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})
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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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