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)
Explanation opens after your attempt
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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यदि \(A=\{2,3,5\}\) और \(B=\{4,6,10\}\) है तो नियम (f(x)=2x) से बना संबंध कौन सा है?
If \(A=\{2,3,5\}\) and \(B=\{4,6,10\}\), which relation is formed by the rule (f(x)=2x)?
#functions
#rule
#ordered-pairs
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A \(f=\{(2,4),(3,6),(5,10)\}\)
B \(f=\{(2,6),(3,4),(5,10)\}\)
C \(f=\{(2,4),(3,10),(5,6)\}\)
D \(f=\{(2,2),(3,3),(5,5)\}\)
Explanation opens after your attempt
Correct Answer
A. \(f=\{(2,4),(3,6),(5,10)\}\)
Step 1
Concept
For each \(x\in A\), taking (2x) gives the correct pairs. Check every input separately while applying a rule.
Step 2
Why this answer is correct
The correct answer is A. \(f=\{(2,4),(3,6),(5,10)\}\). For each \(x\in A\), taking (2x) gives the correct pairs. Check every input separately while applying a rule.
Step 3
Exam Tip
हर \(x\in A\) के लिए (2x) लेने पर सही युग्म मिलते हैं। नियम लागू करते समय प्रत्येक इनपुट अलग से जांचें।
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संबंध \(R=\{(1,4),(2,4),(3,4)\}\) किस प्रकार का फलन है?
What type of function is the relation \(R=\{(1,4),(2,4),(3,4)\}\)?
#functions
#constant-function
#classification
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A स्थिर फलन / Constant function
B तत्समक फलन / Identity function
C फलन नहीं / Not a function
D रिक्त फलन / Empty function
Explanation opens after your attempt
Correct Answer
A. स्थिर फलन / Constant function
Step 1
Concept
All inputs have image (4), so it is a constant function. Same output does not make a function wrong.
Step 2
Why this answer is correct
The correct answer is A. स्थिर फलन / Constant function. All inputs have image (4), so it is a constant function. Same output does not make a function wrong.
Step 3
Exam Tip
सभी इनपुट का प्रतिबिंब (4) है इसलिए यह स्थिर फलन है। समान आउटपुट फलन को गलत नहीं बनाता।
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यदि \(f=\{(0,2),(1,3),(2,4)\}\) है तो (f(2)) का मान क्या है?
If \(f=\{(0,2),(1,3),(2,4)\}\), what is the value of (f(2))?
#functions
#function-value
#ordered-pairs
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A (4)
B (2)
C (3)
D (0)
Explanation opens after your attempt
Step 1
Concept
From the pair ((2,4)), the image of (2) is (4). In an ordered pair the first component is the input.
Step 2
Why this answer is correct
The correct answer is A. (4). From the pair ((2,4)), the image of (2) is (4). In an ordered pair the first component is the input.
Step 3
Exam Tip
युग्म ((2,4)) से (2) का प्रतिबिंब (4) है। क्रमित युग्म में पहला अवयव इनपुट होता है।
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यदि \(A=\{1,4,9\}\) और (f(x)=\sqrt{x}) है तो (f(A)) क्या होगा?
If \(A=\{1,4,9\}\) and (f(x)=\sqrt{x}), what is (f(A))?
#functions
#image-set
#square-root
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A ({1,2,3})
B ({1,4,9})
C ({2,4,6})
D ({0,1,2})
Explanation opens after your attempt
Correct Answer
A. ({1,2,3})
Step 1
Concept
\(\sqrt{1}=1\), \(\sqrt{4}=2\), and \(\sqrt{9}=3\). The set of obtained values is (f(A)).
Step 2
Why this answer is correct
The correct answer is A. ({1,2,3}). \(\sqrt{1}=1\), \(\sqrt{4}=2\), and \(\sqrt{9}=3\). The set of obtained values is (f(A)).
Step 3
Exam Tip
\(\sqrt{1}=1\), \(\sqrt{4}=2\), और \(\sqrt{9}=3\) हैं। प्राप्त मानों का समुच्चय (f(A)) है।
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कौन सा संबंध \(A=\{a,b,c\}\) से \(B=\{1,2\}\) में फलन नहीं है?
Which relation from \(A=\{a,b,c\}\) to \(B=\{1,2\}\) is not a function?
#functions
#not-function
#duplicate-input
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A ({(a,1),(b,2),(c,1),(c,2)})
B ({(a,1),(b,1),(c,2)})
C ({(a,2),(b,2),(c,2)})
D ({(a,1),(b,2),(c,1)})
Explanation opens after your attempt
Correct Answer
A. ({(a,1),(b,2),(c,1),(c,2)})
Step 1
Concept
Here (c) has two images (1) and (2). One input with two different outputs is not allowed in a function.
Step 2
Why this answer is correct
The correct answer is A. ({(a,1),(b,2),(c,1),(c,2)}). Here (c) has two images (1) and (2). One input with two different outputs is not allowed in a function.
Step 3
Exam Tip
यहां (c) के दो प्रतिबिंब (1) और (2) हैं। एक इनपुट के दो अलग आउटपुट फलन में मान्य नहीं हैं।
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यदि \(f:A\to B\) में \(A=\{1,2,3,4\}\) है तो (f) के ग्राफ में कितने क्रमित युग्म होंगे?
If \(f:A\to B\) and \(A=\{1,2,3,4\}\), how many ordered pairs will be in the graph of (f)?
#functions
#graph
#counting
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A (4)
B (2)
C (8)
D (1)
Explanation opens after your attempt
Step 1
Concept
The graph of a function has exactly one pair for each domain element. So the total number of pairs is (|A|=4).
Step 2
Why this answer is correct
The correct answer is A. (4). The graph of a function has exactly one pair for each domain element. So the total number of pairs is (|A|=4).
Step 3
Exam Tip
फलन के ग्राफ में प्रांत के हर तत्व के लिए ठीक एक युग्म होता है। इसलिए कुल युग्म (|A|=4) होंगे।
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किस विकल्प में \(A=\{3,6\}\) के प्रत्येक तत्व का ठीक एक प्रतिबिंब है?
In which option does each element of \(A=\{3,6\}\) have exactly one image?
#functions
#identify-function
#domain
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A ({(3,9),(6,12)})
B ({(3,9),(3,12)})
C ({(6,12)})
D ({(3,9),(6,12),(6,9)})
Explanation opens after your attempt
Correct Answer
A. ({(3,9),(6,12)})
Step 1
Concept
In the first option, (3) and (6) each appear exactly once as first components. This is the necessary condition for a function.
Step 2
Why this answer is correct
The correct answer is A. ({(3,9),(6,12)}). In the first option, (3) and (6) each appear exactly once as first components. This is the necessary condition for a function.
Step 3
Exam Tip
पहले विकल्प में (3) और (6) दोनों ठीक एक बार प्रथम अवयव हैं। यही फलन की आवश्यक शर्त है।
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यदि (f(x)=x-2 -1) है तो (f(3)) का मान क्या है?
If (f(x)=x-2 -1), what is the value of (f(3))?
#functions
#evaluation
#quadratic
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A (8)
B (6)
C (9)
D (10)
Explanation opens after your attempt
Step 1
Concept
(f(3)=32 -1=8). Evaluate the power first and then subtract.
Step 2
Why this answer is correct
The correct answer is A. (8). (f(3)=32 -1=8). Evaluate the power first and then subtract.
Step 3
Exam Tip
(f(3)=32 -1=8) है। पहले घात निकालें फिर घटाएं।
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यदि (g(x)=\frac{x}{2}) और (x=8) है तो (g(8)) क्या होगा?
If (g(x)=\frac{x}{2}) and (x=8), what is (g(8))?
#functions
#evaluation
#fraction
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A (4)
B (8)
C (16)
D (2)
Explanation opens after your attempt
Step 1
Concept
(g(8)=\frac{8}{2}=4). To find a function value, put the given (x) in the rule.
Step 2
Why this answer is correct
The correct answer is A. (4). (g(8)=\frac{8}{2}=4). To find a function value, put the given (x) in the rule.
Step 3
Exam Tip
(g(8)=\frac{8}{2}=4) है। फलन मान निकालते समय दिए गए (x) को नियम में रखें।
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फलन \(f=\{(2,7),(4,9),(6,11)\}\) का परिसर क्या है?
What is the range of the function \(f=\{(2,7),(4,9),(6,11)\}\)?
#functions
#range
#ordered-pairs
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A ({7,9,11})
B ({2,4,6})
C ({2,7,11})
D ({4,9})
Explanation opens after your attempt
Correct Answer
A. ({7,9,11})
Step 1
Concept
The range is the set of second components of ordered pairs. Hence the range is ({7,9,11}).
Step 2
Why this answer is correct
The correct answer is A. ({7,9,11}). The range is the set of second components of ordered pairs. Hence the range is ({7,9,11}).
Step 3
Exam Tip
परिसर क्रमित युग्मों के द्वितीय अवयवों का समुच्चय है। इसलिए परिसर ({7,9,11}) है।
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फलन \(f=\{(5,1),(6,1),(7,2)\}\) का प्रांत क्या है?
What is the domain of the function \(f=\{(5,1),(6,1),(7,2)\}\)?
#functions
#domain
#ordered-pairs
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A ({5,6,7})
B ({1,2})
C ({5,1,2})
D ({6,7,1})
Explanation opens after your attempt
Correct Answer
A. ({5,6,7})
Step 1
Concept
The domain is formed by all first components. Here the first components are (5), (6), and (7).
Step 2
Why this answer is correct
The correct answer is A. ({5,6,7}). The domain is formed by all first components. Here the first components are (5), (6), and (7).
Step 3
Exam Tip
प्रांत सभी प्रथम अवयवों से बनता है। यहां प्रथम अवयव (5), (6), और (7) हैं।
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यदि \(A=\{1,2,3\}\) और \(B=\{0,1\}\) है तो (A) से (B) में कुल फलनों की संख्या कितनी है?
If \(A=\{1,2,3\}\) and \(B=\{0,1\}\), how many functions are there from (A) to (B)?
#functions
#counting-functions
#finite-sets
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A (8)
B (6)
C (5)
D (3)
Explanation opens after your attempt
Step 1
Concept
The number of functions is \(|B|^{|A|}=2^3=8\). The base is the number of elements in the codomain.
Step 2
Why this answer is correct
The correct answer is A. (8). The number of functions is \(|B|^{|A|}=2^3=8\). The base is the number of elements in the codomain.
Step 3
Exam Tip
कुल फलनों की संख्या \(|B|^{|A|}=2^3=8\) होती है। आधार सहप्रांत के तत्वों की संख्या होती है।
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यदि (|A|=2) और (|B|=4) है तो (A) से (B) में कुल फलन कितने होंगे?
If (|A|=2) and (|B|=4), how many functions are there from (A) to (B)?
#functions
#counting
#formula
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A \(4^2\)
B \(2^4\)
C (4+2)
D (4-2)
Explanation opens after your attempt
Correct Answer
A. \(4^2\)
Step 1
Concept
By the formula \(|B|^{|A|}\), the answer is \(4^2\). The number of domain elements goes in the exponent.
Step 2
Why this answer is correct
The correct answer is A. \(4^2\). By the formula \(|B|^{|A|}\), the answer is \(4^2\). The number of domain elements goes in the exponent.
Step 3
Exam Tip
सूत्र \(|B|^{|A|}\) से उत्तर \(4^2\) है। प्रांत की संख्या घात में जाती है।
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कौन सा कथन \(f:A\to B\) के लिए सही है?
Which statement is correct for \(f:A\to B\)?
#functions
#notation
#domain-codomain
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A हर \(x\in A\) के लिए \(f(x)\in B\) होता है / For every \(x\in A\), \(f(x)\in B\)
B हर \(x\in B\) के लिए \(f(x)\in A\) होता है / For every \(x\in B\), \(f(x)\in A\)
C (A) हमेशा \(\emptyset\) होता है / (A) is always \(\emptyset\)
D (B) हमेशा (A) के बराबर होता है / (B) is always equal to (A)
Explanation opens after your attempt
Correct Answer
A. हर \(x\in A\) के लिए \(f(x)\in B\) होता है / For every \(x\in A\), \(f(x)\in B\)
Step 1
Concept
In \(f:A\to B\), the input comes from (A) and the output lies in (B). Reading the notation is very important in exams.
Step 2
Why this answer is correct
The correct answer is A. हर \(x\in A\) के लिए \(f(x)\in B\) होता है / For every \(x\in A\), \(f(x)\in B\). In \(f:A\to B\), the input comes from (A) and the output lies in (B). Reading the notation is very important in exams.
Step 3
Exam Tip
\(f:A\to B\) में इनपुट (A) से और आउटपुट (B) में होता है। संकेत पढ़ना परीक्षा में बहुत जरूरी है।
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यदि \(R\subseteq A\times B\) है और (A) का कोई तत्व किसी भी तत्व से नहीं जुड़ा है तो (R) फलन क्यों नहीं होगा?
If \(R\subseteq A\times B\) and some element of (A) is not related to any element, why will (R) not be a function?
#functions
#existence-condition
#not-function
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A क्योंकि हर \(x\in A\) का प्रतिबिंब होना चाहिए / Because every \(x\in A\) must have an image
B क्योंकि हर \(y\in B\) का प्रतिबिंब होना चाहिए / Because every \(y\in B\) must have an image
C क्योंकि \(A\times B\) खाली होना चाहिए / Because \(A\times B\) must be empty
D क्योंकि सभी युग्म समान होने चाहिए / Because all pairs must be same
Explanation opens after your attempt
Correct Answer
A. क्योंकि हर \(x\in A\) का प्रतिबिंब होना चाहिए / Because every \(x\in A\) must have an image
Step 1
Concept
No element of the domain can be left out in a function. Every \(x\in A\) must have exactly one image.
Step 2
Why this answer is correct
The correct answer is A. क्योंकि हर \(x\in A\) का प्रतिबिंब होना चाहिए / Because every \(x\in A\) must have an image. No element of the domain can be left out in a function. Every \(x\in A\) must have exactly one image.
Step 3
Exam Tip
फलन में प्रांत का कोई तत्व छूट नहीं सकता। हर \(x\in A\) का ठीक एक प्रतिबिंब जरूरी है।
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यदि \((p,5)\in f\) और \((p,7)\in f\) हैं तो (f) फलन कब हो सकता है?
If \((p,5)\in f\) and \((p,7)\in f\), when can (f) be a function?
#functions
#uniqueness
#reasoning
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A जब (5=7) हो / When (5=7)
B जब (p=5) हो / When (p=5)
C जब (p=7) हो / When (p=7)
D जब \(p\notin A\) हो / When \(p\notin A\)
Explanation opens after your attempt
Correct Answer
A. जब (5=7) हो / When (5=7)
Step 1
Concept
Two values for the same input (p) are allowed only if they are the same. Here \(5\ne7\), so generally it is not a function.
Step 2
Why this answer is correct
The correct answer is A. जब (5=7) हो / When (5=7). Two values for the same input (p) are allowed only if they are the same. Here \(5\ne7\), so generally it is not a function.
Step 3
Exam Tip
एक ही इनपुट (p) के दो मान तभी मान्य होंगे जब वे समान हों। यहां \(5\ne7\), इसलिए सामान्यतः यह फलन नहीं है।
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किस विकल्प में (R) संबंध तो है पर फलन नहीं है?
In which option is (R) a relation but not a function?
#functions
#relation-not-function
#ordered-pairs
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A \(R=\{(1,2),(1,3)\}\)
B \(R=\{(1,2),(2,3)\}\)
C \(R=\{(1,1),(2,2)\}\)
D \(R=\{(3,4),(4,4)\}\)
Explanation opens after your attempt
Correct Answer
A. \(R=\{(1,2),(1,3)\}\)
Step 1
Concept
In \(R=\{(1,2),(1,3)\}\), (1) is related to two different values. So it is a relation but not a function.
Step 2
Why this answer is correct
The correct answer is A. \(R=\{(1,2),(1,3)\}\). In \(R=\{(1,2),(1,3)\}\), (1) is related to two different values. So it is a relation but not a function.
Step 3
Exam Tip
\(R=\{(1,2),(1,3)\}\) में (1) दो अलग मानों से जुड़ा है। इसलिए यह संबंध है लेकिन फलन नहीं है।
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यदि \(I_A\) तत्समक फलन है और \(A=\{4,5\}\) है तो (I_A(5)) क्या है?
If \(I_A\) is the identity function and \(A=\{4,5\}\), what is (I_A(5))?
#functions
#identity-function
#evaluation
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A (5)
B (4)
C (0)
D (1)
Explanation opens after your attempt
Step 1
Concept
In the identity function, (I_A(x)=x). Therefore (I_A(5)=5).
Step 2
Why this answer is correct
The correct answer is A. (5). In the identity function, (I_A(x)=x). Therefore (I_A(5)=5).
Step 3
Exam Tip
तत्समक फलन में (I_A(x)=x) होता है। इसलिए (I_A(5)=5) है।
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यदि \(A=\{0,1,2\}\) है तो (A) पर तत्समक फलन कौन सा है?
If \(A=\{0,1,2\}\), which is the identity function on (A)?
#functions
#identity-function
#finite-set
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A ({(0,0),(1,1),(2,2)})
B ({(0,1),(1,2),(2,0)})
C ({(0,0),(1,0),(2,0)})
D ({(0,2),(1,1),(2,0)})
Explanation opens after your attempt
Correct Answer
A. ({(0,0),(1,1),(2,2)})
Step 1
Concept
In an identity function, every 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. ({(0,0),(1,1),(2,2)}). In an identity function, every element maps to itself. Therefore all pairs are of the form ((x,x)).
Step 3
Exam Tip
तत्समक फलन में प्रत्येक तत्व स्वयं से जुड़ता है। इसलिए सभी युग्म ((x,x)) रूप में हैं।
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यदि (f(x)=10) सभी \(x\in A\) के लिए है तो (f) का परिसर क्या होगा?
If (f(x)=10) for all \(x\in A\), what will be the range of (f)?
#functions
#constant-function
#range
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A ({10})
B (A)
C \(\emptyset\)
D \({x:x\in A}\)
Explanation opens after your attempt
Step 1
Concept
All inputs have value (10), so only (10) is obtained. The range does not list repetitions.
Step 2
Why this answer is correct
The correct answer is A. ({10}). All inputs have value (10), so only (10) is obtained. The range does not list repetitions.
Step 3
Exam Tip
सभी इनपुट का मान (10) है इसलिए केवल (10) प्राप्त होता है। परिसर दोहराव नहीं लिखता।
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कौन सा विकल्प \(A=\{1,2,3\}\) से \(B=\{a\}\) में संभव फलन है?
Which option is a possible function from \(A=\{1,2,3\}\) to \(B=\{a\}\)?
#functions
#singleton-codomain
#constant
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A ({(1,a),(2,a),(3,a)})
B ({(1,a),(2,a)})
C ({(1,a),(1,a),(2,a)})
D ({(a,1),(a,2),(a,3)})
Explanation opens after your attempt
Correct Answer
A. ({(1,a),(2,a),(3,a)})
Step 1
Concept
Since (B) has only (a), every domain element maps to (a). All (1), (2), and (3) must be included.
Step 2
Why this answer is correct
The correct answer is A. ({(1,a),(2,a),(3,a)}). Since (B) has only (a), every domain element maps to (a). All (1), (2), and (3) must be included.
Step 3
Exam Tip
(B) में केवल (a) है इसलिए हर प्रांत तत्व (a) से जुड़ता है। सभी (1), (2), और (3) शामिल होने चाहिए।
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यदि \(f=\{(1,3),(2,5),(3,7)\}\) है तो कौन सा नियम इस फलन को दर्शाता है?
If \(f=\{(1,3),(2,5),(3,7)\}\), which rule represents this function?
#functions
#rule-identification
#linear
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A (f(x)=2x+1)
B (f(x)=x+1)
C (f(x)=x-2 )
D (f(x)=3x)
Explanation opens after your attempt
Correct Answer
A. (f(x)=2x+1)
Step 1
Concept
Putting (x=1,2,3) in (2x+1) gives (3,5,7). Check all pairs to identify the rule.
Step 2
Why this answer is correct
The correct answer is A. (f(x)=2x+1). Putting (x=1,2,3) in (2x+1) gives (3,5,7). Check all pairs to identify the rule.
Step 3
Exam Tip
(x=1,2,3) रखने पर (2x+1) से (3,5,7) मिलते हैं। नियम पहचानने के लिए सभी युग्म जांचें।
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यदि \(f=\{(1,1),(2,8),(3,27)\}\) है तो कौन सा नियम सही है?
If \(f=\{(1,1),(2,8),(3,27)\}\), which rule is correct?
#functions
#rule-identification
#cube
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A (f(x)=x-3 )
B (f(x)=x-2 )
C (f(x)=x+3)
D (f(x)=3x)
Explanation opens after your attempt
Correct Answer
A. (f(x)=x-3 )
Step 1
Concept
\(1^3=1\), \(2^3=8\), and \(3^3=27\). Hence the rule is (f(x)=x-3 ).
Step 2
Why this answer is correct
The correct answer is A. (f(x)=x-3 ). \(1^3=1\), \(2^3=8\), and \(3^3=27\). Hence the rule is (f(x)=x-3 ).
Step 3
Exam Tip
\(1^3=1\), \(2^3=8\), और \(3^3=27\) हैं। इसलिए नियम (f(x)=x-3 ) है।
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यदि \(A=\{1,2,3\}\) और (f(x)=x+4) है तो फलन का ग्राफ कौन सा होगा?
If \(A=\{1,2,3\}\) and (f(x)=x+4), which will be the graph of the function?
#functions
#graph
#linear-rule
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A ({(1,5),(2,6),(3,7)})
B ({(1,4),(2,5),(3,6)})
C ({(5,1),(6,2),(7,3)})
D ({(1,1),(2,2),(3,3)})
Explanation opens after your attempt
Correct Answer
A. ({(1,5),(2,6),(3,7)})
Step 1
Concept
Applying (x+4), the values for (1,2,3) are (5,6,7). In the graph we write pairs ((x,f(x))).
Step 2
Why this answer is correct
The correct answer is A. ({(1,5),(2,6),(3,7)}). Applying (x+4), the values for (1,2,3) are (5,6,7). In the graph we write pairs ((x,f(x))).
Step 3
Exam Tip
(x+4) लगाने पर (1,2,3) के मान (5,6,7) हैं। ग्राफ में युग्म ((x,f(x))) लिखते हैं।
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यदि \(A=\{2,4,6\}\) और (f(x)=x-2) है तो (f(A)) क्या है?
If \(A=\{2,4,6\}\) and (f(x)=x-2), what is (f(A))?
#functions
#image-set
#subtraction-rule
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A ({0,2,4})
B ({2,4,6})
C ({4,6,8})
D ({0,1,2})
Explanation opens after your attempt
Correct Answer
A. ({0,2,4})
Step 1
Concept
(2-2=0), (4-2=2), and (6-2=4). This is (f(A)).
Step 2
Why this answer is correct
The correct answer is A. ({0,2,4}). (2-2=0), (4-2=2), and (6-2=4). This is (f(A)).
Step 3
Exam Tip
(2-2=0), (4-2=2), और (6-2=4) हैं। यही (f(A)) है।
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किस विकल्प में सभी प्रथम अवयव अलग हैं और कोई प्रांत तत्व नहीं छूटा है, इसलिए वह फलन है?
Which option has all first components distinct and no domain element missing, so it is a function?
#functions
#identify-function
#first-components
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A ({(1,9),(2,8),(3,7)})
B ({(1,9),(1,8),(3,7)})
C ({(1,9),(2,8)}) जब प्रांत ({1,2,3}) है / ({(1,9),(2,8)}) when domain is ({1,2,3})
D ({(2,8),(3,7),(3,9)})
Explanation opens after your attempt
Correct Answer
A. ({(1,9),(2,8),(3,7)})
Step 1
Concept
In the first option, (1), (2), and (3) each appear exactly once. A function needs exactly one output for each input.
Step 2
Why this answer is correct
The correct answer is A. ({(1,9),(2,8),(3,7)}). In the first option, (1), (2), and (3) each appear exactly once. A function needs exactly one output for each input.
Step 3
Exam Tip
पहले विकल्प में (1), (2), और (3) सभी ठीक एक बार आए हैं। फलन के लिए प्रत्येक इनपुट का एक ही आउटपुट होना चाहिए।
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यदि \(f:A\to B\) एक फलन है और \(a\in A\), तो (a) के लिए कितने (f(a)) हो सकते हैं?
If \(f:A\to B\) is a function and \(a\in A\), how many values of (f(a)) can there be?
#functions
#uniqueness
#basic
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A ठीक (1) / Exactly (1)
B (0)
C कम से कम (2) / At least (2)
D अनंत / Infinite
Explanation opens after your attempt
Correct Answer
A. ठीक (1) / Exactly (1)
Step 1
Concept
In a function, every input has exactly one value. This is the uniqueness condition.
Step 2
Why this answer is correct
The correct answer is A. ठीक (1) / Exactly (1). In a function, every input has exactly one value. This is the uniqueness condition.
Step 3
Exam Tip
फलन में हर इनपुट का ठीक एक मान होता है। यही अद्वितीयता की शर्त है।
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यदि \(A=\{1,2\}\) और \(B=\{3,4,5\}\) है तो \(A\times B\) के किस उपसमुच्चय से फलन बनता है?
If \(A=\{1,2\}\) and \(B=\{3,4,5\}\), which subset of \(A\times B\) forms a function?
#functions
#cartesian-product
#subset
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A ({(1,3),(2,5)})
B ({(1,3),(1,4)})
C ({(2,3),(2,4)})
D ({(1,3),(2,4),(2,5)})
Explanation opens after your attempt
Correct Answer
A. ({(1,3),(2,5)})
Step 1
Concept
In the first subset, (1) and (2) each have one image. In the others an input has two different values or is missing.
Step 2
Why this answer is correct
The correct answer is A. ({(1,3),(2,5)}). In the first subset, (1) and (2) each have one image. In the others an input has two different values or is missing.
Step 3
Exam Tip
पहले उपसमुच्चय में (1) और (2) का एक-एक प्रतिबिंब है। बाकी में कोई इनपुट दो अलग मानों से जुड़ा है या छूट गया है।
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यदि \(f=\{(1,2),(2,4),(3,6),(4,8)\}\) है तो (f(4)) क्या होगा?
If \(f=\{(1,2),(2,4),(3,6),(4,8)\}\), what is (f(4))?
#functions
#function-value
#table
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A (8)
B (4)
C (6)
D (2)
Explanation opens after your attempt
Step 1
Concept
The pair ((4,8)) shows that the image of (4) is (8). Choose the pair with the related first component.
Step 2
Why this answer is correct
The correct answer is A. (8). The pair ((4,8)) shows that the image of (4) is (8). Choose the pair with the related first component.
Step 3
Exam Tip
युग्म ((4,8)) बताता है कि (4) का प्रतिबिंब (8) है। संबंधित प्रथम अवयव वाला युग्म चुनें।
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नीचे दिए गए किस संबंध में (2) का कोई प्रतिबिंब नहीं है, इसलिए वह \(A=\{1,2,3\}\) से फलन नहीं है?
In which relation does (2) have no image, so it is not a function from \(A=\{1,2,3\}\)?
#functions
#missing-image
#not-function
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A ({(1,a),(3,b)})
B ({(1,a),(2,a),(3,b)})
C ({(1,b),(2,b),(3,b)})
D ({(1,a),(2,b),(3,a)})
Explanation opens after your attempt
Correct Answer
A. ({(1,a),(3,b)})
Step 1
Concept
In the first option, (2) does not appear as a first component. Every element of the domain must be included.
Step 2
Why this answer is correct
The correct answer is A. ({(1,a),(3,b)}). In the first option, (2) does not appear as a first component. Every element of the domain must be included.
Step 3
Exam Tip
पहले विकल्प में (2) प्रथम अवयव के रूप में नहीं आया है। प्रांत का हर तत्व शामिल होना चाहिए।
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यदि (f(x)=2x-3) है तो (f(5)) क्या है?
If (f(x)=2x-3), what is (f(5))?
#functions
#evaluation
#linear
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A (7)
B (10)
C (5)
D (3)
Explanation opens after your attempt
Step 1
Concept
(f(5)=2\cdot5-3=7). In a linear rule, multiply first.
Step 2
Why this answer is correct
The correct answer is A. (7). (f(5)=2\cdot5-3=7). In a linear rule, multiply first.
Step 3
Exam Tip
(f(5)=2\cdot5-3=7) है। रैखिक नियम में गुणा पहले करें।
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यदि (f(x)=x-2 ) और (f(a)=16) है, सरल धनात्मक मान के लिए (a) क्या होगा?
If (f(x)=x-2 ) and (f(a)=16), what is (a) for the simple positive value?
#functions
#evaluation
#preimage-basic
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A (4)
B (8)
C (2)
D (16)
Explanation opens after your attempt
Step 1
Concept
For the positive value, \(a^2=16\) gives (a=4). Read the positive condition carefully.
Step 2
Why this answer is correct
The correct answer is A. (4). For the positive value, \(a^2=16\) gives (a=4). Read the positive condition carefully.
Step 3
Exam Tip
धनात्मक मान के लिए \(a^2=16\) से (a=4) मिलता है। प्रश्न में धनात्मक शर्त को ध्यान से पढ़ें।
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यदि \(f=\{(1,b),(2,c),(3,b)\}\) है तो किस तत्व का प्रतिबिंब (c) है?
If \(f=\{(1,b),(2,c),(3,b)\}\), which element has image (c)?
#functions
#preimage
#ordered-pairs
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A (2)
B (1)
C (3)
D (b)
Explanation opens after your attempt
Step 1
Concept
In the pair ((2,c)), the image of (2) is (c). Identify the first component from the second component.
Step 2
Why this answer is correct
The correct answer is A. (2). In the pair ((2,c)), the image of (2) is (c). Identify the first component from the second component.
Step 3
Exam Tip
युग्म ((2,c)) में (2) का प्रतिबिंब (c) है। द्वितीय अवयव से प्रथम अवयव पहचानें।
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फलन के संदर्भ में प्रतिबिंब किसे कहते हैं?
In the context of a function, what is an image?
#functions
#image
#terminology
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A किसी \(x\in A\) के लिए (f(x)) / (f(x)) for some \(x\in A\)
B केवल (A) का नाम / Only the name of (A)
C केवल (B) की संख्या / Only the number of (B)
D कोई असंबंधित युग्म / An unrelated pair
Explanation opens after your attempt
Correct Answer
A. किसी \(x\in A\) के लिए (f(x)) / (f(x)) for some \(x\in A\)
Step 1
Concept
If (x) is an input, then (f(x)) is its image. The image always lies inside the codomain.
Step 2
Why this answer is correct
The correct answer is A. किसी \(x\in A\) के लिए (f(x)) / (f(x)) for some \(x\in A\). If (x) is an input, then (f(x)) is its image. The image always lies inside the codomain.
Step 3
Exam Tip
यदि (x) इनपुट है तो (f(x)) उसका प्रतिबिंब है। प्रतिबिंब हमेशा सहप्रांत के अंदर होता है।
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फलन \(f:A\to B\) में पूर्व प्रतिबिंब किससे संबंधित होता है?
In a function \(f:A\to B\), a preimage is related to what?
#functions
#preimage
#terminology
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A प्रांत (A) के तत्व से / An element of domain (A)
B केवल सहप्रांत (B) से / Only codomain (B)
C केवल परिसर से / Only range
D कार्टेशियन गुणनफल के आकार से / Size of Cartesian product
Explanation opens after your attempt
Correct Answer
A. प्रांत (A) के तत्व से / An element of domain (A)
Step 1
Concept
A preimage is the input that maps to an output. It is an element of the domain (A).
Step 2
Why this answer is correct
The correct answer is A. प्रांत (A) के तत्व से / An element of domain (A). A preimage is the input that maps to an output. It is an element of the domain (A).
Step 3
Exam Tip
पूर्व प्रतिबिंब वह इनपुट है जो किसी आउटपुट तक जाता है। यह प्रांत (A) का तत्व होता है।
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कौन सा विकल्प \(f:A\to B\) के परिसर के बारे में सही है?
Which option is correct about the range of \(f:A\to B\)?
#functions
#range
#codomain
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A परिसर (B) का उपसमुच्चय होता है / The range is a subset of (B)
B परिसर हमेशा (A) के बराबर होता है / The range is always equal to (A)
C परिसर हमेशा \(A\times B\) होता है / The range is always \(A\times B\)
D परिसर हमेशा \(\emptyset\) होता है / The range is always \(\emptyset\)
Explanation opens after your attempt
Correct Answer
A. परिसर (B) का उपसमुच्चय होता है / The range is a subset of (B)
Step 1
Concept
The obtained values of the function lie in (B), so the range is a subset of (B). It need not be the whole of (B).
Step 2
Why this answer is correct
The correct answer is A. परिसर (B) का उपसमुच्चय होता है / The range is a subset of (B). The obtained values of the function lie in (B), so the range is a subset of (B). It need not be the whole of (B).
Step 3
Exam Tip
फलन के प्राप्त मान (B) में होते हैं, इसलिए परिसर (B) का उपसमुच्चय होता है। यह जरूरी नहीं कि पूरा (B) हो।
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यदि \(A=\{1,2\}\), \(B=\{3,4\}\), और \(f=\{(1,3),(2,3)\}\) है तो परिसर क्या है?
If \(A=\{1,2\}\), \(B=\{3,4\}\), and \(f=\{(1,3),(2,3)\}\), what is the range?
#functions
#range
#many-one
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A ({3})
B ({3,4})
C ({1,2})
D ({4})
Explanation opens after your attempt
Step 1
Concept
Both inputs have obtained value (3). Therefore (3) is written only once in the range.
Step 2
Why this answer is correct
The correct answer is A. ({3}). Both inputs have obtained value (3). Therefore (3) is written only once in the range.
Step 3
Exam Tip
दोनों इनपुट का प्राप्त मान (3) है। इसलिए परिसर में (3) केवल एक बार लिखा जाएगा।
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यदि \(f:A\to B\) और \(A=\emptyset\) है, तो (f) में कितने क्रमित युग्म होंगे?
If \(f:A\to B\) and \(A=\emptyset\), how many ordered pairs will (f) contain?
#functions
#empty-function
#ordered-pairs
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A (0)
B (1)
C (|B|)
D (2)
Explanation opens after your attempt
Step 1
Concept
An empty domain has no input, so the graph has no pair. Still it is considered the empty function.
Step 2
Why this answer is correct
The correct answer is A. (0). An empty domain has no input, so the graph has no pair. Still it is considered the empty function.
Step 3
Exam Tip
रिक्त प्रांत में कोई इनपुट नहीं है, इसलिए ग्राफ में कोई युग्म नहीं होगा। फिर भी यह रिक्त फलन माना जाता है।
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यदि \(A=\{1,2\}\) और \(B=\emptyset\) है तो (A) से (B) में कितने फलन बन सकते हैं?
If \(A=\{1,2\}\) and \(B=\emptyset\), how many functions can be formed from (A) to (B)?
#functions
#empty-codomain
#counting
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A (0)
B (1)
C (2)
D (4)
Explanation opens after your attempt
Step 1
Concept
Elements of a non-empty domain cannot get any image in \(\emptyset\). Therefore no function can be formed.
Step 2
Why this answer is correct
The correct answer is A. (0). Elements of a non-empty domain cannot get any image in \(\emptyset\). Therefore no function can be formed.
Step 3
Exam Tip
गैर-रिक्त प्रांत के तत्वों को \(\emptyset\) में कोई प्रतिबिंब नहीं मिल सकता। इसलिए कोई फलन नहीं बनेगा।
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यदि \(A=\{2\}\) और \(B=\{5,6\}\) है तो (A) से (B) में कितने फलन होंगे?
If \(A=\{2\}\) and \(B=\{5,6\}\), how many functions are there from (A) to (B)?
#functions
#counting
#singleton-domain
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A (2)
B (1)
C (0)
D (4)
Explanation opens after your attempt
Step 1
Concept
For one input (2), there are (2) choices in (B). Therefore the total number of functions is \(2^1=2\).
Step 2
Why this answer is correct
The correct answer is A. (2). For one input (2), there are (2) choices in (B). Therefore the total number of functions is \(2^1=2\).
Step 3
Exam Tip
एक इनपुट (2) के लिए (B) में (2) विकल्प हैं। इसलिए कुल \(2^1=2\) फलन हैं।
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कौन सा विकल्प बताता है कि फलन में दो अलग इनपुट एक ही आउटपुट रख सकते हैं?
Which option shows that two different inputs can have the same output in a function?
#functions
#many-one
#valid-function
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A ({(4,1),(5,1),(6,2)})
B ({(4,1),(4,2),(6,2)})
C ({(5,1),(5,2)})
D ({(4,1)}) जब प्रांत ({4,5}) है / ({(4,1)}) when domain is ({4,5})
Explanation opens after your attempt
Correct Answer
A. ({(4,1),(5,1),(6,2)})
Step 1
Concept
Both (4) and (5) have image (1), and every input has only one value. This is a valid many-one function.
Step 2
Why this answer is correct
The correct answer is A. ({(4,1),(5,1),(6,2)}). Both (4) and (5) have image (1), and every input has only one value. This is a valid many-one function.
Step 3
Exam Tip
(4) और (5) दोनों का प्रतिबिंब (1) है और हर इनपुट का केवल एक मान है। यह वैध बहु-एक फलन है।
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यदि किसी आरेख में (a) से (1) और (2) दोनों की ओर तीर हैं, तो वह फलन क्यों नहीं है?
If in a mapping diagram arrows from (a) go to both (1) and (2), why is it not a function?
#functions
#mapping-diagram
#not-function
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A क्योंकि (a) के दो प्रतिबिंब हैं / Because (a) has two images
B क्योंकि (1) और (2) सहप्रांत में हैं / Because (1) and (2) are in the codomain
C क्योंकि (a) प्रांत में है / Because (a) is in the domain
D क्योंकि कोई तीर नहीं है / Because there is no arrow
Explanation opens after your attempt
Correct Answer
A. क्योंकि (a) के दो प्रतिबिंब हैं / Because (a) has two images
Step 1
Concept
Two different arrows from one domain element break the function condition. Exactly one arrow must leave each input.
Step 2
Why this answer is correct
The correct answer is A. क्योंकि (a) के दो प्रतिबिंब हैं / Because (a) has two images. Two different arrows from one domain element break the function condition. Exactly one arrow must leave each input.
Step 3
Exam Tip
एक प्रांत तत्व से दो अलग तीर निकलना फलन की शर्त तोड़ता है। हर इनपुट से ठीक एक तीर होना चाहिए।
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यदि आरेख में \(a\to 3\), \(b\to 3\), और \(c\to 4\) है, तो यह फलन है या नहीं?
If a diagram has \(a\to 3\), \(b\to 3\), and \(c\to 4\), is it a function or not?
#functions
#mapping-diagram
#many-one
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A हाँ, यह फलन है / Yes, it is a function
B नहीं, क्योंकि दो इनपुट का समान आउटपुट है / No, because two inputs have same output
C नहीं, क्योंकि (3) दो बार आया है / No, because (3) appears twice
D नहीं, क्योंकि (4) अकेला है / No, because (4) is alone
Explanation opens after your attempt
Correct Answer
A. हाँ, यह फलन है / Yes, it is a function
Step 1
Concept
Each input (a), (b), and (c) has exactly one image. Two inputs having the same output is allowed.
Step 2
Why this answer is correct
The correct answer is A. हाँ, यह फलन है / Yes, it is a function. Each input (a), (b), and (c) has exactly one image. Two inputs having the same output is allowed.
Step 3
Exam Tip
हर इनपुट (a), (b), और (c) का ठीक एक प्रतिबिंब है। दो इनपुट का समान आउटपुट होना मान्य है।
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यदि (f(x)=x) सभी \(x\in A\) के लिए है तो (f) किस प्रकार का फलन है?
If (f(x)=x) for all \(x\in A\), what type of function is (f)?
#functions
#identity-function
#definition
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A तत्समक फलन / Identity function
B स्थिर फलन / Constant function
C रिक्त संबंध / Empty relation
D फलन नहीं / Not a function
Explanation opens after your attempt
Correct Answer
A. तत्समक फलन / Identity function
Step 1
Concept
Every element maps to itself, so it is the identity function. It is also denoted by \(I_A\).
Step 2
Why this answer is correct
The correct answer is A. तत्समक फलन / Identity function. Every element maps to itself, so it is the identity function. It is also denoted by \(I_A\).
Step 3
Exam Tip
हर तत्व स्वयं से जुड़ता है इसलिए यह तत्समक फलन है। इसे \(I_A\) से भी दर्शाते हैं।
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कौन सा विकल्प फलन की सही परिभाषा के सबसे निकट है?
Which option is closest to the correct definition of a function?
#functions
#definition
#basic
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A ऐसा संबंध जिसमें हर \(x\in A\) का ठीक एक \(y\in B\) हो / A relation in which every \(x\in A\) has exactly one \(y\in B\)
B ऐसा संबंध जिसमें हर \(x\in A\) के कई \(y\in B\) हों / A relation in which every \(x\in A\) has many \(y\in B\)
C ऐसा संबंध जिसमें (A) का कोई तत्व न हो / A relation in which (A) has no element
D ऐसा संबंध जिसमें केवल (B) लिखा हो / A relation in which only (B) is written
Explanation opens after your attempt
Correct Answer
A. ऐसा संबंध जिसमें हर \(x\in A\) का ठीक एक \(y\in B\) हो / A relation in which every \(x\in A\) has exactly one \(y\in B\)
Step 1
Concept
In a function, every \(x\in A\) is associated with exactly one element of (B). This is the key point of the definition.
Step 2
Why this answer is correct
The correct answer is A. ऐसा संबंध जिसमें हर \(x\in A\) का ठीक एक \(y\in B\) हो / A relation in which every \(x\in A\) has exactly one \(y\in B\). In a function, every \(x\in A\) is associated with exactly one element of (B). This is the key point of the definition.
Step 3
Exam Tip
फलन में हर \(x\in A\) को (B) के ठीक एक तत्व से जोड़ा जाता है। यही परिभाषा की मुख्य बात है।
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यदि \(R=\{(1,2),(2,3),(3,4)\}\), \(A=\{1,2,3\}\), और \(B=\{2,3,4,5\}\) है तो सही कथन क्या है?
If \(R={(1,2),(2,3),(3,4)\), \(A=\{1,2,3\}\), and \(B=\{2,3,4,5\}\), which statement is correct?
#functions
#codomain
#range
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A (R) (A) से (B) में फलन है / (R) is a function from (A) to (B)
B (R) फलन नहीं है क्योंकि (5) का पूर्व प्रतिबिंब नहीं है / (R) is not a function because (5) has no preimage
C (R) फलन नहीं है क्योंकि (1) छूट गया है / (R) is not a function because (1) is missing
D (R) (B) से (A) में फलन है / (R) is a function from (B) to (A)
Explanation opens after your attempt
Correct Answer
A. (R) (A) से (B) में फलन है / (R) is a function from (A) to (B)
Step 1
Concept
The elements (1), (2), and (3) of (A) each have exactly one image in (B). It is not necessary to use every element of the codomain.
Step 2
Why this answer is correct
The correct answer is A. (R) (A) से (B) में फलन है / (R) is a function from (A) to (B). The elements (1), (2), and (3) of (A) each have exactly one image in (B). It is not necessary to use every element of the codomain.
Step 3
Exam Tip
(A) के (1), (2), और (3) का ठीक एक प्रतिबिंब (B) में है। सहप्रांत के हर तत्व का प्रयोग होना जरूरी नहीं।
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यदि (f(x)=x+1) और प्रांत ({0,2,4}) है तो परिसर क्या है?
If (f(x)=x+1) and the domain is ({0,2,4}), what is the range?
#functions
#range
#evaluation
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A ({1,3,5})
B ({0,2,4})
C ({2,4,6})
D ({1,2,3})
Explanation opens after your attempt
Correct Answer
A. ({1,3,5})
Step 1
Concept
(0+1=1), (2+1=3), and (4+1=5). Hence the range is ({1,3,5}).
Step 2
Why this answer is correct
The correct answer is A. ({1,3,5}). (0+1=1), (2+1=3), and (4+1=5). Hence the range is ({1,3,5}).
Step 3
Exam Tip
(0+1=1), (2+1=3), और (4+1=5) हैं। इसलिए परिसर ({1,3,5}) है।
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किस संबंध का ग्राफ (y=x+2) नियम से \(A=\{1,3\}\) पर बनता है?
Which relation is the graph of the rule (y=x+2) on \(A=\{1,3\}\)?
#functions
#graph
#linear-rule
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A ({(1,3),(3,5)})
B ({(1,2),(3,2)})
C ({(3,1),(5,3)})
D ({(1,1),(3,3)})
Explanation opens after your attempt
Correct Answer
A. ({(1,3),(3,5)})
Step 1
Concept
For (x=1), (y=3), and for (x=3), (y=5). Remember the graph is written in the form ((x,y)).
Step 2
Why this answer is correct
The correct answer is A. ({(1,3),(3,5)}). For (x=1), (y=3), and for (x=3), (y=5). Remember the graph is written in the form ((x,y)).
Step 3
Exam Tip
(x=1) पर (y=3) और (x=3) पर (y=5) है। ग्राफ में ((x,y)) रूप का ध्यान रखें।
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यदि \(f:A\to B\) में \(A=\{m,n\}\) और \(B=\{r,s,t\}\) है तो किसी फलन में (m) के लिए कितने विकल्प हैं?
If \(f:A\to B\), \(A=\{m,n\}\), and \(B=\{r,s,t\}\), how many choices are there for (m) in a function?
#functions
#counting
#choices
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A (3)
B (2)
C (1)
D (6)
Explanation opens after your attempt
Step 1
Concept
The image of (m) can be any one of the (3) elements of (B). In counting, each domain element gets as many choices as the codomain size.
Step 2
Why this answer is correct
The correct answer is A. (3). The image of (m) can be any one of the (3) elements of (B). In counting, each domain element gets as many choices as the codomain size.
Step 3
Exam Tip
(m) का प्रतिबिंब (B) के किसी भी (3) तत्वों में से एक हो सकता है। गिनती में हर प्रांत तत्व के लिए सहप्रांत की संख्या विकल्प देती है।
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फलन की जांच में सबसे पहले किस बात को देखना चाहिए?
What should be checked first while testing a function?
#functions
#exam-tip
#function-test
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A हर प्रांत तत्व ठीक एक बार इनपुट के रूप में आया है या नहीं / Whether each domain element appears exactly once as an input
B सहप्रांत के सभी तत्व बराबर हैं या नहीं / Whether all codomain elements are equal
C परिसर खाली है या नहीं / Whether the range is empty
D सभी आउटपुट अलग हैं या नहीं / Whether all outputs are different
Explanation opens after your attempt
Correct Answer
A. हर प्रांत तत्व ठीक एक बार इनपुट के रूप में आया है या नहीं / Whether each domain element appears exactly once as an input
Step 1
Concept
For a function, every domain element must have exactly one image. So checking first components is most useful.
Step 2
Why this answer is correct
The correct answer is A. हर प्रांत तत्व ठीक एक बार इनपुट के रूप में आया है या नहीं / Whether each domain element appears exactly once as an input. For a function, every domain element must have exactly one image. So checking first components is most useful.
Step 3
Exam Tip
फलन के लिए हर प्रांत तत्व का ठीक एक प्रतिबिंब जरूरी है। इसलिए प्रथम अवयवों की जांच सबसे उपयोगी होती है।
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यदि A={4,6,8} और B={2,3,4} है, तो नियम f(x)=
2
x
से कौन सा फलन बनेगा?
If A={4,6,8} and B={2,3,4}, which function is formed by the rule f(x)=
2
x
?
#functions
#rule
#ordered-pairs
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A f={(4,2),(6,3),(8,4)}
B f={(4,3),(6,2),(8,4)}
C f={(2,4),(3,6),(4,8)}
D f={(4,4),(6,6),(8,8)}
Explanation opens after your attempt
Correct Answer
A. f={(4,2),(6,3),(8,4)}
Step 1
Concept
Applying
2
x
to each x∈A gives 2,3,4. In rule based questions do not reverse input and output.
Step 2
Why this answer is correct
The correct answer is A. f={(4,2),(6,3),(8,4)}. Applying
2
x
to each x∈A gives 2,3,4. In rule based questions do not reverse input and output.
Step 3
Exam Tip
हर x∈A पर
2
x
लगाने से 2,3,4 मिलते हैं। नियम आधारित प्रश्नों में इनपुट और आउटपुट का क्रम न बदलें।
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यदि \(A=\{1,2\}\) और \(B=\{a,b\}\) हों, तो संबंध \(R=\{(1,a),(2,b)\}\) को (A) से (B) में फलन क्यों कहा जाएगा?
If \(A=\{1,2\}\) and \(B=\{a,b\}\), why is the relation \(R=\{(1,a),(2,b)\}\) called a function from (A) to (B)?
#relations-functions
#function-as-relation
#class-11-math
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A क्योंकि (A) के प्रत्येक तत्व की (B) में ठीक एक छवि है / Because every element of (A) has exactly one image in (B)
B क्योंकि (B) के प्रत्येक तत्व की (A) में दो छवियां हैं / Because every element of (B) has two images in (A)
C क्योंकि (1) की दो छवियां (a) और (b) हैं / Because (1) has two images (a) and (b)
D क्योंकि (2) की कोई छवि नहीं है / Because (2) has no image
Explanation opens after your attempt
Correct Answer
A. क्योंकि (A) के प्रत्येक तत्व की (B) में ठीक एक छवि है / Because every element of (A) has exactly one image in (B)
Step 1
Concept
Each element (1) and (2) of (A) has exactly one image in (B). In exams, first count the image of every domain element to test a function.
Step 2
Why this answer is correct
The correct answer is A. क्योंकि (A) के प्रत्येक तत्व की (B) में ठीक एक छवि है / Because every element of (A) has exactly one image in (B). Each element (1) and (2) of (A) has exactly one image in (B). In exams, first count the image of every domain element to test a function.
Step 3
Exam Tip
प्रत्येक (A) के तत्व (1) और (2) की (B) में ठीक एक-एक छवि है। परीक्षा में फलन जांचते समय पहले डोमेन के हर तत्व की छवि गिनें।
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यदि R={(2,5),(4,5),(6,7)} है, तो R के बारे में सही कथन क्या है?
If R={(2,5),(4,5),(6,7)}, which statement about R is correct?
#functions
#many-one
#valid-function
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A R फलन है / R is a function
B R फलन नहीं है / R is not a function
C R में 2 के दो प्रतिबिंब हैं / 2 has two images in R
D R में कोई क्रमित युग्म नहीं है / R has no ordered pair
Explanation opens after your attempt
Correct Answer
A. R फलन है / R is a function
Step 1
Concept
Each first component 2,4,6 appears exactly once. Two inputs having the same output 5 is allowed in a function.
Step 2
Why this answer is correct
The correct answer is A. R फलन है / R is a function. Each first component 2,4,6 appears exactly once. Two inputs having the same output 5 is allowed in a function.
Step 3
Exam Tip
हर प्रथम अवयव 2,4,6 ठीक एक बार आया है। दो इनपुट का समान आउटपुट 5 होना फलन में सही है।
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कौन सा संबंध A={5,7} से B={1,2,3} में फलन नहीं है?
Which relation from A={5,7} to B={1,2,3} is not a function?
#functions
#not-function
#duplicate-input
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A {(5,1),(7,2),(7,3)}
B {(5,1),(7,1)}
C {(5,2),(7,3)}
D {(5,3),(7,2)}
Explanation opens after your attempt
Correct Answer
A. {(5,1),(7,2),(7,3)}
Step 1
Concept
In the first option, 7 has two different images 2 and 3. One input with two different outputs does not form a function.
Step 2
Why this answer is correct
The correct answer is A. {(5,1),(7,2),(7,3)}. In the first option, 7 has two different images 2 and 3. One input with two different outputs does not form a function.
Step 3
Exam Tip
पहले विकल्प में 7 के दो अलग प्रतिबिंब 2 और 3 हैं। एक इनपुट के दो अलग आउटपुट फलन नहीं बनाते।
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यदि f={(−1,1),(0,0),(1,1)} है, तो f(−1) का मान क्या है?
If f={(−1,1),(0,0),(1,1)}, what is the value of f(−1)?
#functions
#function-value
#negative-input
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A 1
B 0
C −1
D 2
Explanation opens after your attempt
Step 1
Concept
From the pair (−1,1), the image of −1 is 1. Read negative inputs the same way as positive inputs.
Step 2
Why this answer is correct
The correct answer is A. 1. From the pair (−1,1), the image of −1 is 1. Read negative inputs the same way as positive inputs.
Step 3
Exam Tip
युग्म (−1,1) से −1 का प्रतिबिंब 1 है। ऋणात्मक इनपुट को भी उसी तरह पढ़ें जैसे धनात्मक इनपुट को पढ़ते हैं।
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यदि f(x)=x
2
+2 और x=2 है, तो f(2) क्या होगा?
If f(x)=x
2
+2 and x=2, what is f(2)?
#functions
#evaluation
#quadratic
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A 6
B 4
C 8
D 2
Explanation opens after your attempt
Step 1
Concept
f(2)=2
2
+2=6. In function evaluation, solve the power before addition.
Step 2
Why this answer is correct
The correct answer is A. 6. f(2)=2
2
+2=6. In function evaluation, solve the power before addition.
Step 3
Exam Tip
f(2)=2
2
+2=6 है। फलन मान में घात को जोड़ से पहले हल करें।
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फलन f={(3,10),(5,12),(7,14)} का प्रांत क्या है?
What is the domain of the function f={(3,10),(5,12),(7,14)}?
#functions
#domain
#ordered-pairs
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A {3,5,7}
B {10,12,14}
C {3,10,14}
D {5,12}
Explanation opens after your attempt
Correct Answer
A. {3,5,7}
Step 1
Concept
The domain is formed by the first components of ordered pairs. Hence the domain is {3,5,7}.
Step 2
Why this answer is correct
The correct answer is A. {3,5,7}. The domain is formed by the first components of ordered pairs. Hence the domain is {3,5,7}.
Step 3
Exam Tip
प्रांत क्रमित युग्मों के प्रथम अवयवों से बनता है। इसलिए प्रांत {3,5,7} है।
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फलन g={(1,4),(2,6),(3,8),(4,10)} का परिसर क्या है?
What is the range of the function g={(1,4),(2,6),(3,8),(4,10)}?
#functions
#range
#ordered-pairs
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A {4,6,8,10}
B {1,2,3,4}
C {1,4,10}
D {2,6,8}
Explanation opens after your attempt
Correct Answer
A. {4,6,8,10}
Step 1
Concept
The range is the set of all second components. Here the obtained values are 4,6,8,10.
Step 2
Why this answer is correct
The correct answer is A. {4,6,8,10}. The range is the set of all second components. Here the obtained values are 4,6,8,10.
Step 3
Exam Tip
परिसर सभी द्वितीय अवयवों का समुच्चय है। यहां प्राप्त मान 4,6,8,10 हैं।
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यदि A={1,2,3,4} और B={0,1} है, तो A से B में कुल फलनों की संख्या क्या है?
If A={1,2,3,4} and B={0,1}, what is the total number of functions from A to B?
#functions
#counting-functions
#formula
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A 2 4
B 4 2
C 2+4
D 4−2
Explanation opens after your attempt
Step 1
Concept
The number of functions is ∣B∣
∣A∣
=2
4
. Remember that the base comes from the codomain and the exponent from the domain.
Step 2
Why this answer is correct
The correct answer is A. 2 4. The number of functions is ∣B∣
∣A∣
=2
4
. Remember that the base comes from the codomain and the exponent from the domain.
Step 3
Exam Tip
कुल फलनों की संख्या ∣B∣
∣A∣
=2
4
होती है। याद रखें आधार सहप्रांत और घात प्रांत से आता है।
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यदि ∣A∣=3 और ∣B∣=4 है, तो A से B में कितने फलन होंगे?
If ∣A∣=3 and ∣B∣=4, how many functions are there from A to B?
#functions
#counting
#finite-sets
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A 4 3
B 3 4
C 4⋅3
D 4+3
Explanation opens after your attempt
Step 1
Concept
Each domain element has 4 choices in B. Therefore total functions are ∣B∣
∣A∣
=4
3
.
Step 2
Why this answer is correct
The correct answer is A. 4 3. Each domain element has 4 choices in B. Therefore total functions are ∣B∣
∣A∣
=4
3
.
Step 3
Exam Tip
हर प्रांत तत्व के लिए B में 4 विकल्प हैं। इसलिए कुल फलन ∣B∣
∣A∣
=4
3
होंगे।
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यदि A={u,v,w} और B={9} है, तो A से B में कौन सा फलन संभव है?
If A={u,v,w} and B={9}, which function is possible from A to B?
#functions
#singleton-codomain
#constant
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A {(u,9),(v,9),(w,9)}
B {(u,9),(v,9)}
C {(9,u),(9,v),(9,w)}
D {(u,9),(u,9),(v,9)}
Explanation opens after your attempt
Correct Answer
A. {(u,9),(v,9),(w,9)}
Step 1
Concept
The codomain has only 9, so every domain element maps to 9. No domain element should be left out.
Step 2
Why this answer is correct
The correct answer is A. {(u,9),(v,9),(w,9)}. The codomain has only 9, so every domain element maps to 9. No domain element should be left out.
Step 3
Exam Tip
सहप्रांत में केवल 9 है, इसलिए हर प्रांत तत्व का प्रतिबिंब 9 होगा। कोई प्रांत तत्व छूटना नहीं चाहिए।
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यदि f(x)=x−4 और प्रांत A={4,5,9} है, तो f(A) क्या है?
If f(x)=x−4 and the domain is A={4,5,9}, what is f(A)?
#functions
#image-set
#evaluation
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A {0,1,5}
B {4,5,9}
C {8,9,13}
D {1,4,5}
Explanation opens after your attempt
Correct Answer
A. {0,1,5}
Step 1
Concept
4−4=0, 5−4=1, and 9−4=5. Hence f(A)={0,1,5}.
Step 2
Why this answer is correct
The correct answer is A. {0,1,5}. 4−4=0, 5−4=1, and 9−4=5. Hence f(A)={0,1,5}.
Step 3
Exam Tip
4−4=0, 5−4=1, और 9−4=5 हैं। इसलिए f(A)={0,1,5} है।
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यदि f(x)=3x और A={0,1,2} है, तो फलन का ग्राफ कौन सा है?
If f(x)=3x and A={0,1,2}, which is the graph of the function?
#functions
#graph
#linear-rule
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A {(0,0),(1,3),(2,6)}
B {(0,3),(1,6),(2,9)}
C {(0,0),(3,1),(6,2)}
D {(0,1),(1,2),(2,3)}
Explanation opens after your attempt
Correct Answer
A. {(0,0),(1,3),(2,6)}
Step 1
Concept
Applying 3x gives values 0,3,6. The graph is written as (x,f(x)).
Step 2
Why this answer is correct
The correct answer is A. {(0,0),(1,3),(2,6)}. Applying 3x gives values 0,3,6. The graph is written as (x,f(x)).
Step 3
Exam Tip
3x लगाने पर मान 0,3,6 मिलते हैं। ग्राफ को (x,f(x)) के रूप में लिखते हैं।
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कौन सा विकल्प A={2,3,4} पर तत्समक फलन को दर्शाता है?
Which option represents the identity function on A={2,3,4}?
#functions
#identity-function
#finite-set
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A {(2,2),(3,3),(4,4)}
B {(2,3),(3,4),(4,2)}
C {(2,2),(3,2),(4,2)}
D {(2,4),(3,3),(4,2)}
Explanation opens after your attempt
Correct Answer
A. {(2,2),(3,3),(4,4)}
Step 1
Concept
In an identity function, every 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. {(2,2),(3,3),(4,4)}. In an identity function, every 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) के रूप में हैं।
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यदि I
A
तत्समक फलन है और A={8,9} है, तो I
A
(8) क्या है?
If I
A
is the identity function and A={8,9}, what is I
A
(8)?
#functions
#identity-function
#evaluation
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A 8
B 9
C 0
D 1
Explanation opens after your attempt
Step 1
Concept
In the identity function, I
A
(x)=x. Hence I
A
(8)=8.
Step 2
Why this answer is correct
The correct answer is A. 8. In the identity function, I
A
(x)=x. Hence I
A
(8)=8.
Step 3
Exam Tip
तत्समक फलन में I
A
(x)=x होता है। इसलिए I
A
(8)=8 है।
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यदि f(x)=6 सभी x∈A के लिए है, तो f के बारे में कौन सा कथन सही है?
If f(x)=6 for all x∈A, which statement about f is correct?
#functions
#constant-function
#definition
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A f स्थिर फलन है / f is a constant function
B f तत्समक फलन है / f is an identity function
C f फलन नहीं है / f is not a function
D f का प्रांत ∅ ही है / The domain of f is only ∅
Explanation opens after your attempt
Correct Answer
A. f स्थिर फलन है / f is a constant function
Step 1
Concept
All inputs have the same value 6, so it is a constant function. A constant function is also a valid function.
Step 2
Why this answer is correct
The correct answer is A. f स्थिर फलन है / f is a constant function. All inputs have the same value 6, so it is a constant function. A constant function is also a valid function.
Step 3
Exam Tip
सभी इनपुट का मान समान 6 है, इसलिए यह स्थिर फलन है। स्थिर फलन भी वैध फलन होता है।
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यदि f={(a,0),(b,0),(c,0),(d,0)} है, तो f का परिसर क्या है?
If f={(a,0),(b,0),(c,0),(d,0)}, what is the range of f?
#functions
#range
#constant-function
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A {0}
B {a,b,c,d}
C {a,0,d}
D ∅
Explanation opens after your attempt
Step 1
Concept
The second component of all pairs is 0. Repetition is not written in the range, so {0} is correct.
Step 2
Why this answer is correct
The correct answer is A. {0}. The second component of all pairs is 0. Repetition is not written in the range, so {0} is correct.
Step 3
Exam Tip
सभी युग्मों का द्वितीय अवयव 0 है। परिसर में दोहराव नहीं लिखा जाता, इसलिए {0} सही है।
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यदि R={(1,2),(2,2),(3,3),(4,4)} है, तो किस इनपुट का प्रतिबिंब 3 है?
If R={(1,2),(2,2),(3,3),(4,4)}, which input has image 3?
#functions
#preimage
#ordered-pairs
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A 3
B 1
C 2
D 4
Explanation opens after your attempt
Step 1
Concept
In the pair (3,3), the second component is 3. Therefore the input with image 3 is 3.
Step 2
Why this answer is correct
The correct answer is A. 3. In the pair (3,3), the second component is 3. Therefore the input with image 3 is 3.
Step 3
Exam Tip
युग्म (3,3) में द्वितीय अवयव 3 है। इसलिए 3 का प्रतिबिंब 3 है।
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यदि f={(10,1),(20,2),(30,3)} है, तो f(20) क्या है?
If f={(10,1),(20,2),(30,3)}, what is f(20)?
#functions
#function-value
#table
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A 2
B 20
C 10
D 3
Explanation opens after your attempt
Step 1
Concept
From the pair (20,2), the image of 20 is 2. Choose the pair whose input is given.
Step 2
Why this answer is correct
The correct answer is A. 2. From the pair (20,2), the image of 20 is 2. Choose the pair whose input is given.
Step 3
Exam Tip
युग्म (20,2) से 20 का प्रतिबिंब 2 है। फलन मान के लिए वही युग्म चुनें जिसमें इनपुट दिया हो।
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यदि f(x)=
2
x+1
है, तो f(5) का मान क्या है?
If f(x)=
2
x+1
, what is the value of f(5)?
#functions
#evaluation
#fraction-rule
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A 3
B 2
C 2 5
D 6
Explanation opens after your attempt
Step 1
Concept
f(5)=
2
5+1
=3. In fraction rules, pay attention to brackets.
Step 2
Why this answer is correct
The correct answer is A. 3. f(5)=
2
5+1
=3. In fraction rules, pay attention to brackets.
Step 3
Exam Tip
f(5)=
2
5+1
=3 है। भिन्न वाले नियम में कोष्ठक का ध्यान रखें।
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कौन सा विकल्प बताता है कि संबंध फलन नहीं है क्योंकि A={1,2,3} का 2 छूट गया है?
Which option shows that the relation is not a function because 2 of A={1,2,3} is missing?
#functions
#missing-domain
#not-function
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A {(1,5),(3,7)}
B {(1,5),(2,6),(3,7)}
C {(1,5),(2,5),(3,5)}
D {(1,7),(2,6),(3,5)}
Explanation opens after your attempt
Correct Answer
A. {(1,5),(3,7)}
Step 1
Concept
In the first option, 2 is not present as a first component. In a function every domain element must be included.
Step 2
Why this answer is correct
The correct answer is A. {(1,5),(3,7)}. In the first option, 2 is not present as a first component. In a function every domain element must be included.
Step 3
Exam Tip
पहले विकल्प में 2 प्रथम अवयव के रूप में नहीं है। फलन में प्रांत का हर तत्व शामिल होना चाहिए।
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यदि f:A→B एक फलन है, तो f किसका उपसमुच्चय होता है?
If f:A→B is a function, then f is a subset of what?
#functions
#cartesian-product
#relation
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A A×B
B B×A
C A∪B
D A−B
Explanation opens after your attempt
Step 1
Concept
Every relation from A to B is a subset of A×B. A function is also such a special relation.
Step 2
Why this answer is correct
The correct answer is A. A×B. Every relation from A to B is a subset of A×B. A function is also such a special relation.
Step 3
Exam Tip
A से B में हर संबंध A×B का उपसमुच्चय होता है। फलन भी ऐसा ही विशेष संबंध है।
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यदि A={1,2} और B={x,y,z} है, तो A×B में से कौन सा उपसमुच्चय फलन है?
If A={1,2} and B={x,y,z}, which subset of A×B is a function?
#functions
#cartesian-product
#subset
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A {(1,x),(2,z)}
B {(1,x),(1,y)}
C {(2,z)}
D {(1,x),(2,y),(2,z)}
Explanation opens after your attempt
Correct Answer
A. {(1,x),(2,z)}
Step 1
Concept
In the first option, both 1 and 2 have exactly one image. In the other options, an input is missing or mapped twice.
Step 2
Why this answer is correct
The correct answer is A. {(1,x),(2,z)}. In the first option, both 1 and 2 have exactly one image. In the other options, an input is missing or mapped twice.
Step 3
Exam Tip
पहले विकल्प में 1 और 2 दोनों का ठीक एक प्रतिबिंब है। अन्य विकल्पों में कोई इनपुट छूटा है या दो बार जुड़ा है।
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यदि f:A→B है और A={r,s,t} है, तो f के ग्राफ में कितने युग्म होंगे?
If f:A→B and A={r,s,t}, how many pairs will the graph of f contain?
#functions
#graph
#counting
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A 3
B 2
C 1
D ∣B∣
Explanation opens after your attempt
Step 1
Concept
The graph has one pair for each element of the domain. Therefore the number of pairs is ∣A∣=3.
Step 2
Why this answer is correct
The correct answer is A. 3. The graph has one pair for each element of the domain. Therefore the number of pairs is ∣A∣=3.
Step 3
Exam Tip
ग्राफ में प्रत्येक प्रांत तत्व के लिए एक युग्म होता है। इसलिए युग्मों की संख्या ∣A∣=3 है।
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यदि A=∅ और B={5,6,7} है, तो A से B में कितने फलन होंगे?
If A=∅ and B={5,6,7}, how many functions are there from A to B?
#functions
#empty-domain
#counting
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A 1
B 0
C 3
D 7
Explanation opens after your attempt
Step 1
Concept
From an empty domain there is one empty function. The formula also gives ∣B∣
0
=1.
Step 2
Why this answer is correct
The correct answer is A. 1. From an empty domain there is one empty function. The formula also gives ∣B∣
0
=1.
Step 3
Exam Tip
रिक्त प्रांत से एक रिक्त फलन होता है। सूत्र से भी ∣B∣
0
=1 मिलता है।
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यदि A={0} और B=∅ है, तो A से B में कितने फलन होंगे?
If A={0} and B=∅, how many functions are there from A to B?
#functions
#empty-codomain
#not-possible
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A 0
B 1
C 2
D अनंत / Infinite
Explanation opens after your attempt
Step 1
Concept
The domain contains 0 but the codomain is empty, so 0 cannot get an image. Hence no function is possible.
Step 2
Why this answer is correct
The correct answer is A. 0. The domain contains 0 but the codomain is empty, so 0 cannot get an image. Hence no function is possible.
Step 3
Exam Tip
प्रांत में 0 है लेकिन सहप्रांत खाली है, इसलिए 0 का कोई प्रतिबिंब नहीं मिल सकता। अतः कोई फलन नहीं बनेगा।
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किस विकल्प में f फलन है लेकिन एक-एक होना जरूरी नहीं दिखाता?
Which option shows f is a function but not necessarily one-one?
#functions
#many-one
#concept
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A {(1,a),(2,a),(3,c)}
B {(1,a),(1,b),(3,c)}
C {(2,a),(2,c)}
D {(1,a),(3,c)} जब प्रांत {1,2,3} है / {(1,a),(3,c)} when domain is {1,2,3}
Explanation opens after your attempt
Correct Answer
A. {(1,a),(2,a),(3,c)}
Step 1
Concept
1 and 2 have the same image a, yet every input has only one value. Therefore it is a function.
Step 2
Why this answer is correct
The correct answer is A. {(1,a),(2,a),(3,c)}. 1 and 2 have the same image a, yet every input has only one value. Therefore it is a function.
Step 3
Exam Tip
1 और 2 का समान प्रतिबिंब a है, फिर भी हर इनपुट का केवल एक मान है। इसलिए यह फलन है।
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यदि f:A→B में (k,4)∈f और (k,m)∈f हैं, तो फलन होने के लिए क्या होना चाहिए?
If f:A→B has (k,4)∈f and (k,m)∈f, what must be true for it to be a function?
#functions
#uniqueness
#reasoning
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A m=4
B k=4
C m=k
D m =4
Explanation opens after your attempt
Step 1
Concept
Two values for the same input k are possible only when both are equal. Therefore m=4 must hold.
Step 2
Why this answer is correct
The correct answer is A. m=4. Two values for the same input k are possible only when both are equal. Therefore m=4 must hold.
Step 3
Exam Tip
एक ही इनपुट k के दो मान तभी संभव हैं जब दोनों समान हों। इसलिए m=4 होना चाहिए।
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यदि f={(1,2),(2,4),(3,6)} है, तो कौन सा नियम f को दर्शाता है?
If f={(1,2),(2,4),(3,6)}, which rule represents f?
#functions
#rule-identification
#linear
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A f(x)=2x
B f(x)=x+2
C f(x)=x 2
D f(x)=3x
Explanation opens after your attempt
Correct Answer
A. f(x)=2x
Step 1
Concept
For x=1,2,3, 2x gives 2,4,6. Check all pairs while identifying a rule.
Step 2
Why this answer is correct
The correct answer is A. f(x)=2x. For x=1,2,3, 2x gives 2,4,6. Check all pairs while identifying a rule.
Step 3
Exam Tip
x=1,2,3 पर 2x से 2,4,6 मिलते हैं। नियम पहचानते समय सभी युग्म जांचें।
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यदि f={(2,4),(3,9),(5,25)} है, तो कौन सा नियम सही है?
If f={(2,4),(3,9),(5,25)}, which rule is correct?
#functions
#rule-identification
#square
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A f(x)=x 2
B f(x)=2x
C f(x)=x+2
D f(x)=x 3
Explanation opens after your attempt
Correct Answer
A. f(x)=x 2
Step 1
Concept
2
2
=4, 3
2
=9, and 5
2
=25. Therefore the correct rule is f(x)=x
2
.
Step 2
Why this answer is correct
The correct answer is A. f(x)=x 2. 2
2
=4, 3
2
=9, and 5
2
=25. Therefore the correct rule is f(x)=x
2
.
Step 3
Exam Tip
2
2
=4, 3
2
=9, और 5
2
=25 हैं। इसलिए सही नियम f(x)=x
2
है।
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यदि f(x)=x+7 और f(a)=12 है, तो a क्या होगा?
If f(x)=x+7 and f(a)=12, what is a?
#functions
#preimage
#equation
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A 5
B 7
C 12
D 19
Explanation opens after your attempt
Step 1
Concept
From a+7=12, we get a=5. In such questions set f(a) equal to the given value.
Step 2
Why this answer is correct
The correct answer is A. 5. From a+7=12, we get a=5. In such questions set f(a) equal to the given value.
Step 3
Exam Tip
a+7=12 से a=5 मिलता है। ऐसे प्रश्नों में f(a) को दिए गए मान के बराबर रखें।
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यदि h(x)=4x−1 है, तो h(2) क्या होगा?
If h(x)=4x−1, what is h(2)?
#functions
#evaluation
#linear-function
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A 7
B 8
C 6
D 3
Explanation opens after your attempt
Step 1
Concept
h(2)=4⋅2−1=7. In a linear function, multiply and subtract in the correct order.
Step 2
Why this answer is correct
The correct answer is A. 7. h(2)=4⋅2−1=7. In a linear function, multiply and subtract in the correct order.
Step 3
Exam Tip
h(2)=4⋅2−1=7 है। रैखिक फलन में सही क्रम से गुणा और घटाव करें।
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यदि p(x)=x
3
−1 है, तो p(2) का मान क्या है?
If p(x)=x
3
−1, what is the value of p(2)?
#functions
#evaluation
#cubic
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A 7
B 8
C 6
D 9
Explanation opens after your attempt
Step 1
Concept
p(2)=2
3
−1=7. Find the cube first and then subtract 1.
Step 2
Why this answer is correct
The correct answer is A. 7. p(2)=2
3
−1=7. Find the cube first and then subtract 1.
Step 3
Exam Tip
p(2)=2
3
−1=7 है। घन का मान निकालकर फिर 1 घटाएं।
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यदि किसी तालिका में x=4 के सामने y=6 और y=9 दोनों लिखे हैं, तो तालिका फलन क्यों नहीं दिखाती?
If a table shows both y=6 and y=9 for x=4, why does the table not show a function?
#functions
#table
#not-function
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A क्योंकि x=4 के दो अलग मान हैं / Because x=4 has two different values
B क्योंकि y=6 छोटा है / Because y=6 is small
C क्योंकि y=9 बड़ा है / Because y=9 is large
D क्योंकि x=4 धनात्मक है / Because x=4 is positive
Explanation opens after your attempt
Correct Answer
A. क्योंकि x=4 के दो अलग मान हैं / Because x=4 has two different values
Step 1
Concept
Two different y values for the same x break the function condition. Always check repeated inputs in a table.
Step 2
Why this answer is correct
The correct answer is A. क्योंकि x=4 के दो अलग मान हैं / Because x=4 has two different values. Two different y values for the same x break the function condition. Always check repeated inputs in a table.
Step 3
Exam Tip
एक ही x के लिए दो अलग y मान फलन की शर्त तोड़ते हैं। तालिका में दोहराए गए इनपुट को हमेशा जांचें।
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यदि A={1,2,3} से B={4,5,6} में 1→4, 2→5, 3→6 है, तो यह क्या है?
If from A={1,2,3} to B={4,5,6}, 1→4, 2→5, 3→6, what is it?
#functions
#mapping-diagram
#function
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A फलन / Function
B फलन नहीं / Not a function
C रिक्त संबंध / Empty relation
D केवल सहप्रांत / Only codomain
Explanation opens after your attempt
Correct Answer
A. फलन / Function
Step 1
Concept
Exactly one arrow leaves each domain element. Therefore it is a function.
Step 2
Why this answer is correct
The correct answer is A. फलन / Function. Exactly one arrow leaves each domain element. Therefore it is a function.
Step 3
Exam Tip
हर प्रांत तत्व से ठीक एक तीर निकलता है। इसलिए यह फलन है।
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यदि आरेख में p→1, p→2, और q→3 है, तो सही निष्कर्ष क्या है?
If a diagram has p→1, p→2, and q→3, what is the correct conclusion?
#functions
#mapping-diagram
#duplicate-image
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A यह फलन नहीं है / It is not a function
B यह स्थिर फलन है / It is a constant function
C यह तत्समक फलन है / It is an identity function
D यह हमेशा फलन है / It is always a function
Explanation opens after your attempt
Correct Answer
A. यह फलन नहीं है / It is not a function
Step 1
Concept
Two arrows leave p, so p has two images. One input cannot have two images in a function.
Step 2
Why this answer is correct
The correct answer is A. यह फलन नहीं है / It is not a function. Two arrows leave p, so p has two images. One input cannot have two images in a function.
Step 3
Exam Tip
p से दो तीर निकल रहे हैं, इसलिए p के दो प्रतिबिंब हैं। एक इनपुट के दो प्रतिबिंब फलन में नहीं हो सकते।
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यदि A={2,4,6} और f(x)=x+1 है, तो कौन सा युग्म फलन के ग्राफ में होगा?
If A={2,4,6} and f(x)=x+1, which pair will be in the graph of the function?
#functions
#graph
#pair-selection
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A (4,5)
B (4,4)
C (5,4)
D (6,8)
Explanation opens after your attempt
Step 1
Concept
f(4)=4+1=5, so (4,5) will be in the graph. In a graph pair the first place is for the input.
Step 2
Why this answer is correct
The correct answer is A. (4,5). f(4)=4+1=5, so (4,5) will be in the graph. In a graph pair the first place is for the input.
Step 3
Exam Tip
f(4)=4+1=5, इसलिए (4,5) ग्राफ में होगा। ग्राफ में पहला स्थान इनपुट का होता है।
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यदि f:A→B है, तो परिसर और सहप्रांत के बारे में सही कथन कौन सा है?
If f:A→B, which statement about range and codomain is correct?
#functions
#range
#codomain
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A परिसर B का उपसमुच्चय होता है / The range is a subset of B
B B हमेशा परिसर का उपसमुच्चय होता है / B is always a subset of the range
C परिसर हमेशा A होता है / The range is always A
D परिसर हमेशा A×B होता है / The range is always A×B
Explanation opens after your attempt
Correct Answer
A. परिसर B का उपसमुच्चय होता है / The range is a subset of B
Step 1
Concept
All obtained values of a function lie in B. Therefore the range is a subset of B.
Step 2
Why this answer is correct
The correct answer is A. परिसर B का उपसमुच्चय होता है / The range is a subset of B. All obtained values of a function lie in B. Therefore the range is a subset of B.
Step 3
Exam Tip
फलन के सभी प्राप्त मान B में होते हैं। इसलिए परिसर B का उपसमुच्चय होता है।
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यदि f={(1,8),(2,8),(3,9)} है, तो किसका प्रतिबिंब 9 है?
If f={(1,8),(2,8),(3,9)}, whose image is 9?
#functions
#image
#preimage
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A 3
B 1
C 2
D 8
Explanation opens after your attempt
Step 1
Concept
In the pair (3,9), the image of 3 is 9. In such questions identify the first component by looking at the second component.
Step 2
Why this answer is correct
The correct answer is A. 3. In the pair (3,9), the image of 3 is 9. In such questions identify the first component by looking at the second component.
Step 3
Exam Tip
युग्म (3,9) में 3 का प्रतिबिंब 9 है। ऐसे प्रश्नों में द्वितीय अवयव देखकर प्रथम अवयव पहचानें।
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यदि A={1,2,3} और f={(1,4),(2,5)} है, तो f A से फलन क्यों नहीं है?
If A={1,2,3} and f={(1,4),(2,5)}, why is f not a function from A?
#functions
#missing-domain
#reasoning
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A क्योंकि 3 का कोई प्रतिबिंब नहीं है / Because 3 has no image
B क्योंकि 1 का प्रतिबिंब 4 है / Because 1 has image 4
C क्योंकि 2 का प्रतिबिंब 5 है / Because 2 has image 5
D क्योंकि 4 और 5 अलग हैं / Because 4 and 5 are different
Explanation opens after your attempt
Correct Answer
A. क्योंकि 3 का कोई प्रतिबिंब नहीं है / Because 3 has no image
Step 1
Concept
The element 3 of domain A is not the first component in any pair. Every domain element must have an image in a function.
Step 2
Why this answer is correct
The correct answer is A. क्योंकि 3 का कोई प्रतिबिंब नहीं है / Because 3 has no image. The element 3 of domain A is not the first component in any pair. Every domain element must have an image in a function.
Step 3
Exam Tip
प्रांत A का 3 किसी भी युग्म में प्रथम अवयव नहीं है। फलन में हर प्रांत तत्व का प्रतिबिंब होना चाहिए।
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यदि f:A→B है और x∈A, तो f(x) को क्या कहा जाता है?
If f:A→B and x∈A, what is f(x) called?
#functions
#image
#terminology
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A x का प्रतिबिंब / Image of x
B x का प्रांत / Domain of x
C x का सहप्रांत / Codomain of x
D x का कार्तीय गुणनफल / Cartesian product of x
Explanation opens after your attempt
Correct Answer
A. x का प्रतिबिंब / Image of x
Step 1
Concept
f(x) is the image of the input x. This value lies in the codomain B.
Step 2
Why this answer is correct
The correct answer is A. x का प्रतिबिंब / Image of x. f(x) is the image of the input x. This value lies in the codomain B.
Step 3
Exam Tip
f(x) उस इनपुट x का प्रतिबिंब होता है। यह मान सहप्रांत B में होता है।
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यदि किसी आउटपुट y∈B के लिए f(x)=y है, तो x को क्या कहा जा सकता है?
If f(x)=y for an output y∈B, what can x be called?
#functions
#preimage
#terminology
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A y का पूर्व प्रतिबिंब / Preimage of y
B y का सहप्रांत / Codomain of y
C y का परिसर / Range of y
D y का रिक्त समुच्चय / Empty set of y
Explanation opens after your attempt
Correct Answer
A. y का पूर्व प्रतिबिंब / Preimage of y
Step 1
Concept
The input that maps to an output is called its preimage. A preimage comes from the domain.
Step 2
Why this answer is correct
The correct answer is A. y का पूर्व प्रतिबिंब / Preimage of y. The input that maps to an output is called its preimage. A preimage comes from the domain.
Step 3
Exam Tip
जो इनपुट किसी आउटपुट तक जाता है, वह उसका पूर्व प्रतिबिंब कहलाता है। पूर्व प्रतिबिंब प्रांत से आता है।
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यदि A={1,2} और B={3,4} है, तो A से B में कुल फलनों की संख्या कितनी है?
If A={1,2} and B={3,4}, how many functions are there from A to B?
#functions
#counting-functions
#basic
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A 4
B 2
C 8
D 1
Explanation opens after your attempt
Step 1
Concept
Each domain element has 2 choices, so the total number of functions is 2
2
=4. Multiply the choices in counting.
Step 2
Why this answer is correct
The correct answer is A. 4. Each domain element has 2 choices, so the total number of functions is 2
2
=4. Multiply the choices in counting.
Step 3
Exam Tip
प्रत्येक प्रांत तत्व के लिए 2 विकल्प हैं, इसलिए कुल 2
2
=4 फलन हैं। गिनती में विकल्पों को गुणा करें।
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कौन सा विकल्प A={1,2,3} से B={a,b} में स्थिर फलन है?
Which option is a constant function from A={1,2,3} to B={a,b}?
#functions
#constant-function
#finite-set
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A {(1,a),(2,a),(3,a)}
B {(1,a),(2,b),(3,a)}
C {(1,a),(2,b),(3,b)}
D {(1,a),(1,b),(3,a)}
Explanation opens after your attempt
Correct Answer
A. {(1,a),(2,a),(3,a)}
Step 1
Concept
In the first option, all inputs have image a. Therefore it is a constant function.
Step 2
Why this answer is correct
The correct answer is A. {(1,a),(2,a),(3,a)}. In the first option, all inputs have image a. Therefore it is a constant function.
Step 3
Exam Tip
पहले विकल्प में सभी इनपुट का प्रतिबिंब a है। इसलिए यह स्थिर फलन है।
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यदि R={(1,1),(2,2),(2,3)} है, तो R फलन क्यों नहीं है?
If R={(1,1),(2,2),(2,3)}, why is R not a function?
#functions
#not-function
#uniqueness
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A क्योंकि 2 के दो प्रतिबिंब हैं / Because 2 has two images
B क्योंकि 1 का प्रतिबिंब 1 है / Because 1 has image 1
C क्योंकि 3 द्वितीय अवयव है / Because 3 is a second component
D क्योंकि R में तीन युग्म हैं / Because R has three pairs
Explanation opens after your attempt
Correct Answer
A. क्योंकि 2 के दो प्रतिबिंब हैं / Because 2 has two images
Step 1
Concept
2 is related to two different second components 2 and 3. This breaks the uniqueness condition of a function.
Step 2
Why this answer is correct
The correct answer is A. क्योंकि 2 के दो प्रतिबिंब हैं / Because 2 has two images. 2 is related to two different second components 2 and 3. This breaks the uniqueness condition of a function.
Step 3
Exam Tip
2 दो अलग द्वितीय अवयवों 2 और 3 से जुड़ा है। यह फलन की अद्वितीयता शर्त तोड़ता है।
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यदि A={2,3,4} और R={(x,y):y=x+2, x∈A} है, तो R क्या है?
If A={2,3,4} and R={(x,y):y=x+2, x∈A}, what is R?
#functions
#set-builder
#graph
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A {(2,4),(3,5),(4,6)}
B {(2,2),(3,3),(4,4)}
C {(4,2),(5,3),(6,4)}
D {(2,5),(3,6),(4,7)}
Explanation opens after your attempt
Correct Answer
A. {(2,4),(3,5),(4,6)}
Step 1
Concept
From y=x+2, for x=2,3,4 we get 4,5,6. Apply the rule while converting set builder form into pairs.
Step 2
Why this answer is correct
The correct answer is A. {(2,4),(3,5),(4,6)}. From y=x+2, for x=2,3,4 we get 4,5,6. Apply the rule while converting set builder form into pairs.
Step 3
Exam Tip
y=x+2 से x=2,3,4 पर 4,5,6 मिलते हैं। सेट बिल्डर रूप को युग्मों में बदलते समय नियम लगाएं।
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यदि f(x)=∣x∣ और A={−2,0,3} है, तो f(A) क्या होगा?
If f(x)=∣x∣ and A={−2,0,3}, what is f(A)?
#functions
#modulus
#image-set
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A {0,2,3}
B {−2,0,3}
C {2,0,−3}
D {2,3,5}
Explanation opens after your attempt
Correct Answer
A. {0,2,3}
Step 1
Concept
∣−2∣=2, ∣0∣=0, and ∣3∣=3. Values in a set are written without repetition.
Step 2
Why this answer is correct
The correct answer is A. {0,2,3}. ∣−2∣=2, ∣0∣=0, and ∣3∣=3. Values in a set are written without repetition.
Step 3
Exam Tip
∣−2∣=2, ∣0∣=0, और ∣3∣=3 हैं। समुच्चय में मानों को दोहराए बिना लिखते हैं।
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यदि f(x)=x
2
और A={−1,0,1} है, तो यह फलन किस बात का उदाहरण दिखाता है?
If f(x)=x
2
and A={−1,0,1}, what idea does this function show?
#functions
#many-one
#square-function
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A दो अलग इनपुट का समान आउटपुट हो सकता है / Two different inputs can have the same output
B एक इनपुट के दो अलग आउटपुट हो सकते हैं / One input can have two different outputs
C प्रांत खाली होना चाहिए / The domain must be empty
D सहप्रांत हमेशा प्रांत के बराबर होता है / The codomain is always equal to the domain
Explanation opens after your attempt
Correct Answer
A. दो अलग इनपुट का समान आउटपुट हो सकता है / Two different inputs can have the same output
Step 1
Concept
Both −1 and 1 give value 1. Therefore having the same output is acceptable in a function.
Step 2
Why this answer is correct
The correct answer is A. दो अलग इनपुट का समान आउटपुट हो सकता है / Two different inputs can have the same output. Both −1 and 1 give value 1. Therefore having the same output is acceptable in a function.
Step 3
Exam Tip
−1 और 1 दोनों का मान 1 आता है। इसलिए समान आउटपुट होना फलन में स्वीकार्य है।
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कौन सा विकल्प फलन की जांच के लिए सबसे उपयोगी है?
Which option is most useful for testing whether a relation is a function?
#functions
#function-test
#exam-tip
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A हर प्रथम अवयव का ठीक एक द्वितीय अवयव है या नहीं / Whether each first component has exactly one second component
B हर द्वितीय अवयव अलग है या नहीं / Whether each second component is different
C सभी संख्याएं बड़ी हैं या नहीं / Whether all numbers are large
D समुच्चय में केवल 0 है या नहीं / Whether the set has only 0
Explanation opens after your attempt
Correct Answer
A. हर प्रथम अवयव का ठीक एक द्वितीय अवयव है या नहीं / Whether each first component has exactly one second component
Step 1
Concept
Checking first components is most important while testing a function. No first component should be linked with two different second components.
Step 2
Why this answer is correct
The correct answer is A. हर प्रथम अवयव का ठीक एक द्वितीय अवयव है या नहीं / Whether each first component has exactly one second component. Checking first components is most important while testing a function. No first component should be linked with two different second components.
Step 3
Exam Tip
फलन की जांच में प्रथम अवयवों को देखना सबसे जरूरी है। कोई प्रथम अवयव दो अलग द्वितीय अवयवों से नहीं जुड़ना चाहिए।
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फलन को संबंध का विशेष रूप क्यों कहा जाता है?
Why is a function called a special form of relation?
#functions
#special-relation
#summary
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A क्योंकि इसमें प्रत्येक x∈A का B में ठीक एक प्रतिबिंब होता है / Because each x∈A has exactly one image in B
B क्योंकि इसमें हर x∈A के कई प्रतिबिंब होते हैं / Because each x∈A has many images
C क्योंकि इसमें कोई क्रमित युग्म नहीं होता / Because it has no ordered pair
D क्योंकि इसमें प्रांत हमेशा ∅ होता है / Because the domain is always ∅
Explanation opens after your attempt
Correct Answer
A. क्योंकि इसमें प्रत्येक x∈A का B में ठीक एक प्रतिबिंब होता है / Because each x∈A has exactly one image in B
Step 1
Concept
Every function is a relation, but every relation is not a function. The condition of exactly one image is necessary for a function.
Step 2
Why this answer is correct
The correct answer is A. क्योंकि इसमें प्रत्येक x∈A का B में ठीक एक प्रतिबिंब होता है / Because each x∈A has exactly one image in B. Every function is a relation, but every relation is not a function. The condition of exactly one image is necessary for a function.
Step 3
Exam Tip
हर फलन संबंध है, पर हर संबंध फलन नहीं होता। फलन बनने के लिए ठीक एक प्रतिबिंब की शर्त जरूरी है।
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