\), लेकिन (|2|=2) और (|-2|=2)। चरण 3: एक समान प्रतिबिंब मिलते ही एक-एकता टूट जाती है।

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MCQ Data:
Class: Class 12
Subject: Mathematics
Chapter: Relations and Functions
Topic: One-one function
Difficulty: Expert
Level: 22

Question Hindi:
यदि (f:\mathbb{Z}\to\mathbb{Z}) तथा (f(n)=|n|) है, तो (f) एक-एक क्यों नहीं है?

Question English:
If (f:\mathbb{Z}\to\mathbb{Z}) and (f(n)=|n|), why is (f) not one-one?

Options:
A. क्योंकि (f(0)) परिभाषित नहीं है / Because (f(0)) is not defined
B. क्योंकि हर मान ऋणात्मक है / Because every value is negative
C. क्योंकि (f(2)=f(-2)) / Because (f(2)=f(-2))
D. क्योंकि (f(1)\neq f(-1)) / Because (f(1)\neq f(-1))

Correct Answer:
C. क्योंकि (f(2)=f(-2)) / Because (f(2)=f(-2))

Explanation Hindi:
चरण 1: निरपेक्ष मान विपरीत पूर्णांकों को समान मान देता है। चरण 2: (2\neq-2), लेकिन (|2|=2) और (|-2|=2)। चरण 3: एक समान प्रतिबिंब मिलते ही एक-एकता टूट जाती है।

Explanation English:
Step 1: Absolute value gives the same value for opposite integers. Step 2: (2\neq-2), but (|2|=2) and (|-2|=2). Step 3: One repeated image is enough to break injectivity.

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

यदि \(f:\mathbb{R}\to\mathbb{R}\) तथा (f(x)=\ln\(x^2+1\)) है, तो (f) के लिए सही कथन चुनिए।

If \(f:\mathbb{R}\to\mathbb{R}\) and (f(x)=\ln\(x^2+1\)), choose the correct statement about (f).

Explanation opens after your attempt

Correct Answer

D. (f) एक-एक नहीं है क्योंकि (f(1)=f(-1))(f) is not one-one because (f(1)=f(-1))

Step 1

Concept

\(x^2\) gives the same value for (x) and (-x).

Step 2

Why this answer is correct

Hence (f(1)=\ln2) and (f(-1)=\ln2), while \(1\neq-1\).

Step 3

Exam Tip

When an even power appears inside, check injectivity carefully. चरण 1: \(x^2\) में (x) और (-x) समान मान देते हैं। चरण 2: इसलिए (f(1)=\ln2) और (f(-1)=\ln2), जबकि \(1\neq-1\)। चरण 3: अंदर सम घात हो तो एक-एकता को खास ध्यान से जांचें।

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MCQ Data:
Class: Class 12
Subject: Mathematics
Chapter: Relations and Functions
Topic: One-one function
Difficulty: Expert
Level: 22

Question Hindi:
यदि (f:\mathbb{R}\to\mathbb{R}) तथा (f(x)=\ln(x^2+1)) है, तो (f) के लिए सही कथन चुनिए।

Question English:
If (f:\mathbb{R}\to\mathbb{R}) and (f(x)=\ln(x^2+1)), choose the correct statement about (f).

Options:
A. (f) एक-एक है / (f) is one-one
B. (f) परिभाषित नहीं है / (f) is not defined
C. (f) केवल धनात्मक आगतों पर परिभाषित है / (f) is defined only for positive inputs
D. (f) एक-एक नहीं है क्योंकि (f(1)=f(-1)) / (f) is not one-one because (f(1)=f(-1))

Correct Answer:
D. (f) एक-एक नहीं है क्योंकि (f(1)=f(-1)) / (f) is not one-one because (f(1)=f(-1))

Explanation Hindi:
चरण 1: (x^2) में (x) और (-x) समान मान देते हैं। चरण 2: इसलिए (f(1)=\ln2) और (f(-1)=\ln2), जबकि (1\neq-1)। चरण 3: अंदर सम घात हो तो एक-एकता को खास ध्यान से जांचें।

Explanation English:
Step 1: (x^2) gives the same value for (x) and (-x). Step 2: Hence (f(1)=\ln2) and (f(-1)=\ln2), while (1\neq-1). Step 3: When an even power appears inside, check injectivity carefully.

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

यदि \(f:\mathbb{R}\to\mathbb{R}\) तथा (f(x)=x^-3-3x) है, तो (f) के बारे में सही कथन चुनिए।

If \(f:\mathbb{R}\to\mathbb{R}\) and (f(x)=x^-3-3x), choose the correct statement about (f).

Explanation opens after your attempt

Correct Answer

C. (f) एक-एक नहीं है क्योंकि (f(0)=f\(\sqrt{3}\))(f) is not one-one because (f(0)=f\(\sqrt{3}\))

Step 1

Concept

To disprove injectivity it is enough to show two different inputs with the same output.

Step 2

Why this answer is correct

(f(0)=0) and (f\(\sqrt{3}\)=3\sqrt{3}-3\sqrt{3}=0), while \(0\neq\sqrt{3}\).

Step 3

Exam Tip

Do not decide by degree alone; test special values. चरण 1: एक-एकता को गलत सिद्ध करने के लिए दो अलग आगतों का समान मान दिखाना पर्याप्त है। चरण 2: (f(0)=0) और (f\(\sqrt{3}\)=3\sqrt{3}-3\sqrt{3}=0), जबकि \(0\neq\sqrt{3}\)। चरण 3: बहुपद में केवल घात देखकर निर्णय न करें, विशेष मानों से जांच करें।

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MCQ Data:
Class: Class 12
Subject: Mathematics
Chapter: Relations and Functions
Topic: One-one function
Difficulty: Expert
Level: 22

Question Hindi:
यदि (f:\mathbb{R}\to\mathbb{R}) तथा (f(x)=x^3-3x) है, तो (f) के बारे में सही कथन चुनिए।

Question English:
If (f:\mathbb{R}\to\mathbb{R}) and (f(x)=x^3-3x), choose the correct statement about (f).

Options:
A. (f) एक-एक है / (f) is one-one
B. (f) केवल आच्छादक है इसलिए एक-एक भी है / (f) is onto only so it is also one-one
C. (f) एक-एक नहीं है क्योंकि (f(0)=f(\sqrt{3})) / (f) is not one-one because (f(0)=f(\sqrt{3}))
D. (f) किसी भी वास्तविक संख्या पर परिभाषित नहीं है / (f) is not defined for any real number

Correct Answer:
C. (f) एक-एक नहीं है क्योंकि (f(0)=f(\sqrt{3})) / (f) is not one-one because (f(0)=f(\sqrt{3}))

Explanation Hindi:
चरण 1: एक-एकता को गलत सिद्ध करने के लिए दो अलग आगतों का समान मान दिखाना पर्याप्त है। चरण 2: (f(0)=0) और (f(\sqrt{3})=3\sqrt{3}-3\sqrt{3}=0), जबकि (0\neq\sqrt{3})। चरण 3: बहुपद में केवल घात देखकर निर्णय न करें, विशेष मानों से जांच करें।

Explanation English:
Step 1: To disprove injectivity it is enough to show two different inputs with the same output. Step 2: (f(0)=0) and (f(\sqrt{3})=3\sqrt{3}-3\sqrt{3}=0), while (0\neq\sqrt{3}). Step 3: Do not decide by degree alone; test special values.

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1. 0-2 sec: Premium Muft Shiksha intro with logo glow and title: "MCQ Challenge".
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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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7. 30-35 sec: CTA: "More free MCQs ke liye visit karein MuftShiksha.com".

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- Formula/Math/Chemistry text must stay exactly readable; use LaTeX-style clean rendering where needed.

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Question Hard Mathematics Relations and Functions One-one function Class 12 Level 24

यदि \(f:\mathbb{R}\to\mathbb{R}\) तथा (f(x)=x^-3+kx) है, तो (f) के एक-एक होने के लिए (k) की कौन सी शर्त पर्याप्त है?

If \(f:\mathbb{R}\to\mathbb{R}\) and (f(x)=x^-3+kx), which condition on (k) is sufficient for (f) to be one-one?

Explanation opens after your attempt

Correct Answer

A. \(k\ge0\)

Step 1

Concept

For injectivity it is useful to check whether the function keeps moving in one direction.

Step 2

Why this answer is correct

When \(k\ge0\), both \(x^3\) and (kx) contribute an increasing effect, so different (x)-values give different outputs.

Step 3

Exam Tip

In parameter-based questions always check how the parameter changes increasing or decreasing behavior. चरण 1: एक-एकता के लिए यह देखना उपयोगी है कि फलन लगातार एक ही दिशा में बढ़ रहा है या नहीं। चरण 2: जब \(k\ge0\), तब \(x^3\) और (kx) दोनों बढ़ने वाले प्रभाव देते हैं, इसलिए अलग-अलग (x) के लिए अलग-अलग मान मिलते हैं। चरण 3: ऐसे प्रश्नों में पैरामीटर का मान फलन के बढ़ने या घटने को कैसे बदलता है, यह जरूर जांचें।

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

Question Hindi:
यदि (f:\mathbb{R}\to\mathbb{R}) तथा (f(x)=x^3+kx) है, तो (f) के एक-एक होने के लिए (k) की कौन सी शर्त पर्याप्त है?

Question English:
If (f:\mathbb{R}\to\mathbb{R}) and (f(x)=x^3+kx), which condition on (k) is sufficient for (f) to be one-one?

Options:
A. (k\ge0)
B. (k<0)
C. (k=-3)
D. (k) कोई भी वास्तविक संख्या हो / (k) may be any real number

Correct Answer:
A. (k\ge0)

Explanation Hindi:
चरण 1: एक-एकता के लिए यह देखना उपयोगी है कि फलन लगातार एक ही दिशा में बढ़ रहा है या नहीं। चरण 2: जब (k\ge0), तब (x^3) और (kx) दोनों बढ़ने वाले प्रभाव देते हैं, इसलिए अलग-अलग (x) के लिए अलग-अलग मान मिलते हैं। चरण 3: ऐसे प्रश्नों में पैरामीटर का मान फलन के बढ़ने या घटने को कैसे बदलता है, यह जरूर जांचें।

Explanation English:
Step 1: For injectivity it is useful to check whether the function keeps moving in one direction. Step 2: When (k\ge0), both (x^3) and (kx) contribute an increasing effect, so different (x)-values give different outputs. Step 3: In parameter-based questions always check how the parameter changes increasing or decreasing behavior.

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Question Hard Mathematics Relations and Functions One-one function Class 12 Level 24

एक-एक फलन की पहचान के लिए रेखाचित्र से जुड़ा सही नियम कौन सा है?

Explanation opens after your attempt

Correct Answer

A. हर क्षैतिज रेखा रेखाचित्र को अधिकतम एक बिंदु पर काटेEvery horizontal line cuts the graph at at most one point

Step 1

Concept

Injectivity means the same (y)-value must not occur at two different (x)-values.

Step 2

Why this answer is correct

A horizontal line represents a fixed (y)-value.

Step 3

Exam Tip

Hence the horizontal line test is a quick way to check one-one nature. चरण 1: एक-एकता में एक ही (y)-मान दो अलग (x)-मानों पर नहीं आना चाहिए। चरण 2: क्षैतिज रेखा समान (y)-मान को दिखाती है। चरण 3: इसलिए क्षैतिज रेखा परीक्षण एक-एकता जांचने का तेज तरीका है।

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

Question Hindi:
एक-एक फलन की पहचान के लिए रेखाचित्र से जुड़ा सही नियम कौन सा है?

Question English:
Which graph-related rule correctly identifies a one-one function?

Options:
A. हर क्षैतिज रेखा रेखाचित्र को अधिकतम एक बिंदु पर काटे / Every horizontal line cuts the graph at at most one point
B. हर ऊर्ध्वाधर रेखा रेखाचित्र को दो बिंदुओं पर काटे / Every vertical line cuts the graph at two points
C. हर क्षैतिज रेखा रेखाचित्र को कम से कम दो बिंदुओं पर काटे / Every horizontal line cuts the graph at at least two points
D. रेखाचित्र का परवलय होना जरूरी है / The graph must be a parabola

Correct Answer:
A. हर क्षैतिज रेखा रेखाचित्र को अधिकतम एक बिंदु पर काटे / Every horizontal line cuts the graph at at most one point

Explanation Hindi:
चरण 1: एक-एकता में एक ही (y)-मान दो अलग (x)-मानों पर नहीं आना चाहिए। चरण 2: क्षैतिज रेखा समान (y)-मान को दिखाती है। चरण 3: इसलिए क्षैतिज रेखा परीक्षण एक-एकता जांचने का तेज तरीका है।

Explanation English:
Step 1: Injectivity means the same (y)-value must not occur at two different (x)-values. Step 2: A horizontal line represents a fixed (y)-value. Step 3: Hence the horizontal line test is a quick way to check one-one nature.

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Question Hard Mathematics Relations and Functions One-one function Class 12 Level 24

यदि \(f=\{(1,4),(2,5),(3,5)\}\) है, जहां \(f:{1,2,3}\to{4,5,6}\), तो सही कथन चुनिए।

If \(f=\{(1,4),(2,5),(3,5)\}\), where \(f:{1,2,3}\to{4,5,6}\), choose the correct statement.

Explanation opens after your attempt

Correct Answer

A. (f) एक-एक नहीं है(f) is not one-one

Step 1

Concept

Injectivity requires no repetition among images.

Step 2

Why this answer is correct

Both (2) and (3) have image (5).

Step 3

Exam Tip

In ordered-pair questions quickly check the second components. चरण 1: एक-एकता में प्रतिबिंबों की पुनरावृत्ति नहीं होनी चाहिए। चरण 2: (2) और (3) दोनों का प्रतिबिंब (5) है। चरण 3: क्रमित युग्मों में दूसरे अवयवों को देखकर जल्दी जांच करें।

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

Question Hindi:
यदि (f={(1,4),(2,5),(3,5)}) है, जहां (f:{1,2,3}\to{4,5,6}), तो सही कथन चुनिए।

Question English:
If (f={(1,4),(2,5),(3,5)}), where (f:{1,2,3}\to{4,5,6}), choose the correct statement.

Options:
A. (f) एक-एक नहीं है / (f) is not one-one
B. (f) एक-एक है / (f) is one-one
C. (f) फलन नहीं है / (f) is not a function
D. (f) का प्रांत रिक्त है / The domain of (f) is empty

Correct Answer:
A. (f) एक-एक नहीं है / (f) is not one-one

Explanation Hindi:
चरण 1: एक-एकता में प्रतिबिंबों की पुनरावृत्ति नहीं होनी चाहिए। चरण 2: (2) और (3) दोनों का प्रतिबिंब (5) है। चरण 3: क्रमित युग्मों में दूसरे अवयवों को देखकर जल्दी जांच करें।

Explanation English:
Step 1: Injectivity requires no repetition among images. Step 2: Both (2) and (3) have image (5). Step 3: In ordered-pair questions quickly check the second components.

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Question Hard Mathematics Relations and Functions One-one function Class 12 Level 24

फलन \(f:\mathbb{R}\to\mathbb{R}\) को (f(x)=x^-2) यदि (x<0) और (f(x)=x+1) यदि \(x\ge0\) से परिभाषित किया गया है। (f) के लिए सही कथन क्या है?

The function \(f:\mathbb{R}\to\mathbb{R}\) is defined by (f(x)=x^-2) if (x<0) and (f(x)=x+1) if \(x\ge0\). What is the correct statement about (f)?

Explanation opens after your attempt

Correct Answer

A. (f) एक-एक नहीं है(f) is not one-one

Step 1

Concept

The ranges of the two parts can overlap.

Step 2

Why this answer is correct

(f(-2)=4) and (f(3)=4), while \(-2\neq3\).

Step 3

Exam Tip

In a piecewise function same value from different pieces breaks injectivity. चरण 1: दोनों भागों के परास में समान मान आ सकते हैं। चरण 2: (f(-2)=4) और (f(3)=4), जबकि \(-2\neq3\)। चरण 3: टुकड़ों वाले फलन में अलग भागों से समान मान मिलना एक-एकता तोड़ देता है।

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

Question Hindi:
फलन (f:\mathbb{R}\to\mathbb{R}) को (f(x)=x^2) यदि (x<0) और (f(x)=x+1) यदि (x\ge0) से परिभाषित किया गया है। (f) के लिए सही कथन क्या है?

Question English:
The function (f:\mathbb{R}\to\mathbb{R}) is defined by (f(x)=x^2) if (x<0) and (f(x)=x+1) if (x\ge0). What is the correct statement about (f)?

Options:
A. (f) एक-एक नहीं है / (f) is not one-one
B. (f) एक-एक है / (f) is one-one
C. (f) परिभाषित नहीं है / (f) is not defined
D. (f) का परास रिक्त है / The range of (f) is empty

Correct Answer:
A. (f) एक-एक नहीं है / (f) is not one-one

Explanation Hindi:
चरण 1: दोनों भागों के परास में समान मान आ सकते हैं। चरण 2: (f(-2)=4) और (f(3)=4), जबकि (-2\neq3)। चरण 3: टुकड़ों वाले फलन में अलग भागों से समान मान मिलना एक-एकता तोड़ देता है।

Explanation English:
Step 1: The ranges of the two parts can overlap. Step 2: (f(-2)=4) and (f(3)=4), while (-2\neq3). Step 3: In a piecewise function same value from different pieces breaks injectivity.

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Question Hard Mathematics Relations and Functions One-one function Class 12 Level 24

यदि कोई फलन \(f:A\to B\) एक-एक है, तो उसके प्रतिलोम संबंध के बारे में सही कथन क्या है?

If a function \(f:A\to B\) is one-one, what is the correct statement about its inverse relation?

Explanation opens after your attempt

Correct Answer

A. प्रतिलोम संबंध (f(A)) से (A) तक फलन होता हैThe inverse relation is a function from (f(A)) to (A)

Step 1

Concept

In a one-one function every image has only one original input.

Step 2

Why this answer is correct

So when reversed, each element of (f(A)) gets exactly one element of (A).

Step 3

Exam Tip

Injectivity is the key condition for an inverse to be a function. चरण 1: एक-एक फलन में हर प्रतिबिंब का केवल एक मूल आगत होता है। चरण 2: इसलिए उल्टा करने पर (f(A)) के हर तत्व को (A) का एक ही तत्व मिलेगा। चरण 3: प्रतिलोम फलन के लिए एक-एकता मुख्य शर्त है।

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

Question Hindi:
यदि कोई फलन (f:A\to B) एक-एक है, तो उसके प्रतिलोम संबंध के बारे में सही कथन क्या है?

Question English:
If a function (f:A\to B) is one-one, what is the correct statement about its inverse relation?

Options:
A. प्रतिलोम संबंध (f(A)) से (A) तक फलन होता है / The inverse relation is a function from (f(A)) to (A)
B. प्रतिलोम संबंध कभी फलन नहीं होता / The inverse relation is never a function
C. प्रतिलोम संबंध केवल तब बनता है जब (A) रिक्त हो / The inverse relation exists only when (A) is empty
D. प्रतिलोम संबंध हमेशा स्थिर होता है / The inverse relation is always constant

Correct Answer:
A. प्रतिलोम संबंध (f(A)) से (A) तक फलन होता है / The inverse relation is a function from (f(A)) to (A)

Explanation Hindi:
चरण 1: एक-एक फलन में हर प्रतिबिंब का केवल एक मूल आगत होता है। चरण 2: इसलिए उल्टा करने पर (f(A)) के हर तत्व को (A) का एक ही तत्व मिलेगा। चरण 3: प्रतिलोम फलन के लिए एक-एकता मुख्य शर्त है।

Explanation English:
Step 1: In a one-one function every image has only one original input. Step 2: So when reversed, each element of (f(A)) gets exactly one element of (A). Step 3: Injectivity is the key condition for an inverse to be a function.

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Question Hard Mathematics Relations and Functions One-one function Class 12 Level 24

यदि \(g\circ f\) एक-एक है, तो (g) के बारे में कौन सा कथन सही है?

If \(g\circ f\) is one-one, which statement about (g) is correct?

Explanation opens after your attempt

Correct Answer

A. (g) का एक-एक होना जरूरी नहीं है(g) need not be one-one

Step 1

Concept

\(g\circ f\) only uses the behavior of (g) on the range of (f).

Step 2

Why this answer is correct

(g) may fail to be one-one on its full domain but still behave distinctly on the range of (f).

Step 3

Exam Tip

In composition questions distinguish the whole domain from the used part. चरण 1: \(g\circ f\) केवल (f) के परास पर (g) के व्यवहार को देखता है। चरण 2: (g) पूरे प्रांत पर एक-एक न हो, फिर भी (f) के परास पर अलग मान दे सकता है। चरण 3: संयोजन वाले प्रश्नों में पूरे प्रांत और प्रयुक्त भाग में अंतर समझें।

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

Question Hindi:
यदि (g\circ f) एक-एक है, तो (g) के बारे में कौन सा कथन सही है?

Question English:
If (g\circ f) is one-one, which statement about (g) is correct?

Options:
A. (g) का एक-एक होना जरूरी नहीं है / (g) need not be one-one
B. (g) हमेशा एक-एक होगा / (g) will always be one-one
C. (g) हमेशा स्थिर होगा / (g) will always be constant
D. (g) फलन नहीं हो सकता / (g) cannot be a function

Correct Answer:
A. (g) का एक-एक होना जरूरी नहीं है / (g) need not be one-one

Explanation Hindi:
चरण 1: (g\circ f) केवल (f) के परास पर (g) के व्यवहार को देखता है। चरण 2: (g) पूरे प्रांत पर एक-एक न हो, फिर भी (f) के परास पर अलग मान दे सकता है। चरण 3: संयोजन वाले प्रश्नों में पूरे प्रांत और प्रयुक्त भाग में अंतर समझें।

Explanation English:
Step 1: (g\circ f) only uses the behavior of (g) on the range of (f). Step 2: (g) may fail to be one-one on its full domain but still behave distinctly on the range of (f). Step 3: In composition questions distinguish the whole domain from the used part.

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Question Hard Mathematics Relations and Functions One-one function Class 12 Level 24

यदि \(f:\mathbb{R}\to\mathbb{R}\) तथा (f(x)=\frac{x}{1+x^-2}) है, तो (f) के बारे में सही कथन चुनिए।

If \(f:\mathbb{R}\to\mathbb{R}\) and (f(x)=\frac{x}{1+x^-2}), choose the correct statement about (f).

Explanation opens after your attempt

Correct Answer

A. (f) एक-एक नहीं है क्योंकि (f(2)=f\left\(\frac{1}{2}\right\))(f) is not one-one because (f(2)=f\left\(\frac{1}{2}\right\))

Step 1

Concept

Injectivity can be checked by finding repeated images.

Step 2

Why this answer is correct

(f(2)=\frac{2}{5}) and (f\left\(\frac{1}{2}\right\)=\frac{2}{5}), while \(2\neq \frac{1}{2}\).

Step 3

Exam Tip

Do not assume a rational expression is one-one without testing. चरण 1: समान प्रतिबिंब खोजकर एक-एकता की जांच की जा सकती है। चरण 2: (f(2)=\frac{2}{5}) और (f\left\(\frac{1}{2}\right\)=\frac{2}{5}) है, जबकि \(2\neq \frac{1}{2}\)। चरण 3: भिन्न रूप देखकर तुरंत एक-एक न मानें।

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

Question Hindi:
यदि (f:\mathbb{R}\to\mathbb{R}) तथा (f(x)=\frac{x}{1+x^2}) है, तो (f) के बारे में सही कथन चुनिए।

Question English:
If (f:\mathbb{R}\to\mathbb{R}) and (f(x)=\frac{x}{1+x^2}), choose the correct statement about (f).

Options:
A. (f) एक-एक नहीं है क्योंकि (f(2)=f\left(\frac{1}{2}\right)) / (f) is not one-one because (f(2)=f\left(\frac{1}{2}\right))
B. (f) एक-एक है / (f) is one-one
C. (f) परिभाषित नहीं है / (f) is not defined
D. (f) स्थिर है / (f) is constant

Correct Answer:
A. (f) एक-एक नहीं है क्योंकि (f(2)=f\left(\frac{1}{2}\right)) / (f) is not one-one because (f(2)=f\left(\frac{1}{2}\right))

Explanation Hindi:
चरण 1: समान प्रतिबिंब खोजकर एक-एकता की जांच की जा सकती है। चरण 2: (f(2)=\frac{2}{5}) और (f\left(\frac{1}{2}\right)=\frac{2}{5}) है, जबकि (2\neq \frac{1}{2})। चरण 3: भिन्न रूप देखकर तुरंत एक-एक न मानें।

Explanation English:
Step 1: Injectivity can be checked by finding repeated images. Step 2: (f(2)=\frac{2}{5}) and (f\left(\frac{1}{2}\right)=\frac{2}{5}), while (2\neq \frac{1}{2}). Step 3: Do not assume a rational expression is one-one without testing.

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Question Hard Mathematics Relations and Functions One-one function Class 12 Level 24

यदि \(f:\mathbb{R}-{0}\to\mathbb{R}\) तथा (f(x)=\frac{1}{x}) है, तो (f) एक-एक है या नहीं?

If \(f:\mathbb{R}-{0}\to\mathbb{R}\) and (f(x)=\frac{1}{x}), is (f) one-one?

Explanation opens after your attempt

Correct Answer

A. हाँYes

Step 1

Concept

Zero is removed from the domain, so the function is well-defined.

Step 2

Why this answer is correct

From \(\frac{1}{x_1}=\frac{1}{x_2}\), we get \(x_1=x_2\).

Step 3

Exam Tip

Negative and positive parts together do not destroy injectivity here. चरण 1: प्रांत में शून्य को हटाया गया है, इसलिए फलन ठीक से परिभाषित है। चरण 2: \(\frac{1}{x_1}=\frac{1}{x_2}\) से \(x_1=x_2\) मिलता है। चरण 3: ऋणात्मक और धनात्मक दोनों भाग मिलकर भी यहां एक-एकता नहीं बिगाड़ते।

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

Question Hindi:
यदि (f:\mathbb{R}-{0}\to\mathbb{R}) तथा (f(x)=\frac{1}{x}) है, तो (f) एक-एक है या नहीं?

Question English:
If (f:\mathbb{R}-{0}\to\mathbb{R}) and (f(x)=\frac{1}{x}), is (f) one-one?

Options:
A. हाँ / Yes
B. नहीं / No
C. केवल (x>0) के लिए / Only for (x>0)
D. केवल (x<0) के लिए / Only for (x<0)

Correct Answer:
A. हाँ / Yes

Explanation Hindi:
चरण 1: प्रांत में शून्य को हटाया गया है, इसलिए फलन ठीक से परिभाषित है। चरण 2: (\frac{1}{x_1}=\frac{1}{x_2}) से (x_1=x_2) मिलता है। चरण 3: ऋणात्मक और धनात्मक दोनों भाग मिलकर भी यहां एक-एकता नहीं बिगाड़ते।

Explanation English:
Step 1: Zero is removed from the domain, so the function is well-defined. Step 2: From (\frac{1}{x_1}=\frac{1}{x_2}), we get (x_1=x_2). Step 3: Negative and positive parts together do not destroy injectivity here.

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Question Hard Mathematics Relations and Functions Functions Class 12 Level 21

यदि \(f:\mathbb{R}\to\mathbb{R}\) और (f(x)=x^-3-3x), तो (f) एकैकी क्यों नहीं है?

If \(f:\mathbb{R}\to\mathbb{R}\) and (f(x)=x^-3-3x), why is (f) not one-one?

Explanation opens after your attempt

Correct Answer

A. क्योंकि (f(0)=f\(\sqrt{3}\))Because (f(0)=f\(\sqrt{3}\))

Step 1

Concept

(f(0)=0).

Step 2

Why this answer is correct

(f\(\sqrt{3}\)=\(\sqrt{3}\)³-3\sqrt{3}=3\sqrt{3}-3\sqrt{3}=0), while \(0\neq \sqrt{3}\).

Step 3

Exam Tip

If two distinct inputs give the same image, the function is not one-one. चरण 1: (f(0)=0) है। चरण 2: (f\(\sqrt{3}\)=\(\sqrt{3}\)³-3\sqrt{3}=3\sqrt{3}-3\sqrt{3}=0), जबकि \(0\neq \sqrt{3}\)। चरण 3: दो अलग इनपुट की समान छवि मिलने पर फलन एकैकी नहीं होता।

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MCQ Data:
Class: Class 12
Subject: Mathematics
Chapter: Relations and Functions
Topic: Functions
Difficulty: Hard
Level: 21

Question Hindi:
यदि (f:\mathbb{R}\to\mathbb{R}) और (f(x)=x^3-3x), तो (f) एकैकी क्यों नहीं है?

Question English:
If (f:\mathbb{R}\to\mathbb{R}) and (f(x)=x^3-3x), why is (f) not one-one?

Options:
A. क्योंकि (f(0)=f(\sqrt{3})) / Because (f(0)=f(\sqrt{3}))
B. क्योंकि (f(1)=f(2)) / Because (f(1)=f(2))
C. क्योंकि (f(x)) परिभाषित नहीं है / Because (f(x)) is not defined
D. क्योंकि परास खाली है / Because the range is empty

Correct Answer:
A. क्योंकि (f(0)=f(\sqrt{3})) / Because (f(0)=f(\sqrt{3}))

Explanation Hindi:
चरण 1: (f(0)=0) है। चरण 2: (f(\sqrt{3})=(\sqrt{3})^3-3\sqrt{3}=3\sqrt{3}-3\sqrt{3}=0), जबकि (0\neq \sqrt{3})। चरण 3: दो अलग इनपुट की समान छवि मिलने पर फलन एकैकी नहीं होता।

Explanation English:
Step 1: (f(0)=0). Step 2: (f(\sqrt{3})=(\sqrt{3})^3-3\sqrt{3}=3\sqrt{3}-3\sqrt{3}=0), while (0\neq \sqrt{3}). Step 3: If two distinct inputs give the same image, the function is not one-one.

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Question Hard Mathematics Relations and Functions Functions Class 12 Level 21

यदि \(f:\mathbb{R}\to\mathbb{R}\) और (f(x)=x^-2), तो (f) एकैकी क्यों नहीं है?

If \(f:\mathbb{R}\to\mathbb{R}\) and (f(x)=x^-2), why is (f) not one-one?

Explanation opens after your attempt

Correct Answer

A. क्योंकि (f(1)=f(-1))Because (f(1)=f(-1))

Step 1

Concept

In a one-one function, distinct inputs must have distinct images.

Step 2

Why this answer is correct

Here \(1\neq -1\), but (f(1)=1) and (f(-1)=1).

Step 3

Exam Tip

Once two different inputs give the same output, the function is not one-one. चरण 1: एकैकी फलन में अलग इनपुट की छवियाँ अलग होनी चाहिए। चरण 2: यहाँ \(1\neq -1\), लेकिन (f(1)=1) और (f(-1)=1)। चरण 3: एक ही छवि देने वाले दो अलग इनपुट मिलते ही फलन एकैकी नहीं रहता।

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MCQ Data:
Class: Class 12
Subject: Mathematics
Chapter: Relations and Functions
Topic: Functions
Difficulty: Hard
Level: 21

Question Hindi:
यदि (f:\mathbb{R}\to\mathbb{R}) और (f(x)=x^2), तो (f) एकैकी क्यों नहीं है?

Question English:
If (f:\mathbb{R}\to\mathbb{R}) and (f(x)=x^2), why is (f) not one-one?

Options:
A. क्योंकि (f(1)=f(-1)) / Because (f(1)=f(-1))
B. क्योंकि (f(0)=1) / Because (f(0)=1)
C. क्योंकि (f(x)) हमेशा ऋणात्मक है / Because (f(x)) is always negative
D. क्योंकि (f(x)) परिभाषित नहीं है / Because (f(x)) is not defined

Correct Answer:
A. क्योंकि (f(1)=f(-1)) / Because (f(1)=f(-1))

Explanation Hindi:
चरण 1: एकैकी फलन में अलग इनपुट की छवियाँ अलग होनी चाहिए। चरण 2: यहाँ (1\neq -1), लेकिन (f(1)=1) और (f(-1)=1)। चरण 3: एक ही छवि देने वाले दो अलग इनपुट मिलते ही फलन एकैकी नहीं रहता।

Explanation English:
Step 1: In a one-one function, distinct inputs must have distinct images. Step 2: Here (1\neq -1), but (f(1)=1) and (f(-1)=1). Step 3: Once two different inputs give the same output, the function is not one-one.

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5. 18-23 sec: Reveal correct answer with green glow and subtle success sound.
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Search Class 10 Questions

यदि (f:\(0,\infty\)\to\mathbb{R}) तथा (f(x)=\log x) है, तो (f) के बारे में सही निष्कर्ष क्या है?

Concept

Why this answer is correct

Exam Tip

यदि \(f:\mathbb{R}\to\mathbb{R}\) तथा (f(x)=x-3-6x-2+9x) है, तो कौन सा विकल्प (f) के एक-एक न होने को सिद्ध करता है?

Concept

Why this answer is correct

Exam Tip

यदि \(f:\mathbb{R}\to\mathbb{R}\) तथा (f(x)=2x-3-5) है, तो (f) के लिए सही कथन चुनिए।

Concept

Why this answer is correct

Exam Tip

यदि \(g\circ f\) एक-एक है, तो (g) के बारे में कौन सा कथन सबसे सटीक है?

Concept

Why this answer is correct

Exam Tip

यदि \(f:A\to B\) एक-एक है, तो निम्न में से कौन सा कथन इसके बराबर अर्थ रखता है?

Concept

Why this answer is correct

Exam Tip

यदि \(f:\mathbb{R}\to\mathbb{R}\) तथा (f(x)=x-5-x) है, तो कौन सा विकल्प (f) के एक-एक न होने को सिद्ध करता है?

Concept

Why this answer is correct

Exam Tip

यदि \(f:\left[0,\frac{\pi}{2}\right\)\to\mathbb{R}) तथा (f(x)=\tan x) है, तो सही कथन चुनिए।

Concept

Why this answer is correct

Exam Tip

फलन \(f:\mathbb{R}\to\mathbb{R}\) को (f(x)=x+1) यदि (x<0) और (f(x)=x-2) यदि \(x\ge0\) से परिभाषित किया गया है। सही कथन क्या है?

Concept

Why this answer is correct

Exam Tip

यदि \(f:\mathbb{R}\to\mathbb{R}\) तथा (f(x)=\frac{x}{1+x-2}) है, तो (f) के बारे में सही कथन क्या है?

Concept

Why this answer is correct

Exam Tip

एक-एक फलन की रेखाचित्र द्वारा जांच के लिए कौन सा नियम सही है?

Concept

Why this answer is correct

Exam Tip

किस स्थिति में \(f:A\to B\) का प्रतिलोम संबंध (f(A)\to A) एक फलन बनेगा?

Concept

Why this answer is correct

Exam Tip

यदि \(f:\mathbb{Z}\to\mathbb{Z}\) तथा (f(n)=|n|) है, तो (f) एक-एक क्यों नहीं है?

Concept

Why this answer is correct

Exam Tip

यदि \(f:\mathbb{R}\to\mathbb{R}\) तथा (f(x)=\ln\(x^2+1\)) है, तो (f) के लिए सही कथन चुनिए।

Concept

Why this answer is correct

Exam Tip

यदि \(f:\mathbb{R}\to\mathbb{R}\) तथा (f(x)=x-3-3x) है, तो (f) के बारे में सही कथन चुनिए।

Concept

Why this answer is correct

Exam Tip

यदि \(f:\mathbb{R}\to\mathbb{R}\) तथा (f(x)=x-3+kx) है, तो (f) के एक-एक होने के लिए (k) की कौन सी शर्त पर्याप्त है?

Concept

Why this answer is correct

Exam Tip

एक-एक फलन की पहचान के लिए रेखाचित्र से जुड़ा सही नियम कौन सा है?

Concept

Why this answer is correct

Exam Tip

यदि \(f=\{(1,4),(2,5),(3,5)\}\) है, जहां \(f:{1,2,3}\to{4,5,6}\), तो सही कथन चुनिए।

Concept

Why this answer is correct

Exam Tip

फलन \(f:\mathbb{R}\to\mathbb{R}\) को (f(x)=x-2) यदि (x<0) और (f(x)=x+1) यदि \(x\ge0\) से परिभाषित किया गया है। (f) के लिए सही कथन क्या है?

Concept

Why this answer is correct

Exam Tip

यदि कोई फलन \(f:A\to B\) एक-एक है, तो उसके प्रतिलोम संबंध के बारे में सही कथन क्या है?

Concept

Why this answer is correct

Exam Tip

यदि \(g\circ f\) एक-एक है, तो (g) के बारे में कौन सा कथन सही है?

Concept

Why this answer is correct

यदि \(f:\mathbb{R}\to\mathbb{R}\) तथा (f(x)=x^-3-6x^-2+9x) है, तो कौन सा विकल्प (f) के एक-एक न होने को सिद्ध करता है?

यदि \(f:\mathbb{R}\to\mathbb{R}\) तथा (f(x)=2x^-3-5) है, तो (f) के लिए सही कथन चुनिए।

यदि \(f:\mathbb{R}\to\mathbb{R}\) तथा (f(x)=x^-5-x) है, तो कौन सा विकल्प (f) के एक-एक न होने को सिद्ध करता है?

फलन \(f:\mathbb{R}\to\mathbb{R}\) को (f(x)=x+1) यदि (x<0) और (f(x)=x^-2) यदि \(x\ge0\) से परिभाषित किया गया है। सही कथन क्या है?

यदि \(f:\mathbb{R}\to\mathbb{R}\) तथा (f(x)=\frac{x}{1+x^-2}) है, तो (f) के बारे में सही कथन क्या है?

यदि \(f:\mathbb{R}\to\mathbb{R}\) तथा (f(x)=x^-3-3x) है, तो (f) के बारे में सही कथन चुनिए।

यदि \(f:\mathbb{R}\to\mathbb{R}\) तथा (f(x)=x^-3+kx) है, तो (f) के एक-एक होने के लिए (k) की कौन सी शर्त पर्याप्त है?

फलन \(f:\mathbb{R}\to\mathbb{R}\) को (f(x)=x^-2) यदि (x<0) और (f(x)=x+1) यदि \(x\ge0\) से परिभाषित किया गया है। (f) के लिए सही कथन क्या है?

यदि \(f:\mathbb{R}\to\mathbb{R}\) तथा (f(x)=\frac{x}{1+x^-2}) है, तो (f) के बारे में सही कथन चुनिए।

यदि \(f:\mathbb{R}\to\mathbb{R}\) और (f(x)=x^-3-3x), तो (f) एकैकी क्यों नहीं है?

यदि \(f:\mathbb{R}\to\mathbb{R}\) और (f(x)=x^-2), तो (f) एकैकी क्यों नहीं है?