यदि \(A=\{2,4,6\}\) और \(B=\{1,3,5\}\) हैं, तो \(A\times B\) में वह युग्म कौन सा है जिसका पहला अवयव (4) और दूसरा अवयव (5) है?
If \(A=\{2,4,6\}\) and \(B=\{1,3,5\}\), which pair in \(A\times B\) has first component (4) and second component (5)?
#cartesian-product
#ordered-pair
#basics
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A ((4,5))
B ((5,4))
C ((4,3))
D ((6,5))
Explanation opens after your attempt
Correct Answer
A. ((4,5))
Step 1
Concept
In \(A\times B\), the first component comes from (A) and the second from (B). Changing order changes the ordered pair.
Step 2
Why this answer is correct
The correct answer is A. ((4,5)). In \(A\times B\), the first component comes from (A) and the second from (B). Changing order changes the ordered pair.
Step 3
Exam Tip
\(A\times B\) में पहला अवयव (A) से और दूसरा अवयव (B) से आता है। क्रमित युग्म में क्रम बदलने से उत्तर बदल जाता है।
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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) (4) से विभाज्य 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) divisible by (4)?
#cartesian-product
#divisibility
#counting
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A (4)
B (5)
C (6)
D (7)
Explanation opens after your attempt
Step 1
Concept
The pairs with sum divisible by (4) are ((1,3),(2,2),(3,1),(4,4),(5,3)). In such questions, count using remainders.
Step 2
Why this answer is correct
The correct answer is B. (5). The pairs with sum divisible by (4) are ((1,3),(2,2),(3,1),(4,4),(5,3)). In such questions, count using remainders.
Step 3
Exam Tip
योग (4) से विभाज्य होने वाले युग्म ((1,3),(2,2),(3,1),(4,4),(5,3)) हैं। ऐसे प्रश्नों में शेषफल के आधार पर गिनती करें।
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\(यदि (A={x:x\in\mathbb{N},2\le x\le6}) और (B={y:y\in\mathbb{N},y\) is odd\(,y<8}) हैं, तो (|A\times B|) क्या है\)?
\(If (A={x:x\in\mathbb{N},2\le x\le6}) and (B={y:y\in\mathbb{N},y\) is odd\(,y<8}), what is (|A\times B|)\)?
#set-builder
#cardinality
#cartesian-product
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A (15)
B (20)
C (25)
D (30)
Explanation opens after your attempt
Step 1
Concept
Here \(A=\{2,3,4,5,6\}\) and \(B=\{1,3,5,7\}\), so \(|A\times B|=5\cdot4=20\). First count both sets.
Step 2
Why this answer is correct
The correct answer is B. (20). Here \(A=\{2,3,4,5,6\}\) and \(B=\{1,3,5,7\}\), so \(|A\times B|=5\cdot4=20\). First count both sets.
Step 3
Exam Tip
यहाँ \(A=\{2,3,4,5,6\}\) और \(B=\{1,3,5,7\}\), इसलिए \(|A\times B|=5\cdot4=20\)। पहले दोनों समुच्चयों की संख्या गिनें।
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यदि \(A=\{1,2,3,4\}\), \(B=\{2,3,4,5,6\}\) और \(R=\{(a,b):b\ge a+2\}\) है, तो \(R\subseteq A\times B\) में कितने अवयव हैं?
If \(A=\{1,2,3,4\}\), \(B=\{2,3,4,5,6\}\), and \(R=\{(a,b):b\ge a+2\}\), how many elements are in \(R\subseteq A\times B\)?
#relation-subset
#inequality
#cartesian-product
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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 counts of (b) are (4,3,2,1), so there are (10) pairs. In inequalities, check the boundary for each first component.
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), so there are (10) pairs. In inequalities, check the boundary for each first component.
Step 3
Exam Tip
(a=1,2,3,4) के लिए (b) के क्रमशः (4,3,2,1) मान मिलते हैं, इसलिए कुल (10) युग्म हैं। असमानता में हर पहले अवयव पर सीमा अलग जाँचें।
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यदि \(A=\{1,2,3\}\), \(B=\{4,5\}\) और \(C=\{6,7,8\}\) हैं, तो \(|A\times B\times C|\) का मान क्या होगा?
If \(A=\{1,2,3\}\), \(B=\{4,5\}\), and \(C=\{6,7,8\}\), what is \(|A\times B\times C|\)?
#ordered-triple
#cardinality
#hard
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A (8)
B (12)
C (18)
D (24)
Explanation opens after your attempt
Step 1
Concept
For product of three sets, the cardinality is \(3\cdot2\cdot3=18\). For ordered triples, multiply the sizes of all sets.
Step 2
Why this answer is correct
The correct answer is C. (18). For product of three sets, the cardinality is \(3\cdot2\cdot3=18\). For ordered triples, multiply the sizes of all sets.
Step 3
Exam Tip
तीन समुच्चयों के गुणनफल में कार्डिनलिटी \(3\cdot2\cdot3=18\) होती है। क्रमित त्रिक में सभी समुच्चयों के आकार गुणा करें।
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यदि \(A=\{a,b,c\}\) और \(B=\{1,2\}\) हैं, तो \(B\times A\) में कितने अवयव होंगे?
If \(A=\{a,b,c\}\) and \(B=\{1,2\}\), how many elements will \(B\times A\) have?
#reverse-product
#cardinality
#sets
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A (5)
B (6)
C (8)
D (9)
Explanation opens after your attempt
Step 1
Concept
\(|B\times A|=|B|\cdot|A|=2\cdot3=6\). Reversing order changes pairs but not the cardinality.
Step 2
Why this answer is correct
The correct answer is B. (6). \(|B\times A|=|B|\cdot|A|=2\cdot3=6\). Reversing order changes pairs but not the cardinality.
Step 3
Exam Tip
\(|B\times A|=|B|\cdot|A|=2\cdot3=6\)। क्रम बदलने से युग्म बदलते हैं, पर कार्डिनलिटी समान रहती है।
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यदि \(A=\{1,2,3,4\}\) और \(B=\{2,4,6\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (a+b=8) है?
If \(A=\{1,2,3,4\}\) and \(B=\{2,4,6\}\), how many pairs ((a,b)) in \(A\times B\) satisfy (a+b=8)?
#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 the pairs ((2,6)) and ((4,4)). For each (a), check (b=8-a).
Step 2
Why this answer is correct
The correct answer is B. (2). The condition gives the pairs ((2,6)) and ((4,4)). For each (a), check (b=8-a).
Step 3
Exam Tip
शर्त से युग्म ((2,6)) और ((4,4)) मिलते हैं। हर (a) के लिए (b=8-a) जाँचें।
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यदि \(A=\{0,1,2,3\}\) और \(B=\{1,2,3,4\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (a<b) है?
If \(A=\{0,1,2,3\}\) and \(B=\{1,2,3,4\}\), how many pairs ((a,b)) in \(A\times B\) satisfy (a<b)?
#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=0,1,2,3), the counts of (b) are (4,3,2,1), totaling (10). Count row-wise in inequality questions.
Step 2
Why this answer is correct
The correct answer is C. (10). For (a=0,1,2,3), the counts of (b) are (4,3,2,1), totaling (10). Count row-wise in inequality questions.
Step 3
Exam Tip
(a=0,1,2,3) के लिए (b) के क्रमशः (4,3,2,1) मान मिलते हैं, कुल (10)। असमानता में पंक्ति के अनुसार गिनती करें।
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यदि \(A=\{1,2,3,4,5\}\) और \(B=\{1,2,3\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें \(a\ge2b\) है?
If \(A=\{1,2,3,4,5\}\) and \(B=\{1,2,3\}\), how many pairs ((a,b)) in \(A\times B\) satisfy \(a\ge2b\)?
#inequality
#twice
#counting
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A (3)
B (4)
C (5)
D (6)
Explanation opens after your attempt
Step 1
Concept
For (b=1), (a=2,3,4,5); for (b=2,3), counts are (2,0), so total is (6). The correct option should be (D).
Step 2
Why this answer is correct
The correct answer is B. (4). For (b=1), (a=2,3,4,5); for (b=2,3), counts are (2,0), so total is (6). The correct option should be (D).
Step 3
Exam Tip
(b=1) पर (a=2,3,4,5) मिलते हैं और (b=2,3) पर क्रमशः (2,0) मान मिलते हैं, कुल (4+2+0=6) नहीं बल्कि (6) है। सही विकल्प (D) होना चाहिए।
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यदि \(A=\{1,2,3,4\}\) और \(B=\{1,4,9,16\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें \(b=a^2\) है?
If \(A=\{1,2,3,4\}\) and \(B=\{1,4,9,16\}\), how many pairs ((a,b)) in \(A\times B\) satisfy \(b=a^2\)?
#square-rule
#function-rule
#counting
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A (2)
B (3)
C (4)
D (8)
Explanation opens after your attempt
Step 1
Concept
For every \(a\in A\), \(a^2\in B\), so (4) pairs are formed. In rule-based questions, check all first components.
Step 2
Why this answer is correct
The correct answer is C. (4). For every \(a\in A\), \(a^2\in B\), so (4) pairs are formed. In rule-based questions, check all first components.
Step 3
Exam Tip
हर \(a\in A\) के लिए \(a^2\in B\) है, इसलिए (4) युग्म बनते हैं। नियम आधारित प्रश्न में सभी पहले अवयव जाँचें।
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यदि \(A=\{1,2,3\}\) और \(B=\{2,3,4,5\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (a) (b) को विभाजित करता है?
If \(A=\{1,2,3\}\) and \(B=\{2,3,4,5\}\), how many pairs ((a,b)) in \(A\times B\) have (a) dividing (b)?
#divisibility
#relation
#counting
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A (5)
B (6)
C (7)
D (8)
Explanation opens after your attempt
Step 1
Concept
For (a=1), there are (4) pairs; for (a=2), (2); and for (a=3), (1), totaling (7). Check each divisor separately.
Step 2
Why this answer is correct
The correct answer is C. (7). For (a=1), there are (4) pairs; for (a=2), (2); and for (a=3), (1), totaling (7). Check each divisor separately.
Step 3
Exam Tip
(a=1) से (4), (a=2) से (2), और (a=3) से (1) युग्म मिलते हैं, कुल (7)। विभाज्यता में प्रत्येक भाजक को अलग जाँचें।
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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 (\gcd(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 (\gcd(a,b)=2)?
#gcd
#counting
#ordered-pair
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A (2)
B (3)
C (4)
D (5)
Explanation opens after your attempt
Step 1
Concept
The pairs are ((2,2),(2,4),(4,2)), so the number is (3). In \(\gcd\) questions, check common factors.
Step 2
Why this answer is correct
The correct answer is B. (3). The pairs are ((2,2),(2,4),(4,2)), so the number is (3). In \(\gcd\) questions, check common factors.
Step 3
Exam Tip
ऐसे युग्म ((2,2),(2,4),(4,2)) हैं, इसलिए संख्या (3) है। \(\gcd\) वाले प्रश्न में सामान्य गुणनखंड जाँचें।
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यदि (|A|=3), (|B|=4) और \(R\subseteq A\times B\), तो (R) में ठीक (3) अवयव चुनने के कितने तरीके हैं?
If (|A|=3), (|B|=4), and \(R\subseteq A\times B\), in how many ways can (R) have exactly (3) elements?
#combinations
#relations
#counting
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A (12)
B (36)
C (220)
D (4096)
Explanation opens after your attempt
Step 1
Concept
Since \(|A\times B|=12\), the number of ways to choose exactly (3) pairs is \(\binom{12}{3}=220\). Use combinations when an exact size is asked.
Step 2
Why this answer is correct
The correct answer is C. (220). Since \(|A\times B|=12\), the number of ways to choose exactly (3) pairs is \(\binom{12}{3}=220\). Use combinations when an exact size is asked.
Step 3
Exam Tip
\(|A\times B|=12\), इसलिए ठीक (3) युग्म चुनने के तरीके \(\binom{12}{3}=220\) हैं। ठीक संख्या पूछी हो तो संयोजन लगाएँ।
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यदि (|A|=2) और (|B|=5) हैं, तो (A) से (B) तक संभव संबंधों की संख्या क्या है?
If (|A|=2) and (|B|=5), what is the number of possible relations from (A) to (B)?
#relations
#power-set
#cardinality
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A (10)
B (25)
C (32)
D (1024)
Explanation opens after your attempt
Step 1
Concept
A relation is a subset of \(A\times B\), and \(|A\times B|=10\), so the number is \(2^{10}=1024\). Number of relations uses a power of (2).
Step 2
Why this answer is correct
The correct answer is D. (1024). A relation is a subset of \(A\times B\), and \(|A\times B|=10\), so the number is \(2^{10}=1024\). Number of relations uses a power of (2).
Step 3
Exam Tip
संबंध \(A\times B\) का उपसमुच्चय है और \(|A\times B|=10\), इसलिए संख्या \(2^{10}=1024\) है। संबंधों की संख्या के लिए घात (2) लगती है।
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यदि \(A=\{1,2,3\}\), \(B=\{2,3,4\}\) और \(C=\{3,4,5\}\) हैं, तो (|A\times\(B\cup C\)|) क्या है?
If \(A=\{1,2,3\}\), \(B=\{2,3,4\}\), and \(C=\{3,4,5\}\), what is (|A\times\(B\cup C\)|)?
#union
#cardinality
#cartesian-product
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A (9)
B (12)
C (15)
D (18)
Explanation opens after your attempt
Step 1
Concept
\(B\cup C={2,3,4,5}\), so (|A\times\(B\cup C\)|=3\cdot4=12). Find the union first, then multiply.
Step 2
Why this answer is correct
The correct answer is B. (12). \(B\cup C={2,3,4,5}\), so (|A\times\(B\cup C\)|=3\cdot4=12). Find the union first, then multiply.
Step 3
Exam Tip
\(B\cup C={2,3,4,5}\), इसलिए (|A\times\(B\cup C\)|=3\cdot4=12)। पहले संघ निकालें फिर गुणा करें।
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यदि \(A=\{1,2,3,4\}\), \(B=\{1,2,5\}\) और \(C=\{2,4,5\}\) हैं, तो (|A\times\(B\cap C\)|) क्या होगा?
If \(A=\{1,2,3,4\}\), \(B=\{1,2,5\}\), and \(C=\{2,4,5\}\), what is (|A\times\(B\cap C\)|)?
#intersection
#cardinality
#set-operation
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A (4)
B (8)
C (12)
D (16)
Explanation opens after your attempt
Step 1
Concept
\(B\cap C={2,5}\), so the cardinality is \(4\cdot2=8\). Simplify the intersection first.
Step 2
Why this answer is correct
The correct answer is B. (8). \(B\cap C={2,5}\), so the cardinality is \(4\cdot2=8\). Simplify the intersection first.
Step 3
Exam Tip
\(B\cap C={2,5}\), इसलिए कार्डिनलिटी \(4\cdot2=8\) है। प्रतिच्छेद को पहले सरल करना जरूरी है।
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यदि \(A=\{0,1,2\}\), \(B=\{1,2,3\}\) और \(C=\{2,3,4\}\) हैं, तो \(A\times(B-C)\) क्या है?
If \(A=\{0,1,2\}\), \(B=\{1,2,3\}\), and \(C=\{2,3,4\}\), what is \(A\times(B-C)\)?
#difference
#set-operation
#cartesian-product
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A ({(0,1),(1,1),(2,1)})
B ({(0,2),(1,2),(2,2)})
C ({(0,3),(1,3),(2,3)})
D \(\varnothing\)
Explanation opens after your attempt
Correct Answer
A. ({(0,1),(1,1),(2,1)})
Step 1
Concept
(B-C={1}), so \(A\times(B-C)={(0,1),(1,1),(2,1)}\). In difference, take only remaining elements of (B).
Step 2
Why this answer is correct
The correct answer is A. ({(0,1),(1,1),(2,1)}). (B-C={1}), so \(A\times(B-C)={(0,1),(1,1),(2,1)}\). In difference, take only remaining elements of (B).
Step 3
Exam Tip
(B-C={1}), इसलिए \(A\times(B-C)={(0,1),(1,1),(2,1)}\)। अंतर में केवल (B) के बचे अवयव लें।
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यदि \(A=\{1,2\}\), \(B=\{2,3\}\) और \(C=\{3,4\}\) हैं, तो (\(A\times B\)\cap\(A\times C\)) की कार्डिनलिटी क्या है?
If \(A=\{1,2\}\), \(B=\{2,3\}\), and \(C=\{3,4\}\), what is the cardinality of (\(A\times B\)\cap\(A\times C\))?
#intersection-property
#cardinality
#identity
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A (0)
B (2)
C (4)
D (6)
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 \(2\cdot1=2\).
Step 2
Why this answer is correct
The correct answer is B. (2). (\(A\times B\)\cap\(A\times C\)=A\times\(B\cap C\)), and \(B\cap C={3}\). Hence the cardinality is \(2\cdot1=2\).
Step 3
Exam Tip
(\(A\times B\)\cap\(A\times C\)=A\times\(B\cap C\)) और \(B\cap C={3}\)। इसलिए कार्डिनलिटी \(2\cdot1=2\) है।
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यदि \(A=\{1,2,3\}\), \(B=\{4,5\}\) और \(C=\{5,6\}\) हैं, तो (\(A\times B\)\cup\(A\times C\)) की कार्डिनलिटी क्या है?
If \(A=\{1,2,3\}\), \(B=\{4,5\}\), and \(C=\{5,6\}\), what is the cardinality of (\(A\times B\)\cup\(A\times C\))?
#union-property
#set-identity
#cardinality
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A (6)
B (7)
C (9)
D (12)
Explanation opens after your attempt
Step 1
Concept
(\(A\times B\)\cup\(A\times C\)=A\times\(B\cup C\)), and \(B\cup C={4,5,6}\). Thus there are \(3\cdot3=9\) elements.
Step 2
Why this answer is correct
The correct answer is C. (9). (\(A\times B\)\cup\(A\times C\)=A\times\(B\cup C\)), and \(B\cup C={4,5,6}\). Thus there are \(3\cdot3=9\) elements.
Step 3
Exam Tip
(\(A\times B\)\cup\(A\times C\)=A\times\(B\cup C\)) और \(B\cup C={4,5,6}\)। इसलिए \(3\cdot3=9\) अवयव हैं।
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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|=1) 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|=1)?
#absolute-difference
#counting
#ordered-pair
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A (4)
B (5)
C (6)
D (8)
Explanation opens after your attempt
Step 1
Concept
The pairs are ((1,2),(2,1),(2,3),(3,2),(3,4),(4,3)), totaling (6). Reversed ordered pairs are counted separately.
Step 2
Why this answer is correct
The correct answer is C. (6). The pairs are ((1,2),(2,1),(2,3),(3,2),(3,4),(4,3)), totaling (6). Reversed ordered pairs are counted separately.
Step 3
Exam Tip
युग्म ((1,2),(2,1),(2,3),(3,2),(3,4),(4,3)) हैं, कुल (6)। उलटे क्रमित युग्म अलग गिने जाते हैं।
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यदि \(A=\{0,1,2,3\}\) और \(B=\{0,1,2,3\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (a+b) सम है?
If \(A=\{0,1,2,3\}\) 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 (6)
B (8)
C (10)
D (12)
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 होता है जब दोनों अवयवों की समता समान हो। गिनती \(2\cdot2+2\cdot2=8\) है।
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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 \(a+b\le6\) है?
If \(A=\{1,2,3,4,5\}\) and \(B=\{1,2,3,4\}\), how many pairs ((a,b)) in \(A\times B\) satisfy \(a+b\le6\)?
#sum-inequality
#counting
#hard
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A (12)
B (14)
C (15)
D (16)
Explanation opens after your attempt
Step 1
Concept
For (a=1,2,3,4,5), the counts of (b) are (4,4,3,2,1), totaling (14). Include equality in a boundary inequality.
Step 2
Why this answer is correct
The correct answer is B. (14). For (a=1,2,3,4,5), the counts of (b) are (4,4,3,2,1), totaling (14). Include equality in a boundary inequality.
Step 3
Exam Tip
(a=1,2,3,4,5) के लिए (b) के (4,4,3,2,1) मान हैं, कुल (14)। सीमा सहित असमानता में बराबरी भी गिनें।
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यदि \(A=\{1,2,3,4\}\) और \(B=\{2,3,4,5\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (ab) सम है?
If \(A=\{1,2,3,4\}\) and \(B=\{2,3,4,5\}\), how many pairs ((a,b)) in \(A\times B\) have (ab) even?
#product-parity
#complement-counting
#ordered-pair
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A (10)
B (11)
C (12)
D (13)
Explanation opens after your attempt
Step 1
Concept
There are (16) total pairs, and (ab) is odd only when both are odd, giving \(2\cdot2=4\). Hence even products are (16-4=12).
Step 2
Why this answer is correct
The correct answer is C. (12). There are (16) total pairs, and (ab) is odd only when both are odd, giving \(2\cdot2=4\). Hence even products are (16-4=12).
Step 3
Exam Tip
कुल (16) युग्म हैं और (ab) विषम केवल तब है जब दोनों विषम हों, ऐसे \(2\cdot2=4\) हैं। इसलिए सम गुणनफल (16-4=12) हैं।
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यदि \(A=\{1,2,3,4\}\) और \(B=\{1,2,4,8\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (b=2a) है?
If \(A=\{1,2,3,4\}\) and \(B=\{1,2,4,8\}\), how many pairs ((a,b)) in \(A\times B\) satisfy (b=2a)?
#linear-rule
#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 ((1,2)) and ((2,4)); for (a=3,4), (b) is not in (B). Hence there are (2) pairs.
Step 2
Why this answer is correct
The correct answer is B. (2). The condition gives ((1,2)) and ((2,4)); for (a=3,4), (b) is not in (B). Hence there are (2) pairs.
Step 3
Exam Tip
शर्त से ((1,2)) और ((2,4)) मिलते हैं; (a=3,4) पर (b) (B) में नहीं है। इसलिए (2) युग्म हैं।
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यदि \(A=\{1,2,3,4,5\}\) और \(B=\{0,1,2\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (a-2b=1) है?
If \(A=\{1,2,3,4,5\}\) and \(B=\{0,1,2\}\), how many pairs ((a,b)) in \(A\times B\) satisfy (a-2b=1)?
#linear-equation
#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+1), so for (b=0,1,2), (a=1,3,5). There are (3) pairs in total.
Step 2
Why this answer is correct
The correct answer is C. (3). The condition gives (a=2b+1), so for (b=0,1,2), (a=1,3,5). There are (3) pairs in total.
Step 3
Exam Tip
शर्त (a=2b+1) देती है, इसलिए (b=0,1,2) पर (a=1,3,5) मिलते हैं। कुल (3) युग्म 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 (b-a) विषम है?
If \(A=\{1,2,3,4\}\) and \(B=\{1,2,3,4,5\}\), how many pairs ((a,b)) in \(A\times B\) have (b-a) odd?
#parity
#difference
#counting
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A (8)
B (9)
C (10)
D (11)
Explanation opens after your attempt
Step 1
Concept
The difference is odd when (a) and (b) have opposite parity. The count is \(2\cdot3+2\cdot2=10\).
Step 2
Why this answer is correct
The correct answer is C. (10). The difference is odd when (a) and (b) have opposite parity. The count is \(2\cdot3+2\cdot2=10\).
Step 3
Exam Tip
अंतर विषम तब होता है जब (a) और (b) की समता अलग हो। गिनती \(2\cdot3+2\cdot2=10\) 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=b) नहीं 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 \(a\ne b\)?
#not-equal
#complement-counting
#diagonal
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A (15)
B (20)
C (25)
D (30)
Explanation opens after your attempt
Step 1
Concept
There are (25) total pairs and (5) equal pairs, so (25-5=20). Complement counting is a quick method.
Step 2
Why this answer is correct
The correct answer is B. (20). There are (25) total pairs and (5) equal pairs, so (25-5=20). Complement counting is a quick method.
Step 3
Exam Tip
कुल (25) युग्म हैं और बराबर युग्म (5) हैं, इसलिए (25-5=20)। पूरक गिनती तेज तरीका है।
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यदि \(A=\{-2,-1,0,1,2\}\) और \(B=\{0,1,4\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें \(a^2=b\) है?
If \(A=\{-2,-1,0,1,2\}\) and \(B=\{0,1,4\}\), how many pairs ((a,b)) in \(A\times B\) satisfy \(a^2=b\)?
#square-condition
#negative-numbers
#counting
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A (3)
B (4)
C (5)
D (6)
Explanation opens after your attempt
Step 1
Concept
For every \(a\in A\), \(a^2\) is in (B), so (5) pairs are formed. Watch squares of negative numbers carefully.
Step 2
Why this answer is correct
The correct answer is C. (5). For every \(a\in A\), \(a^2\) is in (B), so (5) pairs are formed. Watch squares of negative numbers carefully.
Step 3
Exam Tip
हर \(a\in A\) के लिए \(a^2\) (B) में है, इसलिए (5) युग्म बनते हैं। ऋणात्मक संख्याओं के वर्ग को ध्यान से देखें।
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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) अभाज्य है?
If \(A=\{1,2,3,4\}\) and \(B=\{1,2,3\}\), 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 all sums gives (7) pairs with prime sum. In such questions, look for prime sums like (2,3,5,7).
Step 2
Why this answer is correct
The correct answer is C. (7). Checking all sums gives (7) pairs with prime sum. In such questions, look for prime sums like (2,3,5,7).
Step 3
Exam Tip
सभी योग जाँचने पर अभाज्य योग वाले (7) युग्म मिलते हैं। ऐसे प्रश्नों में (2,3,5,7) जैसे अभाज्य योग देखें।
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यदि \(A=\{1,2,3\}\) और \(B=\{2,4,6,8\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें \(\frac{b}{a}=2\) है?
If \(A=\{1,2,3\}\) and \(B=\{2,4,6,8\}\), how many pairs ((a,b)) in \(A\times B\) satisfy \(\frac{b}{a}=2\)?
#fraction-condition
#linear-rule
#counting
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A (2)
B (3)
C (4)
D (6)
Explanation opens after your attempt
Step 1
Concept
The condition is (b=2a), so the pairs are ((1,2),(2,4),(3,6)). Convert the fraction into a simple equation.
Step 2
Why this answer is correct
The correct answer is B. (3). The condition is (b=2a), so the pairs are ((1,2),(2,4),(3,6)). Convert the fraction into a simple equation.
Step 3
Exam Tip
शर्त (b=2a) है, इसलिए ((1,2),(2,4),(3,6)) मिलते हैं। भिन्न को सरल समीकरण में बदलें।
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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 (ab>8) है?
If \(A=\{1,2,3,4\}\) and \(B=\{1,2,3,4\}\), how many pairs ((a,b)) in \(A\times B\) satisfy (ab>8)?
#product-inequality
#counting
#hard
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A (2)
B (3)
C (4)
D (5)
Explanation opens after your attempt
Step 1
Concept
The satisfying pairs are ((3,3),(3,4),(4,3),(4,4)), totaling (4). The correct option should be (C).
Step 2
Why this answer is correct
The correct answer is B. (3). The satisfying pairs are ((3,3),(3,4),(4,3),(4,4)), totaling (4). The correct option should be (C).
Step 3
Exam Tip
संतुष्ट युग्म ((3,3),(3,4),(4,3),(4,4)) नहीं बल्कि ((3,3),(3,4),(4,3),(4,4)) कुल (4) हैं। सही विकल्प (C) होना चाहिए।
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यदि \(A=\{1,2,3\}\) और \(B=\{1,2,3,4\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (a+b) (3) से विभाज्य है?
If \(A=\{1,2,3\}\) and \(B=\{1,2,3,4\}\), how many pairs ((a,b)) in \(A\times B\) have (a+b) divisible by (3)?
#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 (3) are ((1,2),(2,1),(2,4),(3,3)). Hence there are (4) pairs.
Step 2
Why this answer is correct
The correct answer is B. (4). The pairs with sum divisible by (3) are ((1,2),(2,1),(2,4),(3,3)). Hence there are (4) pairs.
Step 3
Exam Tip
योग (3) से विभाज्य होने वाले युग्म ((1,2),(2,1),(2,4),(3,3)) हैं। इसलिए कुल (4) युग्म 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 (b) (a) का गुणज है?
If \(A=\{1,2,3,4\}\) and \(B=\{1,2,3,4,5\}\), how many pairs ((a,b)) in \(A\times B\) have (b) as a multiple of (a)?
#multiples
#counting
#relation
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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 counts of (b) are (5,2,1,1), totaling (9). The correct option should be (B).
Step 2
Why this answer is correct
The correct answer is C. (10). For (a=1,2,3,4), the counts of (b) are (5,2,1,1), totaling (9). The correct option should be (B).
Step 3
Exam Tip
(a=1,2,3,4) के लिए (b) के (5,2,1,1) मान हैं, कुल (9) नहीं बल्कि (9) हैं। सही विकल्प (B) होना चाहिए।
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यदि \(A=\{1,2\}\), \(B=\{3,4\}\) और \(A\times C={(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)}\), तो (C) क्या है?
If \(A=\{1,2\}\), \(B=\{3,4\}\), and \(A\times C={(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)}\), what is (C)?
#identify-set
#cartesian-product
#reasoning
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A ({1,2})
B ({3,4})
C ({3,4,5})
D ({1,2,3,4,5})
Explanation opens after your attempt
Correct Answer
C. ({3,4,5})
Step 1
Concept
The set of second components is \(C=\{3,4,5\}\). When identifying original sets, observe the positions.
Step 2
Why this answer is correct
The correct answer is C. ({3,4,5}). The set of second components is \(C=\{3,4,5\}\). When identifying original sets, observe the positions.
Step 3
Exam Tip
दूसरे अवयवों का समुच्चय \(C=\{3,4,5\}\) है। कार्तीय गुणनफल से मूल समुच्चय पहचानते समय स्थान देखें।
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यदि \(A\times B=\varnothing\) और \(B=\{7,8\}\), तो (A) के बारे में क्या सही है?
If \(A\times B=\varnothing\) and \(B=\{7,8\}\), what is true about (A)?
#empty-set
#cartesian-product
#concept
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A \(A=\varnothing\)
B (A=B)
C \(A=\{0\}\)
D \(A\subseteq B\) अवश्य है / \(A\subseteq B\) must hold
Explanation opens after your attempt
Correct Answer
A. \(A=\varnothing\)
Step 1
Concept
Since (B) is non-empty and \(A\times B=\varnothing\), we must have \(A=\varnothing\). An empty product means at least one set is empty.
Step 2
Why this answer is correct
The correct answer is A. \(A=\varnothing\). Since (B) is non-empty and \(A\times B=\varnothing\), we must have \(A=\varnothing\). An empty product means at least one set is empty.
Step 3
Exam Tip
क्योंकि (B) रिक्त नहीं है और \(A\times B=\varnothing\), इसलिए \(A=\varnothing\)। रिक्त गुणनफल में कम से कम एक समुच्चय रिक्त होता है।
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यदि \(A=\{1,2,3\}\), \(B=\{4,5,6\}\) हैं, तो \(A\times B\) में ((3,6)), ((4,5)), ((2,4)) में से कौन सा युग्म नहीं है?
If \(A=\{1,2,3\}\), \(B=\{4,5,6\}\), which among ((3,6)), ((4,5)), and ((2,4)) is not in \(A\times B\)?
#membership
#error-check
#ordered-pair
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A ((3,6))
B ((4,5))
C ((2,4))
D तीनों हैं / All three are present
Explanation opens after your attempt
Correct Answer
B. ((4,5))
Step 1
Concept
In ((4,5)), the first component (4) is not in (A). Check the first and second positions separately.
Step 2
Why this answer is correct
The correct answer is B. ((4,5)). In ((4,5)), the first component (4) is not in (A). Check the first and second positions separately.
Step 3
Exam Tip
((4,5)) में पहला अवयव (4) है, जो (A) में नहीं है। सदस्यता में पहला और दूसरा स्थान अलग-अलग जाँचें।
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यदि \(A=\{1,2,3\}\) और \(B=\{3,4,5\}\) हैं, तो कथन \(A\times B=B\times A\) कैसा है?
If \(A=\{1,2,3\}\) and \(B=\{3,4,5\}\), how is the statement \(A\times B=B\times A\)?
#non-commutative
#counterexample
#cartesian-product
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A सत्य / True
B असत्य / False
C हमेशा सत्य / Always true
D निर्धारित नहीं / Not determined
Explanation opens after your attempt
Correct Answer
B. असत्य / False
Step 1
Concept
For example, \((1,4)\in A\times B\), but \((1,4)\notin B\times A\). Therefore the equality is false.
Step 2
Why this answer is correct
The correct answer is B. असत्य / False. For example, \((1,4)\in A\times B\), but \((1,4)\notin B\times A\). Therefore the equality is false.
Step 3
Exam Tip
उदाहरण के लिए \((1,4)\in A\times B\), पर \((1,4)\notin B\times A\) है। इसलिए बराबरी गलत है।
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यदि \(A=\{1,2,3\}\) है और \(A\times B\) में कुल (15) अवयव हैं, तो (|B|) क्या होगा?
If \(A=\{1,2,3\}\) and \(A\times B\) has (15) elements, what is (|B|)?
#unknown-cardinality
#formula
#sets
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A (3)
B (4)
C (5)
D (12)
Explanation opens after your attempt
Step 1
Concept
Since (|A|=3) and \(3\cdot|B|=15\), (|B|=5). Divide to find the unknown cardinality.
Step 2
Why this answer is correct
The correct answer is C. (5). Since (|A|=3) and \(3\cdot|B|=15\), (|B|=5). Divide to find the unknown cardinality.
Step 3
Exam Tip
(|A|=3) और \(3\cdot|B|=15\), इसलिए (|B|=5)। अज्ञात कार्डिनलिटी के लिए भाग करें।
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यदि (A) में (4) अवयव और (B) में (3) अवयव हैं, तो \(A\times B\) के गैर-रिक्त उपसमुच्चयों की संख्या क्या है?
If (A) has (4) elements and (B) has (3) elements, what is the number of non-empty subsets of \(A\times B\)?
#non-empty-subsets
#power-set
#counting
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A (12)
B (64)
C (255)
D (4095)
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 D. (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 \(A=\{1,2,3\}\) और \(B=\{1,2,3\}\) हैं, तो \(A\times B\) के उपसमुच्चय \(R=\{(a,b):a+b=4\}\) में कितने अवयव हैं?
If \(A=\{1,2,3\}\) and \(B=\{1,2,3\}\), how many elements are in the subset \(R=\{(a,b):a+b=4\}\) of \(A\times B\)?
#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 ((1,3),(2,2),(3,1)), so there are (3) elements. A relation is a selected part of \(A\times B\).
Step 2
Why this answer is correct
The correct answer is B. (3). The condition gives ((1,3),(2,2),(3,1)), so there are (3) elements. A relation is a selected part of \(A\times B\).
Step 3
Exam Tip
शर्त से ((1,3),(2,2),(3,1)) मिलते हैं, इसलिए (3) अवयव हैं। संबंध \(A\times B\) का चुना हुआ भाग होता है।
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यदि \(A=\{1,2,3,4\}\) और \(B=\{1,3,5\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (a+b) विषम है?
If \(A=\{1,2,3,4\}\) and \(B=\{1,3,5\}\), how many pairs ((a,b)) in \(A\times B\) have (a+b) odd?
#odd-sum
#parity
#counting
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A (3)
B (4)
C (6)
D (9)
Explanation opens after your attempt
Step 1
Concept
All elements of (B) are odd, so the sum is odd only when (a) is even. There are (2) even values of (a) and (3) values in (B), totaling (6).
Step 2
Why this answer is correct
The correct answer is C. (6). All elements of (B) are odd, so the sum is odd only when (a) is even. There are (2) even values of (a) and (3) values in (B), totaling (6).
Step 3
Exam Tip
(B) के सभी अवयव विषम हैं, इसलिए योग विषम तभी होगा जब (a) सम हो। (a) के (2) सम मान और (B) के (3) मान हैं, कुल (6)।
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यदि \(A=\{1,2,3,4\}\) और \(B=\{2,4,6,8\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (a+b=10) है?
If \(A=\{1,2,3,4\}\) and \(B=\{2,4,6,8\}\), how many pairs ((a,b)) in \(A\times B\) satisfy (a+b=10)?
#sum-condition
#counting
#sets
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A (1)
B (2)
C (3)
D (4)
Explanation opens after your attempt
Step 1
Concept
The condition gives ((2,8)) and ((4,6)). Check whether (b=10-a) lies in (B).
Step 2
Why this answer is correct
The correct answer is B. (2). The condition gives ((2,8)) and ((4,6)). Check whether (b=10-a) lies in (B).
Step 3
Exam Tip
शर्त से ((2,8)) और ((4,6)) मिलते हैं। (b=10-a) (B) में है या नहीं, यह जाँचें।
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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=6) और (a<b) दोनों हैं?
If \(A=\{1,2,3,4,5\}\) and \(B=\{1,2,3,4,5\}\), how many pairs ((a,b)) in \(A\times B\) satisfy both (a+b=6) and (a<b)?
#combined-condition
#sum-inequality
#counting
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A (2)
B (3)
C (4)
D (5)
Explanation opens after your attempt
Step 1
Concept
Among pairs with sum (6), only ((1,5)) and ((2,4)) have (a<b). Apply combined conditions one by one.
Step 2
Why this answer is correct
The correct answer is A. (2). Among pairs with sum (6), only ((1,5)) and ((2,4)) have (a<b). Apply combined conditions one by one.
Step 3
Exam Tip
योग (6) वाले युग्मों में (a<b) के लिए ((1,5)) और ((2,4)) ही मिलते हैं। संयुक्त शर्तों को एक-एक करके लगाएँ।
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यदि \(A=\{1,2,3\}\), \(B=\{1,2,3\}\) और \(R=\{(a,b):a\le b\}\), तो (R) में कितने अवयव हैं?
If \(A=\{1,2,3\}\), \(B=\{1,2,3\}\), and \(R=\{(a,b):a\le b\}\), how many elements are in (R)?
#relation-subset
#less-than-equal
#counting
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A (3)
B (5)
C (6)
D (9)
Explanation opens after your attempt
Step 1
Concept
For (a=1,2,3), the counts of (b) are (3,2,1), totaling (6). The symbol \(\le\) includes equality.
Step 2
Why this answer is correct
The correct answer is C. (6). For (a=1,2,3), the counts of (b) are (3,2,1), totaling (6). The symbol \(\le\) includes equality.
Step 3
Exam Tip
(a=1,2,3) के लिए (b) के (3,2,1) मान मिलते हैं, कुल (6)। \(\le\) में बराबरी भी शामिल होती है।
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यदि \(A=\{2,3,4\}\) और \(B=\{4,6,8,9,12\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (b) (a) से विभाज्य है?
If \(A=\{2,3,4\}\) and \(B=\{4,6,8,9,12\}\), 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)
Explanation opens after your attempt
Step 1
Concept
For (a=2,3,4), the counts of (b) are (4,3,2), totaling (9). Count multiples of each (a) inside (B).
Step 2
Why this answer is correct
The correct answer is C. (9). For (a=2,3,4), the counts of (b) are (4,3,2), totaling (9). Count multiples of each (a) inside (B).
Step 3
Exam Tip
(a=2,3,4) 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 \(A=\{1,2,3,4\}\) और \(B=\{1,2,3,4,5,6\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें \(ab\le8\) है?
If \(A=\{1,2,3,4\}\) and \(B=\{1,2,3,4,5,6\}\), how many pairs ((a,b)) in \(A\times B\) satisfy \(ab\le8\)?
#product-inequality
#counting
#hard
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A (13)
B (14)
C (15)
D (16)
Explanation opens after your attempt
Step 1
Concept
For (a=1,2,3,4), the counts of (b) are (6,4,2,2), totaling (14). In product conditions, the limit changes for each (a).
Step 2
Why this answer is correct
The correct answer is B. (14). For (a=1,2,3,4), the counts of (b) are (6,4,2,2), totaling (14). In product conditions, the limit changes for each (a).
Step 3
Exam Tip
(a=1,2,3,4) के लिए (b) के (6,4,2,2) मान मिलते हैं, कुल (14)। गुणन शर्त में प्रत्येक (a) पर सीमा अलग होती है।
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यदि \(A=\{1,2,3\}\), \(B=\{4,5\}\) और \(C=\{6,7\}\) हैं, तो (|\(A\times B\)\times C|) क्या है?
If \(A=\{1,2,3\}\), \(B=\{4,5\}\), and \(C=\{6,7\}\), what is (|\(A\times B\)\times C|)?
#nested-product
#cardinality
#ordered-pair
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A (7)
B (10)
C (12)
D (24)
Explanation opens after your attempt
Step 1
Concept
\(|A\times B|=3\cdot2=6\), and then \(6\cdot2=12\). Cardinality still multiplies in nested products.
Step 2
Why this answer is correct
The correct answer is C. (12). \(|A\times B|=3\cdot2=6\), and then \(6\cdot2=12\). Cardinality still multiplies in nested products.
Step 3
Exam Tip
\(|A\times B|=3\cdot2=6\) और फिर \(6\cdot2=12\)। नेस्टेड गुणनफल में भी कार्डिनलिटी गुणा होती है।
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यदि \(A=\{1,2,3,4\}\) और \(B=\{1,2,3,4\}\) हैं, तो \(A\times B\) में विकर्ण युग्मों ((a,a)) के अतिरिक्त कितने युग्म हैं?
If \(A=\{1,2,3,4\}\) and \(B=\{1,2,3,4\}\), 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 (8)
B (10)
C (12)
D (16)
Explanation opens after your attempt
Step 1
Concept
There are (16) total pairs and (4) diagonal pairs, so (16-4=12). Removing the diagonal leaves pairs with \(a\ne b\).
Step 2
Why this answer is correct
The correct answer is C. (12). There are (16) total pairs and (4) diagonal pairs, so (16-4=12). Removing the diagonal leaves pairs with \(a\ne b\).
Step 3
Exam Tip
कुल (16) युग्म हैं और विकर्ण युग्म (4) हैं, इसलिए (16-4=12)। विकर्ण हटाने पर \(a\ne b\) वाले युग्म बचते हैं।
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यदि \(A=\{1,2,3,4,5\}\) 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 \(A=\{1,2,3,4,5\}\) 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 (8)
B (10)
C (12)
D (14)
Explanation opens after your attempt
Step 1
Concept
The sum is divisible by (2) when both 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 divisible by (2) when both have the same parity. The count is \(3\cdot2+2\cdot2=10\).
Step 3
Exam Tip
योग (2) से विभाज्य तब है जब दोनों की समता समान हो। गिनती \(3\cdot2+2\cdot2=10\) है।
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यदि \(A=\{1,2,3\}\), \(B=\{1,2,3,4\}\) और \(S=\{(a,b):b>a+1\}\) है, तो \(S\subseteq A\times B\) में कितने अवयव हैं?
If \(A=\{1,2,3\}\), \(B=\{1,2,3,4\}\), and \(S=\{(a,b):b>a+1\}\), how many elements are in \(S\subseteq A\times B\)?
#relation-subset
#inequality
#counting
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A (2)
B (3)
C (4)
D (5)
Explanation opens after your attempt
Step 1
Concept
For (a=1), (b=3,4); for (a=2), (b=4), giving total (3). Apply the condition to each first component.
Step 2
Why this answer is correct
The correct answer is B. (3). For (a=1), (b=3,4); for (a=2), (b=4), giving total (3). Apply the condition to each first component.
Step 3
Exam Tip
(a=1) पर (b=3,4) और (a=2) पर (b=4) मिलता है, कुल (3)। शर्त को प्रत्येक पहले अवयव पर लगाएँ।
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