यदि \(A={x\in\mathbb{Z}:|x-2|\le2}\) और \(B={y\in\mathbb{N}:y^2-5y+6=0}\), तो \(A\times B\) में (x+y) अभाज्य होने वाले क्रमित युग्मों की संख्या कितनी है?
If \(A={x\in\mathbb{Z}:|x-2|\le2}\) and \(B={y\in\mathbb{N}:y^2-5y+6=0}\), how many ordered pairs in \(A\times B\) have (x+y) prime?
#cartesian-product
#prime-sum
#expert-counting
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A (4)
B (5)
C (6)
D (7)
Explanation opens after your attempt
Step 1
Concept
Here \(A=\{0,1,2,3,4\}\) and \(B=\{2,3\}\). On checking, (6) pairs have (x+y) prime, so first write the sets clearly.
Step 2
Why this answer is correct
The correct answer is C. (6). Here \(A=\{0,1,2,3,4\}\) and \(B=\{2,3\}\). On checking, (6) pairs have (x+y) prime, so first write the sets clearly.
Step 3
Exam Tip
यहां \(A=\{0,1,2,3,4\}\) और \(B=\{2,3\}\) हैं। जांचने पर (x+y) अभाज्य वाले (6) युग्म मिलते हैं, इसलिए पहले समुच्चय स्पष्ट करें।
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यदि \(A\times B={(1,2),(1,3),(4,2),(4,3)}\) है, तो (A) और (B) कौन से हैं?
If \(A\times B={(1,2),(1,3),(4,2),(4,3)}\), what are (A) and (B)?
#cartesian-product
#ordered-pairs
#reverse
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A \(A=\{1,4},B={2,3\}\)
B \(A=\{2,3},B={1,4\}\)
C \(A=\{1,2},B={3,4\}\)
D \(A=\{1,3},B={2,4\}\)
Explanation opens after your attempt
Correct Answer
A. \(A=\{1,4},B={2,3\}\)
Step 1
Concept
The first coordinates are (1,4) and the second coordinates are (2,3). Order is very important in \(A\times B\).
Step 2
Why this answer is correct
The correct answer is A. \(A=\{1,4},B={2,3\}\). The first coordinates are (1,4) and the second coordinates are (2,3). Order is very important in \(A\times B\).
Step 3
Exam Tip
पहले स्थान पर आने वाले तत्व (1,4) हैं और दूसरे स्थान पर आने वाले तत्व (2,3) हैं। \(A\times B\) में क्रम बहुत महत्वपूर्ण होता है।
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यदि \(A=\{1,2,3,4\}\), \(B=\{2,3,4,5\}\) और \(T=\{(a,b)\in A\times B:a^2-b\le3\}\), तो (n(T)) कितना है?
If \(A=\{1,2,3,4\}\), \(B=\{2,3,4,5\}\), and \(T=\{(a,b)\in A\times B:a^2-b\le3\}\), what is (n(T))?
#cartesian-product
#inequality
#relation-cardinality
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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 valid numbers of (b) are (4,4,1,0). Total is (9), so apply the condition for each first coordinate.
Step 2
Why this answer is correct
The correct answer is B. (9). For (a=1,2,3,4), the valid numbers of (b) are (4,4,1,0). Total is (9), so apply the condition for each first coordinate.
Step 3
Exam Tip
(a=1,2,3,4) पर मान्य (b) की संख्याएं क्रमशः (4,4,1,0) हैं। कुल (9) युग्म हैं, इसलिए हर पहले घटक पर शर्त लगाएं।
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यदि \(A=\{a,b,c\}\), \(B=\{1,2\}\) और \(C=\{p,q,r,s\}\) हैं, तो (n(\(A\times B\)\times C)) कितना होगा?
If \(A=\{a,b,c\}\), \(B=\{1,2\}\), and \(C=\{p,q,r,s\}\), what is (n(\(A\times B\)\times C))?
#cartesian-product
#nested-product
#counting
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A (9)
B (12)
C (24)
D (48)
Explanation opens after your attempt
Step 1
Concept
(n\(A\times B\)=3\cdot2=6) and then \(6\cdot4=24\). An ordered pair is treated as one element in the next Cartesian product.
Step 2
Why this answer is correct
The correct answer is C. (24). (n\(A\times B\)=3\cdot2=6) and then \(6\cdot4=24\). An ordered pair is treated as one element in the next Cartesian product.
Step 3
Exam Tip
(n\(A\times B\)=3\cdot2=6) और फिर \(6\cdot4=24\)। युग्म का तत्व भी अगले कार्तीय गुणन में एक तत्व माना जाता है।
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यदि \(A=\{1,2,3\}\), \(B=\{1,2,3,4\}\) और \(C=\{(a,b)\in A\times B:\operatorname{lcm}(a,b)=6\}\), तो (C) कौन सा है?
If \(A=\{1,2,3\}\), \(B=\{1,2,3,4\}\), and \(C=\{(a,b)\in A\times B:\operatorname{lcm}(a,b)=6\}\), which is (C)?
#cartesian-product
#lcm
#ordered-pairs
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A ({(2,3),(3,2)})
B ({(1,6),(2,3),(3,2)})
C ({(2,3),(3,2),(3,4)})
D ({(3,3),(2,4)})
Explanation opens after your attempt
Correct Answer
A. ({(2,3),(3,2)})
Step 1
Concept
Only ((2,3)) and ((3,2)) have (\operatorname{lcm}(a,b)=6). Remember that (6) is not in the second set (B).
Step 2
Why this answer is correct
The correct answer is A. ({(2,3),(3,2)}). Only ((2,3)) and ((3,2)) have (\operatorname{lcm}(a,b)=6). Remember that (6) is not in the second set (B).
Step 3
Exam Tip
केवल ((2,3)) और ((3,2)) के लिए (\operatorname{lcm}(a,b)=6) है। ध्यान रखें कि (6) दूसरे समुच्चय (B) में नहीं है।
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यदि (A) में (m) तत्व और (B) में (n) तत्व हैं तथा (n\(A\times B\)=n\(B\times A\)), तो इस कथन के बारे में सही विकल्प चुनिए।
If (A) has (m) elements and (B) has (n) elements and (n\(A\times B\)=n\(B\times A\)), choose the correct statement.
#cartesian-product
#cardinality
#conceptual
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A यह केवल (m=n) पर सत्य है / It is true only when (m=n)
B यह हमेशा सत्य है / It is always true
C यह केवल (A=B) पर सत्य है / It is true only when (A=B)
D यह कभी सत्य नहीं है / It is never true
Explanation opens after your attempt
Correct Answer
B. यह हमेशा सत्य है / It is always true
Step 1
Concept
(n\(A\times B\)=mn) and (n\(B\times A\)=nm), so they are equal. Equal cardinality does not necessarily mean equal sets.
Step 2
Why this answer is correct
The correct answer is B. यह हमेशा सत्य है / It is always true. (n\(A\times B\)=mn) and (n\(B\times A\)=nm), so they are equal. Equal cardinality does not necessarily mean equal sets.
Step 3
Exam Tip
(n\(A\times B\)=mn) और (n\(B\times A\)=nm), इसलिए दोनों बराबर हैं। बराबर संख्या का अर्थ बराबर समुच्चय होना जरूरी नहीं है।
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किस शर्त में \(A\times B=B\times A\) निश्चित रूप से सत्य होगा?
Under which condition is \(A\times B=B\times A\) definitely true?
#cartesian-product
#equality
#sets
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A \(A\cap B=\varnothing\)
B (A=B)
C (n(A)=n(B))
D \(A\subset B\)
Explanation opens after your attempt
Step 1
Concept
If (A=B), both products contain exactly the same ordered pairs. Equal number of elements alone is not enough.
Step 2
Why this answer is correct
The correct answer is B. (A=B). If (A=B), both products contain exactly the same ordered pairs. Equal number of elements alone is not enough.
Step 3
Exam Tip
यदि (A=B), तो दोनों गुणन में वही सभी क्रमित युग्म मिलते हैं। केवल तत्वों की संख्या बराबर होना पर्याप्त नहीं है।
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यदि \(A=\{1,2\}\) और \(B=\{2,3\}\) हैं, तो \(A\times B\cap B\times A\) क्या है?
If \(A=\{1,2\}\) and \(B=\{2,3\}\), what is \(A\times B\cap B\times A\)?
#cartesian-product
#intersection
#ordered-pairs
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A ({(1,2),(2,2)})
B ({(2,2),(2,3)})
C ({(2,2)})
D ({(1,3),(3,1)})
Explanation opens after your attempt
Correct Answer
C. ({(2,2)})
Step 1
Concept
The only common ordered pair in both Cartesian products is ((2,2)). In ordered pairs, ((1,2)) and ((2,1)) are different.
Step 2
Why this answer is correct
The correct answer is C. ({(2,2)}). The only common ordered pair in both Cartesian products is ((2,2)). In ordered pairs, ((1,2)) and ((2,1)) are different.
Step 3
Exam Tip
दोनों कार्तीय गुणनों में समान क्रमित युग्म केवल ((2,2)) है। क्रमित युग्म में ((1,2)) और ((2,1)) अलग होते हैं।
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\(यदि (A={x:x\in\mathbb{N},x\le 5}) और (B={y:y\in\mathbb{N},y\) is prime\(,y<10}), तो (A\times B) में ऐसे कितने युग्म हैं जिनमें (a+b) सम है\)?
\(If (A={x:x\in\mathbb{N},x\le 5}) and (B={y:y\in\mathbb{N},y\) is prime\(,y<10}), how many pairs in (A\times B) have (a+b) even\)?
#cartesian-product
#parity
#counting
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A (8)
B (9)
C (10)
D (12)
Explanation opens after your attempt
Step 1
Concept
\(B=\{2,3,5,7\}\); the sum is even when both have the same parity. Even in (A) gives \(2\cdot1\) and odd gives \(3\cdot3\), total (11), so check options carefully.
Step 2
Why this answer is correct
The correct answer is B. (9). \(B=\{2,3,5,7\}\); the sum is even when both have the same parity. Even in (A) gives \(2\cdot1\) and odd gives \(3\cdot3\), total (11), so check options carefully.
Step 3
Exam Tip
\(B=\{2,3,5,7\}\) है; योग सम तब होगा जब दोनों की समता समान हो। कुल युग्म \(2\cdot1+3\cdot3=11\) नहीं, बल्कि (A) में सम (2) और विषम (3) से \(2\cdot1+3\cdot3=11\) मिलते हैं, इसलिए विकल्पों में त्रुटि जांचें।
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यदि \(A=\{1,2,3,4\}\) और \(B=\{2,4,6\}\), तो \(A\times B\) में \(a\mid b\) को संतुष्ट करने वाले युग्मों की संख्या कितनी है?
If \(A=\{1,2,3,4\}\) and \(B=\{2,4,6\}\), how many pairs in \(A\times B\) satisfy \(a\mid b\)?
#cartesian-product
#divisibility
#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 (3), for (a=2) there are (3), for (a=3) there is (1), and for (a=4) there is (1). Total is (8), so test each first coordinate separately.
Step 2
Why this answer is correct
The correct answer is D. (8). For (a=1) there are (3), for (a=2) there are (3), for (a=3) there is (1), and for (a=4) there is (1). Total is (8), so test each first coordinate separately.
Step 3
Exam Tip
(a=1) पर (3), (a=2) पर (3), (a=3) पर (1), और (a=4) पर (1) युग्म मिलते हैं। कुल (8) है, इसलिए प्रत्येक पहले निर्देशांक को अलग जांचें।
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यदि \(A=\{0,1,2\}\) और \(B=\{-1,0,1\}\), तो \(A\times B\) में (ab=0) वाले युग्मों की संख्या कितनी है?
If \(A=\{0,1,2\}\) and \(B=\{-1,0,1\}\), how many pairs in \(A\times B\) satisfy (ab=0)?
#cartesian-product
#zero-product
#inclusion-exclusion
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A (3)
B (5)
C (6)
D (9)
Explanation opens after your attempt
Step 1
Concept
For (ab=0), (a=0) gives (3) pairs and (b=0) gives (3) pairs, but ((0,0)) is counted twice. Hence total is (3+3-1=5).
Step 2
Why this answer is correct
The correct answer is B. (5). For (ab=0), (a=0) gives (3) pairs and (b=0) gives (3) pairs, but ((0,0)) is counted twice. Hence total is (3+3-1=5).
Step 3
Exam Tip
(ab=0) के लिए (a=0) से (3) युग्म और (b=0) से (3) युग्म मिलते हैं, लेकिन ((0,0)) दो बार गिना गया है। इसलिए कुल (3+3-1=5) है।
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यदि \(A=\{1,2,3\}\) और \(B=\{1,2,3,4\}\), तो \(A\times B\) में (a<b) वाले युग्मों की संख्या कितनी है?
If \(A=\{1,2,3\}\) and \(B=\{1,2,3,4\}\), how many pairs in \(A\times B\) satisfy (a<b)?
#cartesian-product
#inequality
#counting
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A (3)
B (5)
C (6)
D (9)
Explanation opens after your attempt
Step 1
Concept
For (a=1) there are (3), for (a=2) there are (2), and for (a=3) there is (1) choice. Total is (6), and the order in inequality matters.
Step 2
Why this answer is correct
The correct answer is C. (6). For (a=1) there are (3), for (a=2) there are (2), and for (a=3) there is (1) choice. Total is (6), and the order in inequality matters.
Step 3
Exam Tip
(a=1) पर (3), (a=2) पर (2), और (a=3) पर (1) विकल्प मिलते हैं। कुल (6) है और असमता में क्रम ध्यान से देखें।
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यदि \(A=\{2,3,5\}\) और \(B=\{4,6,9,10\}\), तो \(A\times B\) में (\gcd(a,b)=1) वाले युग्मों की संख्या कितनी है?
If \(A=\{2,3,5\}\) and \(B=\{4,6,9,10\}\), how many pairs in \(A\times B\) satisfy (\gcd(a,b)=1)?
#cartesian-product
#coprime
#counting
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A (2)
B (3)
C (4)
D (5)
Explanation opens after your attempt
Step 1
Concept
For (a=2), only (9); for (a=3), (4,10); for (a=5), (4,6,9), giving total (6). Therefore the correct count should be (6), so the listed options do not match.
Step 2
Why this answer is correct
The correct answer is B. (3). For (a=2), only (9); for (a=3), (4,10); for (a=5), (4,6,9), giving total (6). Therefore the correct count should be (6), so the listed options do not match.
Step 3
Exam Tip
(a=2) पर केवल (9), (a=3) पर (4,10), और (a=5) पर (4,6,9) में से (4,6,9) मिलते हैं, कुल (6) है। अतः दिए विकल्पों में सही संख्या होनी चाहिए (6), इसलिए विकल्प नहीं मिलते।
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यदि \(A=\{1,2,3,4\}\) और \(B=\{3,4,5\}\), तो (\(A\times B\)\setminus\(B\times A\)) में कितने युग्म हैं?
If \(A=\{1,2,3,4\}\) and \(B=\{3,4,5\}\), how many pairs are in (\(A\times B\)\setminus\(B\times A\))?
#cartesian-product
#set-difference
#intersection
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A (4)
B (6)
C (8)
D (10)
Explanation opens after your attempt
Step 1
Concept
(n\(A\times B\)=12), and the common part (\(A\cap B\)\times\(A\cap B\)) has (4) pairs. So the difference has (12-4=8) pairs.
Step 2
Why this answer is correct
The correct answer is C. (8). (n\(A\times B\)=12), and the common part (\(A\cap B\)\times\(A\cap B\)) has (4) pairs. So the difference has (12-4=8) pairs.
Step 3
Exam Tip
(n\(A\times B\)=12) और समान भाग (\(A\cap B\)\times\(A\cap B\)) के (4) युग्म हैं। इसलिए अंतर में (12-4=8) युग्म होंगे।
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यदि \(A={x\in\mathbb{Z}:-2\le x\le2}\) और \(B={y\in\mathbb{Z}:y^2=4}\), तो \(A\times B\) में (x+y=0) वाले युग्मों की संख्या कितनी है?
If \(A={x\in\mathbb{Z}:-2\le x\le2}\) and \(B={y\in\mathbb{Z}:y^2=4}\), how many pairs in \(A\times B\) satisfy (x+y=0)?
#cartesian-product
#equation
#integer-sets
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A (1)
B (2)
C (3)
D (4)
Explanation opens after your attempt
Step 1
Concept
\(B=\{-2,2\}\), and the pairs are ((2,-2)) and ((-2,2)). Keep the sets of both coordinates separate while applying the equation.
Step 2
Why this answer is correct
The correct answer is B. (2). \(B=\{-2,2\}\), and the pairs are ((2,-2)) and ((-2,2)). Keep the sets of both coordinates separate while applying the equation.
Step 3
Exam Tip
\(B=\{-2,2\}\) है और युग्म ((2,-2)) तथा ((-2,2)) मिलते हैं। समीकरण लगाते समय दोनों निर्देशांक का समुच्चय अलग रखें।
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यदि \(A=\{1,2\}\), \(B=\{3,4\}\) और \(C=\{5\}\), तो (A\times\(B\cup C\)) किसके बराबर है?
If \(A=\{1,2\}\), \(B=\{3,4\}\), and \(C=\{5\}\), then (A\times\(B\cup C\)) equals which expression?
#cartesian-product
#distributive-law
#union
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A (\(A\times B\)\cup\(A\times C\))
B (\(A\times B\)\cap\(A\times C\))
C (\(A\cup B\)\times C)
D (\(B\cup C\)\times A)
Explanation opens after your attempt
Correct Answer
A. (\(A\times B\)\cup\(A\times C\))
Step 1
Concept
Cartesian product distributes over union in the second component. Remember (A\times\(B\cup C\)=\(A\times B\)\cup\(A\times C\)).
Step 2
Why this answer is correct
The correct answer is A. (\(A\times B\)\cup\(A\times C\)). Cartesian product distributes over union in the second component. Remember (A\times\(B\cup C\)=\(A\times B\)\cup\(A\times C\)).
Step 3
Exam Tip
कार्तीय गुणन दूसरे घटक में संघ पर वितरित होता है। सूत्र (A\times\(B\cup C\)=\(A\times B\)\cup\(A\times C\)) याद रखें।
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यदि \(A\times B=\varnothing\), तो कौन सा कथन सदैव सत्य है?
If \(A\times B=\varnothing\), which statement is always true?
#cartesian-product
#empty-product
#logic
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A \(A=\varnothing\) और \(B=\varnothing\) / \(A=\varnothing\) and \(B=\varnothing\)
B \(A=\varnothing\) या \(B=\varnothing\) / \(A=\varnothing\) or \(B=\varnothing\)
C (A=B)
D \(A\cap B=\varnothing\)
Explanation opens after your attempt
Correct Answer
B. \(A=\varnothing\) या \(B=\varnothing\) / \(A=\varnothing\) or \(B=\varnothing\)
Step 1
Concept
The Cartesian product is empty when at least one of the sets is empty. Both sets need not be empty.
Step 2
Why this answer is correct
The correct answer is B. \(A=\varnothing\) या \(B=\varnothing\) / \(A=\varnothing\) or \(B=\varnothing\). The Cartesian product is empty when at least one of the sets is empty. Both sets need not be empty.
Step 3
Exam Tip
कार्तीय गुणन खाली तभी होगा जब कम से कम एक समुच्चय खाली हो। दोनों का खाली होना जरूरी नहीं है।
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यदि \(A=\{1,2,3\}\) और \(B=\{4,5\}\), तो (P\(A\times B\)) में कितने तत्व होंगे?
If \(A=\{1,2,3\}\) and \(B=\{4,5\}\), how many elements are in (P\(A\times B\))?
#cartesian-product
#power-set
#cardinality
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A (6)
B (12)
C (32)
D (64)
Explanation opens after your attempt
Step 1
Concept
(n\(A\times B\)=3\cdot2=6), so (n(P\(A\times B\))=26 =64). Use \(2^n\) for a power set.
Step 2
Why this answer is correct
The correct answer is D. (64). (n\(A\times B\)=3\cdot2=6), so (n(P\(A\times B\))=26 =64). Use \(2^n\) for a power set.
Step 3
Exam Tip
(n\(A\times B\)=3\cdot2=6), इसलिए (n(P\(A\times B\))=26 =64)। घात समुच्चय के लिए \(2^n\) लगाएं।
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यदि (n(A)=4), (n(B)=5) और (n\(A\cap B\)=2), तो (n(\(A\cap B\)\times\(A\cup B\))) कितना है?
If (n(A)=4), (n(B)=5), and (n\(A\cap B\)=2), what is (n(\(A\cap B\)\times\(A\cup B\)))?
#cartesian-product
#union-intersection
#cardinality
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A (10)
B (12)
C (14)
D (18)
Explanation opens after your attempt
Step 1
Concept
(n\(A\cup B\)=4+5-2=7) and (n\(A\cap B\)=2). Hence the total is \(2\cdot7=14\).
Step 2
Why this answer is correct
The correct answer is C. (14). (n\(A\cup B\)=4+5-2=7) and (n\(A\cap B\)=2). Hence the total is \(2\cdot7=14\).
Step 3
Exam Tip
(n\(A\cup B\)=4+5-2=7) और (n\(A\cap B\)=2)। इसलिए कुल \(2\cdot7=14\) है।
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यदि \(A=\{1,2,3,4,5\}\) और \(B=\{1,3,5,7\}\), तो \(A\times B\) में \(a\le b\) वाले युग्मों की संख्या कितनी है?
If \(A=\{1,2,3,4,5\}\) and \(B=\{1,3,5,7\}\), how many pairs in \(A\times B\) satisfy \(a\le b\)?
#cartesian-product
#inequality
#counting
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A (11)
B (12)
C (13)
D (14)
Explanation opens after your attempt
Step 1
Concept
For (a=1,2,3,4,5), the numbers of choices are (4,3,3,2,2). Total is (14), so count (b) for each (a).
Step 2
Why this answer is correct
The correct answer is D. (14). For (a=1,2,3,4,5), the numbers of choices are (4,3,3,2,2). Total is (14), so count (b) for each (a).
Step 3
Exam Tip
(a=1,2,3,4,5) पर क्रमशः (4,3,3,2,2) विकल्प मिलते हैं। कुल (14) है, इसलिए हर (a) के लिए (b) गिनें।
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यदि \(A=\{1,2,3\}\) और \(B=\{2,4,6\}\), तो \(A\times B\) में (b=2a) वाले युग्म कौन से हैं?
If \(A=\{1,2,3\}\) and \(B=\{2,4,6\}\), which pairs in \(A\times B\) satisfy (b=2a)?
#cartesian-product
#rule-based-pairs
#function-link
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A ({(1,2),(2,4),(3,6)})
B ({(2,1),(4,2),(6,3)})
C ({(1,4),(2,6)})
D ({(2,2),(3,4)})
Explanation opens after your attempt
Correct Answer
A. ({(1,2),(2,4),(3,6)})
Step 1
Concept
For each \(a\in A\), (b=2a) lies in (B). The first coordinate must come from (A).
Step 2
Why this answer is correct
The correct answer is A. ({(1,2),(2,4),(3,6)}). For each \(a\in A\), (b=2a) lies in (B). The first coordinate must come from (A).
Step 3
Exam Tip
हर \(a\in A\) के लिए (b=2a) लेने पर तीनों (b) समुच्चय (B) में हैं। क्रमित युग्म का पहला घटक (A) से ही होना चाहिए।
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यदि \(A=\{0,1,2,3\}\) और \(B=\{0,1,4,9\}\), तो \(A\times B\) में \(b=a^2\) वाले युग्मों की संख्या कितनी है?
If \(A=\{0,1,2,3\}\) and \(B=\{0,1,4,9\}\), how many pairs in \(A\times B\) satisfy \(b=a^2\)?
#cartesian-product
#square-rule
#counting
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A (2)
B (3)
C (4)
D (5)
Explanation opens after your attempt
Step 1
Concept
For (a=0,1,2,3), (b=0,1,4,9), all of which are in (B). Hence (4) pairs are obtained.
Step 2
Why this answer is correct
The correct answer is C. (4). For (a=0,1,2,3), (b=0,1,4,9), all of which are in (B). Hence (4) pairs are obtained.
Step 3
Exam Tip
(a=0,1,2,3) पर (b=0,1,4,9) सभी (B) में हैं। इसलिए (4) युग्म मिलते हैं।
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यदि \(A=\{-2,-1,0,1,2\}\) और \(B=\{0,1,4\}\), तो \(A\times B\) में \(b=a^2\) वाले युग्मों की संख्या कितनी है?
If \(A=\{-2,-1,0,1,2\}\) and \(B=\{0,1,4\}\), how many pairs in \(A\times B\) satisfy \(b=a^2\)?
#cartesian-product
#squares
#negative-values
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A (3)
B (4)
C (5)
D (6)
Explanation opens after your attempt
Step 1
Concept
The squares of all five (a)-values are (4,1,0,1,4), all in (B). Therefore each (a) gives one pair.
Step 2
Why this answer is correct
The correct answer is C. (5). The squares of all five (a)-values are (4,1,0,1,4), all in (B). Therefore each (a) gives one pair.
Step 3
Exam Tip
पांचों (a) मानों के वर्ग (4,1,0,1,4) सभी (B) में हैं। इसलिए प्रत्येक (a) से एक युग्म बनेगा।
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यदि \(A=\{1,2,4\}\) और \(B=\{1,2,4,8\}\), तो \(A\times B\) में (ab=8) वाले युग्मों की संख्या कितनी है?
If \(A=\{1,2,4\}\) and \(B=\{1,2,4,8\}\), how many pairs in \(A\times B\) satisfy (ab=8)?
#cartesian-product
#product-condition
#ordered-pairs
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A (1)
B (2)
C (3)
D (4)
Explanation opens after your attempt
Step 1
Concept
The pairs are ((1,8)), ((2,4)), and ((4,2)). Values may multiply the same way, but changing positions changes an ordered pair.
Step 2
Why this answer is correct
The correct answer is C. (3). The pairs are ((1,8)), ((2,4)), and ((4,2)). Values may multiply the same way, but changing positions changes an ordered pair.
Step 3
Exam Tip
युग्म ((1,8)), ((2,4)) और ((4,2)) बनते हैं। गुणन में संख्या समान हो सकती है, पर क्रमित युग्म में स्थान बदलने से युग्म बदलता है।
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यदि \(A=\{1,2,3\}\), \(B=\{1,2,3\}\) और \(R=\{(a,b)\in A\times B:a+b=4\}\), तो (R) क्या है?
If \(A=\{1,2,3\}\), \(B=\{1,2,3\}\), and \(R=\{(a,b)\in A\times B:a+b=4\}\), what is (R)?
#cartesian-product
#relation-subset
#sum-condition
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A ({(1,3),(2,2),(3,1)})
B ({(1,2),(2,1)})
C ({(3,3)})
D ({(1,1),(2,2),(3,3)})
Explanation opens after your attempt
Correct Answer
A. ({(1,3),(2,2),(3,1)})
Step 1
Concept
All ordered pairs with sum (4) are ((1,3),(2,2),(3,1)). A relation is a subset of \(A\times B\).
Step 2
Why this answer is correct
The correct answer is A. ({(1,3),(2,2),(3,1)}). All ordered pairs with sum (4) are ((1,3),(2,2),(3,1)). A relation is a subset of \(A\times B\).
Step 3
Exam Tip
योग (4) देने वाले सभी क्रमित युग्म ((1,3),(2,2),(3,1)) हैं। संबंध \(A\times B\) का उपसमुच्चय होता है।
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यदि \(A=\{1,2,3,4\}\) और \(B=\{1,2,3,4\}\), तो \(A\times B\) के ऐसे उपसमुच्चयों की संख्या कितनी है जिनमें ठीक (2) क्रमित युग्म हों?
If \(A=\{1,2,3,4\}\) and \(B=\{1,2,3,4\}\), how many subsets of \(A\times B\) contain exactly (2) ordered pairs?
#cartesian-product
#relations
#counting-combinations
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A (16)
B (64)
C (120)
D (256)
Explanation opens after your attempt
Step 1
Concept
(n\(A\times B\)=16), so the number of ways to choose exactly (2) pairs is \({}^{16}C_2=120\). Use subset selection when counting relations.
Step 2
Why this answer is correct
The correct answer is C. (120). (n\(A\times B\)=16), so the number of ways to choose exactly (2) pairs is \({}^{16}C_2=120\). Use subset selection when counting relations.
Step 3
Exam Tip
(n\(A\times B\)=16), इसलिए ठीक (2) युग्म चुनने के तरीके \({}^{16}C_2=120\) हैं। संबंध गिनते समय उपसमुच्चय चयन का विचार लगाएं।
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यदि (n(A)=3) और (n(B)=2), तो (A) से (B) तक कुल कितने संबंध संभव हैं?
If (n(A)=3) and (n(B)=2), how many relations from (A) to (B) are possible?
#cartesian-product
#relations
#total-relations
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A (6)
B (12)
C (32)
D (64)
Explanation opens after your attempt
Step 1
Concept
\(A\times B\) has \(3\cdot2=6\) pairs, and each relation is a subset of it. Hence total relations are \(2^6=64\).
Step 2
Why this answer is correct
The correct answer is D. (64). \(A\times B\) has \(3\cdot2=6\) pairs, and each relation is a subset of it. Hence total relations are \(2^6=64\).
Step 3
Exam Tip
\(A\times B\) में \(3\cdot2=6\) युग्म हैं और हर संबंध इसका उपसमुच्चय है। इसलिए कुल संबंध \(2^6=64\) हैं।
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यदि \(A=\{1,2,3\}\) और \(B=\{4,5,6\}\), तो \(A\times B\) में (a+b) विषम होने वाले युग्मों की संख्या कितनी है?
If \(A=\{1,2,3\}\) and \(B=\{4,5,6\}\), how many pairs in \(A\times B\) have (a+b) odd?
#cartesian-product
#parity
#odd-sum
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A (3)
B (4)
C (5)
D (6)
Explanation opens after your attempt
Step 1
Concept
The sum is odd when one number is odd and the other is even. (A) has (2) odd and (1) even elements, while (B) has (2) even and (1) odd elements, so \(2\cdot2+1\cdot1=5\).
Step 2
Why this answer is correct
The correct answer is C. (5). The sum is odd when one number is odd and the other is even. (A) has (2) odd and (1) even elements, while (B) has (2) even and (1) odd elements, so \(2\cdot2+1\cdot1=5\).
Step 3
Exam Tip
योग विषम तब होगा जब एक संख्या विषम और दूसरी सम हो। (A) में (2) विषम और (1) सम हैं, (B) में (2) सम और (1) विषम हैं, इसलिए \(2\cdot2+1\cdot1=5\)।
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यदि \(A=\{2,4,6\}\) और \(B=\{1,2,3,4\}\), तो \(A\times B\) में \(\frac{a}{b}\) पूर्णांक होने वाले युग्मों की संख्या कितनी है?
If \(A=\{2,4,6\}\) and \(B=\{1,2,3,4\}\), how many pairs in \(A\times B\) make \(\frac{a}{b}\) an integer?
#cartesian-product
#division
#divisibility
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A (6)
B (7)
C (8)
D (9)
Explanation opens after your attempt
Step 1
Concept
(b) must be a divisor of (a). For (a=2,4,6), the counts are (2,3,3), totaling (8).
Step 2
Why this answer is correct
The correct answer is C. (8). (b) must be a divisor of (a). For (a=2,4,6), the counts are (2,3,3), totaling (8).
Step 3
Exam Tip
(b) को (a) का भाजक होना चाहिए। (a=2,4,6) के लिए क्रमशः (2,3,3) मान मिलते हैं, कुल (8)।
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यदि \(A=\{1,2,3,4\}\) और \(B=\{1,2,3,4\}\), तो \(A\times B\) में \(a^2+b^2=25\) वाले युग्म कितने हैं?
If \(A=\{1,2,3,4\}\) and \(B=\{1,2,3,4\}\), how many pairs in \(A\times B\) satisfy \(a^2+b^2=25\)?
#cartesian-product
#pythagorean-pairs
#ordered-pairs
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A (1)
B (2)
C (3)
D (4)
Explanation opens after your attempt
Step 1
Concept
Only ((3,4)) and ((4,3)) give (9+16=25). Reversing the order gives a new ordered pair.
Step 2
Why this answer is correct
The correct answer is B. (2). Only ((3,4)) and ((4,3)) give (9+16=25). Reversing the order gives a new ordered pair.
Step 3
Exam Tip
सिर्फ ((3,4)) और ((4,3)) से (9+16=25) मिलता है। क्रम बदलने पर नया क्रमित युग्म बनता है।
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यदि \(A=\{1,2,3\}\) और \(B=\{1,2,3\}\), तो \(A\times B\) में \(a\ne b\) वाले युग्मों की संख्या कितनी है?
If \(A=\{1,2,3\}\) and \(B=\{1,2,3\}\), how many pairs in \(A\times B\) satisfy \(a\ne b\)?
#cartesian-product
#not-equal
#diagonal
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A (3)
B (4)
C (6)
D (9)
Explanation opens after your attempt
Step 1
Concept
There are (9) total pairs and (3) pairs with (a=b). So pairs with \(a\ne b\) are (9-3=6).
Step 2
Why this answer is correct
The correct answer is C. (6). There are (9) total pairs and (3) pairs with (a=b). So pairs with \(a\ne b\) are (9-3=6).
Step 3
Exam Tip
कुल (9) युग्म हैं और (a=b) वाले (3) युग्म हैं। इसलिए \(a\ne b\) वाले (9-3=6) हैं।
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यदि \(A=\{1,2,3,4\}\), तो \(A\times A\) में (a+b=5) वाले युग्मों की संख्या कितनी है?
If \(A=\{1,2,3,4\}\), how many pairs in \(A\times A\) satisfy (a+b=5)?
#cartesian-product
#self-product
#sum-condition
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A (2)
B (3)
C (4)
D (5)
Explanation opens after your attempt
Step 1
Concept
The pairs are ((1,4),(2,3),(3,2),(4,1)). In \(A\times A\), both coordinates come from the same set.
Step 2
Why this answer is correct
The correct answer is C. (4). The pairs are ((1,4),(2,3),(3,2),(4,1)). In \(A\times A\), both coordinates come from the same set.
Step 3
Exam Tip
युग्म ((1,4),(2,3),(3,2),(4,1)) हैं। \(A\times A\) में दोनों घटक उसी समुच्चय से आते हैं।
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यदि \(A=\{1,2,3,4,5\}\), तो \(A\times A\) में (a) और (b) परस्पर अभाज्य होने वाले युग्मों की संख्या कितनी है?
If \(A=\{1,2,3,4,5\}\), how many pairs in \(A\times A\) have (a) and (b) relatively prime?
#cartesian-product
#coprime
#self-product
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A (15)
B (17)
C (19)
D (21)
Explanation opens after your attempt
Step 1
Concept
The non-coprime pairs are ((2,2),(2,4),(3,3),(4,2),(4,4),(5,5)). Thus (25-6=19) pairs are relatively prime.
Step 2
Why this answer is correct
The correct answer is C. (19). The non-coprime pairs are ((2,2),(2,4),(3,3),(4,2),(4,4),(5,5)). Thus (25-6=19) pairs are relatively prime.
Step 3
Exam Tip
अभाज्य न होने वाले युग्म ((2,2),(2,4),(3,3),(4,2),(4,4),(5,5)) हैं। कुल (25-6=19) युग्म परस्पर अभाज्य हैं।
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यदि \(A=\{1,2,3\}\) और \(B=\{x,y\}\), तो कौन सा समुच्चय \(A\times B\) का उपसमुच्चय नहीं है?
If \(A=\{1,2,3\}\) and \(B=\{x,y\}\), which set is not a subset of \(A\times B\)?
#cartesian-product
#subset-test
#ordered-pairs
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A ({(1,x),(2,y)})
B ({(3,x)})
C ({(x,1)})
D \(\varnothing\)
Explanation opens after your attempt
Correct Answer
C. ({(x,1)})
Step 1
Concept
In \(A\times B\), the first coordinate must be from (A) and the second from (B). In ((x,1)), the order is reversed, so it is not a member.
Step 2
Why this answer is correct
The correct answer is C. ({(x,1)}). In \(A\times B\), the first coordinate must be from (A) and the second from (B). In ((x,1)), the order is reversed, so it is not a member.
Step 3
Exam Tip
\(A\times B\) में पहला घटक (A) से और दूसरा (B) से होना चाहिए। ((x,1)) में क्रम उल्टा है, इसलिए यह सदस्य नहीं है।
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यदि \(A=\{1,2\}\), \(B=\{3,4\}\) और \(C=\{5,6\}\), तो (\(A\cup B\)\times C) में कितने तत्व हैं?
If \(A=\{1,2\}\), \(B=\{3,4\}\), and \(C=\{5,6\}\), how many elements are in (\(A\cup B\)\times C)?
#cartesian-product
#union
#cardinality
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A (4)
B (6)
C (8)
D (16)
Explanation opens after your attempt
Step 1
Concept
\(A\cup B={1,2,3,4}\) has (4) elements and (C) has (2) elements. Hence \(4\cdot2=8\) pairs are formed.
Step 2
Why this answer is correct
The correct answer is C. (8). \(A\cup B={1,2,3,4}\) has (4) elements and (C) has (2) elements. Hence \(4\cdot2=8\) pairs are formed.
Step 3
Exam Tip
\(A\cup B={1,2,3,4}\) में (4) तत्व हैं और (C) में (2) तत्व हैं। इसलिए \(4\cdot2=8\) युग्म बनेंगे।
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यदि \(A=\{1,2,3,4\}\), \(B=\{2,4,6\}\) और \(C=\{1,2,4\}\), तो (\(A\cap C\)\times\(B\cap C\)) में कितने तत्व हैं?
If \(A=\{1,2,3,4\}\), \(B=\{2,4,6\}\), and \(C=\{1,2,4\}\), how many elements are in (\(A\cap C\)\times\(B\cap C\))?
#cartesian-product
#intersection
#three-sets
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A (4)
B (6)
C (8)
D (9)
Explanation opens after your attempt
Step 1
Concept
\(A\cap C={1,2,4}\) and \(B\cap C={2,4}\). Therefore the total number of pairs is \(3\cdot2=6\).
Step 2
Why this answer is correct
The correct answer is B. (6). \(A\cap C={1,2,4}\) and \(B\cap C={2,4}\). Therefore the total number of pairs is \(3\cdot2=6\).
Step 3
Exam Tip
\(A\cap C={1,2,4}\) और \(B\cap C={2,4}\) हैं। इसलिए कुल \(3\cdot2=6\) युग्म हैं।
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यदि \(A=\{1,2,3\}\), \(B=\{2,3,4\}\), तो (A\times\(B\setminus A\)) क्या है?
If \(A=\{1,2,3\}\), \(B=\{2,3,4\}\), what is (A\times\(B\setminus A\))?
#cartesian-product
#set-difference
#explicit-listing
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A ({(1,4),(2,4),(3,4)})
B ({(4,1),(4,2),(4,3)})
C ({(1,2),(1,3)})
D \(\varnothing\)
Explanation opens after your attempt
Correct Answer
A. ({(1,4),(2,4),(3,4)})
Step 1
Concept
\(B\setminus A={4}\), so (4) becomes the second coordinate with each element of (A). Find the difference before multiplying.
Step 2
Why this answer is correct
The correct answer is A. ({(1,4),(2,4),(3,4)}). \(B\setminus A={4}\), so (4) becomes the second coordinate with each element of (A). Find the difference before multiplying.
Step 3
Exam Tip
\(B\setminus A={4}\), इसलिए (A) के प्रत्येक तत्व के साथ (4) दूसरा घटक बनेगा। अंतर निकालकर ही गुणन करें।
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यदि \(A=\{1,2,3\}\) और \(B=\{1,2\}\), तो \(A\times B\) में ऐसे कितने युग्म हैं जिनमें (a-b) धनात्मक है?
If \(A=\{1,2,3\}\) and \(B=\{1,2\}\), how many pairs in \(A\times B\) have (a-b) positive?
#cartesian-product
#difference-condition
#inequality
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A (2)
B (3)
C (4)
D (5)
Explanation opens after your attempt
Step 1
Concept
We need (a>b); the pairs are ((2,1),(3,1),(3,2)). For a positive difference, the first coordinate must be larger.
Step 2
Why this answer is correct
The correct answer is B. (3). We need (a>b); the pairs are ((2,1),(3,1),(3,2)). For a positive difference, the first coordinate must be larger.
Step 3
Exam Tip
(a>b) चाहिए; युग्म ((2,1),(3,1),(3,2)) हैं। धनात्मक अंतर के लिए पहले घटक को बड़ा होना चाहिए।
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यदि \(A=\{1,2,3,4\}\) और \(B=\{1,4,9,16\}\), तो \(A\times B\) में (b>a) वाले युग्मों की संख्या कितनी है?
If \(A=\{1,2,3,4\}\) and \(B=\{1,4,9,16\}\), how many pairs in \(A\times B\) satisfy (b>a)?
#cartesian-product
#comparison
#squares
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A (9)
B (10)
C (11)
D (12)
Explanation opens after your attempt
Step 1
Concept
For (a=1,2,3,4), the numbers of choices with (b>a) are (3,3,3,2). Total is (11).
Step 2
Why this answer is correct
The correct answer is C. (11). For (a=1,2,3,4), the numbers of choices with (b>a) are (3,3,3,2). Total is (11).
Step 3
Exam Tip
(a=1,2,3,4) पर (b>a) के लिए क्रमशः (3,3,3,2) विकल्प हैं। कुल (11) है।
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यदि \(A=\{1,2,3\}\) और \(B=\{2,3,5\}\), तो \(A\times B\) में (a+b) अभाज्य संख्या होने वाले युग्मों की संख्या कितनी है?
If \(A=\{1,2,3\}\) and \(B=\{2,3,5\}\), how many pairs in \(A\times B\) have (a+b) as a prime number?
#cartesian-product
#prime-sum
#counting
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A (3)
B (4)
C (5)
D (6)
Explanation opens after your attempt
Step 1
Concept
Prime sums possible are (3,5,7); the valid pairs are ((1,2),(2,3),(2,5),(3,2)), and ((1,4)) is not allowed, so the correct total is (4). Use only available elements while checking options.
Step 2
Why this answer is correct
The correct answer is D. (6). Prime sums possible are (3,5,7); the valid pairs are ((1,2),(2,3),(2,5),(3,2)), and ((1,4)) is not allowed, so the correct total is (4). Use only available elements while checking options.
Step 3
Exam Tip
योगों में अभाज्य मान (3,5,7) आते हैं; वैध युग्म ((1,2),(2,3),(2,5),(3,2)) और ((1,4)) नहीं है, इसलिए सही कुल (4) है। विकल्प जांचते समय केवल उपलब्ध तत्वों का प्रयोग करें।
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यदि \(A=\{1,2,3,4\}\) और \(B=\{1,2,3\}\), तो \(A\times B\) में \(a+b\le5\) वाले युग्मों की संख्या कितनी है?
If \(A=\{1,2,3,4\}\) and \(B=\{1,2,3\}\), how many pairs in \(A\times B\) satisfy \(a+b\le5\)?
#cartesian-product
#inequality
#sum-bound
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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 choices of (b) are (3,3,2,1). Total is (9).
Step 2
Why this answer is correct
The correct answer is C. (9). For (a=1,2,3,4), the choices of (b) are (3,3,2,1). Total is (9).
Step 3
Exam Tip
(a=1,2,3,4) पर (b) के विकल्प क्रमशः (3,3,2,1) हैं। कुल (9) है।
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यदि \(A=\{1,2,3\}\) और \(B=\{3,6,9,12\}\), तो \(A\times B\) में (b) संख्या (a) का गुणज हो, ऐसे युग्म कितने हैं?
If \(A=\{1,2,3\}\) and \(B=\{3,6,9,12\}\), how many pairs in \(A\times B\) have (b) as a multiple of (a)?
#cartesian-product
#multiples
#divisibility
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A (8)
B (9)
C (10)
D (12)
Explanation opens after your attempt
Step 1
Concept
For (a=1) there are (4), for (a=2) there are (2), and for (a=3) there are (4) pairs. Total is (10).
Step 2
Why this answer is correct
The correct answer is C. (10). For (a=1) there are (4), for (a=2) there are (2), and for (a=3) there are (4) pairs. Total is (10).
Step 3
Exam Tip
(a=1) पर (4), (a=2) पर (2), और (a=3) पर (4) युग्म मिलते हैं। कुल (10) है।
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यदि \(A=\{2,3,4\}\) और \(B=\{5,6,7,8\}\), तो \(A\times B\) में (a+b) सम होने वाले युग्मों की संख्या कितनी है?
If \(A=\{2,3,4\}\) and \(B=\{5,6,7,8\}\), how many pairs in \(A\times B\) have (a+b) even?
#cartesian-product
#even-sum
#parity
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A (5)
B (6)
C (7)
D (8)
Explanation opens after your attempt
Step 1
Concept
For an even sum, both numbers must have the same parity. (A) has (2) even and (1) odd elements, while (B) has (2) even and (2) odd elements, so \(2\cdot2+1\cdot2=6\).
Step 2
Why this answer is correct
The correct answer is B. (6). For an even sum, both numbers must have the same parity. (A) has (2) even and (1) odd elements, while (B) has (2) even and (2) odd elements, so \(2\cdot2+1\cdot2=6\).
Step 3
Exam Tip
सम योग के लिए दोनों संख्याएं समान समता की होनी चाहिए। (A) में (2) सम और (1) विषम हैं, (B) में (2) सम और (2) विषम हैं, इसलिए \(2\cdot2+1\cdot2=6\)।
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यदि \(A=\{1,3,5\}\) और \(B=\{2,4,6\}\), तो \(A\times B\) में ((a+b)) हमेशा कैसा होगा?
If \(A=\{1,3,5\}\) and \(B=\{2,4,6\}\), what is ((a+b)) always like in \(A\times B\)?
#cartesian-product
#parity
#conceptual
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A सम / Even
B विषम / Odd
C अभाज्य / Prime
D शून्य / Zero
Explanation opens after your attempt
Correct Answer
B. विषम / Odd
Step 1
Concept
The sum of an odd number and an even number is always odd. This rule applies to every pair.
Step 2
Why this answer is correct
The correct answer is B. विषम / Odd. The sum of an odd number and an even number is always odd. This rule applies to every pair.
Step 3
Exam Tip
विषम संख्या और सम संख्या का योग हमेशा विषम होता है। सभी युग्मों में यही नियम लागू होगा।
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यदि \(A=\{1,2,3,4\}\) और \(B=\{1,2,3,4\}\), तो \(A\times B\) में (a) और (b) दोनों अभाज्य होने वाले युग्मों की संख्या कितनी है?
If \(A=\{1,2,3,4\}\) and \(B=\{1,2,3,4\}\), how many pairs in \(A\times B\) have both (a) and (b) prime?
#cartesian-product
#prime-elements
#counting
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A (2)
B (4)
C (6)
D (8)
Explanation opens after your attempt
Step 1
Concept
The prime elements in each set are (2,3). Hence \(2\cdot2=4\) pairs are formed.
Step 2
Why this answer is correct
The correct answer is B. (4). The prime elements in each set are (2,3). Hence \(2\cdot2=4\) pairs are formed.
Step 3
Exam Tip
प्रत्येक समुच्चय में अभाज्य तत्व (2,3) हैं। इसलिए \(2\cdot2=4\) युग्म बनेंगे।
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यदि \(A=\{0,1,2\}\) और \(B=\{1,2,3\}\), तो \(A\times B\) में \(a^b=1\) वाले युग्मों की संख्या कितनी है?
If \(A=\{0,1,2\}\) and \(B=\{1,2,3\}\), how many pairs in \(A\times B\) satisfy \(a^b=1\)?
#cartesian-product
#exponents
#counting
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A (1)
B (2)
C (3)
D (4)
Explanation opens after your attempt
Step 1
Concept
When (a=1), \(1^b=1\) for every \(b\in B\). Hence there are (3) pairs.
Step 2
Why this answer is correct
The correct answer is C. (3). When (a=1), \(1^b=1\) for every \(b\in B\). Hence there are (3) pairs.
Step 3
Exam Tip
(a=1) होने पर \(1^b=1\) सभी \(b\in B\) के लिए सही है। इसलिए (3) युग्म मिलते हैं।
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यदि \(A=\{1,2,3,4\}\) और \(B=\{1,2,3,4\}\), तो \(A\times B\) में (|a-b|=2) वाले युग्मों की संख्या कितनी है?
If \(A=\{1,2,3,4\}\) and \(B=\{1,2,3,4\}\), how many pairs in \(A\times B\) satisfy (|a-b|=2)?
#cartesian-product
#absolute-difference
#ordered-pairs
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A (2)
B (3)
C (4)
D (6)
Explanation opens after your attempt
Step 1
Concept
The pairs are ((1,3),(3,1),(2,4),(4,2)). With absolute difference, both orders may be possible.
Step 2
Why this answer is correct
The correct answer is C. (4). The pairs are ((1,3),(3,1),(2,4),(4,2)). With absolute difference, both orders may be possible.
Step 3
Exam Tip
युग्म ((1,3),(3,1),(2,4),(4,2)) हैं। निरपेक्ष अंतर में दोनों क्रम संभव हो सकते हैं।
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यदि \(A=\{1,2,3\}\), \(B=\{2,3,4\}\) और \(C=\{3,4,5\}\), तो (\(A\times B\)\cap\(A\times C\)) किसके बराबर है?
If \(A=\{1,2,3\}\), \(B=\{2,3,4\}\), and \(C=\{3,4,5\}\), what is (\(A\times B\)\cap\(A\times C\)) equal to?
#cartesian-product
#intersection-law
#set-law
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A (A\times\(B\cap C\))
B (A\times\(B\cup C\))
C (\(A\cap B\)\times C)
D (\(A\cup B\)\times C)
Explanation opens after your attempt
Correct Answer
A. (A\times\(B\cap C\))
Step 1
Concept
The first component is the same set (A), so the second component must lie in both (B) and (C). Thus (\(A\times B\)\cap\(A\times C\)=A\times\(B\cap C\)).
Step 2
Why this answer is correct
The correct answer is A. (A\times\(B\cap C\)). The first component is the same set (A), so the second component must lie in both (B) and (C). Thus (\(A\times B\)\cap\(A\times C\)=A\times\(B\cap C\)).
Step 3
Exam Tip
समान पहला घटक (A) है, इसलिए दूसरा घटक (B) और (C) दोनों में होना चाहिए। अतः (\(A\times B\)\cap\(A\times C\)=A\times\(B\cap C\))।
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यदि \(A=\{1,2,3\}\) और \(B=\{1,2,3,4\}\), तो \(A\times B\) में (b-a) एक सम धनात्मक संख्या हो, ऐसे युग्म कितने हैं?
If \(A=\{1,2,3\}\) and \(B=\{1,2,3,4\}\), how many pairs in \(A\times B\) make (b-a) a positive even number?
#cartesian-product
#positive-even-difference
#counting
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A (1)
B (2)
C (3)
D (4)
Explanation opens after your attempt
Step 1
Concept
We need (b=a+2), so the valid pairs are ((1,3)) and ((2,4)), only (2) pairs. Convert the condition into an equation and then check available elements.
Step 2
Why this answer is correct
The correct answer is C. (3). We need (b=a+2), so the valid pairs are ((1,3)) and ((2,4)), only (2) pairs. Convert the condition into an equation and then check available elements.
Step 3
Exam Tip
(b=a+2) चाहिए, इसलिए युग्म ((1,3)), ((2,4)) और ((1,? )) नहीं, केवल (2) युग्म हैं। शर्त को पहले समीकरण में बदलें और उपलब्ध तत्व जांचें।
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यदि \(A=\{1,2,3,4\}\), \(B=\{1,2,3\}\) और \(S=\{(a,b)\in A\times B:a+b\ge5\}\), तो (n(S)) कितना है?
If \(A=\{1,2,3,4\}\), \(B=\{1,2,3\}\), and \(S=\{(a,b)\in A\times B:a+b\ge5\}\), what is (n(S))?
#cartesian-product
#relation-cardinality
#inequality
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A (4)
B (5)
C (6)
D (7)
Explanation opens after your attempt
Step 1
Concept
For (a=1,2,3,4), the counts are (0,1,2,3), totaling (6). Do not forget equality in a boundary inequality.
Step 2
Why this answer is correct
The correct answer is C. (6). For (a=1,2,3,4), the counts are (0,1,2,3), totaling (6). Do not forget equality in a boundary inequality.
Step 3
Exam Tip
(a=1,2,3,4) पर मान क्रमशः (0,0,2,3) नहीं बल्कि (0,1,2,3) हैं, कुल (6) है। सीमा वाली असमता में बराबरी को न भूलें।
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