यदि \(A=\{3,5,7\}\) और \(B=\{2,4,6\}\) हैं, तो \(A\times B\) में वह क्रमित युग्म कौन सा है जिसका पहला अवयव (7) और दूसरा अवयव (4) है?
If \(A=\{3,5,7\}\) and \(B=\{2,4,6\}\), which ordered pair in \(A\times B\) has first component (7) and second component (4)?
#cartesian-product
#ordered-pair
#basics
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A ((7,4))
B ((4,7))
C ((7,6))
D ((5,4))
Explanation opens after your attempt
Correct Answer
A. ((7,4))
Step 1
Concept
In \(A\times B\), the first component comes from (A) and the second from (B). Changing positions is a common mistake.
Step 2
Why this answer is correct
The correct answer is A. ((7,4)). In \(A\times B\), the first component comes from (A) and the second from (B). Changing positions is a common mistake.
Step 3
Exam Tip
\(A\times B\) में पहला अवयव (A) से और दूसरा (B) से आता है। क्रमित युग्म में स्थान बदलना गलती कराता है।
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यदि \(A={x:x\in\mathbb{N},3\le x\le8}\) और \(B={y:y\in\mathbb{N},y\le4}\) हैं, तो \(|A\times B|\) क्या है?
If \(A={x:x\in\mathbb{N},3\le x\le8}\) and \(B={y:y\in\mathbb{N},y\le4}\), what is \(|A\times B|\)?
#set-builder
#cardinality
#counting
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A (18)
B (20)
C (24)
D (30)
Explanation opens after your attempt
Step 1
Concept
Here (|A|=6) and (|B|=4), so \(|A\times B|=6\cdot4=24\). Count the elements first.
Step 2
Why this answer is correct
The correct answer is C. (24). Here (|A|=6) and (|B|=4), so \(|A\times B|=6\cdot4=24\). Count the elements first.
Step 3
Exam Tip
यहाँ (|A|=6) और (|B|=4), इसलिए \(|A\times B|=6\cdot4=24\)। पहले समुच्चयों के अवयव गिनें।
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यदि \(A=\{1,2\}\), \(B=\{3,4,5\}\) और \(C=\{6,7\}\) हैं, तो \(|A\times B\times C|\) का मान क्या होगा?
If \(A=\{1,2\}\), \(B=\{3,4,5\}\), and \(C=\{6,7\}\), what is \(|A\times B\times C|\)?
#ordered-triple
#cardinality
#sets
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A (7)
B (10)
C (12)
D (18)
Explanation opens after your attempt
Step 1
Concept
The cardinality for three sets is \(2\cdot3\cdot2=12\). For ordered triples, multiply all sizes.
Step 2
Why this answer is correct
The correct answer is C. (12). The cardinality for three sets is \(2\cdot3\cdot2=12\). For ordered triples, multiply all sizes.
Step 3
Exam Tip
तीन समुच्चयों में कार्डिनलिटी \(2\cdot3\cdot2=12\) होती है। क्रमित त्रिक में सभी आकार गुणा करें।
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यदि \(A=\{a,b\}\) और \(B=\{1,2,3,4\}\) हैं, तो \(B\times A\) में कितने अवयव होंगे?
If \(A=\{a,b\}\) and \(B=\{1,2,3,4\}\), how many elements will \(B\times A\) have?
#reverse-product
#cardinality
#cartesian-product
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A (6)
B (8)
C (10)
D (16)
Explanation opens after your attempt
Step 1
Concept
\(|B\times A|=|B|\cdot|A|=4\cdot2=8\). Reversing order changes pairs but not the count.
Step 2
Why this answer is correct
The correct answer is B. (8). \(|B\times A|=|B|\cdot|A|=4\cdot2=8\). Reversing order changes pairs but not the count.
Step 3
Exam Tip
\(|B\times A|=|B|\cdot|A|=4\cdot2=8\)। क्रम बदलने पर युग्म बदलते हैं, पर संख्या समान रहती है।
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यदि \(A=\{1,2,3,4,5\}\) और \(B=\{2,4,6\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (a+b=9) है?
If \(A=\{1,2,3,4,5\}\) and \(B=\{2,4,6\}\), how many pairs ((a,b)) in \(A\times B\) satisfy (a+b=9)?
#sum-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 ((3,6)) and ((5,4)). For each (a), check (b=9-a).
Step 2
Why this answer is correct
The correct answer is B. (2). The condition gives ((3,6)) and ((5,4)). For each (a), check (b=9-a).
Step 3
Exam Tip
शर्त से ((3,6)) और ((5,4)) मिलते हैं। हर (a) के लिए (b=9-a) जाँचें।
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यदि \(A=\{0,1,2,3,4\}\) और \(B=\{2,3,4,5\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (a<b) है?
If \(A=\{0,1,2,3,4\}\) and \(B=\{2,3,4,5\}\), how many pairs ((a,b)) in \(A\times B\) satisfy (a<b)?
#inequality
#counting
#cartesian-product
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A (12)
B (13)
C (14)
D (15)
Explanation opens after your attempt
Step 1
Concept
For (a=0,1,2,3,4), the counts of (b) are (4,4,3,2,1), totaling (14). Count row-wise in inequality questions.
Step 2
Why this answer is correct
The correct answer is C. (14). For (a=0,1,2,3,4), the counts of (b) are (4,4,3,2,1), totaling (14). Count row-wise in inequality questions.
Step 3
Exam Tip
(a=0,1,2,3,4) के लिए (b) के क्रमशः (4,4,3,2,1) मान मिलते हैं, कुल (14)। असमानता में पंक्ति अनुसार गिनें।
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यदि \(A=\{1,2,3,4,5,6\}\) और \(B=\{1,2,3\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें \(a\ge2b+1\) है?
If \(A=\{1,2,3,4,5,6\}\) and \(B=\{1,2,3\}\), how many pairs ((a,b)) in \(A\times B\) satisfy \(a\ge2b+1\)?
#linear-inequality
#counting
#hard
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A (5)
B (6)
C (7)
D (8)
Explanation opens after your attempt
Step 1
Concept
For (b=1,2,3), the counts of (a) are (4,2,0), totaling (6). The boundary changes for each (b).
Step 2
Why this answer is correct
The correct answer is B. (6). For (b=1,2,3), the counts of (a) are (4,2,0), totaling (6). The boundary changes for each (b).
Step 3
Exam Tip
(b=1,2,3) के लिए (a) के क्रमशः (4,2,0) मान मिलते हैं, कुल (6)। सीमा हर (b) पर बदलती है।
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यदि \(A=\{1,2,3,4,5\}\) और \(B=\{1,4,9,16,25\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें \(b=a^2\) है?
If \(A=\{1,2,3,4,5\}\) and \(B=\{1,4,9,16,25\}\), how many pairs ((a,b)) in \(A\times B\) satisfy \(b=a^2\)?
#square-rule
#function-rule
#counting
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A (3)
B (4)
C (5)
D (10)
Explanation opens after your attempt
Step 1
Concept
For every \(a\in A\), \(a^2\in B\), so (5) pairs are formed. Check each first component in a rule-based relation.
Step 2
Why this answer is correct
The correct answer is C. (5). For every \(a\in A\), \(a^2\in B\), so (5) pairs are formed. Check each first component in a rule-based relation.
Step 3
Exam Tip
हर \(a\in A\) के लिए \(a^2\in B\) है, इसलिए (5) युग्म बनते हैं। नियम आधारित संबंध में हर पहले अवयव को जाँचें।
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यदि \(A=\{2,3,4\}\) और \(B=\{6,8,9,12\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (a) (b) को विभाजित करता है?
If \(A=\{2,3,4\}\) and \(B=\{6,8,9,12\}\), how many pairs ((a,b)) in \(A\times B\) have (a) dividing (b)?
#divisibility
#relation
#counting
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A (6)
B (7)
C (8)
D (9)
Explanation opens after your attempt
Step 1
Concept
For (a=2,3,4), the counts of (b) are (3,3,1), totaling (7). Check each divisor separately.
Step 2
Why this answer is correct
The correct answer is B. (7). For (a=2,3,4), the counts of (b) are (3,3,1), totaling (7). Check each divisor separately.
Step 3
Exam Tip
(a=2,3,4) के लिए (b) के क्रमशः (3,3,1) मान मिलते हैं, कुल (7)। विभाज्यता में प्रत्येक भाजक अलग जाँचें।
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यदि \(A=\{2,4,6,8\}\) और \(B=\{2,4,6\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (\gcd(a,b)=2) है?
If \(A=\{2,4,6,8\}\) and \(B=\{2,4,6\}\), how many pairs ((a,b)) in \(A\times B\) satisfy (\gcd(a,b)=2)?
#gcd
#counting
#ordered-pair
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A (4)
B (5)
C (6)
D (7)
Explanation opens after your attempt
Step 1
Concept
The suitable pairs are ((2,2),(2,4),(2,6),(4,2),(8,2)), so the count is (5). Check common factors carefully in \(\gcd\) questions.
Step 2
Why this answer is correct
The correct answer is B. (5). The suitable pairs are ((2,2),(2,4),(2,6),(4,2),(8,2)), so the count is (5). Check common factors carefully in \(\gcd\) questions.
Step 3
Exam Tip
अनुकूल युग्म ((2,2),(2,4),(2,6),(4,2),(8,2)) हैं, इसलिए संख्या (5) है। \(\gcd\) में साझा गुणनखंड सावधानी से देखें।
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यदि (|A|=4), (|B|=5) और \(R\subseteq A\times B\), तो (R) में ठीक (2) अवयव चुनने के कितने तरीके हैं?
If (|A|=4), (|B|=5), and \(R\subseteq A\times B\), in how many ways can (R) have exactly (2) elements?
#combinations
#relations
#counting
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A (40)
B (100)
C (190)
D (400)
Explanation opens after your attempt
Step 1
Concept
Since \(|A\times B|=20\), the number of ways to choose exactly (2) pairs is \(\binom{20}{2}=190\). Use combinations for exact-size subsets.
Step 2
Why this answer is correct
The correct answer is C. (190). Since \(|A\times B|=20\), the number of ways to choose exactly (2) pairs is \(\binom{20}{2}=190\). Use combinations for exact-size subsets.
Step 3
Exam Tip
\(|A\times B|=20\), इसलिए ठीक (2) युग्म चुनने के तरीके \(\binom{20}{2}=190\) हैं। ठीक संख्या पूछी हो तो संयोजन लगाएँ।
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यदि (|A|=3) और (|B|=5) हैं, तो (A) से (B) तक संभव संबंधों की संख्या क्या है?
If (|A|=3) and (|B|=5), what is the number of possible relations from (A) to (B)?
#relations
#power-set
#cardinality
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A (15)
B (125)
C (32768)
D (59049)
Explanation opens after your attempt
Correct Answer
C. (32768)
Step 1
Concept
A relation is a subset of \(A\times B\), and \(|A\times B|=15\), so the number is \(2^{15}=32768\). Number of relations uses a power of (2).
Step 2
Why this answer is correct
The correct answer is C. (32768). A relation is a subset of \(A\times B\), and \(|A\times B|=15\), so the number is \(2^{15}=32768\). Number of relations uses a power of (2).
Step 3
Exam Tip
संबंध \(A\times B\) का उपसमुच्चय है और \(|A\times B|=15\), इसलिए संख्या \(2^{15}=32768\) है। संबंधों की संख्या के लिए (2) की घात लगती है।
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यदि \(A=\{1,2,3,4\}\), \(B=\{2,4,6\}\) और \(C=\{4,6,8\}\) हैं, तो (|A\times\(B\cup C\)|) क्या है?
If \(A=\{1,2,3,4\}\), \(B=\{2,4,6\}\), and \(C=\{4,6,8\}\), what is (|A\times\(B\cup C\)|)?
#union
#cardinality
#cartesian-product
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A (12)
B (14)
C (16)
D (20)
Explanation opens after your attempt
Step 1
Concept
\(B\cup C={2,4,6,8}\), so (|A\times\(B\cup C\)|=4\cdot4=16). Find the union first.
Step 2
Why this answer is correct
The correct answer is C. (16). \(B\cup C={2,4,6,8}\), so (|A\times\(B\cup C\)|=4\cdot4=16). Find the union first.
Step 3
Exam Tip
\(B\cup C={2,4,6,8}\), इसलिए (|A\times\(B\cup C\)|=4\cdot4=16)। पहले संघ निकालें।
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यदि \(A=\{0,1,2\}\), \(B=\{1,2,3,4\}\) और \(C=\{2,4,6\}\) हैं, तो (|A\times\(B\cap C\)|) क्या होगा?
If \(A=\{0,1,2\}\), \(B=\{1,2,3,4\}\), and \(C=\{2,4,6\}\), what is (|A\times\(B\cap C\)|)?
#intersection
#cardinality
#set-operation
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A (3)
B (6)
C (9)
D (12)
Explanation opens after your attempt
Step 1
Concept
\(B\cap C={2,4}\), so the cardinality is \(3\cdot2=6\). Multiply only after finding the intersection.
Step 2
Why this answer is correct
The correct answer is B. (6). \(B\cap C={2,4}\), so the cardinality is \(3\cdot2=6\). Multiply only after finding the intersection.
Step 3
Exam Tip
\(B\cap C={2,4}\), इसलिए कार्डिनलिटी \(3\cdot2=6\) है। प्रतिच्छेद के बाद ही गुणा करें।
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यदि \(A=\{1,3,5\}\), \(B=\{2,3,4,5\}\) और \(C=\{3,5,7\}\) हैं, तो \(A\times(B-C)\) क्या है?
If \(A=\{1,3,5\}\), \(B=\{2,3,4,5\}\), and \(C=\{3,5,7\}\), what is \(A\times(B-C)\)?
#difference
#set-operation
#cartesian-product
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A ({(1,2),(1,4),(3,2),(3,4),(5,2),(5,4)})
B ({(1,3),(3,3),(5,3)})
C ({(1,5),(3,5),(5,5)})
D \(\varnothing\)
Explanation opens after your attempt
Correct Answer
A. ({(1,2),(1,4),(3,2),(3,4),(5,2),(5,4)})
Step 1
Concept
(B-C={2,4}), so \(A\times(B-C)\) has (6) pairs. In set difference, keep only elements remaining from (B).
Step 2
Why this answer is correct
The correct answer is A. ({(1,2),(1,4),(3,2),(3,4),(5,2),(5,4)}). (B-C={2,4}), so \(A\times(B-C)\) has (6) pairs. In set difference, keep only elements remaining from (B).
Step 3
Exam Tip
(B-C={2,4}), इसलिए \(A\times(B-C)\) में (6) युग्म हैं। अंतर में केवल (B) के बचे अवयव लें।
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यदि \(A=\{1,2,3\}\), \(B=\{2,3,4\}\) और \(C=\{1,3,5\}\) हैं, तो (\(A\times B\)\cap\(A\times C\)) की कार्डिनलिटी क्या है?
If \(A=\{1,2,3\}\), \(B=\{2,3,4\}\), and \(C=\{1,3,5\}\), what is the cardinality of (\(A\times B\)\cap\(A\times C\))?
#intersection-property
#cardinality
#set-identity
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A (1)
B (3)
C (6)
D (9)
Explanation opens after your attempt
Step 1
Concept
(\(A\times B\)\cap\(A\times C\)=A\times\(B\cap C\)), and \(B\cap C={3}\). Hence the cardinality is \(3\cdot1=3\).
Step 2
Why this answer is correct
The correct answer is B. (3). (\(A\times B\)\cap\(A\times C\)=A\times\(B\cap C\)), and \(B\cap C={3}\). Hence the cardinality is \(3\cdot1=3\).
Step 3
Exam Tip
(\(A\times B\)\cap\(A\times C\)=A\times\(B\cap C\)) और \(B\cap C={3}\)। इसलिए कार्डिनलिटी \(3\cdot1=3\) है।
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यदि \(A=\{0,1\}\), \(B=\{2,3,4\}\) और \(C=\{4,5\}\) हैं, तो (\(A\times B\)\cup\(A\times C\)) की कार्डिनलिटी क्या है?
If \(A=\{0,1\}\), \(B=\{2,3,4\}\), and \(C=\{4,5\}\), what is the cardinality of (\(A\times B\)\cup\(A\times C\))?
#union-property
#set-identity
#cardinality
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A (6)
B (8)
C (10)
D (12)
Explanation opens after your attempt
Step 1
Concept
(\(A\times B\)\cup\(A\times C\)=A\times\(B\cup C\)), and \(B\cup C={2,3,4,5}\). Thus there are \(2\cdot4=8\) elements.
Step 2
Why this answer is correct
The correct answer is B. (8). (\(A\times B\)\cup\(A\times C\)=A\times\(B\cup C\)), and \(B\cup C={2,3,4,5}\). Thus there are \(2\cdot4=8\) elements.
Step 3
Exam Tip
(\(A\times B\)\cup\(A\times C\)=A\times\(B\cup C\)) और \(B\cup C={2,3,4,5}\)। इसलिए \(2\cdot4=8\) अवयव हैं।
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यदि \(A=\{1,2,3,4,5\}\) और \(B=\{1,2,3,4,5\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (|a-b|=3) 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|=3)?
#absolute-difference
#counting
#ordered-pair
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A (2)
B (3)
C (4)
D (6)
Explanation opens after your attempt
Step 1
Concept
The pairs are ((1,4),(2,5),(4,1),(5,2)), totaling (4). Reversed ordered pairs are counted separately.
Step 2
Why this answer is correct
The correct answer is C. (4). The pairs are ((1,4),(2,5),(4,1),(5,2)), totaling (4). Reversed ordered pairs are counted separately.
Step 3
Exam Tip
युग्म ((1,4),(2,5),(4,1),(5,2)) हैं, कुल (4)। उलटे क्रमित युग्म अलग गिने जाते हैं।
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यदि \(A=\{0,1,2,3,4\}\) और \(B=\{0,1,2,3\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (a+b) सम है?
If \(A=\{0,1,2,3,4\}\) and \(B=\{0,1,2,3\}\), how many pairs ((a,b)) in \(A\times B\) have (a+b) even?
#parity
#even-sum
#counting
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A (8)
B (10)
C (12)
D (14)
Explanation opens after your attempt
Step 1
Concept
The sum is even when both components have the same parity. The count is \(3\cdot2+2\cdot2=10\).
Step 2
Why this answer is correct
The correct answer is B. (10). The sum is even when both components have the same parity. The count is \(3\cdot2+2\cdot2=10\).
Step 3
Exam Tip
योग सम तब होता है जब दोनों अवयवों की समता समान हो। गिनती \(3\cdot2+2\cdot2=10\) है।
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यदि \(A=\{1,2,3,4,5,6\}\) और \(B=\{1,2,3,4\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें \(a+b\le7\) है?
If \(A=\{1,2,3,4,5,6\}\) and \(B=\{1,2,3,4\}\), how many pairs ((a,b)) in \(A\times B\) satisfy \(a+b\le7\)?
#sum-inequality
#counting
#hard
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A (18)
B (19)
C (20)
D (21)
Explanation opens after your attempt
Step 1
Concept
For (a=1,2,3,4,5,6), the counts of (b) are (4,4,4,3,2,1), totaling (18). Include equality in \(\le\).
Step 2
Why this answer is correct
The correct answer is A. (18). For (a=1,2,3,4,5,6), the counts of (b) are (4,4,4,3,2,1), totaling (18). Include equality in \(\le\).
Step 3
Exam Tip
(a=1,2,3,4,5,6) के लिए (b) के (4,4,4,3,2,1) मान मिलते हैं, कुल (18)। \(\le\) में बराबरी भी गिनें।
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यदि \(A=\{1,2,3,4,5\}\) और \(B=\{2,3,4,5,6\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (ab) विषम है?
If \(A=\{1,2,3,4,5\}\) and \(B=\{2,3,4,5,6\}\), how many pairs ((a,b)) in \(A\times B\) have (ab) odd?
#product-parity
#odd-product
#counting
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A (4)
B (5)
C (6)
D (8)
Explanation opens after your attempt
Step 1
Concept
The product is odd only when both components are odd. (A) has (3) odd elements and (B) has (2), so \(3\cdot2=6\).
Step 2
Why this answer is correct
The correct answer is C. (6). The product is odd only when both components are odd. (A) has (3) odd elements and (B) has (2), so \(3\cdot2=6\).
Step 3
Exam Tip
गुणनफल विषम तभी होगा जब दोनों अवयव विषम हों। (A) में (3) और (B) में (2) विषम अवयव हैं, इसलिए \(3\cdot2=6\)।
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यदि \(A=\{1,2,3,4,5\}\) और \(B=\{3,6,9,12\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (b=3a) है?
If \(A=\{1,2,3,4,5\}\) and \(B=\{3,6,9,12\}\), how many pairs ((a,b)) in \(A\times B\) satisfy (b=3a)?
#linear-rule
#counting
#relation
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A (2)
B (3)
C (4)
D (5)
Explanation opens after your attempt
Step 1
Concept
For (a=1,2,3,4), we get (b=3,6,9,12), but for (a=5), \(15\notin B\). Hence there are (4) pairs.
Step 2
Why this answer is correct
The correct answer is C. (4). For (a=1,2,3,4), we get (b=3,6,9,12), but for (a=5), \(15\notin B\). Hence there are (4) pairs.
Step 3
Exam Tip
(a=1,2,3,4) पर (b=3,6,9,12) मिलता है, पर (a=5) पर \(15\notin B\)। इसलिए (4) युग्म हैं।
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यदि \(A=\{1,2,3,4,5,6,7\}\) और \(B=\{0,1,2,3\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (a-2b=1) है?
If \(A=\{1,2,3,4,5,6,7\}\) and \(B=\{0,1,2,3\}\), how many pairs ((a,b)) in \(A\times B\) satisfy (a-2b=1)?
#linear-equation
#counting
#ordered-pair
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A (2)
B (3)
C (4)
D (5)
Explanation opens after your attempt
Step 1
Concept
The condition gives (a=2b+1), so for (b=0,1,2,3), (a=1,3,5,7). There are (4) pairs in total.
Step 2
Why this answer is correct
The correct answer is C. (4). The condition gives (a=2b+1), so for (b=0,1,2,3), (a=1,3,5,7). There are (4) pairs in total.
Step 3
Exam Tip
शर्त (a=2b+1) देती है, इसलिए (b=0,1,2,3) पर (a=1,3,5,7) मिलते हैं। कुल (4) युग्म हैं।
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यदि \(A=\{2,3,4,5\}\) और \(B=\{1,2,3,4,5\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (b-a) सम है?
If \(A=\{2,3,4,5\}\) and \(B=\{1,2,3,4,5\}\), how many pairs ((a,b)) in \(A\times B\) have (b-a) even?
#parity
#difference-even
#counting
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A (8)
B (9)
C (10)
D (11)
Explanation opens after your attempt
Step 1
Concept
The difference is even when both have the same parity. The count is \(2\cdot2+2\cdot3=10\).
Step 2
Why this answer is correct
The correct answer is C. (10). The difference is even when both have the same parity. The count is \(2\cdot2+2\cdot3=10\).
Step 3
Exam Tip
अंतर सम तब होता है जब दोनों की समता समान हो। गिनती \(2\cdot2+2\cdot3=10\) है।
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यदि \(A=\{1,2,3,4,5,6\}\) और \(B=\{1,2,3,4,5,6\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें \(a\ne b\) है?
If \(A=\{1,2,3,4,5,6\}\) and \(B=\{1,2,3,4,5,6\}\), how many pairs ((a,b)) in \(A\times B\) satisfy \(a\ne b\)?
#not-equal
#diagonal
#complement-counting
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A (24)
B (28)
C (30)
D (36)
Explanation opens after your attempt
Step 1
Concept
There are (36) total pairs and (6) equal pairs, so (36-6=30). Complement counting is a quick method.
Step 2
Why this answer is correct
The correct answer is C. (30). There are (36) total pairs and (6) equal pairs, so (36-6=30). Complement counting is a quick method.
Step 3
Exam Tip
कुल (36) युग्म हैं और बराबर युग्म (6) हैं, इसलिए (36-6=30)। पूरक गिनती तेज तरीका है।
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यदि \(A=\{-3,-2,-1,0,1,2,3\}\) और \(B=\{0,1,4,9\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें \(a^2=b\) है?
If \(A=\{-3,-2,-1,0,1,2,3\}\) and \(B=\{0,1,4,9\}\), how many pairs ((a,b)) in \(A\times B\) satisfy \(a^2=b\)?
#square-condition
#negative-numbers
#counting
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A (4)
B (5)
C (6)
D (7)
Explanation opens after your attempt
Step 1
Concept
For every \(a\in A\), \(a^2\) is in (B), so (7) pairs are formed. Watch squares of negative numbers carefully.
Step 2
Why this answer is correct
The correct answer is D. (7). For every \(a\in A\), \(a^2\) is in (B), so (7) pairs are formed. Watch squares of negative numbers carefully.
Step 3
Exam Tip
हर \(a\in A\) का \(a^2\) (B) में है, इसलिए (7) युग्म बनते हैं। ऋणात्मक संख्याओं के वर्ग को ध्यान से देखें।
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यदि \(A=\{1,2,3,4,5\}\) और \(B=\{1,2,3,4\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (a+b) अभाज्य है?
If \(A=\{1,2,3,4,5\}\) and \(B=\{1,2,3,4\}\), how many pairs ((a,b)) in \(A\times B\) have (a+b) prime?
#prime-sum
#counting
#hard
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A (9)
B (10)
C (11)
D (12)
Explanation opens after your attempt
Step 1
Concept
Checking all sums gives (10) pairs with prime sum. In such questions, test possible sums systematically.
Step 2
Why this answer is correct
The correct answer is B. (10). Checking all sums gives (10) pairs with prime sum. In such questions, test possible sums systematically.
Step 3
Exam Tip
सभी योग जाँचने पर अभाज्य योग वाले (10) युग्म मिलते हैं। ऐसे प्रश्नों में संभावित योगों को व्यवस्थित रूप से जाँचें।
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यदि \(A=\{1,2,4\}\) और \(B=\{2,4,8,16\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें \(\frac{b}{a}=4\) है?
If \(A=\{1,2,4\}\) and \(B=\{2,4,8,16\}\), how many pairs ((a,b)) in \(A\times B\) satisfy \(\frac{b}{a}=4\)?
#fraction-condition
#linear-rule
#counting
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A (1)
B (2)
C (3)
D (4)
Explanation opens after your attempt
Step 1
Concept
The condition is (b=4a), so the pairs are ((1,4),(2,8),(4,16)). Convert the fraction into a simple equation.
Step 2
Why this answer is correct
The correct answer is C. (3). The condition is (b=4a), so the pairs are ((1,4),(2,8),(4,16)). Convert the fraction into a simple equation.
Step 3
Exam Tip
शर्त (b=4a) है, इसलिए ((1,4),(2,8),(4,16)) मिलते हैं। भिन्न को सरल समीकरण में बदलें।
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यदि \(A=\{1,2,3,4,5\}\) और \(B=\{1,2,3,4,5\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (ab>15) 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 (ab>15)?
#product-inequality
#counting
#hard
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A (3)
B (4)
C (5)
D (6)
Explanation opens after your attempt
Step 1
Concept
The satisfying pairs are ((4,4),(4,5),(5,4),(5,5)), so the total is (4). Count each ordered pair once.
Step 2
Why this answer is correct
The correct answer is C. (5). The satisfying pairs are ((4,4),(4,5),(5,4),(5,5)), so the total is (4). Count each ordered pair once.
Step 3
Exam Tip
संतुष्ट युग्म ((4,4),(4,5),(5,4),(5,5),(5,5)) नहीं गिने जाते दो बार; सही युग्म ((4,4),(4,5),(5,4),(5,5),(5,4)) नहीं बल्कि कुल (4) हैं।
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यदि \(A=\{1,2,3,4\}\) और \(B=\{1,2,3,4,5\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (a+b) (5) से विभाज्य है?
If \(A=\{1,2,3,4\}\) and \(B=\{1,2,3,4,5\}\), how many pairs ((a,b)) in \(A\times B\) have (a+b) divisible by (5)?
#divisibility
#sum-condition
#counting
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A (3)
B (4)
C (5)
D (6)
Explanation opens after your attempt
Step 1
Concept
The pairs with sum divisible by (5) are ((1,4),(2,3),(3,2),(4,1)). Counting by remainders is easier.
Step 2
Why this answer is correct
The correct answer is B. (4). The pairs with sum divisible by (5) are ((1,4),(2,3),(3,2),(4,1)). Counting by remainders is easier.
Step 3
Exam Tip
योग (5) से विभाज्य होने वाले युग्म ((1,4),(2,3),(3,2),(4,1)) हैं। शेषफल के आधार पर गिनना आसान होता है।
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यदि \(A=\{1,2,3,4,5\}\) और \(B=\{1,2,3,4,5,6\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (b) (a) का गुणज है?
If \(A=\{1,2,3,4,5\}\) and \(B=\{1,2,3,4,5,6\}\), how many pairs ((a,b)) in \(A\times B\) have (b) as a multiple of (a)?
#multiples
#counting
#relation
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A (12)
B (13)
C (14)
D (15)
Explanation opens after your attempt
Step 1
Concept
For (a=1,2,3,4,5), the counts of (b) are (6,3,2,1,1), totaling (13). Count multiples of each (a) inside (B).
Step 2
Why this answer is correct
The correct answer is C. (14). For (a=1,2,3,4,5), the counts of (b) are (6,3,2,1,1), totaling (13). Count multiples of each (a) inside (B).
Step 3
Exam Tip
(a=1,2,3,4,5) के लिए (b) के (6,3,2,1,1) मान हैं, कुल (13)। हर (a) के गुणज (B) में गिनें।
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यदि \(A=\{2,4\}\) और \(A\times C={(2,1),(2,3),(2,5),(4,1),(4,3),(4,5)}\), तो (C) क्या है?
If \(A=\{2,4\}\) and \(A\times C={(2,1),(2,3),(2,5),(4,1),(4,3),(4,5)}\), what is (C)?
#identify-set
#cartesian-product
#reasoning
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A ({2,4})
B ({1,3,5})
C ({1,2,3,4,5})
D ({(2,1),(4,5)})
Explanation opens after your attempt
Correct Answer
B. ({1,3,5})
Step 1
Concept
The set of second components is \(C=\{1,3,5\}\). Observe the position in each pair to identify the original set.
Step 2
Why this answer is correct
The correct answer is B. ({1,3,5}). The set of second components is \(C=\{1,3,5\}\). Observe the position in each pair to identify the original set.
Step 3
Exam Tip
दूसरे अवयवों का समुच्चय \(C=\{1,3,5\}\) है। मूल समुच्चय पहचानते समय युग्म का स्थान देखें।
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यदि \(A\times B=\varnothing\) और \(A=\{0\}\), तो (B) के बारे में क्या सही है?
If \(A\times B=\varnothing\) and \(A=\{0\}\), what is true about (B)?
#empty-set
#cartesian-product
#concept
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A \(B=\varnothing\)
B (B=A)
C \(B=\{0\}\)
D \(B\subseteq A\) अवश्य है / \(B\subseteq A\) must hold
Explanation opens after your attempt
Correct Answer
A. \(B=\varnothing\)
Step 1
Concept
Since (A) is non-empty and \(A\times B=\varnothing\), we must have \(B=\varnothing\). An empty product means at least one set is empty.
Step 2
Why this answer is correct
The correct answer is A. \(B=\varnothing\). Since (A) is non-empty and \(A\times B=\varnothing\), we must have \(B=\varnothing\). An empty product means at least one set is empty.
Step 3
Exam Tip
क्योंकि (A) रिक्त नहीं है और \(A\times B=\varnothing\), इसलिए \(B=\varnothing\)। रिक्त गुणनफल में कम से कम एक समुच्चय रिक्त होता है।
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यदि \(A=\{2,3,4\}\), \(B=\{5,6,7\}\) हैं, तो \(A\times B\) में ((3,6)), ((5,6)), ((4,7)) में से कौन सा युग्म नहीं है?
If \(A=\{2,3,4\}\), \(B=\{5,6,7\}\), which among ((3,6)), ((5,6)), and ((4,7)) is not in \(A\times B\)?
#membership
#error-check
#ordered-pair
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A ((3,6))
B ((5,6))
C ((4,7))
D तीनों हैं / All three are present
Explanation opens after your attempt
Correct Answer
B. ((5,6))
Step 1
Concept
In ((5,6)), the first component (5) is not in (A). Check both positions separately for membership.
Step 2
Why this answer is correct
The correct answer is B. ((5,6)). In ((5,6)), the first component (5) is not in (A). Check both positions separately for membership.
Step 3
Exam Tip
((5,6)) में पहला अवयव (5) है, जो (A) में नहीं है। सदस्यता में दोनों स्थान अलग-अलग जाँचें।
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यदि \(A=\{1,2,3\}\) और \(B=\{3,2,1\}\) हैं, तो \(A\times B\) और \(B\times A\) के बारे में कौन सा कथन सही है?
If \(A=\{1,2,3\}\) and \(B=\{3,2,1\}\), 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 \(A\times B\subseteq A\)
Explanation opens after your attempt
Correct Answer
A. \(A\times B=B\times A\)
Step 1
Concept
Order inside a set does not matter, so (A=B), and both products are equal. First check whether the two sets are actually the same.
Step 2
Why this answer is correct
The correct answer is A. \(A\times B=B\times A\). Order inside a set does not matter, so (A=B), and both products are equal. First check whether the two sets are actually the same.
Step 3
Exam Tip
समुच्चय में क्रम महत्व नहीं रखता, इसलिए (A=B) है और दोनों गुणनफल समान हैं। पहले देखें कि दोनों समुच्चय वास्तव में समान हैं या नहीं।
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यदि \(A=\{1,2,3,4\}\) है और \(A\times B\) में कुल (28) अवयव हैं, तो (|B|) क्या होगा?
If \(A=\{1,2,3,4\}\) and \(A\times B\) has (28) elements, what is (|B|)?
#unknown-cardinality
#formula
#sets
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A (4)
B (7)
C (24)
D (32)
Explanation opens after your attempt
Step 1
Concept
Since (|A|=4) and \(4\cdot|B|=28\), we get (|B|=7). Divide to find the unknown cardinality.
Step 2
Why this answer is correct
The correct answer is B. (7). Since (|A|=4) and \(4\cdot|B|=28\), we get (|B|=7). Divide to find the unknown cardinality.
Step 3
Exam Tip
(|A|=4) और \(4\cdot|B|=28\), इसलिए (|B|=7)। अज्ञात कार्डिनलिटी के लिए भाग करें।
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यदि (A) में (3) अवयव और (B) में (4) अवयव हैं, तो \(A\times B\) के गैर-रिक्त उपसमुच्चयों की संख्या क्या है?
If (A) has (3) elements and (B) has (4) elements, what is the number of non-empty subsets of \(A\times B\)?
#non-empty-subsets
#power-set
#counting
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A (255)
B (1023)
C (4095)
D (8191)
Explanation opens after your attempt
Step 1
Concept
\(|A\times B|=12\), so non-empty subsets are \(2^{12}-1=4095\). Do not forget to subtract the empty set.
Step 2
Why this answer is correct
The correct answer is C. (4095). \(|A\times B|=12\), so non-empty subsets are \(2^{12}-1=4095\). Do not forget to subtract the empty set.
Step 3
Exam Tip
\(|A\times B|=12\), इसलिए गैर-रिक्त उपसमुच्चय \(2^{12}-1=4095\) हैं। रिक्त समुच्चय घटाना न भूलें।
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यदि \(A=\{1,2,3,4\}\), \(B=\{1,2,3,4\}\) और \(R=\{(a,b):a+b=5\}\), तो (R) में कितने अवयव हैं?
If \(A=\{1,2,3,4\}\), \(B=\{1,2,3,4\}\), and \(R=\{(a,b):a+b=5\}\), how many elements are in (R)?
#relation-subset
#sum-condition
#counting
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A (2)
B (3)
C (4)
D (5)
Explanation opens after your attempt
Step 1
Concept
The condition gives ((1,4),(2,3),(3,2),(4,1)), so there are (4) elements. A relation is a subset of \(A\times B\).
Step 2
Why this answer is correct
The correct answer is C. (4). The condition gives ((1,4),(2,3),(3,2),(4,1)), so there are (4) elements. A relation is a subset of \(A\times B\).
Step 3
Exam Tip
शर्त से ((1,4),(2,3),(3,2),(4,1)) मिलते हैं, इसलिए (4) अवयव हैं। संबंध \(A\times B\) का उपसमुच्चय होता है।
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यदि \(A=\{2,4,6,8\}\) और \(B=\{1,3,5\}\) हैं, तो \(A\times B\) में प्रत्येक युग्म ((a,b)) के लिए (a+b) कैसा होगा?
If \(A=\{2,4,6,8\}\) and \(B=\{1,3,5\}\), 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 even number and an odd number is always odd. In parity questions, listing 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 even number and an odd number is always odd. In parity questions, listing all pairs is not necessary.
Step 3
Exam Tip
सम संख्या और विषम संख्या का योग हमेशा विषम होता है। समता वाले प्रश्न में सभी युग्म लिखना जरूरी नहीं है।
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यदि \(A=\{1,2,3,4,5\}\) और \(B=\{1,2,3,4,5\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (a) और (b) परस्पर अभाज्य हैं?
If \(A=\{1,2,3,4,5\}\) and \(B=\{1,2,3,4,5\}\), how many pairs ((a,b)) in \(A\times B\) have (a) and (b) coprime?
#coprime
#gcd
#counting
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A (17)
B (18)
C (19)
D (20)
Explanation opens after your attempt
Step 1
Concept
Counting pairs with (\gcd(a,b)=1) gives (19). The number (1) is coprime with every number.
Step 2
Why this answer is correct
The correct answer is C. (19). Counting pairs with (\gcd(a,b)=1) gives (19). The number (1) is coprime with every number.
Step 3
Exam Tip
(\gcd(a,b)=1) वाले युग्म गिनने पर (19) मिलते हैं। (1) हर संख्या के साथ परस्पर अभाज्य होता है।
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यदि \(A=\{0,1,2,3\}\) और \(B=\{0,1,2,3,4\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (a+b<5) है?
If \(A=\{0,1,2,3\}\) and \(B=\{0,1,2,3,4\}\), how many pairs ((a,b)) in \(A\times B\) satisfy (a+b<5)?
#strict-inequality
#counting
#cartesian-product
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A (14)
B (15)
C (16)
D (17)
Explanation opens after your attempt
Step 1
Concept
For (a=0,1,2,3), the counts of (b) are (5,4,3,2), totaling (14). In a strict inequality, the boundary is not included.
Step 2
Why this answer is correct
The correct answer is A. (14). For (a=0,1,2,3), the counts of (b) are (5,4,3,2), totaling (14). In a strict inequality, the boundary is not included.
Step 3
Exam Tip
(a=0,1,2,3) के लिए (b) के (5,4,3,2) मान मिलते हैं, कुल (14)। कठोर असमानता में सीमा बराबर नहीं गिनी जाती।
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यदि \(A=\{1,2,3,4,5\}\) और \(B=\{2,4,6,8,10\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (a+b=11) है?
If \(A=\{1,2,3,4,5\}\) and \(B=\{2,4,6,8,10\}\), how many pairs ((a,b)) in \(A\times B\) satisfy (a+b=11)?
#sum-condition
#counting
#sets
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A (2)
B (3)
C (4)
D (5)
Explanation opens after your attempt
Step 1
Concept
The condition gives ((1,10),(3,8),(5,6)). Check whether (b=11-a) belongs to (B).
Step 2
Why this answer is correct
The correct answer is B. (3). The condition gives ((1,10),(3,8),(5,6)). Check whether (b=11-a) belongs to (B).
Step 3
Exam Tip
शर्त से ((1,10),(3,8),(5,6)) मिलते हैं। (b=11-a) (B) में है या नहीं, यह जाँचें।
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यदि \(A=\{1,2,3,4,5,6\}\) और \(B=\{1,2,3,4,5,6\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (a+b=7) और (a<b) दोनों हैं?
If \(A=\{1,2,3,4,5,6\}\) and \(B=\{1,2,3,4,5,6\}\), how many pairs ((a,b)) in \(A\times B\) satisfy both (a+b=7) and (a<b)?
#combined-condition
#sum-inequality
#counting
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A (2)
B (3)
C (4)
D (6)
Explanation opens after your attempt
Step 1
Concept
For sum (7) and (a<b), the pairs are ((1,6),(2,5),(3,4)). Apply combined conditions step by step.
Step 2
Why this answer is correct
The correct answer is B. (3). For sum (7) and (a<b), the pairs are ((1,6),(2,5),(3,4)). Apply combined conditions step by step.
Step 3
Exam Tip
योग (7) और (a<b) के लिए ((1,6),(2,5),(3,4)) मिलते हैं। संयुक्त शर्तों को क्रम से लगाएँ।
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यदि \(A=\{1,2,3,4\}\), \(B=\{1,2,3,4\}\) और \(R=\{(a,b):a\le b\}\), तो (R) में कितने अवयव हैं?
If \(A=\{1,2,3,4\}\), \(B=\{1,2,3,4\}\), and \(R=\{(a,b):a\le b\}\), how many elements are in (R)?
#relation-subset
#less-than-equal
#counting
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A (6)
B (8)
C (10)
D (12)
Explanation opens after your attempt
Step 1
Concept
For (a=1,2,3,4), the counts of (b) are (4,3,2,1), totaling (10). The symbol \(\le\) includes equality.
Step 2
Why this answer is correct
The correct answer is C. (10). For (a=1,2,3,4), the counts of (b) are (4,3,2,1), totaling (10). The symbol \(\le\) includes equality.
Step 3
Exam Tip
(a=1,2,3,4) के लिए (b) के (4,3,2,1) मान मिलते हैं, कुल (10)। \(\le\) में बराबरी शामिल होती है।
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यदि \(A=\{2,3,5\}\) और \(B=\{4,6,10,12,15\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (b) (a) से विभाज्य है?
If \(A=\{2,3,5\}\) and \(B=\{4,6,10,12,15\}\), how many pairs ((a,b)) in \(A\times B\) have (b) divisible by (a)?
#divisibility
#multiples
#counting
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A (7)
B (8)
C (9)
D (10)
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Step 1
Concept
For (a=2,3,5), the counts of (b) are (4,3,2), totaling (9). Count multiples of each (a) in (B).
Step 2
Why this answer is correct
The correct answer is C. (9). For (a=2,3,5), the counts of (b) are (4,3,2), totaling (9). Count multiples of each (a) in (B).
Step 3
Exam Tip
(a=2,3,5) के लिए (b) के क्रमशः (4,3,2) मान मिलते हैं, कुल (9)। हर (a) के गुणज (B) में गिनें।
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यदि \(A=\{1,2,3,4,5\}\) और \(B=\{1,2,3,4,5,6\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें \(ab\le10\) है?
If \(A=\{1,2,3,4,5\}\) and \(B=\{1,2,3,4,5,6\}\), how many pairs ((a,b)) in \(A\times B\) satisfy \(ab\le10\)?
#product-inequality
#counting
#hard
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A (18)
B (19)
C (20)
D (21)
Explanation opens after your attempt
Step 1
Concept
For (a=1,2,3,4,5), the counts of (b) are (6,5,3,2,2), totaling (18). In product conditions, the boundary changes.
Step 2
Why this answer is correct
The correct answer is B. (19). For (a=1,2,3,4,5), the counts of (b) are (6,5,3,2,2), totaling (18). In product conditions, the boundary changes.
Step 3
Exam Tip
(a=1,2,3,4,5) के लिए (b) के (6,5,3,2,2) मान मिलते हैं, कुल (18)। गुणन शर्त में सीमा बदलती है।
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यदि \(A=\{1,2\}\), \(B=\{3,4,5\}\) और \(C=\{6,7,8\}\) हैं, तो (|\(A\times B\)\times C|) क्या है?
If \(A=\{1,2\}\), \(B=\{3,4,5\}\), and \(C=\{6,7,8\}\), what is (|\(A\times B\)\times C|)?
#nested-product
#cardinality
#ordered-pair
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A (12)
B (15)
C (18)
D (24)
Explanation opens after your attempt
Step 1
Concept
\(|A\times B|=2\cdot3=6\), and then \(6\cdot3=18\). Cardinality also multiplies in nested products.
Step 2
Why this answer is correct
The correct answer is C. (18). \(|A\times B|=2\cdot3=6\), and then \(6\cdot3=18\). Cardinality also multiplies in nested products.
Step 3
Exam Tip
\(|A\times B|=2\cdot3=6\) और फिर \(6\cdot3=18\)। नेस्टेड गुणनफल में भी कार्डिनलिटी गुणा होती है।
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यदि \(A=\{1,2,3,4,5\}\) और \(B=\{1,2,3,4,5\}\) हैं, तो \(A\times B\) में विकर्ण युग्मों ((a,a)) के अतिरिक्त कितने युग्म हैं?
If \(A=\{1,2,3,4,5\}\) and \(B=\{1,2,3,4,5\}\), how many pairs in \(A\times B\) are other than the diagonal pairs ((a,a))?
#diagonal-pairs
#not-equal
#counting
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A (15)
B (18)
C (20)
D (25)
Explanation opens after your attempt
Step 1
Concept
There are (25) total pairs and (5) diagonal pairs, so (25-5=20). Removing the diagonal leaves pairs with \(a\ne b\).
Step 2
Why this answer is correct
The correct answer is C. (20). There are (25) total pairs and (5) diagonal pairs, so (25-5=20). Removing the diagonal leaves pairs with \(a\ne b\).
Step 3
Exam Tip
कुल (25) युग्म हैं और विकर्ण युग्म (5) हैं, इसलिए (25-5=20)। विकर्ण हटाने पर \(a\ne b\) वाले युग्म बचते हैं।
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यदि \(A=\{1,2,3,4,5,6\}\) और \(B=\{2,3,4,5\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (a+b) (2) से विभाज्य है?
If \(A=\{1,2,3,4,5,6\}\) and \(B=\{2,3,4,5\}\), how many pairs ((a,b)) in \(A\times B\) have (a+b) divisible by (2)?
#parity
#divisible-by-two
#counting
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A (10)
B (12)
C (14)
D (16)
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Step 1
Concept
The sum is divisible by (2) when both have the same parity. The count is \(3\cdot2+3\cdot2=12\).
Step 2
Why this answer is correct
The correct answer is B. (12). The sum is divisible by (2) when both have the same parity. The count is \(3\cdot2+3\cdot2=12\).
Step 3
Exam Tip
योग (2) से विभाज्य तब है जब दोनों की समता समान हो। गिनती \(3\cdot2+3\cdot2=12\) है।
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यदि \(A=\{1,2,3,4\}\), \(B=\{2,3,4,5,6\}\) और \(S=\{(a,b):b>a+2\}\) है, तो \(S\subseteq A\times B\) में कितने अवयव हैं?
If \(A=\{1,2,3,4\}\), \(B=\{2,3,4,5,6\}\), and \(S=\{(a,b):b>a+2\}\), how many elements are in \(S\subseteq A\times B\)?
#relation-subset
#inequality
#counting
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A (5)
B (6)
C (7)
D (8)
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Step 1
Concept
For (a=1,2,3,4), the counts of (b) are (3,2,1,0), totaling (6). Apply the condition to each first component.
Step 2
Why this answer is correct
The correct answer is B. (6). For (a=1,2,3,4), the counts of (b) are (3,2,1,0), totaling (6). Apply the condition to each first component.
Step 3
Exam Tip
(a=1,2,3,4) के लिए (b) के क्रमशः (3,2,1,0) मान मिलते हैं, कुल (6)। शर्त को हर पहले अवयव पर लगाएँ।
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