A. यह (A) से (B) में फलन है/It is a function from (A) to (B)
Step 1
Concept
Each element of domain (A) has exactly one image. In exams, repetition of the same image 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 domain (A) has exactly one image. In exams, repetition of the same image does not make a function invalid.
Step 3
Exam Tip
प्रांत (A) के प्रत्येक तत्व की ठीक एक छवि है। परीक्षा में समान छवि दोहरना फलन को गलत नहीं करता।
A. क्योंकि (3) की कोई छवि नहीं है/Because (3) has no image
Step 1
Concept
In a function every domain element must have exactly one image but (3) is missing. In exams, compare first components with the whole domain.
Step 2
Why this answer is correct
The correct answer is A. क्योंकि (3) की कोई छवि नहीं है / Because (3) has no image. In a function every domain element must have exactly one image but (3) is missing. In exams, compare first components with the whole domain.
Step 3
Exam Tip
फलन में प्रांत के हर तत्व की ठीक एक छवि होनी चाहिए पर (3) छूट गया है। परीक्षा में पहले घटकों को पूरे प्रांत से मिलाएं।
In a constant function all domain elements map to one chosen element of (B). Therefore the number of constant functions is (3).
Step 2
Why this answer is correct
The correct answer is A. (3). In a constant function all domain elements map to one chosen element of (B). Therefore the number of constant functions is (3).
Step 3
Exam Tip
स्थिर फलन में सभी प्रांत तत्व (B) के एक ही चुने हुए तत्व पर जाते हैं। इसलिए स्थिर फलनों की संख्या (3) है।
\(A\times B\) has \(3\times 4=12\) pairs, so relations are \(2^{12}=4096\). In exams, relations are counted as subsets.
Step 2
Why this answer is correct
The correct answer is A. \(2^{12}=4096\). \(A\times B\) has \(3\times 4=12\) pairs, so relations are \(2^{12}=4096\). In exams, relations are counted as subsets.
Step 3
Exam Tip
\(A\times B\) में \(3\times 4=12\) युग्म हैं इसलिए संबंध \(2^{12}=4096\) होंगे। परीक्षा में संबंध उपसमुच्चयों से गिने जाते हैं।
(2) has two images and after removing ((2,c)) every first component has exactly one image. In exams, remove one pair from the repeated first component.
Step 2
Why this answer is correct
The correct answer is A. ((2,c)). (2) has two images and after removing ((2,c)) every first component has exactly one image. In exams, remove one pair from the repeated first component.
Step 3
Exam Tip
(2) की दो छवियां हैं और ((2,c)) हटाने पर हर पहले घटक की ठीक एक छवि रह जाएगी। परीक्षा में दोहराए पहले घटक से एक युग्म हटाएं।
A. हर पहले घटक की ठीक एक छवि है/Every first component has exactly one image
Step 1
Concept
In the reversed relation, each of (3,5,7) has exactly one image. In exams, test the reversed relation separately by the function condition.
Step 2
Why this answer is correct
The correct answer is A. हर पहले घटक की ठीक एक छवि है / Every first component has exactly one image. In the reversed relation, each of (3,5,7) has exactly one image. In exams, test the reversed relation separately by the function condition.
Step 3
Exam Tip
उल्टे संबंध में (3,5,7) में से प्रत्येक की ठीक एक छवि है। परीक्षा में उल्टा संबंध भी फलन शर्त से अलग जांचें।
A. क्योंकि (p) की दो छवियां (1) और (2) हैं/Because (p) has two images (1) and (2)
Step 1
Concept
In the reversed relation, (p) as a first component is linked to two different images. In exams, watch repeated second components while reversing.
Step 2
Why this answer is correct
The correct answer is A. क्योंकि (p) की दो छवियां (1) और (2) हैं / Because (p) has two images (1) and (2). In the reversed relation, (p) as a first component is linked to two different images. In exams, watch repeated second components while reversing.
Step 3
Exam Tip
उल्टे संबंध में (p) पहले घटक के रूप में दो अलग छवियों से जुड़ा है। परीक्षा में उल्टा करते समय दोहराए दूसरे घटक पर ध्यान दें।
A. क्योंकि (x<1) पर \(\sqrt{x-1}\) वास्तविक नहीं है/Because \(\sqrt{x-1}\) is not real for (x<1)
Step 1
Concept
The domain is \(\mathbb{R}\), but no real image is obtained for (x<1). In exams, check that the expression inside a square root is not negative.
Step 2
Why this answer is correct
The correct answer is A. क्योंकि (x<1) पर \(\sqrt{x-1}\) वास्तविक नहीं है / Because \(\sqrt{x-1}\) is not real for (x<1). The domain is \(\mathbb{R}\), but no real image is obtained for (x<1). In exams, check that the expression inside a square root is not negative.
Step 3
Exam Tip
प्रांत \(\mathbb{R}\) है पर (x<1) के लिए वास्तविक छवि नहीं मिलती। परीक्षा में वर्गमूल के अंदर का मान ऋणात्मक न हो यह जांचें।
A. क्योंकि (x=-3) पर हर (0) हो जाता है/Because the denominator becomes (0) at (x=-3)
Step 1
Concept
(x=-3) is in the domain but (f(-3)) is not defined. In exams, find values that make the denominator (0).
Step 2
Why this answer is correct
The correct answer is A. क्योंकि (x=-3) पर हर (0) हो जाता है / Because the denominator becomes (0) at (x=-3). (x=-3) is in the domain but (f(-3)) is not defined. In exams, find values that make the denominator (0).
Step 3
Exam Tip
(x=-3) प्रांत में है पर (f(-3)) परिभाषित नहीं है। परीक्षा में भिन्न में हर को (0) बनाने वाले मान खोजें।
A. क्योंकि (x=-3) को प्रांत से हटा दिया गया है/Because (x=-3) is removed from the domain
Step 1
Concept
Now \(\frac{1}{x+3}\) is defined for every domain element and gives one value. In exams, changing the domain can make a rule valid.
Step 2
Why this answer is correct
The correct answer is A. क्योंकि (x=-3) को प्रांत से हटा दिया गया है / Because (x=-3) is removed from the domain. Now \(\frac{1}{x+3}\) is defined for every domain element and gives one value. In exams, changing the domain can make a rule valid.
Step 3
Exam Tip
अब हर प्रांत तत्व के लिए \(\frac{1}{x+3}\) परिभाषित है और एक ही मान देता है। परीक्षा में प्रांत बदलने से नियम वैध हो सकता है।
A. क्योंकि (f(3)=4) और \(4\notin{1,2,3}\)/Because (f(3)=4) and \(4\notin{1,2,3}\)
Step 1
Concept
The image of (3) is (4), which is not in the codomain. In exams, check whether every output lies in the codomain.
Step 2
Why this answer is correct
The correct answer is A. क्योंकि (f(3)=4) और \(4\notin{1,2,3}\) / Because (f(3)=4) and \(4\notin{1,2,3}\). The image of (3) is (4), which is not in the codomain. In exams, check whether every output lies in the codomain.
Step 3
Exam Tip
(3) की छवि (4) आती है जो सहप्रांत में नहीं है। परीक्षा में हर आउटपुट सहप्रांत में है या नहीं जांचें।
A. यह फलन है क्योंकि हर \(x\in\mathbb{Z}\) की एक निश्चित छवि है/It is a function because every \(x\in\mathbb{Z}\) has one definite image
Step 1
Concept
For every integer (x), \(x^2\) gives one definite integer. Having the same image does not break the function condition.
Step 2
Why this answer is correct
The correct answer is A. यह फलन है क्योंकि हर \(x\in\mathbb{Z}\) की एक निश्चित छवि है / It is a function because every \(x\in\mathbb{Z}\) has one definite image. For every integer (x), \(x^2\) gives one definite integer. Having the same image does not break the function condition.
Step 3
Exam Tip
हर पूर्णांक (x) के लिए \(x^2\) एक निश्चित पूर्णांक देता है। समान छवि आना फलन की शर्त नहीं तोड़ता।
Two different elements have the same image, so it is a many-one type function. In exams, the same image is allowed.
Step 2
Why this answer is correct
The correct answer is A. अनेक-एक फलन / Many-one function. Two different elements have the same image, so it is a many-one type function. In exams, the same image is allowed.
Step 3
Exam Tip
दो अलग तत्वों की एक ही छवि आ रही है इसलिए यह अनेक-एक प्रकार का फलन है। परीक्षा में समान छवि स्वीकार्य होती है।
In option (A), different domain elements have different images. In exams, images should not repeat for a one-one function.
Step 2
Why this answer is correct
The correct answer is A. ({(1,p),(2,q),(3,r)}). In option (A), different domain elements have different images. In exams, images should not repeat for a one-one function.
Step 3
Exam Tip
विकल्प (A) में अलग-अलग प्रांत तत्वों की छवियां भी अलग-अलग हैं। परीक्षा में एक-एक फलन के लिए छवियों की पुनरावृत्ति न हो।
Every domain element has image (7), so the range is ({7}). In exams, do not write an unattained codomain element in the range.
Step 2
Why this answer is correct
The correct answer is A. ({7}). Every domain element has image (7), so the range is ({7}). In exams, do not write an unattained codomain element in the range.
Step 3
Exam Tip
हर प्रांत तत्व की छवि (7) है इसलिए परिसर ({7}) है। परीक्षा में अप्राप्त सहप्रांत तत्व को परिसर में न लिखें।
A. (8) सहप्रांत में है पर परिसर में नहीं/(8) is in the codomain but not in the range
Step 1
Concept
The images are (2,4,6), so (8) is not obtained. In exams, keep the difference between codomain and range clear.
Step 2
Why this answer is correct
The correct answer is A. (8) सहप्रांत में है पर परिसर में नहीं / (8) is in the codomain but not in the range. The images are (2,4,6), so (8) is not obtained. In exams, keep the difference between codomain and range clear.
Step 3
Exam Tip
छवियां (2,4,6) हैं इसलिए (8) प्राप्त नहीं होता। परीक्षा में सहप्रांत और परिसर का अंतर स्पष्ट रखें।
Each domain element can map to one of the (2) elements of (B). In exams, multiply these choices to count total functions.
Step 2
Why this answer is correct
The correct answer is A. (2). Each domain element can map to one of the (2) elements of (B). In exams, multiply these choices to count total functions.
Step 3
Exam Tip
प्रत्येक प्रांत तत्व (B) के (2) तत्वों में से किसी एक पर जा सकता है। परीक्षा में कुल फलनों के लिए इन पसंदों को गुणा करें।
\(1^3=1\), \(2^3=8\), and \(3^3=27\). In exams, the first component is the input and the second is the output.
Step 2
Why this answer is correct
The correct answer is A. ({(1,1),(2,8),(3,27)}). \(1^3=1\), \(2^3=8\), and \(3^3=27\). In exams, the first component is the input and the second is the output.
Step 3
Exam Tip
\(1^3=1\), \(2^3=8\) और \(3^3=27\) हैं। परीक्षा में पहला घटक इनपुट और दूसरा घटक आउटपुट होता है।
A. (5) सहप्रांत में है पर परिसर में नहीं/(5) is in the codomain but not in the range
Step 1
Concept
The range is ({2,3,4}), so (5) is not obtained. In exams, every element of the codomain need not be in the range.
Step 2
Why this answer is correct
The correct answer is A. (5) सहप्रांत में है पर परिसर में नहीं / (5) is in the codomain but not in the range. The range is ({2,3,4}), so (5) is not obtained. In exams, every element of the codomain need not be in the range.
Step 3
Exam Tip
परिसर ({2,3,4}) है इसलिए (5) प्राप्त नहीं होता। परीक्षा में सहप्रांत का हर तत्व परिसर में हो यह जरूरी नहीं है।
In option (A), (1) is associated with both (a) and (b). In exams, look for the same first component with two different second components.
Step 2
Why this answer is correct
The correct answer is A. ({(1,a),(1,b),(2,c),(3,d),(4,a)}). In option (A), (1) is associated with both (a) and (b). In exams, look for the same first component with two different second components.
Step 3
Exam Tip
विकल्प (A) में (1) को (a) और (b) दोनों से जोड़ा गया है। परीक्षा में समान पहले घटक के दो अलग दूसरे घटक खोजें।
In option (A), there is no pair for (4). In exams, every domain element must be included for a function.
Step 2
Why this answer is correct
The correct answer is A. ({(1,a),(2,b),(3,c)}). In option (A), there is no pair for (4). In exams, every domain element must be included for a function.
Step 3
Exam Tip
विकल्प (A) में (4) के लिए कोई युग्म नहीं है। परीक्षा में फलन के लिए हर प्रांत तत्व शामिल होना चाहिए।
A. क्योंकि प्रांत के हर तत्व की ठीक एक छवि होती है/Because every domain element has exactly one image
Step 1
Concept
A function is a relation in which every domain element is associated with exactly one image. In exams, remember that every relation is not a function.
Step 2
Why this answer is correct
The correct answer is A. क्योंकि प्रांत के हर तत्व की ठीक एक छवि होती है / Because every domain element has exactly one image. A function is a relation in which every domain element is associated with exactly one image. In exams, remember that every relation is not a function.
Step 3
Exam Tip
फलन संबंध है पर इसमें प्रत्येक प्रांत तत्व ठीक एक छवि से जुड़ता है। परीक्षा में याद रखें कि हर संबंध फलन नहीं होता।
In option (A), both (1) and (2) have exactly one image. In exams, check the whole domain before calling a subset a function.
Step 2
Why this answer is correct
The correct answer is A. ({(1,3),(2,5)}). In option (A), both (1) and (2) have exactly one image. In exams, check the whole domain before calling a subset a function.
Step 3
Exam Tip
विकल्प (A) में (1) और (2) दोनों की ठीक एक छवि है। परीक्षा में उपसमुच्चय को फलन कहने से पहले पूरा प्रांत जांचें।
(4) is not in the domain, so (16) is not obtained. In exams, a preimage must come from the domain only.
Step 2
Why this answer is correct
The correct answer is A. इसकी कोई पूर्वछवि नहीं है / It has no preimage. (4) is not in the domain, so (16) is not obtained. In exams, a preimage must come from the domain only.
Step 3
Exam Tip
प्रांत में (4) नहीं है इसलिए (16) प्राप्त नहीं होता। परीक्षा में पूर्वछवि केवल प्रांत से ही ली जाती है।
A. (f) फलन है और परिसर ({2,4,6}) है/(f) is a function and the range is ({2,4,6})
Step 1
Concept
Every domain element has exactly one image and the images are (2,4,6). In exams, check both function validity and range using the rule.
Step 2
Why this answer is correct
The correct answer is A. (f) फलन है और परिसर ({2,4,6}) है / (f) is a function and the range is ({2,4,6}). Every domain element has exactly one image and the images are (2,4,6). In exams, check both function validity and range using the rule.
Step 3
Exam Tip
हर प्रांत तत्व की ठीक एक छवि है और छवियां (2,4,6) हैं। परीक्षा में फलन और परिसर दोनों को नियम से जांचें।
In option (A), every domain element has one image, but (1) and (2) both have image (a). In exams, test function and one-one separately.
Step 2
Why this answer is correct
The correct answer is A. ({(1,a),(2,a),(3,b)}). In option (A), every domain element has one image, but (1) and (2) both have image (a). In exams, test function and one-one separately.
Step 3
Exam Tip
विकल्प (A) में हर प्रांत तत्व की एक छवि है पर (1) और (2) दोनों की छवि (a) है। परीक्षा में फलन और एक-एक की जांच अलग करें।
A. यह फलन है क्योंकि दोहराया युग्म अलग छवि नहीं देता/It is a function because the repeated pair does not give a different image
Step 1
Concept
Repeating the same pair does not create a new different image. In exams, distinguish between a repeated pair and a different image.
Step 2
Why this answer is correct
The correct answer is A. यह फलन है क्योंकि दोहराया युग्म अलग छवि नहीं देता / It is a function because the repeated pair does not give a different image. Repeating the same pair does not create a new different image. In exams, distinguish between a repeated pair and a different image.
Step 3
Exam Tip
समान युग्म दोहराने से कोई नई अलग छवि नहीं बनती। परीक्षा में दोहराए युग्म और अलग छवि में अंतर करें।
(A) contains both even and odd elements, so both images (0) and (1) are obtained. In exams, check all types of inputs.
Step 2
Why this answer is correct
The correct answer is C. ({0,1}). (A) contains both even and odd elements, so both images (0) and (1) are obtained. In exams, check all types of inputs.
Step 3
Exam Tip
(A) में सम और विषम दोनों तत्व हैं इसलिए दोनों छवियां (0) और (1) प्राप्त होती हैं। परीक्षा में सभी प्रकार के इनपुट जांचें।
A function has exactly one pair for every domain element, so the number of pairs will be (3). In exams, pairs in a function equal the number of domain elements.
Step 2
Why this answer is correct
The correct answer is A. (3). A function has exactly one pair for every domain element, so the number of pairs will be (3). In exams, pairs in a function equal the number of domain elements.
Step 3
Exam Tip
फलन में प्रांत के हर तत्व के लिए ठीक एक युग्म होता है इसलिए युग्मों की संख्या (3) होगी। परीक्षा में फलन के युग्म प्रांत के तत्वों के बराबर होते हैं।
The values are (|-2|=2), (|-1|=1), (|0|=0), (|1|=1), and (|2|=2), 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 A. ({0,1,2}). The values are (|-2|=2), (|-1|=1), (|0|=0), (|1|=1), and (|2|=2), so the range is ({0,1,2}). In exams, write only distinct obtained images in the range.
Step 3
Exam Tip
(|-2|=2), (|-1|=1), (|0|=0), (|1|=1) और (|2|=2), इसलिए परिसर ({0,1,2}) है। परीक्षा में परिसर में केवल अलग-अलग प्राप्त छवियां लिखें।
