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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