यदि \(A=\{1,2,3\}\) और \(B=\{a,b\}\) हैं, तो \(A\times B\) में वह क्रमित युग्म कौन सा है जिसका पहला अवयव (3) है और दूसरा अवयव (b) है?
If \(A=\{1,2,3\}\) and \(B=\{a,b\}\), which ordered pair in \(A\times B\) has first component (3) and second component (b)?
#cartesian-product
#ordered-pair
#sets
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A ((3,b))
B ((b,3))
C ((3,a))
D ((2,b))
Explanation opens after your attempt
Correct Answer
A. ((3,b))
Step 1
Concept
In \(A\times B\), the first component comes from (A) and the second from (B). In exams, always check the order.
Step 2
Why this answer is correct
The correct answer is A. ((3,b)). In \(A\times B\), the first component comes from (A) and the second from (B). In exams, always check the order.
Step 3
Exam Tip
\(A\times B\) में पहला अवयव (A) से और दूसरा (B) से आता है। परीक्षा में क्रम का ध्यान रखें।
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यदि \(A=\{1,2,3,4\}\) और \(B=\{1,2,3,4\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (|a-b|=2) है?
If \(A=\{1,2,3,4\}\) and \(B=\{1,2,3,4\}\), how many pairs ((a,b)) in \(A\times B\) satisfy (|a-b|=2)?
#cartesian-product
#absolute-difference
#counting
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A (,2,)
B (,3,)
C (,4,)
D (,6,)
Explanation opens after your attempt
Step 1
Concept
The condition (|a-b|=2) gives the pairs ((1,3),(2,4),(3,1),(4,2)). In ordered pairs, reversed pairs are counted separately.
Step 2
Why this answer is correct
The correct answer is C. (,4,). The condition (|a-b|=2) gives the pairs ((1,3),(2,4),(3,1),(4,2)). In ordered pairs, reversed pairs are counted separately.
Step 3
Exam Tip
शर्त (|a-b|=2) से युग्म ((1,3),(2,4),(3,1),(4,2)) मिलते हैं। क्रमित युग्मों में उलटे युग्म अलग गिने जाते हैं।
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यदि \(A=\{0,1\}\) और \(B=\{2,4,6\}\) हैं, तो \(|A\times B|\) का मान क्या होगा?
If \(A=\{0,1\}\) and \(B=\{2,4,6\}\), what is the value of \(|A\times B|\)?
#cardinality
#cartesian-product
#counting
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A (5)
B (6)
C (8)
D (3)
Explanation opens after your attempt
Step 1
Concept
\(|A\times B|=|A|\cdot |B|=2\cdot3=6\). For cardinality questions, multiply the sizes directly.
Step 2
Why this answer is correct
The correct answer is B. (6). \(|A\times B|=|A|\cdot |B|=2\cdot3=6\). For cardinality questions, multiply the sizes directly.
Step 3
Exam Tip
\(|A\times B|=|A|\cdot |B|=2\cdot3=6\) होता है। कार्डिनलिटी वाले प्रश्नों में सीधे गुणा करें।
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यदि \(A=\{p,q\}\) और \(B=\{r,s\}\) हैं, तो \(B\times A\) कौन सा है?
If \(A=\{p,q\}\) and \(B=\{r,s\}\), which is \(B\times A\)?
#reverse-product
#ordered-pair
#sets
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A ({(p,r),(p,s),(q,r),(q,s)})
B ({(r,p),(r,q),(s,p),(s,q)})
C ({(p,q),(r,s)})
D ({(r,s),(p,q)})
Explanation opens after your attempt
Correct Answer
B. ({(r,p),(r,q),(s,p),(s,q)})
Step 1
Concept
In \(B\times A\), the first component is from (B) and the second from (A). Do not confuse \(A\times B\) with \(B\times A\).
Step 2
Why this answer is correct
The correct answer is B. ({(r,p),(r,q),(s,p),(s,q)}). In \(B\times A\), the first component is from (B) and the second from (A). Do not confuse \(A\times B\) with \(B\times A\).
Step 3
Exam Tip
\(B\times A\) में पहला अवयव (B) से और दूसरा (A) से लिया जाता है। \(A\times B\) और \(B\times A\) को अलग समझें।
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यदि \(A=\{1,2\}\), \(B=\{2,3\}\) और \(A\times B=B\times A\) नहीं है, तो इसका मुख्य कारण क्या है?
If \(A=\{1,2\}\), \(B=\{2,3\}\), and \(A\times B\ne B\times A\), what is the main reason?
#non-commutative
#ordered-pair
#concept
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A क्रमित युग्मों में क्रम महत्त्वपूर्ण होता है / Order matters in ordered pairs
B दोनों समुच्चयों में समान अवयव हैं / Both sets have the same elements
C कार्तीय गुणनफल हमेशा रिक्त होता है / Cartesian product is always empty
D कार्डिनलिटी हमेशा अलग होती है / Cardinalities are always different
Explanation opens after your attempt
Correct Answer
A. क्रमित युग्मों में क्रम महत्त्वपूर्ण होता है / Order matters in ordered pairs
Step 1
Concept
In ordered pairs, ((1,3)) and ((3,1)) are not the same. Hence \(A\times B\) is generally not equal to \(B\times A\).
Step 2
Why this answer is correct
The correct answer is A. क्रमित युग्मों में क्रम महत्त्वपूर्ण होता है / Order matters in ordered pairs. In ordered pairs, ((1,3)) and ((3,1)) are not the same. Hence \(A\times B\) is generally not equal to \(B\times A\).
Step 3
Exam Tip
क्रमित युग्म में ((1,3)) और ((3,1)) समान नहीं होते। इसलिए \(A\times B\) सामान्यतः \(B\times A\) के बराबर नहीं होता।
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यदि \(A=\{1,2\}\), \(B=\{3\}\) और \(C=\{4,5,6\}\) हैं, तो (|\(A\times B\)\times C|) क्या होगा?
If \(A=\{1,2\}\), \(B=\{3\}\), and \(C=\{4,5,6\}\), what is (|\(A\times B\)\times C|)?
#nested-product
#cardinality
#hard
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A (5)
B (6)
C (9)
D (12)
Explanation opens after your attempt
Step 1
Concept
\(|A\times B|=2\cdot1=2\), then (|\(A\times B\)\times C|=2\cdot3=6). Cardinalities multiply in nested products too.
Step 2
Why this answer is correct
The correct answer is B. (6). \(|A\times B|=2\cdot1=2\), then (|\(A\times B\)\times C|=2\cdot3=6). Cardinalities multiply in nested products too.
Step 3
Exam Tip
\(|A\times B|=2\cdot1=2\) और फिर (|\(A\times B\)\times C|=2\cdot3=6)। संयुक्त गुणनफल में भी कार्डिनलिटी गुणा होती है।
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यदि \(A=\{1,2,3\}\) और \(B=\{4,5\}\) हैं, तो \(A\times B\) का उपसमुच्चय \(R=\{(x,y):x+y=7\}\) क्या है?
If \(A=\{1,2,3\}\) and \(B=\{4,5\}\), what is the subset \(R=\{(x,y):x+y=7\}\) of \(A\times B\)?
#relation-subset
#condition
#sum
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A ({(2,5),(3,4)})
B ({(1,5),(2,4)})
C ({(3,5)})
D ({(1,4),(2,5)})
Explanation opens after your attempt
Correct Answer
A. ({(2,5),(3,4)})
Step 1
Concept
Since (2+5=7) and (3+4=7), these pairs belong to (R). Think of all pairs in \(A\times B\) before filtering.
Step 2
Why this answer is correct
The correct answer is A. ({(2,5),(3,4)}). Since (2+5=7) and (3+4=7), these pairs belong to (R). Think of all pairs in \(A\times B\) before filtering.
Step 3
Exam Tip
(2+5=7) और (3+4=7), इसलिए यही युग्म आते हैं। संबंध बनाने से पहले पूरा \(A\times B\) सोचें।
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यदि \(A=\{-1,0,1\}\) और \(B=\{0,1\}\) हैं, तो \(A\times B\) में कितने युग्म ((x,y)) ऐसे हैं जिनके लिए \(x^2=y\) है?
If \(A=\{-1,0,1\}\) and \(B=\{0,1\}\), how many pairs ((x,y)) in \(A\times B\) satisfy \(x^2=y\)?
#condition-counting
#square
#cartesian-product
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A (2)
B (3)
C (4)
D (6)
Explanation opens after your attempt
Step 1
Concept
((-1)2 =1), \(0^2=0\), and \(1^2=1\), so there are (3) pairs. Check (y) for each (x).
Step 2
Why this answer is correct
The correct answer is B. (3). ((-1)2 =1), \(0^2=0\), and \(1^2=1\), so there are (3) pairs. Check (y) for each (x).
Step 3
Exam Tip
((-1)2 =1), \(0^2=0\), और \(1^2=1\), इसलिए (3) युग्म मिलते हैं। प्रत्येक (x) के लिए (y) जाँचें।
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यदि \(A=\{1,2,3,4\}\) और \(B=\{1,2,3\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (a>b) है?
If \(A=\{1,2,3,4\}\) and \(B=\{1,2,3\}\), how many pairs ((a,b)) in \(A\times B\) have (a>b)?
#inequality
#counting
#ordered-pair
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A (5)
B (6)
C (7)
D (8)
Explanation opens after your attempt
Step 1
Concept
For (a=2,3,4), the counts are (1,2,3), giving (6). For inequalities, count using the first component.
Step 2
Why this answer is correct
The correct answer is B. (6). For (a=2,3,4), the counts are (1,2,3), giving (6). For inequalities, count using the first component.
Step 3
Exam Tip
(a=2,3,4) के लिए क्रमशः (1,2,3) मान मिलते हैं, कुल (6)। असमानता में पहले अवयव को आधार बनाकर गिनें।
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यदि \(A=\{1,2,3\}\) है और \(|A\times B|=12\), तो (|B|) क्या होगा?
If \(A=\{1,2,3\}\) and \(|A\times B|=12\), what is (|B|)?
#unknown-cardinality
#formula
#sets
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A (3)
B (4)
C (9)
D (12)
Explanation opens after your attempt
Step 1
Concept
Since (|A|=3) and \(3\cdot |B|=12\), we get (|B|=4). Divide to find the unknown cardinality.
Step 2
Why this answer is correct
The correct answer is B. (4). Since (|A|=3) and \(3\cdot |B|=12\), we get (|B|=4). Divide to find the unknown cardinality.
Step 3
Exam Tip
(|A|=3) और \(3\cdot |B|=12\), इसलिए (|B|=4)। अज्ञात कार्डिनलिटी निकालने में भाग दें।
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यदि (A) में (m) अवयव और (B) में (n) अवयव हैं, तो \(A\times B\) के उपसमुच्चयों की संख्या क्या होगी?
If (A) has (m) elements and (B) has (n) elements, how many subsets does \(A\times B\) have?
#power-set
#cardinality
#general-form
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A (mn)
B \(2^{m+n}\)
C \(2^{mn}\)
D \(m^n\)
Explanation opens after your attempt
Correct Answer
C. \(2^{mn}\)
Step 1
Concept
\(|A\times B|=mn\), so the number of subsets is \(2^{mn}\). First find the cardinality of the base set.
Step 2
Why this answer is correct
The correct answer is C. \(2^{mn}\). \(|A\times B|=mn\), so the number of subsets is \(2^{mn}\). First find the cardinality of the base set.
Step 3
Exam Tip
\(|A\times B|=mn\), इसलिए इसके उपसमुच्चयों की संख्या \(2^{mn}\) होगी। पहले मूल समुच्चय की कार्डिनलिटी निकालें।
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यदि \(A=\{1,2\}\) और \(B=\{x,y,z\}\) हैं, तो \(A\times B\) के सभी संबंधों की संख्या कितनी है?
If \(A=\{1,2\}\) and \(B=\{x,y,z\}\), how many relations from (A) to (B) are possible?
#relations
#counting
#power-set
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A (6)
B (8)
C (32)
D (64)
Explanation opens after your attempt
Step 1
Concept
A relation is any subset of \(A\times B\), and \(|A\times B|=6\). Hence the number of relations is \(2^6=64\).
Step 2
Why this answer is correct
The correct answer is D. (64). A relation is any subset of \(A\times B\), and \(|A\times B|=6\). Hence the number of relations is \(2^6=64\).
Step 3
Exam Tip
संबंध \(A\times B\) का कोई भी उपसमुच्चय है और \(|A\times B|=6\)। इसलिए संबंधों की संख्या \(2^6=64\) है।
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यदि \(A=\{1,2,3\}\) और \(B=\{4,5\}\) हैं, तो \(A\times B\) के कितने गैर-रिक्त उपसमुच्चय हैं?
If \(A=\{1,2,3\}\) and \(B=\{4,5\}\), how many non-empty subsets does \(A\times B\) have?
#non-empty-subsets
#relations
#counting
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A (31)
B (63)
C (64)
D (32)
Explanation opens after your attempt
Step 1
Concept
\(|A\times B|=3\cdot2=6\), so non-empty subsets are \(2^6-1=63\). Do not forget to subtract the empty set.
Step 2
Why this answer is correct
The correct answer is B. (63). \(|A\times B|=3\cdot2=6\), so non-empty subsets are \(2^6-1=63\). Do not forget to subtract the empty set.
Step 3
Exam Tip
\(|A\times B|=3\cdot2=6\), इसलिए गैर-रिक्त उपसमुच्चय \(2^6-1=63\) हैं। रिक्त समुच्चय घटाना न भूलें।
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यदि \(A=\{1,2,3\}\), \(B=\{2,3,4\}\) हैं, तो \(A\times B\) में ((2,2)), ((3,4)), ((4,3)) में से कौन सा युग्म नहीं है?
If \(A=\{1,2,3\}\), \(B=\{2,3,4\}\), which among ((2,2)), ((3,4)), and ((4,3)) is not in \(A\times B\)?
#membership
#ordered-pair
#error-check
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A ((2,2))
B ((3,4))
C ((4,3))
D तीनों हैं / All three are present
Explanation opens after your attempt
Correct Answer
C. ((4,3))
Step 1
Concept
In ((4,3)), the first component (4) is not in (A). Check both positions separately for membership.
Step 2
Why this answer is correct
The correct answer is C. ((4,3)). In ((4,3)), the first component (4) is not in (A). Check both positions separately for membership.
Step 3
Exam Tip
((4,3)) में पहला अवयव (4) है, जो (A) में नहीं है। सदस्यता में दोनों स्थान अलग-अलग जाँचें।
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यदि \(A=\{1,2\}\) है और \(A\times B={(1,a),(1,b),(2,a),(2,b)}\), तो (B) क्या है?
If \(A=\{1,2\}\) and \(A\times B={(1,a),(1,b),(2,a),(2,b)}\), what is (B)?
#identify-set
#cartesian-product
#reasoning
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A ({1,2})
B ({a,b})
C ({(1,a),(2,b)})
D ({1,a,2,b})
Explanation opens after your attempt
Correct Answer
B. ({a,b})
Step 1
Concept
The set of second components is \(B=\{a,b\}\). When identifying original sets, observe the positions.
Step 2
Why this answer is correct
The correct answer is B. ({a,b}). The set of second components is \(B=\{a,b\}\). When identifying original sets, observe the positions.
Step 3
Exam Tip
दूसरे अवयवों का समुच्चय \(B=\{a,b\}\) है। कार्तीय गुणनफल से मूल समुच्चय पहचानते समय स्थान देखें।
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यदि \(A\times B=\varnothing\) और \(A=\{1,2,3\}\), तो (B) के बारे में कौन सा कथन सही है?
If \(A\times B=\varnothing\) and \(A=\{1,2,3\}\), which statement about (B) is correct?
#empty-set
#cartesian-product
#concept
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A \(B=\varnothing\)
B (B=A)
C (B) में (3) अवयव हैं / (B) has (3) elements
D \(B\subset A\) अवश्य है / \(B\subset A\) must hold
Explanation opens after your attempt
Correct Answer
A. \(B=\varnothing\)
Step 1
Concept
If (A) is non-empty and \(A\times B=\varnothing\), then \(B=\varnothing\). A Cartesian product is empty if at least one set is empty.
Step 2
Why this answer is correct
The correct answer is A. \(B=\varnothing\). If (A) is non-empty and \(A\times B=\varnothing\), then \(B=\varnothing\). A Cartesian product is empty if at least one set is empty.
Step 3
Exam Tip
यदि (A) रिक्त नहीं है और \(A\times B=\varnothing\), तो \(B=\varnothing\) होना चाहिए। रिक्त गुणनफल में कम से कम एक समुच्चय रिक्त होता है।
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यदि \(A=\varnothing\) और \(B=\{1,2,3\}\), तो \(A\times B\) क्या होगा?
If \(A=\varnothing\) and \(B=\{1,2,3\}\), what is \(A\times B\)?
#empty-set
#basic-property
#cartesian-product
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A \(\varnothing\)
B ({(0,1),(0,2),(0,3)})
C ({1,2,3})
D \({\varnothing,1,2,3}\)
Explanation opens after your attempt
Correct Answer
A. \(\varnothing\)
Step 1
Concept
The first set is empty, so no ordered pair can be formed. Treat the empty set as a set with (0) elements.
Step 2
Why this answer is correct
The correct answer is A. \(\varnothing\). The first set is empty, so no ordered pair can be formed. Treat the empty set as a set with (0) elements.
Step 3
Exam Tip
पहला समुच्चय रिक्त है, इसलिए कोई क्रमित युग्म बन नहीं सकता। रिक्त समुच्चय को (0) अवयव वाला समुच्चय मानें।
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यदि (|A|=4), (|B|=5) और \(R\subseteq A\times B\), तो (R) में अधिकतम कितने अवयव हो सकते हैं?
If (|A|=4), (|B|=5), and \(R\subseteq A\times B\), what is the maximum possible number of elements in (R)?
#maximum-elements
#relation-subset
#cardinality
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A (9)
B (20)
C (25)
D \(2^{20}\)
Explanation opens after your attempt
Step 1
Concept
Since (R) can be at most the whole \(A\times B\), and \(|A\times B|=20\). If maximum elements are asked, do not write \(2^{20}\).
Step 2
Why this answer is correct
The correct answer is B. (20). Since (R) can be at most the whole \(A\times B\), and \(|A\times B|=20\). If maximum elements are asked, do not write \(2^{20}\).
Step 3
Exam Tip
क्योंकि (R) अधिकतम पूरे \(A\times B\) के बराबर हो सकता है और \(|A\times B|=20\)। अधिकतम अवयव पूछे जाएँ तो \(2^{20}\) नहीं लिखें।
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यदि (|A|=2), (|B|=3) और (|C|=4), तो \(|A\times B\times C|\) क्या होगा?
If (|A|=2), (|B|=3), and (|C|=4), what is \(|A\times B\times C|\)?
#ordered-triple
#three-sets
#cardinality
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A (9)
B (12)
C (24)
D \(2^9\)
Explanation opens after your attempt
Step 1
Concept
For Cartesian product of three sets, the cardinality is \(2\cdot3\cdot4=24\). For ordered triples, multiply all three sizes.
Step 2
Why this answer is correct
The correct answer is C. (24). For Cartesian product of three sets, the cardinality is \(2\cdot3\cdot4=24\). For ordered triples, multiply all three sizes.
Step 3
Exam Tip
तीन समुच्चयों के कार्तीय गुणनफल में कार्डिनलिटी \(2\cdot3\cdot4=24\) है। क्रमित त्रिक के प्रश्न में तीनों आकार गुणा करें।
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यदि \(A=\{1,2\}\), \(B=\{3,4\}\), \(C=\{5\}\) हैं, तो (A\times\(B\cup C\)) में कितने अवयव हैं?
If \(A=\{1,2\}\), \(B=\{3,4\}\), \(C=\{5\}\), how many elements are in (A\times\(B\cup C\))?
#union
#cartesian-product
#cardinality
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A (4)
B (5)
C (6)
D (8)
Explanation opens after your attempt
Step 1
Concept
\(B\cup C={3,4,5}\), so (|A\times\(B\cup C\)|=2\cdot3=6). First evaluate the set inside the brackets.
Step 2
Why this answer is correct
The correct answer is C. (6). \(B\cup C={3,4,5}\), so (|A\times\(B\cup C\)|=2\cdot3=6). First evaluate the set inside the brackets.
Step 3
Exam Tip
\(B\cup C={3,4,5}\), इसलिए (|A\times\(B\cup C\)|=2\cdot3=6)। पहले कोष्ठक के भीतर का समुच्चय निकालें।
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यदि \(A=\{1,2,3\}\), \(B=\{2,3,4\}\), \(C=\{3,4,5\}\) हैं, तो (|A\times\(B\cap C\)|) क्या है?
If \(A=\{1,2,3\}\), \(B=\{2,3,4\}\), \(C=\{3,4,5\}\), what is (|A\times\(B\cap C\)|)?
#intersection
#cardinality
#cartesian-product
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A (3)
B (6)
C (9)
D (12)
Explanation opens after your attempt
Step 1
Concept
\(B\cap C={3,4}\), so (|A\times\(B\cap C\)|=3\cdot2=6). Multiply only after finding the intersection.
Step 2
Why this answer is correct
The correct answer is B. (6). \(B\cap C={3,4}\), so (|A\times\(B\cap C\)|=3\cdot2=6). Multiply only after finding the intersection.
Step 3
Exam Tip
\(B\cap C={3,4}\), इसलिए (|A\times\(B\cap C\)|=3\cdot2=6)। प्रतिच्छेद के बाद ही गुणा करें।
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यदि \(A=\{1,2\}\), \(B=\{2,3\}\), \(C=\{3,4\}\) हैं, तो (\(A\times B\)\cap\(A\times C\)) क्या है?
If \(A=\{1,2\}\), \(B=\{2,3\}\), \(C=\{3,4\}\), what is (\(A\times B\)\cap\(A\times C\))?
#intersection-property
#set-identity
#ordered-pair
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A ({(1,3),(2,3)})
B ({(1,2),(2,2)})
C ({(1,4),(2,4)})
D \(\varnothing\)
Explanation opens after your attempt
Correct Answer
A. ({(1,3),(2,3)})
Step 1
Concept
(\(A\times B\)\cap\(A\times C\)=A\times\(B\cap C\)), and \(B\cap C={3}\). Thus the pairs are ((1,3)) and ((2,3)).
Step 2
Why this answer is correct
The correct answer is A. ({(1,3),(2,3)}). (\(A\times B\)\cap\(A\times C\)=A\times\(B\cap C\)), and \(B\cap C={3}\). Thus the pairs are ((1,3)) and ((2,3)).
Step 3
Exam Tip
(\(A\times B\)\cap\(A\times C\)=A\times\(B\cap C\)) और \(B\cap C={3}\)। इसलिए युग्म ((1,3)) और ((2,3)) हैं।
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यदि \(A=\{0,1\}\), \(B=\{1,2\}\), \(C=\{2,3\}\) हैं, तो \(A\times(B-C)\) क्या है?
If \(A=\{0,1\}\), \(B=\{1,2\}\), \(C=\{2,3\}\), what is \(A\times(B-C)\)?
#difference
#cartesian-product
#set-operation
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A ({(0,1),(1,1)})
B ({(0,2),(1,2)})
C ({(0,1),(0,2),(1,1),(1,2)})
D \(\varnothing\)
Explanation opens after your attempt
Correct Answer
A. ({(0,1),(1,1)})
Step 1
Concept
(B-C={1}), so \(A\times(B-C)={(0,1),(1,1)}\). For difference, keep only elements of (B) not in (C).
Step 2
Why this answer is correct
The correct answer is A. ({(0,1),(1,1)}). (B-C={1}), so \(A\times(B-C)={(0,1),(1,1)}\). For difference, keep only elements of (B) not in (C).
Step 3
Exam Tip
(B-C={1}), इसलिए \(A\times(B-C)={(0,1),(1,1)}\)। अंतर निकालने में केवल (B) के अवयव देखें।
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यदि \(A=\{1,2\}\) और \(B=\{3,4,5\}\), तो \(A\times B\) के कितने उपसमुच्चय ठीक (2) अवयवों वाले हैं?
If \(A=\{1,2\}\) and \(B=\{3,4,5\}\), how many subsets of \(A\times B\) have exactly (2) elements?
#combinations
#subsets
#counting
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A (12)
B (15)
C (20)
D (36)
Explanation opens after your attempt
Step 1
Concept
\(|A\times B|=6\), and choosing (2) elements gives \(\binom{6}{2}=15\). Exactly (2) elements means a combination.
Step 2
Why this answer is correct
The correct answer is B. (15). \(|A\times B|=6\), and choosing (2) elements gives \(\binom{6}{2}=15\). Exactly (2) elements means a combination.
Step 3
Exam Tip
\(|A\times B|=6\) और (2) अवयव चुनने के तरीके \(\binom{6}{2}=15\) हैं। ठीक (2) अवयव का अर्थ संयोजन है।
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यदि \(A=\{1,2,3\}\) और \(B=\{4,5\}\), तो \(A\times B\) के कितने उपसमुच्चय ठीक (1) अवयव वाले संबंध हैं?
If \(A=\{1,2,3\}\) and \(B=\{4,5\}\), how many one-element relations are subsets of \(A\times B\)?
#singleton-relation
#counting
#cartesian-product
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A (3)
B (6)
C (12)
D \(2^6\)
Explanation opens after your attempt
Step 1
Concept
\(|A\times B|=6\), so there are (6) ways to choose a one-element relation. A singleton relation is just one ordered pair.
Step 2
Why this answer is correct
The correct answer is B. (6). \(|A\times B|=6\), so there are (6) ways to choose a one-element relation. A singleton relation is just one ordered pair.
Step 3
Exam Tip
\(|A\times B|=6\), इसलिए (1) अवयव वाला संबंध चुनने के (6) तरीके हैं। एकल संबंध सीधे एक क्रमित युग्म होता है।
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यदि \(A=\{1,2,3,4\}\) और \(B=\{1,2,3,4\}\), तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (a+b) सम है?
If \(A=\{1,2,3,4\}\) and \(B=\{1,2,3,4\}\), how many pairs ((a,b)) in \(A\times B\) have (a+b) even?
#parity
#counting
#condition
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A (6)
B (8)
C (10)
D (16)
Explanation opens after your attempt
Step 1
Concept
The sum is even when both components have the same parity. The count is \(2\cdot2+2\cdot2=8\).
Step 2
Why this answer is correct
The correct answer is B. (8). The sum is even when both components have the same parity. The count is \(2\cdot2+2\cdot2=8\).
Step 3
Exam Tip
योग सम तब है जब दोनों अवयवों की समता समान हो। \(2\cdot2+2\cdot2=8\) युग्म मिलते हैं।
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यदि \(A=\{1,2,3,4,5\}\) और \(B=\{1,2,3,4,5\}\), तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (a+b=6) है?
If \(A=\{1,2,3,4,5\}\) and \(B=\{1,2,3,4,5\}\), how many pairs ((a,b)) in \(A\times B\) satisfy (a+b=6)?
#sum-condition
#ordered-pair
#counting
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A (4)
B (5)
C (6)
D (10)
Explanation opens after your attempt
Step 1
Concept
The pairs are ((1,5),(2,4),(3,3),(4,2),(5,1)), so there are (5). Reversed pairs count separately in ordered pairs.
Step 2
Why this answer is correct
The correct answer is B. (5). The pairs are ((1,5),(2,4),(3,3),(4,2),(5,1)), so there are (5). Reversed pairs count separately in ordered pairs.
Step 3
Exam Tip
युग्म ((1,5),(2,4),(3,3),(4,2),(5,1)) हैं, इसलिए कुल (5) हैं। क्रमित युग्मों में उलटे युग्म अलग गिने जाते हैं।
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यदि \(A=\{1,2,3,4\}\) और \(B=\{2,3,4,5\}\), तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (a) (b) को विभाजित करता है?
If \(A=\{1,2,3,4\}\) and \(B=\{2,3,4,5\}\), how many pairs ((a,b)) in \(A\times B\) have (a) dividing (b)?
#divisibility
#counting
#relation
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A (7)
B (8)
C (9)
D (10)
Explanation opens after your attempt
Step 1
Concept
For (a=1,2,3,4), the counts are (4,2,1,1), totaling (8). In divisibility, check each (a) separately.
Step 2
Why this answer is correct
The correct answer is B. (8). For (a=1,2,3,4), the counts are (4,2,1,1), totaling (8). In divisibility, check each (a) separately.
Step 3
Exam Tip
(a=1) से (4), (a=2) से (2), (a=3) से (1), (a=4) से (1) युग्म मिलते हैं, कुल (8)। विभाज्यता में हर (a) अलग जाँचें।
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यदि \(A=\{1,2,3\}\) और \(B=\{1,4,9\}\), तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें \(b=a^2\) है?
If \(A=\{1,2,3\}\) and \(B=\{1,4,9\}\), how many pairs ((a,b)) in \(A\times B\) satisfy \(b=a^2\)?
#function-rule
#square
#cartesian-product
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A (1)
B (2)
C (3)
D (6)
Explanation opens after your attempt
Step 1
Concept
Since \(1^2=1\), \(2^2=4\), and \(3^2=9\), there are (3) pairs. For rule-based relations, check each first component.
Step 2
Why this answer is correct
The correct answer is C. (3). Since \(1^2=1\), \(2^2=4\), and \(3^2=9\), there are (3) pairs. For rule-based relations, check each first component.
Step 3
Exam Tip
\(1^2=1\), \(2^2=4\), और \(3^2=9\), इसलिए (3) युग्म हैं। नियम आधारित संबंध में हर पहला अवयव जाँचें।
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यदि \(A=\{0,1,2\}\) और \(B=\{0,1,2,3\}\), तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (a+b<3) है?
If \(A=\{0,1,2\}\) and \(B=\{0,1,2,3\}\), how many pairs ((a,b)) in \(A\times B\) satisfy (a+b<3)?
#inequality
#counting
#condition
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A (5)
B (6)
C (7)
D (8)
Explanation opens after your attempt
Step 1
Concept
For (a=0,1,2), the possible values of (b) are (3,2,1), totaling (6). Read the inequality boundary carefully.
Step 2
Why this answer is correct
The correct answer is B. (6). For (a=0,1,2), the possible values of (b) are (3,2,1), totaling (6). Read the inequality boundary carefully.
Step 3
Exam Tip
(a=0,1,2) के लिए (b) के क्रमशः (3,2,1) मान हैं, कुल (6)। असमानता में सीमा को सावधानी से देखें।
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यदि \(A=\{1,2,3\}\) और \(B=\{1,2,3\}\), तो \(A\times B\) में विकर्ण युग्मों ((a,a)) की संख्या कितनी है?
If \(A=\{1,2,3\}\) and \(B=\{1,2,3\}\), how many diagonal pairs ((a,a)) are in \(A\times B\)?
#diagonal-pairs
#ordered-pair
#identity-relation
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A (2)
B (3)
C (6)
D (9)
Explanation opens after your attempt
Step 1
Concept
The diagonal pairs are ((1,1),(2,2),(3,3)), so the count is (3). For equal sets, the diagonal count is usually (|A|).
Step 2
Why this answer is correct
The correct answer is B. (3). The diagonal pairs are ((1,1),(2,2),(3,3)), so the count is (3). For equal sets, the diagonal count is usually (|A|).
Step 3
Exam Tip
विकर्ण युग्म ((1,1),(2,2),(3,3)) हैं, इसलिए संख्या (3) है। समान समुच्चयों में विकर्ण की संख्या सामान्यतः (|A|) होती है।
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यदि \(A=\{1,2,3,4\}\) और \(B=\{1,2,3,4\}\), तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें \(a\ne b\) है?
If \(A=\{1,2,3,4\}\) and \(B=\{1,2,3,4\}\), how many pairs ((a,b)) in \(A\times B\) satisfy \(a\ne b\)?
#not-equal
#complement-counting
#cartesian-product
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A (4)
B (8)
C (12)
D (16)
Explanation opens after your attempt
Step 1
Concept
There are (16) total pairs and (4) equal pairs, so (16-4=12). Complement counting gives a quick answer.
Step 2
Why this answer is correct
The correct answer is C. (12). There are (16) total pairs and (4) equal pairs, so (16-4=12). Complement counting gives a quick answer.
Step 3
Exam Tip
कुल युग्म (16) हैं और बराबर युग्म (4) हैं, इसलिए (16-4=12)। पूरक गिनती जल्दी उत्तर देती है।
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यदि \(A=\{2,4,6\}\) और \(B=\{1,2,3\}\), तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें \(\frac{a}{b}=2\) है?
If \(A=\{2,4,6\}\) and \(B=\{1,2,3\}\), how many pairs ((a,b)) in \(A\times B\) satisfy \(\frac{a}{b}=2\)?
#fraction-condition
#counting
#ordered-pair
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A (1)
B (2)
C (3)
D (4)
Explanation opens after your attempt
Step 1
Concept
The condition gives (a=2b), so the pairs are ((2,1),(4,2),(6,3)). Convert a fraction condition into a simple linear relation.
Step 2
Why this answer is correct
The correct answer is C. (3). The condition gives (a=2b), so the pairs are ((2,1),(4,2),(6,3)). Convert a fraction condition into a simple linear relation.
Step 3
Exam Tip
शर्त (a=2b) देती है, इसलिए ((2,1),(4,2),(6,3)) मिलते हैं। भिन्न को सरल रैखिक संबंध में बदलें।
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यदि \(A=\{1,2,3\}\) और \(B=\{2,4,6,8\}\), तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (b=2a) है?
If \(A=\{1,2,3\}\) and \(B=\{2,4,6,8\}\), how many pairs ((a,b)) in \(A\times B\) satisfy (b=2a)?
#linear-rule
#cartesian-product
#counting
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A (2)
B (3)
C (4)
D (6)
Explanation opens after your attempt
Step 1
Concept
For (a=1,2,3), we get (b=2,4,6), all in (B). Therefore there are (3) pairs.
Step 2
Why this answer is correct
The correct answer is B. (3). For (a=1,2,3), we get (b=2,4,6), all in (B). Therefore there are (3) pairs.
Step 3
Exam Tip
(a=1,2,3) से (b=2,4,6) मिलते हैं, और ये (B) में हैं। इसलिए (3) युग्म हैं।
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यदि \(A=\{1,2,3,4,5\}\) और \(B=\{1,2,3\}\), तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें \(a\le b\) है?
If \(A=\{1,2,3,4,5\}\) and \(B=\{1,2,3\}\), how many pairs ((a,b)) in \(A\times B\) satisfy \(a\le b\)?
#inequality
#less-than-equal
#counting
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A (5)
B (6)
C (7)
D (9)
Explanation opens after your attempt
Step 1
Concept
For (b=1,2,3), the possible values of (a) are (1,2,3), totaling (6). Sometimes counting by the second component is easier.
Step 2
Why this answer is correct
The correct answer is B. (6). For (b=1,2,3), the possible values of (a) are (1,2,3), totaling (6). Sometimes counting by the second component is easier.
Step 3
Exam Tip
(b=1,2,3) के लिए (a) के क्रमशः (1,2,3) मान हैं, कुल (6)। कभी-कभी दूसरे अवयव से गिनना आसान होता है।
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यदि \(A=\{1,2,3\}\) और \(B=\{1,2,3\}\), तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें \(a+b\ge5\) है?
If \(A=\{1,2,3\}\) and \(B=\{1,2,3\}\), how many pairs ((a,b)) in \(A\times B\) satisfy \(a+b\ge5\)?
#greater-than-equal
#sum-condition
#counting
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A (2)
B (3)
C (4)
D (5)
Explanation opens after your attempt
Step 1
Concept
The pairs with sum (5) or more are ((2,3),(3,2),(3,3)), totaling (3). Include equality when the condition has a boundary.
Step 2
Why this answer is correct
The correct answer is B. (3). The pairs with sum (5) or more are ((2,3),(3,2),(3,3)), totaling (3). Include equality when the condition has a boundary.
Step 3
Exam Tip
योग (5) या अधिक के युग्म ((2,3),(3,2),(3,3)) हैं, कुल (3)। सीमा सहित शर्त में बराबरी भी गिनें।
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यदि \(A=\{1,3,5\}\) और \(B=\{2,4,6\}\), तो \(A\times B\) में प्रत्येक युग्म ((a,b)) के लिए (a+b) कैसा होगा?
If \(A=\{1,3,5\}\) and \(B=\{2,4,6\}\), what will (a+b) be for every pair ((a,b)) in \(A\times B\)?
#parity
#reasoning
#cartesian-product
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A हमेशा सम / Always even
B हमेशा विषम / Always odd
C कभी सम कभी विषम / Sometimes even sometimes odd
D हमेशा (0) / Always (0)
Explanation opens after your attempt
Correct Answer
B. हमेशा विषम / Always odd
Step 1
Concept
The sum of an odd number and an even number is always odd. For parity questions, writing all pairs is not necessary.
Step 2
Why this answer is correct
The correct answer is B. हमेशा विषम / Always odd. The sum of an odd number and an even number is always odd. For parity questions, writing all pairs is not necessary.
Step 3
Exam Tip
विषम संख्या और सम संख्या का योग हमेशा विषम होता है। समता वाले प्रश्न में सभी युग्म लिखना आवश्यक नहीं।
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यदि \(A=\{1,2\}\) और \(B=\{1,2\}\), तो \(A\times B\) और \(B\times A\) के बारे में कौन सा कथन सही है?
If \(A=\{1,2\}\) and \(B=\{1,2\}\), which statement about \(A\times B\) and \(B\times A\) is correct?
#equality
#reverse-product
#concept
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A \(A\times B=B\times A\)
B \(A\times B\ne B\times A\)
C \(|A\times B|=0\)
D \(A\times B\subset B\)
Explanation opens after your attempt
Correct Answer
A. \(A\times B=B\times A\)
Step 1
Concept
Here (A=B), so \(A\times B\) and \(B\times A\) are the same. Equality is clear when the two sets are identical.
Step 2
Why this answer is correct
The correct answer is A. \(A\times B=B\times A\). Here (A=B), so \(A\times B\) and \(B\times A\) are the same. Equality is clear when the two sets are identical.
Step 3
Exam Tip
यहाँ (A=B), इसलिए \(A\times B\) और \(B\times A\) समान हैं। सामान्य नियम में बराबरी तभी स्पष्ट होती है जब समुच्चय समान हों।
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यदि \(A=\{1,2,3\}\) और \(B=\{4,5\}\), तो कथन \(A\times B\subseteq B\times A\) के बारे में क्या सही है?
If \(A=\{1,2,3\}\) and \(B=\{4,5\}\), what is true about the statement \(A\times B\subseteq B\times A\)?
#subset-check
#counterexample
#ordered-pair
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A सत्य है / True
B असत्य है / False
C केवल (|A|=|B|) होने पर सत्य / True only when (|A|=|B|)
D हमेशा सत्य / Always true
Explanation opens after your attempt
Correct Answer
B. असत्य है / False
Step 1
Concept
\((1,4)\in A\times B\), but \((1,4)\notin B\times A\) because \(1\notin B\). One counterexample is enough.
Step 2
Why this answer is correct
The correct answer is B. असत्य है / False. \((1,4)\in A\times B\), but \((1,4)\notin B\times A\) because \(1\notin B\). One counterexample is enough.
Step 3
Exam Tip
\((1,4)\in A\times B\), पर \((1,4)\notin B\times A\) क्योंकि \(1\notin B\)। एक प्रतिउदाहरण पर्याप्त है।
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यदि \(A=\{1,2\}\), \(B=\{3,4\}\), \(C=\{5,6\}\) हैं, तो (\(A\cup B\)\times C) की कार्डिनलिटी क्या है?
If \(A=\{1,2\}\), \(B=\{3,4\}\), \(C=\{5,6\}\), what is the cardinality of (\(A\cup B\)\times C)?
#union
#cardinality
#cartesian-product
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A (4)
B (6)
C (8)
D (16)
Explanation opens after your attempt
Step 1
Concept
\(A\cup B={1,2,3,4}\), so (|\(A\cup B\)\times C|=4\cdot2=8). Find the union size first.
Step 2
Why this answer is correct
The correct answer is C. (8). \(A\cup B={1,2,3,4}\), so (|\(A\cup B\)\times C|=4\cdot2=8). Find the union size first.
Step 3
Exam Tip
\(A\cup B={1,2,3,4}\), इसलिए (|\(A\cup B\)\times C|=4\cdot2=8)। पहले संघ की कार्डिनलिटी निकालें।
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यदि \(A=\{1,2,3\}\), \(B=\{2,3,4\}\), \(C=\{1,3,5\}\) हैं, तो (|\(A\cap B\)\times C|) क्या होगा?
If \(A=\{1,2,3\}\), \(B=\{2,3,4\}\), \(C=\{1,3,5\}\), what is (|\(A\cap B\)\times C|)?
#intersection
#cardinality
#set-operation
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A (4)
B (5)
C (6)
D (9)
Explanation opens after your attempt
Step 1
Concept
\(A\cap B={2,3}\) and (|C|=3), so the cardinality is \(2\cdot3=6\). Simplify the intersection first.
Step 2
Why this answer is correct
The correct answer is C. (6). \(A\cap B={2,3}\) and (|C|=3), so the cardinality is \(2\cdot3=6\). Simplify the intersection first.
Step 3
Exam Tip
\(A\cap B={2,3}\) और (|C|=3), इसलिए कार्डिनलिटी \(2\cdot3=6\) है। प्रतिच्छेद को पहले सरल करें।
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यदि \(A=\{1,2\}\), \(B=\{2,3\}\), \(C=\{3,4\}\) हैं, तो (\(A\times B\)\cup\(A\times C\)) की कार्डिनलिटी क्या है?
If \(A=\{1,2\}\), \(B=\{2,3\}\), \(C=\{3,4\}\), what is the cardinality of (\(A\times B\)\cup\(A\times C\))?
#union-property
#cardinality
#set-identity
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A (4)
B (6)
C (8)
D (10)
Explanation opens after your attempt
Step 1
Concept
(\(A\times B\)\cup\(A\times C\)=A\times\(B\cup C\)), and \(B\cup C={2,3,4}\). Hence the cardinality is \(2\cdot3=6\).
Step 2
Why this answer is correct
The correct answer is B. (6). (\(A\times B\)\cup\(A\times C\)=A\times\(B\cup C\)), and \(B\cup C={2,3,4}\). Hence the cardinality is \(2\cdot3=6\).
Step 3
Exam Tip
(\(A\times B\)\cup\(A\times C\)=A\times\(B\cup C\)) और \(B\cup C={2,3,4}\)। इसलिए कार्डिनलिटी \(2\cdot3=6\) है।
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यदि \(A=\{1,2,3\}\) और \(B=\{1,2\}\), तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (a-b=1) है?
If \(A=\{1,2,3\}\) and \(B=\{1,2\}\), how many pairs ((a,b)) in \(A\times B\) satisfy (a-b=1)?
#difference-condition
#counting
#relation
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A (1)
B (2)
C (3)
D (4)
Explanation opens after your attempt
Step 1
Concept
The condition gives (a=b+1), so the pairs are ((2,1)) and ((3,2)). Apply the equation to ordered pairs.
Step 2
Why this answer is correct
The correct answer is B. (2). The condition gives (a=b+1), so the pairs are ((2,1)) and ((3,2)). Apply the equation to ordered pairs.
Step 3
Exam Tip
शर्त से (a=b+1), इसलिए ((2,1)) और ((3,2)) मिलते हैं। समीकरण को युग्मों पर लागू करें।
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यदि \(A=\{1,2,3,4\}\) और \(B=\{1,2,4,8\}\), तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें \(b=2^a\) है?
If \(A=\{1,2,3,4\}\) and \(B=\{1,2,4,8\}\), how many pairs ((a,b)) in \(A\times B\) satisfy \(b=2^a\)?
#exponential-rule
#counting
#cartesian-product
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A (2)
B (3)
C (4)
D (5)
Explanation opens after your attempt
Step 1
Concept
For (a=1,2,3), we get (b=2,4,8), but for (a=4), \(16\notin B\). Hence there are (3) pairs.
Step 2
Why this answer is correct
The correct answer is B. (3). For (a=1,2,3), we get (b=2,4,8), but for (a=4), \(16\notin B\). Hence there are (3) pairs.
Step 3
Exam Tip
(a=1,2,3) से (b=2,4,8) मिलते हैं, पर (a=4) से \(16\notin B\)। इसलिए (3) युग्म हैं।
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यदि \(A=\{1,2,3,4\}\) और \(B=\{2,3,5,7\}\), तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (a+b) अभाज्य है?
If \(A=\{1,2,3,4\}\) and \(B=\{2,3,5,7\}\), how many pairs ((a,b)) in \(A\times B\) have (a+b) prime?
#prime-sum
#counting
#hard
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A (5)
B (6)
C (7)
D (8)
Explanation opens after your attempt
Step 1
Concept
Checking systematically gives (6) pairs with prime sum. In such questions, test all possible sums in an organized way.
Step 2
Why this answer is correct
The correct answer is B. (6). Checking systematically gives (6) pairs with prime sum. In such questions, test all possible sums in an organized way.
Step 3
Exam Tip
जाँचने पर अभाज्य योग वाले (6) युग्म मिलते हैं। ऐसे प्रश्नों में सभी संभावित योगों को व्यवस्थित रूप से जाँचें।
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यदि \(A=\{0,1,2,3\}\) और \(B=\{0,1,2\}\), तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (a+b) विषम है?
If \(A=\{0,1,2,3\}\) and \(B=\{0,1,2\}\), how many pairs ((a,b)) in \(A\times B\) have (a+b) odd?
#odd-sum
#parity
#counting
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A (4)
B (5)
C (6)
D (7)
Explanation opens after your attempt
Step 1
Concept
The sum is odd when one component is even and the other is odd. The count is \(2\cdot1+2\cdot2=6\).
Step 2
Why this answer is correct
The correct answer is C. (6). The sum is odd when one component is even and the other is odd. The count is \(2\cdot1+2\cdot2=6\).
Step 3
Exam Tip
योग विषम तब होता है जब एक अवयव सम और दूसरा विषम हो। गिनती \(2\cdot1+2\cdot2=6\) है।
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यदि \(A=\{1,2,3\}\) और \(B=\{1,2,3,4\}\), तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें \(ab\le6\) है?
If \(A=\{1,2,3\}\) and \(B=\{1,2,3,4\}\), how many pairs ((a,b)) in \(A\times B\) satisfy \(ab\le6\)?
#product-inequality
#counting
#cartesian-product
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A (8)
B (9)
C (10)
D (12)
Explanation opens after your attempt
Step 1
Concept
For (a=1,2,3), the possible values of (b) are (4,3,3), totaling (10). In product conditions, the limit changes for each (a).
Step 2
Why this answer is correct
The correct answer is C. (10). For (a=1,2,3), the possible values of (b) are (4,3,3), totaling (10). In product conditions, the limit changes for each (a).
Step 3
Exam Tip
(a=1,2,3) के लिए (b) के क्रमशः (4,3,3) मान मिलते हैं, कुल (10)। गुणन वाली शर्त में प्रत्येक (a) पर सीमा बदलती है।
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यदि \(A=\{1,2,3,4\}\) और \(B=\{1,2,3,4\}\), तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (a) और (b) परस्पर अभाज्य हैं?
If \(A=\{1,2,3,4\}\) and \(B=\{1,2,3,4\}\), how many pairs ((a,b)) in \(A\times B\) have (a) and (b) coprime?
#coprime
#gcd
#counting
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A (9)
B (10)
C (11)
D (12)
Explanation opens after your attempt
Step 1
Concept
Counting pairs with (\gcd(a,b)=1) gives (11). Remember that (1) is coprime with every number.
Step 2
Why this answer is correct
The correct answer is C. (11). Counting pairs with (\gcd(a,b)=1) gives (11). Remember that (1) is coprime with every number.
Step 3
Exam Tip
(\gcd(a,b)=1) वाले युग्म गिनने पर (11) मिलते हैं। (1) के साथ हर संख्या परस्पर अभाज्य मानी जाती है।
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यदि \(A=\{1,2,3\}\) और \(B=\{2,3,4,5\}\), तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (b-a) सम है?
If \(A=\{1,2,3\}\) and \(B=\{2,3,4,5\}\), how many pairs ((a,b)) in \(A\times B\) have (b-a) even?
#parity
#difference-even
#counting
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A (5)
B (6)
C (7)
D (8)
Explanation opens after your attempt
Step 1
Concept
(b-a) is even when (a) and (b) have the same parity. The count is \(2\cdot2+1\cdot2=6\).
Step 2
Why this answer is correct
The correct answer is B. (6). (b-a) is even when (a) and (b) have the same parity. The count is \(2\cdot2+1\cdot2=6\).
Step 3
Exam Tip
(b-a) सम तब है जब (a) और (b) की समता समान हो। गिनती \(2\cdot2+1\cdot2=6\) है।
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यदि \(A=\{1,2,3,4\}\) और \(B=\{1,2,3,4,5\}\), तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें \(a+b\le5\) है?
If \(A=\{1,2,3,4\}\) and \(B=\{1,2,3,4,5\}\), how many pairs ((a,b)) in \(A\times B\) satisfy \(a+b\le5\)?
#sum-inequality
#counting
#hard
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A (8)
B (9)
C (10)
D (11)
Explanation opens after your attempt
Step 1
Concept
For (a=1,2,3,4), the possible values of (b) are (4,3,2,1), totaling (10). Make rows according to the boundary in such counts.
Step 2
Why this answer is correct
The correct answer is C. (10). For (a=1,2,3,4), the possible values of (b) are (4,3,2,1), totaling (10). Make rows according to the boundary in such counts.
Step 3
Exam Tip
(a=1,2,3,4) के लिए (b) के क्रमशः (4,3,2,1) मान मिलते हैं, कुल (10)। ऐसी गिनती में सीमा के अनुसार पंक्तियाँ बनाएं।
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