A. यह (A) से (B) में फलन है/It is a function from (A) to (B)
Step 1
Concept
Each element of (A) has exactly one image in (B). In exams, repetition of the second component does not make a function invalid.
Step 2
Why this answer is correct
The correct answer is A. यह (A) से (B) में फलन है / It is a function from (A) to (B). Each element of (A) has exactly one image in (B). In exams, repetition of the second component does not make a function invalid.
Step 3
Exam Tip
(A) के प्रत्येक तत्व की (B) में ठीक एक छवि है। परीक्षा में दूसरे घटक के दोहराने से फलन गलत नहीं होता।
A. क्योंकि (1) की दो अलग-अलग छवियां हैं/Because (1) has two different images
Step 1
Concept
In a function, one domain element cannot have two different images. In exams, check repeated first components carefully.
Step 2
Why this answer is correct
The correct answer is A. क्योंकि (1) की दो अलग-अलग छवियां हैं / Because (1) has two different images. In a function, one domain element cannot have two different images. In exams, check repeated first components carefully.
Step 3
Exam Tip
किसी फलन में प्रांत के एक तत्व की दो अलग छवियां नहीं हो सकतीं। परीक्षा में समान पहले घटक को ध्यान से जांचें।
A. क्योंकि (6) की कोई छवि नहीं है/Because (6) has no image
Step 1
Concept
For a function, every element of (A) must have exactly one image. In exams, identify a missing domain element quickly.
Step 2
Why this answer is correct
The correct answer is A. क्योंकि (6) की कोई छवि नहीं है / Because (6) has no image. For a function, every element of (A) must have exactly one image. In exams, identify a missing domain element quickly.
Step 3
Exam Tip
फलन के लिए (A) के हर तत्व की ठीक एक छवि होनी चाहिए। परीक्षा में छूटा हुआ प्रांत तत्व तुरंत पहचानें।
Each of the (2) elements of (A) has (3) choices in (B), so total functions are \(3^2=9\). In exams, remember the formula (n(B)^{n(A)}).
Step 2
Why this answer is correct
The correct answer is B. (9). Each of the (2) elements of (A) has (3) choices in (B), so total functions are \(3^2=9\). In exams, remember the formula (n(B)^{n(A)}).
Step 3
Exam Tip
(A) के (2) तत्वों में से प्रत्येक के लिए (B) की (3) पसंद हैं इसलिए कुल \(3^2=9\) फलन हैं। परीक्षा में सूत्र (n(B)^{n(A)}) याद रखें।
Here (n(A)=3) and (n(B)=2), so total functions are \(2^3=8\). In exams, the base is the number of elements in the codomain.
Step 2
Why this answer is correct
The correct answer is C. (8). Here (n(A)=3) and (n(B)=2), so total functions are \(2^3=8\). In exams, the base is the number of elements in the codomain.
Step 3
Exam Tip
यहां (n(A)=3) और (n(B)=2) है इसलिए कुल फलन \(2^3=8\) हैं। परीक्षा में आधार सहप्रांत के तत्वों की संख्या होती है।
A. यह फलन है और प्रत्येक तत्व की छवि (1) अधिक है/It is a function and each image is (1) more
Step 1
Concept
Every first component has exactly one image and it is (x+1). In exams, identifying the rule from ordered pairs is useful.
Step 2
Why this answer is correct
The correct answer is A. यह फलन है और प्रत्येक तत्व की छवि (1) अधिक है / It is a function and each image is (1) more. Every first component has exactly one image and it is (x+1). In exams, identifying the rule from ordered pairs is useful.
Step 3
Exam Tip
हर पहले घटक की ठीक एक छवि है और वह (x+1) है। परीक्षा में क्रमित युग्मों से नियम पहचानना उपयोगी होता है।
C. किसी \(a\in A\) की दो अलग छवियां हों/Some \(a\in A\) has two different images
Step 1
Concept
In a function, one domain element cannot be associated with two different images. This is the most common exam mistake.
Step 2
Why this answer is correct
The correct answer is C. किसी \(a\in A\) की दो अलग छवियां हों / Some \(a\in A\) has two different images. In a function, one domain element cannot be associated with two different images. This is the most common exam mistake.
Step 3
Exam Tip
फलन में एक प्रांत तत्व को दो अलग छवियों से नहीं जोड़ा जा सकता। परीक्षा में यह सबसे सामान्य गलती होती है।
In option (A), every element (1,2,3) of (A) has exactly one image. In exams, match the list of first components with (A).
Step 2
Why this answer is correct
The correct answer is A. ({(1,a),(2,b),(3,c)}). In option (A), every element (1,2,3) of (A) has exactly one image. In exams, match the list of first components with (A).
Step 3
Exam Tip
विकल्प (A) में (A) के हर तत्व (1,2,3) की ठीक एक छवि है। परीक्षा में पहले घटकों की सूची को (A) से मिलाएं।
For each (x), \(y=x^2\) gives only one image. Two different (x) values having the same image does not invalidate a function.
Step 2
Why this answer is correct
The correct answer is A. यह फलन है / It is a function. For each (x), \(y=x^2\) gives only one image. Two different (x) values having the same image does not invalidate a function.
Step 3
Exam Tip
प्रत्येक (x) के लिए \(y=x^2\) से केवल एक छवि मिलती है। दो अलग (x) की समान छवि होना फलन को गलत नहीं करता।
There are (4) choices for each of the (2) elements of (A), so there are \(4^2=16\) functions. In exams, the exponent is the number of domain elements.
Step 2
Why this answer is correct
The correct answer is A. \(4^2=16\). There are (4) choices for each of the (2) elements of (A), so there are \(4^2=16\) functions. In exams, the exponent is the number of domain elements.
Step 3
Exam Tip
(A) के (2) तत्वों के लिए (B) की (4) पसंद हैं इसलिए \(4^2=16\) फलन हैं। परीक्षा में घात प्रांत के तत्वों की संख्या होती है।
In a constant function, all elements have the same image, chosen from one element of (B). Therefore there are (2) constant functions.
Step 2
Why this answer is correct
The correct answer is B. (2). In a constant function, all elements have the same image, chosen from one element of (B). Therefore there are (2) constant functions.
Step 3
Exam Tip
स्थिर फलन में सभी तत्वों की एक ही छवि होती है और वह (B) के किसी एक तत्व से चुनी जाती है। इसलिए (2) स्थिर फलन हैं।
\(1^2=1\), \(2^2=4\), and \(3^2=9\). In exams, the first component comes from the domain and the second from the image.
Step 2
Why this answer is correct
The correct answer is A. ({(1,1),(2,4),(3,9)}). \(1^2=1\), \(2^2=4\), and \(3^2=9\). In exams, the first component comes from the domain and the second from the image.
Step 3
Exam Tip
\(1^2=1\), \(2^2=4\) और \(3^2=9\) हैं। परीक्षा में पहले घटक प्रांत से और दूसरा घटक छवि से आता है।
In \(f:A\to B\), (A) is the domain and (B) is the codomain. In exams, 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 and (B) is the codomain. In exams, read the notation \(f:A\to B\) carefully.
Step 3
Exam Tip
\(f:A\to B\) में (A) प्रांत और (B) सहप्रांत होता है। परीक्षा में संकेत \(f:A\to B\) को ध्यान से पढ़ें।
The set of actually obtained images is called the range. In exams, the range is always a subset of the codomain.
Step 2
Why this answer is correct
The correct answer is C. परिसर / Range. The set of actually obtained images is called the range. In exams, the range is always a subset of the codomain.
Step 3
Exam Tip
वास्तविक रूप से प्राप्त छवियों का समुच्चय परिसर कहलाता है। परीक्षा में परिसर हमेशा सहप्रांत का उपसमुच्चय होता है।
In option (D), (3) has two images (c) and (d), and (4) has no image. In exams, check both conditions.
Step 2
Why this answer is correct
The correct answer is D. ({(1,a),(2,b),(3,c),(3,d)}). In option (D), (3) has two images (c) and (d), and (4) has no image. In exams, check both conditions.
Step 3
Exam Tip
विकल्प (D) में (3) की दो छवियां (c) और (d) हैं तथा (4) की छवि नहीं है। परीक्षा में दोनों शर्तें जांचें।
In every ordered pair, the second component is (2) times the first. In exams, test the rule on two pairs and confirm with the rest.
Step 2
Why this answer is correct
The correct answer is B. (y=2x). In every ordered pair, the second component is (2) times the first. In exams, test the rule on two pairs and confirm with the rest.
Step 3
Exam Tip
हर क्रमित युग्म में दूसरा घटक पहले घटक का (2) गुना है। परीक्षा में दो युग्मों से नियम जांचकर बाकी पर पुष्टि करें।
\(A\times B\) has \(3\times 2=6\) elements, so the number of relations is \(2^6=64\). In exams, counting relations and functions is different.
Step 2
Why this answer is correct
The correct answer is A. \(2^6=64\). \(A\times B\) has \(3\times 2=6\) elements, so the number of relations is \(2^6=64\). In exams, counting relations and functions is different.
Step 3
Exam Tip
\(A\times B\) में \(3\times 2=6\) तत्व हैं इसलिए संबंधों की संख्या \(2^6=64\) है। परीक्षा में संबंध और फलन की गिनती अलग होती है।
The number of functions is (n(B)^{n(A)}=23=8). In exams, do not confuse it with the number of relations \(2^{n(A)n(B)}\).
Step 2
Why this answer is correct
The correct answer is B. \(2^3=8\). The number of functions is (n(B)^{n(A)}=23=8). In exams, do not confuse it with the number of relations \(2^{n(A)n(B)}\).
Step 3
Exam Tip
फलनों की संख्या (n(B)^{n(A)}=23=8) है। परीक्षा में संबंधों की संख्या \(2^{n(A)n(B)}\) से भ्रम न करें।
B. (f) फलन है क्योंकि हर (x) की एक ही निश्चित छवि है/(f) is a function because every (x) has one definite image
Step 1
Concept
For every real (x), \(x^2\) gives one definite real number. Having the same image does not break the function condition.
Step 2
Why this answer is correct
The correct answer is B. (f) फलन है क्योंकि हर (x) की एक ही निश्चित छवि है / (f) is a function because every (x) has one definite image. For every real (x), \(x^2\) gives one definite real number. Having the same image does not break the function condition.
Step 3
Exam Tip
हर वास्तविक (x) के लिए \(x^2\) एक निश्चित वास्तविक संख्या देता है। समान छवि आना फलन की शर्त नहीं तोड़ता।
A. क्योंकि ऋणात्मक (x) के लिए \(\sqrt{x}\) वास्तविक नहीं है/Because \(\sqrt{x}\) is not real for negative (x)
Step 1
Concept
If the domain is \(\mathbb{R}\), every real (x) must have a real image. Negative (x) values make the rule invalid on the whole domain.
Step 2
Why this answer is correct
The correct answer is A. क्योंकि ऋणात्मक (x) के लिए \(\sqrt{x}\) वास्तविक नहीं है / Because \(\sqrt{x}\) is not real for negative (x). If the domain is \(\mathbb{R}\), every real (x) must have a real image. Negative (x) values make the rule invalid on the whole domain.
Step 3
Exam Tip
यदि प्रांत \(\mathbb{R}\) है तो हर वास्तविक (x) की वास्तविक छवि चाहिए। ऋणात्मक (x) के कारण नियम पूरे प्रांत पर लागू नहीं होता।
A. (x=0) पर \(\frac{1}{x}\) परिभाषित नहीं है/\(\frac{1}{x}\) is not defined at (x=0)
Step 1
Concept
(x=0) is in the domain but (f(0)) is not defined. In exams, check exceptional values of every rule.
Step 2
Why this answer is correct
The correct answer is A. (x=0) पर \(\frac{1}{x}\) परिभाषित नहीं है / \(\frac{1}{x}\) is not defined at (x=0). (x=0) is in the domain but (f(0)) is not defined. In exams, check exceptional values of every rule.
Step 3
Exam Tip
(x=0) प्रांत में है पर (f(0)) परिभाषित नहीं है। परीक्षा में हर नियम के अपवाद मान को जांचें।
A. क्योंकि (0) को प्रांत से हटा दिया गया है/Because (0) is removed from the domain
Step 1
Concept
(0) is not in the domain, and \(\frac{1}{x}\) is defined for every remaining (x). In exams, changing the domain can change validity as a function.
Step 2
Why this answer is correct
The correct answer is A. क्योंकि (0) को प्रांत से हटा दिया गया है / Because (0) is removed from the domain. (0) is not in the domain, and \(\frac{1}{x}\) is defined for every remaining (x). In exams, changing the domain can change validity as a function.
Step 3
Exam Tip
प्रांत में (0) नहीं है और बाकी हर (x) के लिए \(\frac{1}{x}\) परिभाषित है। परीक्षा में प्रांत बदलने से फलन की वैधता बदल सकती है।
A. क्योंकि (x=1) की दो छवियां (y=1) और (y=-1) हैं/Because (x=1) has two images (y=1) and (y=-1)
Step 1
Concept
From \(y^2=1\), both (y=1) and (y=-1) are obtained. In exams, if one first component has more than one image, it is not a function.
Step 2
Why this answer is correct
The correct answer is A. क्योंकि (x=1) की दो छवियां (y=1) और (y=-1) हैं / Because (x=1) has two images (y=1) and (y=-1). From \(y^2=1\), both (y=1) and (y=-1) are obtained. In exams, if one first component has more than one image, it is not a function.
Step 3
Exam Tip
\(y^2=1\) से (y=1) और (y=-1) दोनों मिलते हैं। परीक्षा में एक पहले घटक की एक से अधिक छवि होने पर फलन नहीं होता।
A. यह फलन है क्योंकि दोहराया युग्म नया अलग चित्र नहीं देता/It is a function because the repeated pair does not give a new different image
Step 1
Concept
Repeating the same ordered pair does not create a different image for an element. In exams, look for different images, not mere repetition.
Step 2
Why this answer is correct
The correct answer is A. यह फलन है क्योंकि दोहराया युग्म नया अलग चित्र नहीं देता / It is a function because the repeated pair does not give a new different image. Repeating the same ordered pair does not create a different image for an element. In exams, look for different images, not mere repetition.
Step 3
Exam Tip
समान क्रमित युग्म को दोहराने से किसी तत्व की अलग छवि नहीं बनती। परीक्षा में अलग छवियों को देखें, केवल पुनरावृत्ति को नहीं।
Both even and odd elements are present in (A), so both images (0) and (1) occur. In exams, check all types of domain elements.
Step 2
Why this answer is correct
The correct answer is C. ({0,1}). Both even and odd elements are present in (A), so both images (0) and (1) occur. In exams, check all types of domain elements.
Step 3
Exam Tip
सम और विषम दोनों प्रकार के तत्व (A) में हैं इसलिए छवियां (0) और (1) दोनों मिलती हैं। परीक्षा में सभी प्रकार के प्रांत तत्व देखें।
In option (A), both (1) and (2) have exactly one image. In exams, a subset is a function only when the whole domain is covered.
Step 2
Why this answer is correct
The correct answer is A. ({(1,3),(2,4)}). In option (A), both (1) and (2) have exactly one image. In exams, a subset is a function only when the whole domain is covered.
Step 3
Exam Tip
विकल्प (A) में (1) और (2) दोनों की ठीक एक छवि है। परीक्षा में उपसमुच्चय भी तभी फलन है जब पूरा प्रांत शामिल हो।
(a) is a preimage of (b) because applying (f) to (a) gives (b). In exams, do not reverse image and preimage.
Step 2
Why this answer is correct
The correct answer is B. पूर्वछवि / Preimage. (a) is a preimage of (b) because applying (f) to (a) gives (b). In exams, do not reverse image and preimage.
Step 3
Exam Tip
(a), (b) की पूर्वछवि है क्योंकि (f) से (a) का मान (b) मिलता है। परीक्षा में छवि और पूर्वछवि को उल्टा न करें।
All domain elements have image (a), so it is a constant function. In exams, identify a constant function when all images are the same.
Step 2
Why this answer is correct
The correct answer is A. स्थिर फलन / Constant function. All domain elements have image (a), so it is a constant function. In exams, identify a constant function when all images are the same.
Step 3
Exam Tip
सभी प्रांत तत्वों की छवि (a) है, इसलिए यह स्थिर फलन है। परीक्षा में सभी छवियां समान हों तो स्थिर फलन पहचानें।
A. क्योंकि (1) की अलग छवि नहीं बदली है/Because (1) does not get a different image
Step 1
Concept
((1,2)) is repeated, but (1) does not have another different image. In exams, distinguish a repeated pair from two different images.
Step 2
Why this answer is correct
The correct answer is A. क्योंकि (1) की अलग छवि नहीं बदली है / Because (1) does not get a different image. ((1,2)) is repeated, but (1) does not have another different image. In exams, distinguish a repeated pair from two different images.
Step 3
Exam Tip
((1,2)) दोहराया गया है पर (1) की दूसरी अलग छवि नहीं है। परीक्षा में दोहराए युग्म और दो अलग छवियों में अंतर करें।
यदि \(A=\{1,2,3\}\), \(B=\{2,4,6\}\) और \(f=\{(1,2),(2,4),(3,6)\}\) हो, तो \(f^{-1}\) जैसा उल्टा संबंध ({(2,1),(4,2),(6,3)}) क्या (B) से (A) में फलन है?
A. हां क्योंकि (B) के हर तत्व की ठीक एक छवि है/Yes because every element of (B) has exactly one image
Step 1
Concept
In the reversed relation, each of (2,4,6) has exactly one image. In exams, the inverse relation is a function only when no second component repeats with different preimages.
Step 2
Why this answer is correct
The correct answer is A. हां क्योंकि (B) के हर तत्व की ठीक एक छवि है / Yes because every element of (B) has exactly one image. In the reversed relation, each of (2,4,6) has exactly one image. In exams, the inverse relation is a function only when no second component repeats with different preimages.
Step 3
Exam Tip
उल्टे संबंध में (2,4,6) प्रत्येक की ठीक एक छवि है। परीक्षा में उल्टा संबंध तभी फलन होगा जब कोई दूसरा घटक दोहराकर अलग पूर्वछवि न दे।
A. क्योंकि (a) की दो छवियां (1) और (2) हैं/Because (a) has two images (1) and (2)
Step 1
Concept
In the reversed relation, (a) as a first component is associated with two different images. In exams, watch repeated second components when reversing.
Step 2
Why this answer is correct
The correct answer is A. क्योंकि (a) की दो छवियां (1) और (2) हैं / Because (a) has two images (1) and (2). In the reversed relation, (a) as a first component is associated with two different images. In exams, watch repeated second components when reversing.
Step 3
Exam Tip
उल्टे संबंध में (a) पहले घटक के रूप में दो अलग छवियों से जुड़ता है। परीक्षा में उल्टा करते समय दोहराए दूसरे घटक पर ध्यान दें।
All elements have image (0), so the range is ({0}). In exams, do not include an unattained codomain element in the range.
Step 2
Why this answer is correct
The correct answer is A. ({0}). All elements have image (0), so the range is ({0}). In exams, do not include an unattained codomain element in the range.
Step 3
Exam Tip
सभी तत्वों की छवि (0) है इसलिए परिसर ({0}) है। परीक्षा में सहप्रांत में मौजूद पर अप्राप्त तत्व को परिसर में न लिखें।
In option (A), different domain elements have different images. In exams, check repetition of images while testing one-one behavior.
Step 2
Why this answer is correct
The correct answer is A. ({(1,4),(2,5),(3,6)}). In option (A), different domain elements have different images. In exams, check repetition of images while testing one-one behavior.
Step 3
Exam Tip
विकल्प (A) में अलग-अलग प्रांत तत्वों की छवियां भी अलग-अलग हैं। परीक्षा में एक-एक जांचते समय छवियों की पुनरावृत्ति देखें।
A function must assign exactly one image to each of the (4) elements of (A), so (4) pairs are needed. In exams, the number of pairs in a function equals the number of domain elements.
Step 2
Why this answer is correct
The correct answer is B. (4). A function must assign exactly one image to each of the (4) elements of (A), so (4) pairs are needed. In exams, the number of pairs in a function equals the number of domain elements.
Step 3
Exam Tip
फलन में (A) के हर (4) तत्व की ठीक एक छवि होनी चाहिए, इसलिए (4) युग्म चाहिए। परीक्षा में फलन के युग्मों की संख्या प्रांत के तत्वों के बराबर होती है।
Dividing (1,2,3,4) by (3) gives remainders (1,2,0,1), so the range is ({0,1,2}). In exams, write only distinct obtained images in the range.
Step 2
Why this answer is correct
The correct answer is C. ({0,1,2}). Dividing (1,2,3,4) by (3) gives remainders (1,2,0,1), so the range is ({0,1,2}). In exams, write only distinct obtained images in the range.
Step 3
Exam Tip
(1,2,3,4) को (3) से भाग देने पर शेषफल क्रमशः (1,2,0,1) मिलते हैं, इसलिए परिसर ({0,1,2}) है। परीक्षा में परिसर में केवल अलग-अलग प्राप्त छवियां लिखें।
