किस स्थिति में (A) से (B) में दिया गया संबंध फलन कहलाता है?
When is a relation from (A) to (B) called a function?
#functions
#relation
#definition
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A (A) के हर तत्व का (B) में ठीक एक प्रतिबिंब हो / Each element of (A) has exactly one image in (B)
B (A) के कुछ तत्वों का प्रतिबिंब हो / Some elements of (A) have images
C (B) के हर तत्व का पूर्व प्रतिबिंब हो / Every element of (B) has a preimage
D हर तत्व के दो प्रतिबिंब हों / Every element has two images
Explanation opens after your attempt
Correct Answer
A. (A) के हर तत्व का (B) में ठीक एक प्रतिबिंब हो / Each element of (A) has exactly one image in (B)
Step 1
Concept
In a function each element of (A) is related to exactly one element of (B). In exams focus on the phrase exactly one.
Step 2
Why this answer is correct
The correct answer is A. (A) के हर तत्व का (B) में ठीक एक प्रतिबिंब हो / Each element of (A) has exactly one image in (B). In a function each element of (A) is related to exactly one element of (B). In exams focus on the phrase exactly one.
Step 3
Exam Tip
फलन में (A) के प्रत्येक तत्व से (B) का ठीक एक तत्व जुड़ता है। परीक्षा में ठीक एक शब्द पर ध्यान दें।
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यदि \(A=\{1,2\}\) और \(B=\{3,4\}\) है तो कौन सा संबंध (A) से (B) में फलन है?
If \(A=\{1,2\}\) and \(B=\{3,4\}\), which relation from (A) to (B) is a function?
#functions
#ordered-pairs
#finite-set
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A \(R=\{(1,3),(2,4)\}\)
B \(R=\{(1,3),(1,4)\}\)
C \(R=\{(2,3)\}\)
D \(R=\{(1,3),(2,4),(2,3)\}\)
Explanation opens after your attempt
Correct Answer
A. \(R=\{(1,3),(2,4)\}\)
Step 1
Concept
Here both (1) and (2) have exactly one image. To test a function check the first components.
Step 2
Why this answer is correct
The correct answer is A. \(R=\{(1,3),(2,4)\}\). Here both (1) and (2) have exactly one image. To test a function check the first components.
Step 3
Exam Tip
यहां (1) और (2) दोनों का ठीक एक प्रतिबिंब है। फलन जांचते समय पहले अवयवों को देखें।
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संबंध \(R=\{(1,2),(1,3),(2,4)\}\) \(A=\{1,2\}\) से \(B=\{2,3,4\}\) में फलन क्यों नहीं है?
Why is \(R=\{(1,2),(1,3),(2,4)\}\) not a function from \(A=\{1,2\}\) to \(B=\{2,3,4\}\)?
#functions
#not-function
#ordered-pairs
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A क्योंकि (1) के दो प्रतिबिंब हैं / Because (1) has two images
B क्योंकि (2) का कोई प्रतिबिंब नहीं है / Because (2) has no image
C क्योंकि (B) खाली है / Because (B) is empty
D क्योंकि सभी युग्म समान हैं / Because all pairs are same
Explanation opens after your attempt
Correct Answer
A. क्योंकि (1) के दो प्रतिबिंब हैं / Because (1) has two images
Step 1
Concept
The same first component (1) is related to two different second components. Such a relation is not a function.
Step 2
Why this answer is correct
The correct answer is A. क्योंकि (1) के दो प्रतिबिंब हैं / Because (1) has two images. The same first component (1) is related to two different second components. Such a relation is not a function.
Step 3
Exam Tip
एक ही प्रथम अवयव (1) दो अलग द्वितीय अवयवों से जुड़ा है। ऐसी स्थिति फलन नहीं बनाती।
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यदि \(A=\{a,b,c\}\) और \(R=\{(a,1),(b,1),(c,2)\}\) है तो (R) के बारे में सही कथन क्या है?
If \(A=\{a,b,c\}\) and \(R=\{(a,1),(b,1),(c,2)\}\), which statement about (R) is correct?
#functions
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#concept
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A यह फलन है / It is a function
B यह फलन नहीं है / It is not a function
C यह संबंध नहीं है / It is not a relation
D यह केवल (c) पर परिभाषित है / It is defined only on (c)
Explanation opens after your attempt
Correct Answer
A. यह फलन है / It is a function
Step 1
Concept
Each of (a), (b), and (c) has exactly one image. Two different elements may have the same image in a function.
Step 2
Why this answer is correct
The correct answer is A. यह फलन है / It is a function. Each of (a), (b), and (c) has exactly one image. Two different elements may have the same image in a function.
Step 3
Exam Tip
(a), (b), और (c) में से हर एक का ठीक एक प्रतिबिंब है। दो अलग तत्वों का समान प्रतिबिंब होना फलन में मान्य है।
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फलन के लिए कौन सा कथन हमेशा सत्य है?
Which statement is always true for a function?
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#input-output
#basic
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A प्रत्येक इनपुट का ठीक एक आउटपुट होता है / Every input has exactly one output
B प्रत्येक आउटपुट का ठीक एक इनपुट होता है / Every output has exactly one input
C सभी आउटपुट अलग-अलग होते हैं / All outputs are different
D कोई भी इनपुट छोड़ा जा सकता है / Any input may be left out
Explanation opens after your attempt
Correct Answer
A. प्रत्येक इनपुट का ठीक एक आउटपुट होता है / Every input has exactly one output
Step 1
Concept
A function requires each input to have one output. The outputs need not be distinct.
Step 2
Why this answer is correct
The correct answer is A. प्रत्येक इनपुट का ठीक एक आउटपुट होता है / Every input has exactly one output. A function requires each input to have one output. The outputs need not be distinct.
Step 3
Exam Tip
फलन में इनपुट को एक ही आउटपुट मिलना जरूरी है। आउटपुट अलग हों यह जरूरी नहीं।
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यदि \(f:A\to B\) एक फलन है तो (A) को क्या कहा जाता है?
If \(f:A\to B\) is a function, what is (A) called?
#functions
#domain
#notation
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A प्रांत / Domain
B सहप्रांत / Codomain
C परिसर / Range
D ग्राफ / Graph
Explanation opens after your attempt
Correct Answer
A. प्रांत / Domain
Step 1
Concept
In \(f:A\to B\), (A) is the domain of the function. Read the notation \(f:A\to B\) carefully.
Step 2
Why this answer is correct
The correct answer is A. प्रांत / Domain. In \(f:A\to B\), (A) is the domain of the function. Read the notation \(f:A\to B\) carefully.
Step 3
Exam Tip
\(f:A\to B\) में (A) फलन का प्रांत होता है। संकेत \(f:A\to B\) को ध्यान से पढ़ें।
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यदि \(f:A\to B\) एक फलन है तो (B) को क्या कहा जाता है?
If \(f:A\to B\) is a function, what is (B) called?
#functions
#codomain
#notation
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A सहप्रांत / Codomain
B प्रांत / Domain
C पूर्व प्रतिबिंब / Preimage
D स्वतंत्र चर / Independent variable
Explanation opens after your attempt
Correct Answer
A. सहप्रांत / Codomain
Step 1
Concept
In \(f:A\to B\), (B) is called the codomain. The range may be a subset of the codomain.
Step 2
Why this answer is correct
The correct answer is A. सहप्रांत / Codomain. In \(f:A\to B\), (B) is called the codomain. The range may be a subset of the codomain.
Step 3
Exam Tip
\(f:A\to B\) में (B) सहप्रांत कहलाता है। परिसर सहप्रांत का उपसमुच्चय हो सकता है।
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फलन \(f=\{(1,5),(2,5),(3,6)\}\) का परिसर क्या है?
What is the range of the function \(f=\{(1,5),(2,5),(3,6)\}\)?
#functions
#range
#ordered-pairs
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A ({5,6})
B ({1,2,3})
C ({1,5,6})
D ({2,3,5})
Explanation opens after your attempt
Correct Answer
A. ({5,6})
Step 1
Concept
The range contains only the obtained values, so it is (5) and (6). A repeated value is written once.
Step 2
Why this answer is correct
The correct answer is A. ({5,6}). The range contains only the obtained values, so it is (5) and (6). A repeated value is written once.
Step 3
Exam Tip
परिसर में केवल प्राप्त मान आते हैं इसलिए (5) और (6) हैं। दोहराए गए मान को एक बार लिखते हैं।
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फलन \(f=\{(2,4),(3,9),(4,16)\}\) का प्रांत क्या है?
What is the domain of the function \(f=\{(2,4),(3,9),(4,16)\}\)?
#functions
#domain
#ordered-pairs
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A ({2,3,4})
B ({4,9,16})
C ({2,4,16})
D ({3,9})
Explanation opens after your attempt
Correct Answer
A. ({2,3,4})
Step 1
Concept
The first components of ordered pairs form the domain. Hence the domain is ({2,3,4}).
Step 2
Why this answer is correct
The correct answer is A. ({2,3,4}). The first components of ordered pairs form the domain. Hence the domain is ({2,3,4}).
Step 3
Exam Tip
क्रमित युग्मों के प्रथम अवयव प्रांत बनाते हैं। इसलिए प्रांत ({2,3,4}) है।
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यदि (f(x)=x+2) और प्रांत ({1,2,3}) है तो (f(2)) का मान क्या है?
If (f(x)=x+2) and the domain is ({1,2,3}), what is the value of (f(2))?
#functions
#function-value
#evaluation
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A (4)
B (3)
C (5)
D (2)
Explanation opens after your attempt
Step 1
Concept
Putting (x=2), we get (f(2)=2+2=4). Substitute the given input carefully.
Step 2
Why this answer is correct
The correct answer is A. (4). Putting (x=2), we get (f(2)=2+2=4). Substitute the given input carefully.
Step 3
Exam Tip
(x=2) रखने पर (f(2)=2+2=4) मिलता है। फलन मान में दिए गए इनपुट को सावधानी से रखें।
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यदि (g(x)=2x) और \(A=\{1,3,5\}\) है तो (g(3)) क्या होगा?
If (g(x)=2x) and \(A=\{1,3,5\}\), what is (g(3))?
#functions
#evaluation
#rule
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A (6)
B (3)
C (5)
D (8)
Explanation opens after your attempt
Step 1
Concept
(g(3)=2\cdot3=6). Apply the rule directly in simple function questions.
Step 2
Why this answer is correct
The correct answer is A. (6). (g(3)=2\cdot3=6). Apply the rule directly in simple function questions.
Step 3
Exam Tip
(g(3)=2\cdot3=6) है। सरल फलन में नियम को सीधे लागू करें।
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कौन सा आरेखीय विचार फलन की सही पहचान बताता है?
Which mapping idea correctly identifies a function?
#functions
#mapping-diagram
#concept
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A प्रत्येक प्रांत तत्व से केवल एक तीर निकले / Only one arrow leaves each domain element
B एक प्रांत तत्व से दो तीर निकलें / Two arrows leave one domain element
C कुछ प्रांत तत्वों से कोई तीर न निकले / Some domain elements have no arrow
D हर सहप्रांत तत्व पर तीर जरूर आए / Every codomain element must receive an arrow
Explanation opens after your attempt
Correct Answer
A. प्रत्येक प्रांत तत्व से केवल एक तीर निकले / Only one arrow leaves each domain element
Step 1
Concept
In a function exactly one arrow must leave every domain element. It is not necessary that every codomain element receives an arrow.
Step 2
Why this answer is correct
The correct answer is A. प्रत्येक प्रांत तत्व से केवल एक तीर निकले / Only one arrow leaves each domain element. In a function exactly one arrow must leave every domain element. It is not necessary that every codomain element receives an arrow.
Step 3
Exam Tip
फलन में हर प्रांत तत्व से ठीक एक तीर निकलना चाहिए। सहप्रांत के हर तत्व पर तीर आना आवश्यक नहीं।
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यदि \(A=\{1,2,3\}\) से \(B=\{a,b\}\) में \(1\to a\), \(2\to a\), \(3\to b\) दिया है तो यह क्या है?
If from \(A=\{1,2,3\}\) to \(B=\{a,b\}\), \(1\to a\), \(2\to a\), \(3\to b\) is given, what is it?
#functions
#mapping
#many-one
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A फलन / Function
B फलन नहीं / Not a function
C खाली संबंध / Empty relation
D केवल सहप्रांत / Only codomain
Explanation opens after your attempt
Correct Answer
A. फलन / Function
Step 1
Concept
Every domain element has exactly one image. Having the same image is not an error.
Step 2
Why this answer is correct
The correct answer is A. फलन / Function. Every domain element has exactly one image. Having the same image is not an error.
Step 3
Exam Tip
हर प्रांत तत्व का ठीक एक प्रतिबिंब है। समान प्रतिबिंब होना गलती नहीं है।
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यदि \(A=\{1,2,3\}\) और संबंध में केवल ((1,2)) तथा ((2,3)) हैं तो यह (A) से फलन क्यों नहीं है?
If \(A=\{1,2,3\}\) and the relation contains only ((1,2)) and ((2,3)), why is it not a function from (A)?
#functions
#missing-domain
#not-function
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A क्योंकि (3) का कोई प्रतिबिंब नहीं है / Because (3) has no image
B क्योंकि (1) का दो प्रतिबिंब है / Because (1) has two images
C क्योंकि (2) सहप्रांत में नहीं है / Because (2) is not in codomain
D क्योंकि युग्म क्रमित नहीं हैं / Because pairs are not ordered
Explanation opens after your attempt
Correct Answer
A. क्योंकि (3) का कोई प्रतिबिंब नहीं है / Because (3) has no image
Step 1
Concept
Every element of the domain (A) must have an image. Here (3) is missing.
Step 2
Why this answer is correct
The correct answer is A. क्योंकि (3) का कोई प्रतिबिंब नहीं है / Because (3) has no image. Every element of the domain (A) must have an image. Here (3) is missing.
Step 3
Exam Tip
प्रांत (A) के हर तत्व का प्रतिबिंब होना चाहिए। यहां (3) छूट गया है।
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सभी फलनों के बारे में कौन सा कथन सही है?
Which statement is correct about all functions?
#functions
#relation
#special-relation
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A हर फलन एक संबंध है / Every function is a relation
B हर संबंध एक फलन है / Every relation is a function
C हर फलन खाली होता है / Every function is empty
D हर फलन एक-एक होता है / Every function is one-one
Explanation opens after your attempt
Correct Answer
A. हर फलन एक संबंध है / Every function is a relation
Step 1
Concept
A function is a special type of relation. Every relation is not a function because extra conditions are required.
Step 2
Why this answer is correct
The correct answer is A. हर फलन एक संबंध है / Every function is a relation. A function is a special type of relation. Every relation is not a function because extra conditions are required.
Step 3
Exam Tip
फलन संबंध का विशेष प्रकार है। हर संबंध फलन नहीं होता क्योंकि शर्तें अतिरिक्त हैं।
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यदि \(f=\{(1,2),(2,3),(3,4)\}\) है तो (f(1)) क्या है?
If \(f=\{(1,2),(2,3),(3,4)\}\), what is (f(1))?
#functions
#function-value
#ordered-pairs
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A (2)
B (1)
C (3)
D (4)
Explanation opens after your attempt
Step 1
Concept
From the pair ((1,2)), the image of (1) is (2). In (f(x)), the first component is the input.
Step 2
Why this answer is correct
The correct answer is A. (2). From the pair ((1,2)), the image of (1) is (2). In (f(x)), the first component is the input.
Step 3
Exam Tip
युग्म ((1,2)) से (1) का प्रतिबिंब (2) है। (f(x)) में प्रथम अवयव इनपुट होता है।
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यदि \(h=\{(0,1),(1,2),(2,3)\}\) है तो (h(0)) का मान क्या है?
If \(h=\{(0,1),(1,2),(2,3)\}\), what is the value of (h(0))?
#functions
#evaluation
#ordered-pairs
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A (1)
B (0)
C (2)
D (3)
Explanation opens after your attempt
Step 1
Concept
((0,1)) shows that the image of (0) is (1). Do not reverse the ordered pair while answering.
Step 2
Why this answer is correct
The correct answer is A. (1). ((0,1)) shows that the image of (0) is (1). Do not reverse the ordered pair while answering.
Step 3
Exam Tip
((0,1)) दिखाता है कि (0) का प्रतिबिंब (1) है। युग्म की दिशा बदलकर उत्तर न निकालें।
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कौन सा संबंध \(A=\{1,2\}\) से \(B=\{5,6\}\) में फलन नहीं है?
Which relation from \(A=\{1,2\}\) to \(B=\{5,6\}\) is not a function?
#functions
#not-function
#duplicate-input
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A ({(1,5),(1,6),(2,5)})
B ({(1,5),(2,5)})
C ({(1,6),(2,6)})
D ({(1,5),(2,6)})
Explanation opens after your attempt
Correct Answer
A. ({(1,5),(1,6),(2,5)})
Step 1
Concept
In the first option, (1) has two different images (5) and (6). Hence it is not a function.
Step 2
Why this answer is correct
The correct answer is A. ({(1,5),(1,6),(2,5)}). In the first option, (1) has two different images (5) and (6). Hence it is not a function.
Step 3
Exam Tip
पहले विकल्प में (1) के दो अलग प्रतिबिंब (5) और (6) हैं। इसलिए यह फलन नहीं है।
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यदि \(f:A\to B\) में \(A=\{2,4\}\) और \(B=\{1,3,5\}\) है तो (f) का कौन सा रूप फलन हो सकता है?
If \(f:A\to B\) where \(A=\{2,4\}\) and \(B=\{1,3,5\}\), which form of (f) can be a function?
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#finite-function
#selection
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A ({(2,1),(4,5)})
B ({(2,1),(2,3)})
C ({(4,5)})
D ({(2,1),(4,5),(4,3)})
Explanation opens after your attempt
Correct Answer
A. ({(2,1),(4,5)})
Step 1
Concept
Both (2) and (4) have exactly one image. In the other options an input is missing or repeated with different images.
Step 2
Why this answer is correct
The correct answer is A. ({(2,1),(4,5)}). Both (2) and (4) have exactly one image. In the other options an input is missing or repeated with different images.
Step 3
Exam Tip
(2) और (4) दोनों का ठीक एक प्रतिबिंब है। बाकी विकल्पों में कोई इनपुट छूटा है या दो बार आया है।
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स्थिर फलन में सभी प्रांत तत्व किससे जुड़ते हैं?
In a constant function, all domain elements are mapped to what?
#functions
#constant-function
#concept
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A सहप्रांत के एक ही निश्चित तत्व से / The same fixed element of the codomain
B अलग-अलग तत्वों से / Different elements
C किसी तत्व से नहीं / No element
D दो-दो तत्वों से / Two elements each
Explanation opens after your attempt
Correct Answer
A. सहप्रांत के एक ही निश्चित तत्व से / The same fixed element of the codomain
Step 1
Concept
In a constant function every input has the same value. It is still a valid function.
Step 2
Why this answer is correct
The correct answer is A. सहप्रांत के एक ही निश्चित तत्व से / The same fixed element of the codomain. In a constant function every input has the same value. It is still a valid function.
Step 3
Exam Tip
स्थिर फलन में प्रत्येक इनपुट का मान समान रहता है। यह भी वैध फलन है।
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यदि (f(x)=7) सभी \(x\in A\) के लिए है तो (f) किस प्रकार का फलन है?
If (f(x)=7) for all \(x\in A\), what type of function is (f)?
#functions
#constant-function
#easy
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A स्थिर फलन / Constant function
B तत्समक फलन / Identity function
C रिक्त फलन / Empty function
D बहुमूल्य संबंध / Multivalued relation
Explanation opens after your attempt
Correct Answer
A. स्थिर फलन / Constant function
Step 1
Concept
The function value is (7) for every input, so it is a constant function. In a constant function all images are the same.
Step 2
Why this answer is correct
The correct answer is A. स्थिर फलन / Constant function. The function value is (7) for every input, so it is a constant function. In a constant function all images are the same.
Step 3
Exam Tip
हर इनपुट पर फलन का मान (7) है इसलिए यह स्थिर फलन है। स्थिर फलन में सभी प्रतिबिंब समान होते हैं।
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तत्समक फलन \(I_A:A\to A\) में किसी \(x\in A\) का प्रतिबिंब क्या होता है?
In the identity function \(I_A:A\to A\), what is the image of any \(x\in A\)?
#functions
#identity-function
#definition
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A (x)
B (0)
C (1)
D (A) का कोई दूसरा तत्व / Another element of (A)
Explanation opens after your attempt
Step 1
Concept
In the identity function, (I_A(x)=x). Each element is mapped to itself.
Step 2
Why this answer is correct
The correct answer is A. (x). In the identity function, (I_A(x)=x). Each element is mapped to itself.
Step 3
Exam Tip
तत्समक फलन में (I_A(x)=x) होता है। इसमें प्रत्येक तत्व स्वयं से जुड़ता है।
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यदि \(A=\{1,2,3\}\) है तो तत्समक फलन \(I_A\) कौन सा है?
If \(A=\{1,2,3\}\), which is the identity function \(I_A\)?
#functions
#identity-function
#ordered-pairs
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A ({(1,1),(2,2),(3,3)})
B ({(1,2),(2,3),(3,1)})
C ({(1,1),(2,1),(3,1)})
D ({(1,3),(2,2),(3,1)})
Explanation opens after your attempt
Correct Answer
A. ({(1,1),(2,2),(3,3)})
Step 1
Concept
In the identity function each element maps to itself. Therefore all pairs are of the form ((x,x)).
Step 2
Why this answer is correct
The correct answer is A. ({(1,1),(2,2),(3,3)}). In the identity function each element maps to itself. Therefore all pairs are of the form ((x,x)).
Step 3
Exam Tip
तत्समक फलन में हर तत्व स्वयं से जुड़ता है। इसलिए सभी युग्म ((x,x)) रूप में हैं।
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फलन \(f:A\to B\) का ग्राफ किसका समुच्चय होता है?
The graph of a function \(f:A\to B\) is a set of what?
#functions
#graph
#ordered-pairs
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A क्रमित युग्म ((x,f(x))) / Ordered pairs ((x,f(x)))
B केवल (A) के तत्व / Only elements of (A)
C केवल (B) के तत्व / Only elements of (B)
D केवल संख्याएं (0) और (1) / Only numbers (0) and (1)
Explanation opens after your attempt
Correct Answer
A. क्रमित युग्म ((x,f(x))) / Ordered pairs ((x,f(x)))
Step 1
Concept
The graph of a function is made of all pairs ((x,f(x))). The first component is input and the second is output.
Step 2
Why this answer is correct
The correct answer is A. क्रमित युग्म ((x,f(x))) / Ordered pairs ((x,f(x))). The graph of a function is made of all pairs ((x,f(x))). The first component is input and the second is output.
Step 3
Exam Tip
फलन का ग्राफ सभी ((x,f(x))) युग्मों से बनता है। इसमें पहला अवयव इनपुट और दूसरा आउटपुट होता है।
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यदि (f(x)=x-2 ) और \(A=\{1,2,3\}\) है तो (f) का ग्राफ कौन सा है?
If (f(x)=x-2 ) and \(A=\{1,2,3\}\), which is the graph of (f)?
#functions
#graph
#square-function
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A ({(1,1),(2,4),(3,9)})
B ({(1,2),(2,4),(3,6)})
C ({(1,1),(2,2),(3,3)})
D ({(1,3),(2,2),(3,1)})
Explanation opens after your attempt
Correct Answer
A. ({(1,1),(2,4),(3,9)})
Step 1
Concept
Since \(1^2=1\), \(2^2=4\), and \(3^2=9\), the correct pairs are (\(x,x^2\)).
Step 2
Why this answer is correct
The correct answer is A. ({(1,1),(2,4),(3,9)}). Since \(1^2=1\), \(2^2=4\), and \(3^2=9\), the correct pairs are (\(x,x^2\)).
Step 3
Exam Tip
\(1^2=1\), \(2^2=4\), और \(3^2=9\) हैं। इसलिए सही युग्म (\(x,x^2\)) हैं।
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यदि (f(x)=x-1) और \(A=\{2,3,4\}\) है तो (f(A)) क्या है?
If (f(x)=x-1) and \(A=\{2,3,4\}\), what is (f(A))?
#functions
#image-set
#evaluation
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A ({1,2,3})
B ({2,3,4})
C ({3,4,5})
D ({1,3,5})
Explanation opens after your attempt
Correct Answer
A. ({1,2,3})
Step 1
Concept
Applying the rule to (2,3,4) gives (1,2,3). The set of obtained values is (f(A)).
Step 2
Why this answer is correct
The correct answer is A. ({1,2,3}). Applying the rule to (2,3,4) gives (1,2,3). The set of obtained values is (f(A)).
Step 3
Exam Tip
(2,3,4) पर नियम लगाने से (1,2,3) मिलते हैं। प्राप्त मानों का समुच्चय (f(A)) है।
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किस विकल्प में (x) के एक ही मान के लिए दो अलग (y) मान होने से फलन नहीं बनता?
Which option is not a function because one value of (x) has two different values of (y)?
#functions
#not-function
#input-output
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A ({(2,5),(2,6),(3,7)})
B ({(2,5),(3,5),(4,5)})
C ({(1,1),(2,4),(3,9)})
D ({(0,0),(1,1),(2,2)})
Explanation opens after your attempt
Correct Answer
A. ({(2,5),(2,6),(3,7)})
Step 1
Concept
The same input (2) is related to two outputs (5) and (6). This breaks the condition of a function.
Step 2
Why this answer is correct
The correct answer is A. ({(2,5),(2,6),(3,7)}). The same input (2) is related to two outputs (5) and (6). This breaks the condition of a function.
Step 3
Exam Tip
एक ही इनपुट (2) दो आउटपुट (5) और (6) से जुड़ा है। यह फलन की शर्त तोड़ता है।
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यदि कोई संबंध (A) से (B) में फलन है तो वह किसका उपसमुच्चय होगा?
If a relation from (A) to (B) is a function, it will be a subset of what?
#functions
#cartesian-product
#relation
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A \(A\times B\)
B \(B\times A\)
C \(A\cup B\)
D \(A\cap B\)
Explanation opens after your attempt
Correct Answer
A. \(A\times B\)
Step 1
Concept
A relation from (A) to (B) is always a subset of \(A\times B\). A function is a special form of such a relation.
Step 2
Why this answer is correct
The correct answer is A. \(A\times B\). A relation from (A) to (B) is always a subset of \(A\times B\). A function is a special form of such a relation.
Step 3
Exam Tip
(A) से (B) में संबंध हमेशा \(A\times B\) का उपसमुच्चय होता है। फलन उसी संबंध का विशेष रूप है।
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यदि \(A=\{1,2\}\) और \(B=\{a,b,c\}\) है तो (A) से (B) में फलन में कितने क्रमित युग्म होंगे?
If \(A=\{1,2\}\) and \(B=\{a,b,c\}\), how many ordered pairs will a function from (A) to (B) have?
#functions
#number-of-pairs
#domain
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A (2)
B (3)
C (5)
D (6)
Explanation opens after your attempt
Step 1
Concept
A function has one ordered pair for each element of the domain (A). Hence the number of pairs is (|A|=2).
Step 2
Why this answer is correct
The correct answer is A. (2). A function has one ordered pair for each element of the domain (A). Hence the number of pairs is (|A|=2).
Step 3
Exam Tip
फलन में प्रांत (A) के हर तत्व के लिए एक युग्म होता है। इसलिए युग्मों की संख्या (|A|=2) है।
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यदि (A) में (4) तत्व हैं और \(f:A\to B\) फलन है तो (f) में कितने क्रमित युग्म होंगे?
If (A) has (4) elements and \(f:A\to B\) is a function, how many ordered pairs will (f) have?
#functions
#domain
#counting
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A (4)
B (1)
C (2)
D (8)
Explanation opens after your attempt
Step 1
Concept
A function has exactly one pair for every domain element. Therefore there will be (4) pairs.
Step 2
Why this answer is correct
The correct answer is A. (4). A function has exactly one pair for every domain element. Therefore there will be (4) pairs.
Step 3
Exam Tip
फलन में प्रत्येक प्रांत तत्व के लिए ठीक एक युग्म होता है। इसलिए कुल युग्म (4) होंगे।
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यदि \(A=\{p,q\}\) और \(B=\{0,1\}\) है तो (A) से (B) में कुल कितने फलन बन सकते हैं?
If \(A=\{p,q\}\) and \(B=\{0,1\}\), how many functions can be formed from (A) to (B)?
#functions
#counting-functions
#finite-sets
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A (4)
B (2)
C (3)
D (1)
Explanation opens after your attempt
Step 1
Concept
Each element (p) and (q) has (2) choices, so total functions are \(2^2=4\). Use \(|B|^{|A|}\) for counting.
Step 2
Why this answer is correct
The correct answer is A. (4). Each element (p) and (q) has (2) choices, so total functions are \(2^2=4\). Use \(|B|^{|A|}\) for counting.
Step 3
Exam Tip
हर तत्व (p) और (q) के लिए (2) विकल्प हैं इसलिए कुल \(2^2=4\) फलन हैं। गिनती में \(|B|^{|A|}\) प्रयोग करें।
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यदि (|A|=3) और (|B|=2) है तो (A) से (B) में कुल फलनों की संख्या क्या है?
If (|A|=3) and (|B|=2), what is the total number of functions from (A) to (B)?
#functions
#counting
#formula
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A \(2^3\)
B \(3^2\)
C (2+3)
D (3-2)
Explanation opens after your attempt
Correct Answer
A. \(2^3\)
Step 1
Concept
Each element of (A) has (2) choices in (B). Therefore the number is \(|B|^{|A|}=2^3\).
Step 2
Why this answer is correct
The correct answer is A. \(2^3\). Each element of (A) has (2) choices in (B). Therefore the number is \(|B|^{|A|}=2^3\).
Step 3
Exam Tip
(A) के प्रत्येक तत्व के लिए (B) में (2) विकल्प हैं। इसलिए संख्या \(|B|^{|A|}=2^3\) है।
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यदि (|A|=2) और (|B|=3) है तो (A) से (B) में कुल फलनों की संख्या क्या है?
If (|A|=2) and (|B|=3), what is the total number of functions from (A) to (B)?
#functions
#counting-functions
#formula
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A \(3^2\)
B \(2^3\)
C (3+2)
D \(6^2\)
Explanation opens after your attempt
Correct Answer
A. \(3^2\)
Step 1
Concept
Using the formula \(|B|^{|A|}\), we get \(3^2\). The base is the number of elements in the codomain.
Step 2
Why this answer is correct
The correct answer is A. \(3^2\). Using the formula \(|B|^{|A|}\), we get \(3^2\). The base is the number of elements in the codomain.
Step 3
Exam Tip
सूत्र \(|B|^{|A|}\) लगाने पर \(3^2\) मिलता है। आधार सहप्रांत की संख्या होती है।
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नीचे दिए संबंधों में कौन सा बहु-एक फलन का उदाहरण है?
Which of the following relations is an example of a many-one function?
#functions
#many-one
#example
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A ({(1,a),(2,a),(3,b)})
B ({(1,a),(1,b),(2,c)})
C ({(1,a),(2,b),(2,c)})
D ({(1,a)}) जहां प्रांत ({1,2}) है / ({(1,a)}) with domain ({1,2})
Explanation opens after your attempt
Correct Answer
A. ({(1,a),(2,a),(3,b)})
Step 1
Concept
Two different inputs (1) and (2) map to the same output (a). Still each input has exactly one image.
Step 2
Why this answer is correct
The correct answer is A. ({(1,a),(2,a),(3,b)}). Two different inputs (1) and (2) map to the same output (a). Still each input has exactly one image.
Step 3
Exam Tip
दो अलग इनपुट (1) और (2) एक ही आउटपुट (a) से जुड़ते हैं। फिर भी हर इनपुट का ठीक एक प्रतिबिंब है।
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कौन सा कथन फलन के लिए गलत है?
Which statement is false for a function?
#functions
#false-statement
#concept
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A एक इनपुट के दो अलग आउटपुट हो सकते हैं / One input can have two different outputs
B दो इनपुट का समान आउटपुट हो सकता है / Two inputs can have the same output
C हर इनपुट का आउटपुट होना चाहिए / Every input must have an output
D फलन एक संबंध है / A function is a relation
Explanation opens after your attempt
Correct Answer
A. एक इनपुट के दो अलग आउटपुट हो सकते हैं / One input can have two different outputs
Step 1
Concept
One input cannot have two different outputs in a function. This is the most common mistake.
Step 2
Why this answer is correct
The correct answer is A. एक इनपुट के दो अलग आउटपुट हो सकते हैं / One input can have two different outputs. One input cannot have two different outputs in a function. This is the most common mistake.
Step 3
Exam Tip
एक इनपुट के दो अलग आउटपुट फलन में स्वीकार नहीं होते। यही सबसे सामान्य गलती है।
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यदि \(f=\{(a,2),(b,3),(c,2)\}\) है तो (f(c)) क्या है?
If \(f=\{(a,2),(b,3),(c,2)\}\), what is (f(c))?
#functions
#function-value
#letters
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A (2)
B (3)
C (a)
D (c)
Explanation opens after your attempt
Step 1
Concept
From the pair ((c,2)), the image of (c) is (2). Find the corresponding pair for the function value.
Step 2
Why this answer is correct
The correct answer is A. (2). From the pair ((c,2)), the image of (c) is (2). Find the corresponding pair for the function value.
Step 3
Exam Tip
युग्म ((c,2)) से (c) का प्रतिबिंब (2) है। फलन मान के लिए संबंधित युग्म खोजें।
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यदि (f(x)=3x+1) है तो (f(1)) क्या होगा?
If (f(x)=3x+1), what is (f(1))?
#functions
#evaluation
#linear-function
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A (4)
B (3)
C (1)
D (5)
Explanation opens after your attempt
Step 1
Concept
(f(1)=3\cdot1+1=4). During substitution multiply first and then add.
Step 2
Why this answer is correct
The correct answer is A. (4). (f(1)=3\cdot1+1=4). During substitution multiply first and then add.
Step 3
Exam Tip
(f(1)=3\cdot1+1=4) है। प्रतिस्थापन में पहले गुणा फिर जोड़ करें।
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यदि (f(x)=x-2 +1) है तो (f(2)) का मान क्या है?
If (f(x)=x-2 +1), what is the value of (f(2))?
#functions
#evaluation
#quadratic-rule
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A (5)
B (4)
C (3)
D (6)
Explanation opens after your attempt
Step 1
Concept
(f(2)=22 +1=5). Evaluate the power before addition.
Step 2
Why this answer is correct
The correct answer is A. (5). (f(2)=22 +1=5). Evaluate the power before addition.
Step 3
Exam Tip
(f(2)=22 +1=5) है। घात का मान जोड़ से पहले निकालें।
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यदि (f(x)=x+5) है तो (f(0)) क्या है?
If (f(x)=x+5), what is (f(0))?
#functions
#evaluation
#zero-input
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A (5)
B (0)
C (1)
D (10)
Explanation opens after your attempt
Step 1
Concept
Putting (x=0), we get (f(0)=0+5=5). Substitute zero correctly.
Step 2
Why this answer is correct
The correct answer is A. (5). Putting (x=0), we get (f(0)=0+5=5). Substitute zero correctly.
Step 3
Exam Tip
(x=0) रखने पर (f(0)=0+5=5) मिलता है। शून्य को सही ढंग से प्रतिस्थापित करें।
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कौन सा संबंध \(A=\{1,2,3\}\) से \(B=\{4,5\}\) में फलन है?
Which relation from \(A=\{1,2,3\}\) to \(B=\{4,5\}\) is a function?
#functions
#finite-set
#identify-function
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A ({(1,4),(2,5),(3,4)})
B ({(1,4),(2,5)})
C ({(1,4),(2,5),(2,4)})
D ({(1,4),(1,5),(3,4)})
Explanation opens after your attempt
Correct Answer
A. ({(1,4),(2,5),(3,4)})
Step 1
Concept
All three domain elements (1,2,3) appear exactly once. This is the main test for a function.
Step 2
Why this answer is correct
The correct answer is A. ({(1,4),(2,5),(3,4)}). All three domain elements (1,2,3) appear exactly once. This is the main test for a function.
Step 3
Exam Tip
तीनों प्रांत तत्व (1,2,3) ठीक एक-एक बार आए हैं। यही फलन की मुख्य जांच है।
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फलन \(f:A\to B\) में किसी \(x\in A\) के लिए (f(x)) कहां का तत्व होता है?
For a function \(f:A\to B\), where does (f(x)) belong for \(x\in A\)?
#functions
#codomain
#value
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A (B)
B (A)
C \(A\times B\)
D \(\emptyset\)
Explanation opens after your attempt
Step 1
Concept
In \(f:A\to B\), every function value lies in (B). Hence we can write \(f(x)\in B\).
Step 2
Why this answer is correct
The correct answer is A. (B). In \(f:A\to B\), every function value lies in (B). Hence we can write \(f(x)\in B\).
Step 3
Exam Tip
\(f:A\to B\) में हर फलन मान (B) में होता है। इसलिए \(f(x)\in B\) लिखा जा सकता है।
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यदि \(f:A\to B\) है और \(x\in A\) है तो (x) को क्या माना जाता है?
If \(f:A\to B\) and \(x\in A\), what is (x) considered as?
#functions
#input
#domain
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A इनपुट / Input
B आउटपुट / Output
C सहप्रांत / Codomain
D परिसर / Range
Explanation opens after your attempt
Correct Answer
A. इनपुट / Input
Step 1
Concept
An element of the domain is an input. Its image (f(x)) is the output.
Step 2
Why this answer is correct
The correct answer is A. इनपुट / Input. An element of the domain is an input. Its image (f(x)) is the output.
Step 3
Exam Tip
प्रांत का तत्व इनपुट होता है। उसका प्रतिबिंब (f(x)) आउटपुट होता है।
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यदि कोई संबंध (R) फलन है तो \((a,b)\in R\) और \((a,c)\in R\) होने पर क्या निष्कर्ष होगा?
If a relation (R) is a function and \((a,b)\in R\), \((a,c)\in R\), what conclusion follows?
#functions
#uniqueness
#reasoning
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A (b=c)
B (a=b)
C (a=c)
D \(b\ne c\)
Explanation opens after your attempt
Step 1
Concept
In a function the same input cannot have two different outputs. Therefore (b=c) must hold.
Step 2
Why this answer is correct
The correct answer is A. (b=c). In a function the same input cannot have two different outputs. Therefore (b=c) must hold.
Step 3
Exam Tip
फलन में एक ही इनपुट के दो अलग आउटपुट नहीं हो सकते। इसलिए (b=c) होना चाहिए।
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कौन सा विकल्प फलन की अद्वितीयता शर्त को सही दिखाता है?
Which option correctly shows the uniqueness condition of a function?
#functions
#uniqueness
#definition
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A एक (x) के लिए केवल एक (f(x)) / For one (x), only one (f(x))
B एक (x) के लिए कई (f(x)) / For one (x), many (f(x))
C कुछ (x) के लिए कोई (f(x)) नहीं / For some (x), no (f(x))
D हर (f(x)) के लिए अलग (x) जरूरी / Each (f(x)) must have a different (x)
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Correct Answer
A. एक (x) के लिए केवल एक (f(x)) / For one (x), only one (f(x))
Step 1
Concept
Uniqueness means one input has only one value. This is the basic condition of a function.
Step 2
Why this answer is correct
The correct answer is A. एक (x) के लिए केवल एक (f(x)) / For one (x), only one (f(x)). Uniqueness means one input has only one value. This is the basic condition of a function.
Step 3
Exam Tip
अद्वितीयता का अर्थ है कि एक इनपुट का केवल एक मान हो। यह फलन की मूल शर्त है।
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यदि \(A=\{1,2,3\}\) और \(B=\{0\}\) है तो (A) से (B) में कितने फलन बनेंगे?
If \(A=\{1,2,3\}\) and \(B=\{0\}\), how many functions are possible from (A) to (B)?
#functions
#counting
#constant-case
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A (1)
B (0)
C (3)
D (2)
Explanation opens after your attempt
Step 1
Concept
For each domain element, (B) has only one choice (0). Hence the total number of functions is \(1^3=1\).
Step 2
Why this answer is correct
The correct answer is A. (1). For each domain element, (B) has only one choice (0). Hence the total number of functions is \(1^3=1\).
Step 3
Exam Tip
हर प्रांत तत्व के लिए (B) में केवल (0) एक ही विकल्प है। इसलिए कुल फलन \(1^3=1\) है।
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यदि \(A=\emptyset\) और \(B=\{1,2\}\) है तो (A) से (B) में कितने फलन होते हैं?
If \(A=\emptyset\) and \(B=\{1,2\}\), how many functions are there from (A) to (B)?
#functions
#empty-domain
#counting
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A (1)
B (0)
C (2)
D (4)
Explanation opens after your attempt
Step 1
Concept
From an empty domain, one empty function is considered. Counting also gives \(|B|^0=1\).
Step 2
Why this answer is correct
The correct answer is A. (1). From an empty domain, one empty function is considered. Counting also gives \(|B|^0=1\).
Step 3
Exam Tip
रिक्त प्रांत से एक रिक्त फलन माना जाता है। गिनती से भी \(|B|^0=1\) मिलता है।
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यदि \(A=\{1\}\) और \(B=\{2,3,4\}\) है तो (A) से (B) में कुल कितने फलन बनेंगे?
If \(A=\{1\}\) and \(B=\{2,3,4\}\), how many functions can be formed from (A) to (B)?
#functions
#counting
#singleton-domain
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A (3)
B (1)
C (4)
D (0)
Explanation opens after your attempt
Step 1
Concept
One domain element has (3) choices in (B). Therefore the total number of functions is \(3^1=3\).
Step 2
Why this answer is correct
The correct answer is A. (3). One domain element has (3) choices in (B). Therefore the total number of functions is \(3^1=3\).
Step 3
Exam Tip
एक प्रांत तत्व के लिए (B) में (3) विकल्प हैं। इसलिए कुल \(3^1=3\) फलन हैं।
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यदि \(f=\{(1,0),(2,0),(3,0)\}\) है तो यह किस प्रकार का फलन है?
If \(f=\{(1,0),(2,0),(3,0)\}\), what type of function is it?
#functions
#constant-function
#ordered-pairs
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A स्थिर फलन / Constant function
B तत्समक फलन / Identity function
C फलन नहीं / Not a function
D रिक्त संबंध / Empty relation
Explanation opens after your attempt
Correct Answer
A. स्थिर फलन / Constant function
Step 1
Concept
All inputs have image (0). Hence it is a constant function.
Step 2
Why this answer is correct
The correct answer is A. स्थिर फलन / Constant function. All inputs have image (0). Hence it is a constant function.
Step 3
Exam Tip
सभी इनपुट का प्रतिबिंब (0) है। इसलिए यह स्थिर फलन है।
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किस विकल्प में \(A=\{1,2\}\) के हर तत्व का ठीक एक प्रतिबिंब है?
In which option does every element of \(A=\{1,2\}\) have exactly one image?
#functions
#identify-function
#easy
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A ({(1,a),(2,b)})
B ({(1,a)})
C ({(1,a),(1,b)})
D ({(2,a),(2,b)})
Explanation opens after your attempt
Correct Answer
A. ({(1,a),(2,b)})
Step 1
Concept
In the first option, both (1) and (2) appear once as first components. Therefore it is a function.
Step 2
Why this answer is correct
The correct answer is A. ({(1,a),(2,b)}). In the first option, both (1) and (2) appear once as first components. Therefore it is a function.
Step 3
Exam Tip
पहले विकल्प में (1) और (2) दोनों एक-एक बार प्रथम अवयव हैं। इसलिए यह फलन है।
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फलन को संबंध का विशेष प्रकार क्यों कहा जाता है?
Why is a function called a special type of relation?
#functions
#special-relation
#summary
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A क्योंकि इसमें हर प्रांत तत्व का ठीक एक प्रतिबिंब होता है / Because every domain element has exactly one image
B क्योंकि इसमें कोई क्रमित युग्म नहीं होता / Because it has no ordered pair
C क्योंकि इसमें सहप्रांत खाली होता है / Because the codomain is empty
D क्योंकि इसमें हर आउटपुट समान होना चाहिए / Because every output must be the same
Explanation opens after your attempt
Correct Answer
A. क्योंकि इसमें हर प्रांत तत्व का ठीक एक प्रतिबिंब होता है / Because every domain element has exactly one image
Step 1
Concept
A function is a relation with the extra condition of exactly one image. Hence it is called a special relation.
Step 2
Why this answer is correct
The correct answer is A. क्योंकि इसमें हर प्रांत तत्व का ठीक एक प्रतिबिंब होता है / Because every domain element has exactly one image. A function is a relation with the extra condition of exactly one image. Hence it is called a special relation.
Step 3
Exam Tip
फलन संबंध है लेकिन उस पर ठीक एक प्रतिबिंब की अतिरिक्त शर्त लगती है। इसलिए इसे विशेष संबंध कहते हैं।
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