यदि \(A=\{1,2\}\) और \(B=\{3,4,5\}\) हैं, तो \(A\times B\) में कितने क्रमित युग्म होंगे?
If \(A=\{1,2\}\) and \(B=\{3,4,5\}\), how many ordered pairs are there in \(A\times B\)?
#cartesian-product
#cardinality
#ordered-pairs
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A (6)
B (5)
C (3)
D (2)
Explanation opens after your attempt
Step 1
Concept
(n\(A\times B\)=n(A)n(B)=2\times3=6). In exams, first count elements of both sets.
Step 2
Why this answer is correct
The correct answer is A. (6). (n\(A\times B\)=n(A)n(B)=2\times3=6). In exams, first count elements of both sets.
Step 3
Exam Tip
(n\(A\times B\)=n(A)n(B)=2\times3=6) होता है। परीक्षा में पहले दोनों समुच्चयों के अवयव गिनें।
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यदि \(A=\{a,b\}\) और \(B=\{1,2\}\) हैं, तो \(A\times B\) कौन सा है?
If \(A=\{a,b\}\) and \(B=\{1,2\}\), which one is \(A\times B\)?
#cartesian-product
#listing
#ordered-pairs
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A ({(a,1),(a,2),(b,1),(b,2)})
B ({(1,a),(2,a),(1,b),(2,b)})
C ({(a,b),(1,2)})
D ({(a,1),(b,2)})
Explanation opens after your attempt
Correct Answer
A. ({(a,1),(a,2),(b,1),(b,2)})
Step 1
Concept
In \(A\times B\), the first element comes from (A) and the second from (B). Order is very important in ordered pairs.
Step 2
Why this answer is correct
The correct answer is A. ({(a,1),(a,2),(b,1),(b,2)}). In \(A\times B\), the first element comes from (A) and the second from (B). Order is very important in ordered pairs.
Step 3
Exam Tip
\(A\times B\) में पहला अवयव (A) से और दूसरा (B) से आता है। क्रमित युग्म में क्रम बहुत महत्वपूर्ण है।
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यदि \(A=\{1,2,4\}\) और \(B=\{2,3,4\}\) हैं, तो \(A\times B\) में कितने युग्म ((x,y)) ऐसे हैं कि (x+y) सम है?
If \(A=\{1,2,4\}\) and \(B=\{2,3,4\}\), how many pairs ((x,y)) in \(A\times B\) have (x+y) even?
#cartesian-product
#parity
#application
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A (4)
B (5)
C (6)
D (3)
Explanation opens after your attempt
Step 1
Concept
The sum is even when both components are even or both are odd. Here the valid pairs are ((2,2),(2,4),(4,2),(4,4)).
Step 2
Why this answer is correct
The correct answer is A. (4). The sum is even when both components are even or both are odd. Here the valid pairs are ((2,2),(2,4),(4,2),(4,4)).
Step 3
Exam Tip
योग सम तब होगा जब दोनों घटक सम हों या दोनों विषम हों। यहां सही युग्म ((2,2),(2,4),(4,2),(4,4)) हैं।
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यदि \(A=\{2,4,6\}\) और \(B=\{1,3\}\) हैं, तो \(B\times A\) में कौन सा युग्म आएगा?
If \(A=\{2,4,6\}\) and \(B=\{1,3\}\), which pair will belong to \(B\times A\)?
#cartesian-product
#membership
#ordered-pairs
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A ((1,4))
B ((4,1))
C ((2,3))
D ((6,1))
Explanation opens after your attempt
Correct Answer
A. ((1,4))
Step 1
Concept
In \(B\times A\), the first element must be from (B) and the second from (A). Hence ((1,4)) is correct.
Step 2
Why this answer is correct
The correct answer is A. ((1,4)). In \(B\times A\), the first element must be from (B) and the second from (A). Hence ((1,4)) is correct.
Step 3
Exam Tip
\(B\times A\) में पहला अवयव (B) से और दूसरा (A) से होना चाहिए। इसलिए ((1,4)) सही है।
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यदि \(A=\{0,1,2\}\) और \(B=\{1,2,3\}\) हैं, तो \(A\times B\) में कितने युग्म ((x,y)) ऐसे हैं कि (xy=2)?
If \(A=\{0,1,2\}\) and \(B=\{1,2,3\}\), how many pairs ((x,y)) in \(A\times B\) satisfy (xy=2)?
#cartesian-product
#product-condition
#ordered-pairs
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A (2)
B (1)
C (3)
D (4)
Explanation opens after your attempt
Step 1
Concept
The valid pairs are ((1,2)) and ((2,1)). Remember that these two ordered pairs are considered different.
Step 2
Why this answer is correct
The correct answer is A. (2). The valid pairs are ((1,2)) and ((2,1)). Remember that these two ordered pairs are considered different.
Step 3
Exam Tip
सही युग्म ((1,2)) और ((2,1)) हैं। ध्यान रखें कि दोनों क्रमित युग्म अलग माने जाते हैं।
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यदि \(A=\{0,1\}\), \(B=\{x,y,z\}\) हैं, तो \(A\times B\) में ((1,y)) का स्थान किस कारण सही है?
If \(A=\{0,1\}\), \(B=\{x,y,z\}\), why is ((1,y)) correctly placed in \(A\times B\)?
#cartesian-product
#membership
#concept
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A क्योंकि \(1\in A\) और \(y\in B\) / because \(1\in A\) and \(y\in B\)
B क्योंकि \(1\in B\) और \(y\in A\) / because \(1\in B\) and \(y\in A\)
C क्योंकि (1=y) / because (1=y)
D क्योंकि (A=B) / because (A=B)
Explanation opens after your attempt
Correct Answer
A. क्योंकि \(1\in A\) और \(y\in B\) / because \(1\in A\) and \(y\in B\)
Step 1
Concept
\((1,y)\in A\times B\) only when \(1\in A\) and \(y\in B\). Do not reverse order while checking membership.
Step 2
Why this answer is correct
The correct answer is A. क्योंकि \(1\in A\) और \(y\in B\) / because \(1\in A\) and \(y\in B\). \((1,y)\in A\times B\) only when \(1\in A\) and \(y\in B\). Do not reverse order while checking membership.
Step 3
Exam Tip
\((1,y)\in A\times B\) तभी होगा जब \(1\in A\) और \(y\in B\) हो। सदस्यता जांच में क्रम न बदलें।
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यदि \(A=\{1,2,3\}\) और \(B=\{3,4,5\}\) हैं, तो \(A\times B\) में कितने युग्म ((x,y)) ऐसे हैं कि (y-x=2)?
If \(A=\{1,2,3\}\) and \(B=\{3,4,5\}\), how many pairs ((x,y)) in \(A\times B\) satisfy (y-x=2)?
#cartesian-product
#difference-condition
#counting
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A (3)
B (2)
C (4)
D (1)
Explanation opens after your attempt
Step 1
Concept
The valid pairs are ((1,3),(2,4),(3,5)). In such conditions, choose (x) first and then find (y).
Step 2
Why this answer is correct
The correct answer is A. (3). The valid pairs are ((1,3),(2,4),(3,5)). In such conditions, choose (x) first and then find (y).
Step 3
Exam Tip
सही युग्म ((1,3),(2,4),(3,5)) हैं। ऐसी शर्तों में पहले (x) चुनकर (y) निकालना आसान है।
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यदि \(A=\{1,2,3\}\) और \(B=\varnothing\) हैं, तो \(A\times B\) क्या होगा?
If \(A=\{1,2,3\}\) and \(B=\varnothing\), what is \(A\times B\)?
#cartesian-product
#empty-set
#cardinality
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A \(\varnothing\)
B ({1,2,3})
C ({(1,0),(2,0),(3,0)})
D ({\(\varnothing,1\)})
Explanation opens after your attempt
Correct Answer
A. \(\varnothing\)
Step 1
Concept
The Cartesian product of any set with an empty set is \(\varnothing\). If one set is empty, no ordered pair is formed.
Step 2
Why this answer is correct
The correct answer is A. \(\varnothing\). The Cartesian product of any set with an empty set is \(\varnothing\). If one set is empty, no ordered pair is formed.
Step 3
Exam Tip
किसी भी समुच्चय का रिक्त समुच्चय के साथ कार्तीय गुणन \(\varnothing\) होता है। यदि एक भी समुच्चय रिक्त हो तो कोई क्रमित युग्म नहीं बनता।
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यदि \(A=\{2,3,5\}\) और \(B=\{1,2,3\}\) हैं, तो \(A\times B\) में कितने युग्म ((x,y)) ऐसे हैं कि (x>y)?
If \(A=\{2,3,5\}\) and \(B=\{1,2,3\}\), how many pairs ((x,y)) in \(A\times B\) satisfy (x>y)?
#cartesian-product
#inequality
#reasoning
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A (6)
B (5)
C (4)
D (7)
Explanation opens after your attempt
Step 1
Concept
The valid pairs are ((2,1),(3,1),(3,2),(5,1),(5,2),(5,3)). In inequalities, reversing order can change the answer.
Step 2
Why this answer is correct
The correct answer is A. (6). The valid pairs are ((2,1),(3,1),(3,2),(5,1),(5,2),(5,3)). In inequalities, reversing order can change the answer.
Step 3
Exam Tip
सही युग्म ((2,1),(3,1),(3,2),(5,1),(5,2),(5,3)) हैं। असमानता में क्रम बदलने से उत्तर बदल सकता है।
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यदि (n(A)=4) और (n\(A\times B\)=20) है, तो (n(B)) कितना होगा?
If (n(A)=4) and (n\(A\times B\)=20), what is (n(B))?
#cartesian-product
#cardinality
#numerical
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A (5)
B (16)
C (24)
D (80)
Explanation opens after your attempt
Step 1
Concept
(n\(A\times B\)=n(A)n(B)), so (20=4n(B)) and (n(B)=5). Use the multiplication formula directly in such questions.
Step 2
Why this answer is correct
The correct answer is A. (5). (n\(A\times B\)=n(A)n(B)), so (20=4n(B)) and (n(B)=5). Use the multiplication formula directly in such questions.
Step 3
Exam Tip
(n\(A\times B\)=n(A)n(B)), इसलिए (20=4n(B)) और (n(B)=5)। ऐसे प्रश्नों में गुणन सूत्र सीधे लगाएं।
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यदि \(A=\{1,2\}\), \(B=\{3,4\}\) और \(C=\{5,6\}\) हैं, तो (A\times\(B\cup C\)) में कितने अवयव होंगे?
If \(A=\{1,2\}\), \(B=\{3,4\}\) and \(C=\{5,6\}\), how many elements are in (A\times\(B\cup C\))?
#cartesian-product
#union
#cardinality
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A (8)
B (6)
C (4)
D (10)
Explanation opens after your attempt
Step 1
Concept
\(B\cup C={3,4,5,6}\), so (n(A\times\(B\cup C\))=2\times4=8). Complete the set operation first.
Step 2
Why this answer is correct
The correct answer is A. (8). \(B\cup C={3,4,5,6}\), so (n(A\times\(B\cup C\))=2\times4=8). Complete the set operation first.
Step 3
Exam Tip
\(B\cup C={3,4,5,6}\), इसलिए (n(A\times\(B\cup C\))=2\times4=8)। पहले समुच्चय संक्रिया पूरी करें।
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यदि \(A=\{p,q,r\}\) है और \(A\times A\) बनाया जाता है, तो इसमें कितने क्रमित युग्म होंगे?
If \(A=\{p,q,r\}\) and \(A\times A\) is formed, how many ordered pairs will it contain?
#cartesian-product
#self-product
#cardinality
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A (9)
B (6)
C (3)
D (12)
Explanation opens after your attempt
Step 1
Concept
(n\(A\times A\)=n(A)2 =32 =9). In \(A\times A\), pairs with equal elements are also included.
Step 2
Why this answer is correct
The correct answer is A. (9). (n\(A\times A\)=n(A)2 =32 =9). In \(A\times A\), pairs with equal elements are also included.
Step 3
Exam Tip
(n\(A\times A\)=n(A)2 =32 =9) होता है। \(A\times A\) में समान अवयव वाले युग्म भी शामिल होते हैं।
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यदि \(A=\{1,2,3\}\), \(B=\{2,3,4\}\) और \(C=\{3,4,5\}\) हैं, तो (A\times\(B\cap C\)) में कितने अवयव होंगे?
If \(A=\{1,2,3\}\), \(B=\{2,3,4\}\) and \(C=\{3,4,5\}\), how many elements are in (A\times\(B\cap C\))?
#cartesian-product
#intersection
#cardinality
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A (6)
B (9)
C (3)
D (12)
Explanation opens after your attempt
Step 1
Concept
\(B\cap C={3,4}\), so (n(A\times\(B\cap C\))=3\times2=6). Find the intersection before counting the Cartesian product.
Step 2
Why this answer is correct
The correct answer is A. (6). \(B\cap C={3,4}\), so (n(A\times\(B\cap C\))=3\times2=6). Find the intersection before counting the Cartesian product.
Step 3
Exam Tip
\(B\cap C={3,4}\), इसलिए (n(A\times\(B\cap C\))=3\times2=6)। प्रतिच्छेद निकालकर ही कार्तीय गुणन गिनें।
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यदि \(A=\{1,2\}\) और \(B=\{1,2\}\) हैं, तो \(A\times B\) में कौन सा युग्म अवश्य होगा?
If \(A=\{1,2\}\) and \(B=\{1,2\}\), which pair must be in \(A\times B\)?
#cartesian-product
#self-product
#membership
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A ((2,2))
B ((3,1))
C ((1,3))
D ((0,2))
Explanation opens after your attempt
Correct Answer
A. ((2,2))
Step 1
Concept
Both entries are from their respective sets, so ((2,2)) is included. An ordered pair with equal entries is also valid.
Step 2
Why this answer is correct
The correct answer is A. ((2,2)). Both entries are from their respective sets, so ((2,2)) is included. An ordered pair with equal entries is also valid.
Step 3
Exam Tip
दोनों स्थानों पर अवयव अपने संबंधित समुच्चय से हैं, इसलिए ((2,2)) शामिल है। समान अवयव वाला क्रमित युग्म भी मान्य होता है।
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यदि \(A=\{1,2\}\), \(B=\{2,3\}\) और \(C=\{3,4\}\) हैं, तो (\(A\cup B\)\times C) में कितने अवयव होंगे?
If \(A=\{1,2\}\), \(B=\{2,3\}\) and \(C=\{3,4\}\), how many elements are in (\(A\cup B\)\times C)?
#cartesian-product
#union
#common-error
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A (6)
B (8)
C (4)
D (5)
Explanation opens after your attempt
Step 1
Concept
\(A\cup B={1,2,3}\), so (n(\(A\cup B\)\times C)=3\times2=6). Do not count a common element twice.
Step 2
Why this answer is correct
The correct answer is A. (6). \(A\cup B={1,2,3}\), so (n(\(A\cup B\)\times C)=3\times2=6). Do not count a common element twice.
Step 3
Exam Tip
\(A\cup B={1,2,3}\), इसलिए (n(\(A\cup B\)\times C)=3\times2=6)। समान अवयव को दो बार न गिनें।
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यदि \(A=\{1,2\}\) और \(B=\{3,4\}\) हैं, तो \(A\times B=B\times A\) कब होगा?
If \(A=\{1,2\}\) and \(B=\{3,4\}\), when will \(A\times B=B\times A\)?
#cartesian-product
#noncommutative
#reasoning
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A यह नहीं होगा क्योंकि \(A\ne B\) / it will not happen because \(A\ne B\)
B हमेशा होगा / always happens
C सिर्फ (n(A)=n(B)) होने से होगा / happens only because (n(A)=n(B))
D जब \(A\cap B=\varnothing\) हो / when \(A\cap B=\varnothing\)
Explanation opens after your attempt
Correct Answer
A. यह नहीं होगा क्योंकि \(A\ne B\) / it will not happen because \(A\ne B\)
Step 1
Concept
Having the same number of elements is not enough. Usually \(A\times B=B\times A\) only when (A=B) or one set is empty.
Step 2
Why this answer is correct
The correct answer is A. यह नहीं होगा क्योंकि \(A\ne B\) / it will not happen because \(A\ne B\). Having the same number of elements is not enough. Usually \(A\times B=B\times A\) only when (A=B) or one set is empty.
Step 3
Exam Tip
केवल समान संख्या के अवयव होना पर्याप्त नहीं है। सामान्यतः \(A\times B=B\times A\) तभी जब (A=B) या कोई समुच्चय रिक्त हो।
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यदि \(A=\{1,2,3,4\}\) और \(B=\{2,4,6\}\) हैं, तो \(A\times B\) में कितने युग्म ((x,y)) ऐसे हैं कि (x) संख्या (y) को विभाजित करती है?
If \(A=\{1,2,3,4\}\) and \(B=\{2,4,6\}\), how many pairs ((x,y)) in \(A\times B\) satisfy that (x) divides (y)?
#cartesian-product
#divisibility
#counting
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A (8)
B (6)
C (7)
D (9)
Explanation opens after your attempt
Step 1
Concept
The valid pairs are ((1,2),(1,4),(1,6),(2,2),(2,4),(2,6),(3,6),(4,4)). In divisibility, note carefully which number divides the other.
Step 2
Why this answer is correct
The correct answer is A. (8). The valid pairs are ((1,2),(1,4),(1,6),(2,2),(2,4),(2,6),(3,6),(4,4)). In divisibility, note carefully which number divides the other.
Step 3
Exam Tip
सही युग्म ((1,2),(1,4),(1,6),(2,2),(2,4),(2,6),(3,6),(4,4)) हैं। विभाज्यता में कौन किसे विभाजित कर रहा है यह ध्यान रखें।
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यदि \(A=\{2,3\}\) और \(B=\{5\}\) हैं, तो \(A\times B\) कौन सा है?
If \(A=\{2,3\}\) and \(B=\{5\}\), which is \(A\times B\)?
#cartesian-product
#singleton
#listing
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A ({(2,5),(3,5)})
B ({(5,2),(5,3)})
C ({(2,3),(3,2)})
D ({(2,5)})
Explanation opens after your attempt
Correct Answer
A. ({(2,5),(3,5)})
Step 1
Concept
The first element is taken from (A) and the second from (B). Since (B) has only (5), the second entry in both pairs is (5).
Step 2
Why this answer is correct
The correct answer is A. ({(2,5),(3,5)}). The first element is taken from (A) and the second from (B). Since (B) has only (5), the second entry in both pairs is (5).
Step 3
Exam Tip
पहला अवयव (A) से और दूसरा (B) से लिया जाता है। (B) में केवल (5) है, इसलिए दोनों युग्मों में दूसरा अवयव (5) होगा।
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यदि \(A=\{1,3,5\}\) और \(B=\{2,4,6\}\) हैं, तो \(A\times B\) में कितने युग्म ((x,y)) ऐसे हैं कि (x+y>7)?
If \(A=\{1,3,5\}\) and \(B=\{2,4,6\}\), how many pairs ((x,y)) in \(A\times B\) satisfy (x+y>7)?
#cartesian-product
#inequality
#sum-condition
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A (3)
B (4)
C (5)
D (2)
Explanation opens after your attempt
Step 1
Concept
The valid pairs are ((3,6),(5,4),(5,6)). In boundary inequalities, check whether equality is included or not.
Step 2
Why this answer is correct
The correct answer is A. (3). The valid pairs are ((3,6),(5,4),(5,6)). In boundary inequalities, check whether equality is included or not.
Step 3
Exam Tip
सही युग्म ((3,6),(5,4),(5,6)) हैं। सीमा वाली असमानता में बराबरी शामिल है या नहीं यह जरूर देखें।
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यदि \(A=\{-1,0,1\}\) और \(B=\{2,4\}\) हैं, तो \((-1,4)\in A\times B\) का सत्य मान क्या है?
If \(A=\{-1,0,1\}\) and \(B=\{2,4\}\), what is the truth value of \((-1,4)\in A\times B\)?
#cartesian-product
#membership
#integers
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A सत्य / true
B असत्य / false
C केवल जब (-1=4) / only when (-1=4)
D निर्धारित नहीं / not determined
Explanation opens after your attempt
Correct Answer
A. सत्य / true
Step 1
Concept
Since \(-1\in A\) and \(4\in B\), \((-1,4)\in A\times B\) is true. Check first and second positions separately.
Step 2
Why this answer is correct
The correct answer is A. सत्य / true. Since \(-1\in A\) and \(4\in B\), \((-1,4)\in A\times B\) is true. Check first and second positions separately.
Step 3
Exam Tip
\(-1\in A\) और \(4\in B\), इसलिए \((-1,4)\in A\times B\) सत्य है। पहले और दूसरे स्थान की अलग-अलग जांच करें।
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यदि \(A=\{0,1,2,3\}\) और \(B=\{1,2\}\) हैं, तो \(A\times B\) में कितने युग्म ((x,y)) ऐसे हैं कि (x-y=1)?
If \(A=\{0,1,2,3\}\) and \(B=\{1,2\}\), how many pairs ((x,y)) in \(A\times B\) satisfy (x-y=1)?
#cartesian-product
#difference-condition
#application
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A (2)
B (3)
C (1)
D (4)
Explanation opens after your attempt
Step 1
Concept
The valid pairs are ((2,1)) and ((3,2)). A quick method is to choose elements of (B) and then find (x).
Step 2
Why this answer is correct
The correct answer is A. (2). The valid pairs are ((2,1)) and ((3,2)). A quick method is to choose elements of (B) and then find (x).
Step 3
Exam Tip
सही युग्म ((2,1)) और ((3,2)) हैं। दिए हुए (B) के अवयवों से (x) निकालना तेज तरीका है।
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यदि \(A=\{1,2,3\}\) और \(B=\{2,3,4\}\) हैं, तो \((4,1)\in A\times B\) क्यों असत्य है?
If \(A=\{1,2,3\}\) and \(B=\{2,3,4\}\), why is \((4,1)\in A\times B\) false?
#cartesian-product
#membership
#common-error
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A क्योंकि \(4\notin A\) और \(1\notin B\) / because \(4\notin A\) and \(1\notin B\)
B क्योंकि \(4\in B\) / because \(4\in B\)
C क्योंकि \(1\in A\) / because \(1\in A\)
D क्योंकि \(A\cap B\ne\varnothing\) / because \(A\cap B\ne\varnothing\)
Explanation opens after your attempt
Correct Answer
A. क्योंकि \(4\notin A\) और \(1\notin B\) / because \(4\notin A\) and \(1\notin B\)
Step 1
Concept
In \(A\times B\), the first entry must be from (A) and the second from (B). Here both positions are wrong.
Step 2
Why this answer is correct
The correct answer is A. क्योंकि \(4\notin A\) और \(1\notin B\) / because \(4\notin A\) and \(1\notin B\). In \(A\times B\), the first entry must be from (A) and the second from (B). Here both positions are wrong.
Step 3
Exam Tip
\(A\times B\) में पहले स्थान पर (A) का अवयव होना चाहिए और दूसरे स्थान पर (B) का। यहां दोनों स्थान गलत हैं।
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यदि \(A=\{1,2,3\}\) और \(B=\{1,4,9\}\) हैं, तो \(A\times B\) में कितने युग्म ((x,y)) ऐसे हैं कि \(y=x^2\)?
If \(A=\{1,2,3\}\) and \(B=\{1,4,9\}\), how many pairs ((x,y)) in \(A\times B\) satisfy \(y=x^2\)?
#cartesian-product
#function-pattern
#relation
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A (3)
B (2)
C (6)
D (9)
Explanation opens after your attempt
Step 1
Concept
The valid pairs are ((1,1),(2,4),(3,9)). In a Cartesian product, applying a condition forms a subset relation.
Step 2
Why this answer is correct
The correct answer is A. (3). The valid pairs are ((1,1),(2,4),(3,9)). In a Cartesian product, applying a condition forms a subset relation.
Step 3
Exam Tip
सही युग्म ((1,1),(2,4),(3,9)) हैं। कार्तीय गुणन में दिए संबंध की शर्त लगाकर उपसमुच्चय बनाया जाता है।
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यदि \(A=\{1,2,3,4\}\) और \(B=\{1,2,3,4\}\) हैं, तो \(A\times B\) में कितने युग्म ((x,y)) ऐसे हैं कि (x+y=5)?
If \(A=\{1,2,3,4\}\) and \(B=\{1,2,3,4\}\), how many pairs ((x,y)) in \(A\times B\) satisfy (x+y=5)?
#cartesian-product
#sum-condition
#ordered-pairs
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A (4)
B (3)
C (5)
D (2)
Explanation opens after your attempt
Step 1
Concept
The valid pairs are ((1,4),(2,3),(3,2),(4,1)). In ordered pairs, ((2,3)) and ((3,2)) are different.
Step 2
Why this answer is correct
The correct answer is A. (4). The valid pairs are ((1,4),(2,3),(3,2),(4,1)). In ordered pairs, ((2,3)) and ((3,2)) are different.
Step 3
Exam Tip
सही युग्म ((1,4),(2,3),(3,2),(4,1)) हैं। क्रमित युग्मों में ((2,3)) और ((3,2)) अलग होते हैं।
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यदि \(A={x:x\in\mathbb{Z},-1\le x\le1}\) और \(B=\{0,1\}\) हैं, तो \(A\times B\) में कितने युग्म होंगे?
If \(A={x:x\in\mathbb{Z},-1\le x\le1}\) and \(B=\{0,1\}\), how many pairs are in \(A\times B\)?
#cartesian-product
#set-builder
#integers
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A (6)
B (4)
C (5)
D (2)
Explanation opens after your attempt
Step 1
Concept
\(A=\{-1,0,1\}\) has (3) elements and (B) has (2) elements. Therefore total pairs are \(3\times2=6\).
Step 2
Why this answer is correct
The correct answer is A. (6). \(A=\{-1,0,1\}\) has (3) elements and (B) has (2) elements. Therefore total pairs are \(3\times2=6\).
Step 3
Exam Tip
\(A=\{-1,0,1\}\) में (3) अवयव और (B) में (2) अवयव हैं। इसलिए कुल युग्म \(3\times2=6\) हैं।
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यदि \(A=\{1,2,3,4,5\}\), \(B=\{2,4\}\) और \(C=\{0,1,2\}\) हैं, तो \((A-B)\times C\) में कितने अवयव होंगे?
If \(A=\{1,2,3,4,5\}\), \(B=\{2,4\}\) and \(C=\{0,1,2\}\), how many elements are there in \((A-B)\times C\)?
#cartesian-product
#set-difference
#cardinality
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A (9)
B (6)
C (12)
D (15)
Explanation opens after your attempt
Step 1
Concept
(A-B={1,3,5}), so (n\((A-B)\times C\)=3\times3=9). First find the set difference and then count the Cartesian product.
Step 2
Why this answer is correct
The correct answer is A. (9). (A-B={1,3,5}), so (n\((A-B)\times C\)=3\times3=9). First find the set difference and then count the Cartesian product.
Step 3
Exam Tip
(A-B={1,3,5}), इसलिए (n\((A-B)\times C\)=3\times3=9)। पहले समुच्चय का अंतर निकालें फिर कार्तीय गुणन गिनें।
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यदि \(A=\{1,2\}\), \(B=\{3,4\}\) और \(C=\{5\}\) हैं, तो (\(A\times B\)\times C) में अवयवों की संख्या क्या होगी?
If \(A=\{1,2\}\), \(B=\{3,4\}\) and \(C=\{5\}\), what is the number of elements in (\(A\times B\)\times C)?
#cartesian-product
#nested-product
#cardinality
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A (4)
B (5)
C (8)
D (2)
Explanation opens after your attempt
Step 1
Concept
(n\(A\times B\)=4) and (n(C)=1), so (n(\(A\times B\)\times C)=4). Remember that (\(A\times B\)) itself is a set.
Step 2
Why this answer is correct
The correct answer is A. (4). (n\(A\times B\)=4) and (n(C)=1), so (n(\(A\times B\)\times C)=4). Remember that (\(A\times B\)) itself is a set.
Step 3
Exam Tip
(n\(A\times B\)=4) और (n(C)=1), इसलिए (n(\(A\times B\)\times C)=4)। ध्यान रखें कि (\(A\times B\)) खुद एक समुच्चय है।
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यदि \(A=\{1,2\}\) और \(B=\{a,b,c\}\) हैं, तो \(A\times B\) के सभी युग्मों में पहला घटक कितनी बार (1) होगा?
If \(A=\{1,2\}\) and \(B=\{a,b,c\}\), how many times will the first component be (1) in \(A\times B\)?
#cartesian-product
#pattern
#counting
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A (3)
B (2)
C (1)
D (6)
Explanation opens after your attempt
Step 1
Concept
The element \(1\in A\) pairs with all (3) elements of (B). So the first component (1) appears (3) times.
Step 2
Why this answer is correct
The correct answer is A. (3). The element \(1\in A\) pairs with all (3) elements of (B). So the first component (1) appears (3) times.
Step 3
Exam Tip
\(1\in A\) के साथ (B) के सभी (3) अवयव जुड़ेंगे। इसलिए पहला घटक (1) कुल (3) बार आएगा।
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यदि \(A=\{m,n,o\}\) और \(B=\{7,8\}\) हैं, तो \(A\times B\) के सभी युग्मों में दूसरा घटक (8) कितनी बार आएगा?
If \(A=\{m,n,o\}\) and \(B=\{7,8\}\), how many times will the second component (8) appear in \(A\times B\)?
#cartesian-product
#pattern
#counting
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A (3)
B (2)
C (6)
D (1)
Explanation opens after your attempt
Step 1
Concept
The element \(8\in B\) pairs with all (3) elements of (A). Therefore the second component (8) appears (3) times.
Step 2
Why this answer is correct
The correct answer is A. (3). The element \(8\in B\) pairs with all (3) elements of (A). Therefore the second component (8) appears (3) times.
Step 3
Exam Tip
\(8\in B\) के साथ (A) के सभी (3) अवयव जुड़ेंगे। इसलिए दूसरा घटक (8) कुल (3) बार आएगा।
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यदि \(A=\{1,3,5\}\) और \(B=\{2,4\}\) हैं, तो \(A\times B\) में ऐसा कौन सा युग्म है जिसके दोनों घटकों का योग (7) है?
If \(A=\{1,3,5\}\) and \(B=\{2,4\}\), which pair in \(A\times B\) has sum of components (7)?
#cartesian-product
#application
#sum-condition
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A ((3,4))
B ((5,4))
C ((1,2))
D ((4,3))
Explanation opens after your attempt
Correct Answer
A. ((3,4))
Step 1
Concept
\((3,4)\in A\times B\) and (3+4=7). ((4,3)) is not correct because the first element is not from (A).
Step 2
Why this answer is correct
The correct answer is A. ((3,4)). \((3,4)\in A\times B\) and (3+4=7). ((4,3)) is not correct because the first element is not from (A).
Step 3
Exam Tip
\((3,4)\in A\times B\) और (3+4=7)। ((4,3)) सही नहीं क्योंकि पहला अवयव (A) से नहीं है।
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यदि \(A=\{1,2,3\}\) और \(B=\{1,2,3\}\) हैं, तो \(A\times B\) में कितने युग्म ((x,y)) ऐसे हैं कि (x<y)?
If \(A=\{1,2,3\}\) and \(B=\{1,2,3\}\), how many pairs ((x,y)) in \(A\times B\) satisfy (x<y)?
#cartesian-product
#inequality
#reasoning
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A (3)
B (6)
C (9)
D (2)
Explanation opens after your attempt
Step 1
Concept
The correct pairs are ((1,2),(1,3),(2,3)). Count only after applying the condition.
Step 2
Why this answer is correct
The correct answer is A. (3). The correct pairs are ((1,2),(1,3),(2,3)). Count only after applying the condition.
Step 3
Exam Tip
सही युग्म ((1,2),(1,3),(2,3)) हैं। शर्त लगाने के बाद ही गिनती करें।
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यदि \(A=\{0,1,2\}\) और \(B=\{0,1\}\) हैं, तो \(A\times B\) में कितने युग्म ((x,y)) ऐसे हैं कि (x+y=2)?
If \(A=\{0,1,2\}\) and \(B=\{0,1\}\), how many pairs ((x,y)) in \(A\times B\) satisfy (x+y=2)?
#cartesian-product
#equation-condition
#counting
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A (2)
B (1)
C (3)
D (4)
Explanation opens after your attempt
Step 1
Concept
The correct pairs are ((1,1)) and ((2,0)). Remember the first element must come from (A) and the second from (B).
Step 2
Why this answer is correct
The correct answer is A. (2). The correct pairs are ((1,1)) and ((2,0)). Remember the first element must come from (A) and the second from (B).
Step 3
Exam Tip
सही युग्म ((1,1)) और ((2,0)) हैं। ध्यान रखें कि पहला अवयव (A) से और दूसरा (B) से होना चाहिए।
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यदि \(A=\{1,2,3,4\}\) और \(B=\{2,4\}\) हैं, तो \(A\times B\) में कितने युग्म ((x,y)) ऐसे हैं कि (y) संख्या (x) को विभाजित करती है?
If \(A=\{1,2,3,4\}\) and \(B=\{2,4\}\), how many pairs ((x,y)) in \(A\times B\) satisfy that (y) divides (x)?
#cartesian-product
#divisibility
#application
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A (3)
B (4)
C (2)
D (5)
Explanation opens after your attempt
Step 1
Concept
The correct pairs are ((2,2),(4,2),(4,4)). In divisibility, changing positions changes the meaning.
Step 2
Why this answer is correct
The correct answer is A. (3). The correct pairs are ((2,2),(4,2),(4,4)). In divisibility, changing positions changes the meaning.
Step 3
Exam Tip
सही युग्म ((2,2),(4,2),(4,4)) हैं। विभाज्यता में स्थान बदलने से अर्थ बदल जाता है।
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यदि \(A=\{1,2\}\) और \(B=\{3,4,5\}\) हैं, तो \(A\times B\) का कौन सा उपसमुच्चय एक संबंध हो सकता है?
If \(A=\{1,2\}\) and \(B=\{3,4,5\}\), which subset of \(A\times B\) can be a relation?
#cartesian-product
#relation
#subset
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A ({(1,3),(2,5)})
B ({(3,1),(5,2)})
C ({(1,6)})
D ({(4,2)})
Explanation opens after your attempt
Correct Answer
A. ({(1,3),(2,5)})
Step 1
Concept
A relation from (A) to (B) is any subset of \(A\times B\). Only ({(1,3),(2,5)}) has both pairs in \(A\times B\).
Step 2
Why this answer is correct
The correct answer is A. ({(1,3),(2,5)}). A relation from (A) to (B) is any subset of \(A\times B\). Only ({(1,3),(2,5)}) has both pairs in \(A\times B\).
Step 3
Exam Tip
(A) से (B) में संबंध \(A\times B\) का कोई भी उपसमुच्चय होता है। केवल ({(1,3),(2,5)}) के दोनों युग्म \(A\times B\) में हैं।
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यदि (A) में (3) अवयव और (B) में (2) अवयव हैं, तो (A) से (B) तक कुल कितने संबंध संभव हैं?
If (A) has (3) elements and (B) has (2) elements, how many relations are possible from (A) to (B)?
#cartesian-product
#relations
#counting
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A \(2^6\)
B (6)
C \(3^2\)
D \(2^3\)
Explanation opens after your attempt
Correct Answer
A. \(2^6\)
Step 1
Concept
(n\(A\times B\)=3\times2=6), and every relation is a subset of \(A\times B\). Therefore the number of relations is \(2^6\).
Step 2
Why this answer is correct
The correct answer is A. \(2^6\). (n\(A\times B\)=3\times2=6), and every relation is a subset of \(A\times B\). Therefore the number of relations is \(2^6\).
Step 3
Exam Tip
(n\(A\times B\)=3\times2=6), और हर संबंध \(A\times B\) का उपसमुच्चय है। इसलिए संबंधों की संख्या \(2^6\) होगी।
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यदि \(A=\{1,2,3\}\) और \(B=\{a\}\) हैं, तो (A) से (B) तक कुल संबंधों की संख्या कितनी है?
If \(A=\{1,2,3\}\) and \(B=\{a\}\), how many relations are possible from (A) to (B)?
#cartesian-product
#relations
#singleton
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A (8)
B (3)
C (6)
D (9)
Explanation opens after your attempt
Step 1
Concept
(n\(A\times B\)=3\times1=3), so the number of subsets is \(2^3=8\). For counting relations, remember \(2^{n(A\times B)}\).
Step 2
Why this answer is correct
The correct answer is A. (8). (n\(A\times B\)=3\times1=3), so the number of subsets is \(2^3=8\). For counting relations, remember \(2^{n(A\times B)}\).
Step 3
Exam Tip
(n\(A\times B\)=3\times1=3), इसलिए उपसमुच्चयों की संख्या \(2^3=8\) है। संबंध गिनने में \(2^{n(A\times B)}\) याद रखें।
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यदि \(A=\{1,2\}\) और \(B=\{3,4\}\) हैं, तो \(A\times B\) और \(B\times A\) में कौन सा कथन सही है?
If \(A=\{1,2\}\) and \(B=\{3,4\}\), which statement about \(A\times B\) and \(B\times A\) is correct?
#cartesian-product
#noncommutative
#comparison
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A दोनों में अवयवों की संख्या समान है पर युग्म अलग हैं / both have same number of elements but different pairs
B दोनों हमेशा समान हैं / both are always equal
C दोनों रिक्त हैं / both are empty
D \(A\times B\) में (2) युग्म हैं / \(A\times B\) has (2) pairs
Explanation opens after your attempt
Correct Answer
A. दोनों में अवयवों की संख्या समान है पर युग्म अलग हैं / both have same number of elements but different pairs
Step 1
Concept
Both have \(2\times2=4\) pairs, but changing order changes the pairs. Cartesian product is not commutative.
Step 2
Why this answer is correct
The correct answer is A. दोनों में अवयवों की संख्या समान है पर युग्म अलग हैं / both have same number of elements but different pairs. Both have \(2\times2=4\) pairs, but changing order changes the pairs. Cartesian product is not commutative.
Step 3
Exam Tip
दोनों में \(2\times2=4\) युग्म होंगे, पर क्रम बदलने से युग्म बदल जाते हैं। कार्तीय गुणन क्रमविनिमेय नहीं होता।
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यदि \(A=\{1,2\}\), \(B=\{2,3\}\) हैं, तो \(A\times B\cap B\times A\) में कौन सा युग्म होगा?
If \(A=\{1,2\}\), \(B=\{2,3\}\), which pair belongs to \(A\times B\cap B\times A\)?
#cartesian-product
#intersection
#ordered-pairs
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A ((2,2))
B ((1,3))
C ((3,1))
D ((1,2))
Explanation opens after your attempt
Correct Answer
A. ((2,2))
Step 1
Concept
((2,2)) is in both \(A\times B\) and \(B\times A\). For intersection, the pair must belong to both sets.
Step 2
Why this answer is correct
The correct answer is A. ((2,2)). ((2,2)) is in both \(A\times B\) and \(B\times A\). For intersection, the pair must belong to both sets.
Step 3
Exam Tip
((2,2)) दोनों \(A\times B\) और \(B\times A\) में है। प्रतिच्छेद के लिए युग्म दोनों समुच्चयों में होना चाहिए।
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यदि \(A=\{1,2,3\}\) और \(B=\{4,5\}\) हैं, तो \(A\times B\) में कौन सा युग्म नहीं है?
If \(A=\{1,2,3\}\) and \(B=\{4,5\}\), which pair is not in \(A\times B\)?
#cartesian-product
#membership
#not-belonging
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A ((4,1))
B ((1,4))
C ((2,5))
D ((3,4))
Explanation opens after your attempt
Correct Answer
A. ((4,1))
Step 1
Concept
In ((4,1)), the first element is not from (A) and the second is not from (B). Position checking is necessary in ordered pairs.
Step 2
Why this answer is correct
The correct answer is A. ((4,1)). In ((4,1)), the first element is not from (A) and the second is not from (B). Position checking is necessary in ordered pairs.
Step 3
Exam Tip
((4,1)) में पहला अवयव (A) से नहीं और दूसरा (B) से नहीं है। क्रमित युग्म में स्थान की जांच जरूरी है।
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यदि \(A=\{0,2\}\) और \(B=\{1,3,5\}\) हैं, तो \(A\times B\) में सबसे बड़े घटक-योग वाला युग्म कौन सा है?
If \(A=\{0,2\}\) and \(B=\{1,3,5\}\), which pair in \(A\times B\) has the greatest sum of components?
#cartesian-product
#maximum
#application
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A ((2,5))
B ((0,5))
C ((2,3))
D ((5,2))
Explanation opens after your attempt
Correct Answer
A. ((2,5))
Step 1
Concept
The greatest first component is (2) and the greatest second component is (5), so the maximum sum is (7). ((5,2)) is not in \(A\times B\).
Step 2
Why this answer is correct
The correct answer is A. ((2,5)). The greatest first component is (2) and the greatest second component is (5), so the maximum sum is (7). ((5,2)) is not in \(A\times B\).
Step 3
Exam Tip
सबसे बड़ा पहला घटक (2) और सबसे बड़ा दूसरा घटक (5) है, इसलिए योग (7) अधिकतम है। ((5,2)) \(A\times B\) में नहीं है।
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यदि \(A=\{1,2,3\}\) और \(B=\{1,2\}\) हैं, तो \(A\times B\) में कितने युग्म ((x,y)) ऐसे हैं कि (x=y)?
If \(A=\{1,2,3\}\) and \(B=\{1,2\}\), how many pairs ((x,y)) in \(A\times B\) satisfy (x=y)?
#cartesian-product
#equality-condition
#intersection
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A (2)
B (3)
C (6)
D (1)
Explanation opens after your attempt
Step 1
Concept
The equal-component pairs are ((1,1)) and ((2,2)). Only elements of \(A\cap B\) form such pairs.
Step 2
Why this answer is correct
The correct answer is A. (2). The equal-component pairs are ((1,1)) and ((2,2)). Only elements of \(A\cap B\) form such pairs.
Step 3
Exam Tip
समान घटकों वाले युग्म ((1,1)) और ((2,2)) हैं। केवल \(A\cap B\) के अवयव ऐसे युग्म बनाते हैं।
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यदि \(A=\{2,4,6\}\) और \(B=\{1,2,3\}\) हैं, तो \(A\times B\) में कितने युग्म ((x,y)) ऐसे हैं कि (x=2y)?
If \(A=\{2,4,6\}\) and \(B=\{1,2,3\}\), how many pairs ((x,y)) in \(A\times B\) satisfy (x=2y)?
#cartesian-product
#equation-condition
#application
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A (3)
B (2)
C (6)
D (1)
Explanation opens after your attempt
Step 1
Concept
The correct pairs are ((2,1),(4,2),(6,3)). Apply the condition to every possible pair.
Step 2
Why this answer is correct
The correct answer is A. (3). The correct pairs are ((2,1),(4,2),(6,3)). Apply the condition to every possible pair.
Step 3
Exam Tip
सही युग्म ((2,1),(4,2),(6,3)) हैं। शर्त को हर संभव युग्म पर लागू करें।
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यदि (A=[1,3]) और \(B=\{0\}\) हैं, तो \(A\times B\) का ज्यामितीय रूप कैसा होगा?
If (A=[1,3]) and \(B=\{0\}\), what is the geometric form of \(A\times B\)?
#cartesian-product
#interval
#geometry
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A (x)-अक्ष पर \(1\le x\le3\) वाला रेखाखंड / line segment on (x)-axis with \(1\le x\le3\)
B (y)-अक्ष पर \(1\le y\le3\) वाला रेखाखंड / line segment on (y)-axis with \(1\le y\le3\)
C पूरा तल \(\mathbb{R}^2\) / whole plane \(\mathbb{R}^2\)
D केवल बिंदु ((0,0)) / only point ((0,0))
Explanation opens after your attempt
Correct Answer
A. (x)-अक्ष पर \(1\le x\le3\) वाला रेखाखंड / line segment on (x)-axis with \(1\le x\le3\)
Step 1
Concept
\(A\times B={(x,0):1\le x\le3}\), so it is a line segment on the (x)-axis. In interval questions, write the coordinate form.
Step 2
Why this answer is correct
The correct answer is A. (x)-अक्ष पर \(1\le x\le3\) वाला रेखाखंड / line segment on (x)-axis with \(1\le x\le3\). \(A\times B={(x,0):1\le x\le3}\), so it is a line segment on the (x)-axis. In interval questions, write the coordinate form.
Step 3
Exam Tip
\(A\times B={(x,0):1\le x\le3}\), इसलिए यह (x)-अक्ष पर रेखाखंड है। अंतराल वाले प्रश्नों में निर्देशांक रूप लिखें।
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यदि \(A=\{0\}\) और (B=[-2,2]) हैं, तो \(A\times B\) कौन सा समुच्चय है?
If \(A=\{0\}\) and (B=[-2,2]), which set is \(A\times B\)?
#cartesian-product
#interval
#geometry
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A ({(0,y):-2\le y\le2})
B ({(x,0):-2\le x\le2})
C ({(y,0):-2\le y\le2})
D ({(0,0)})
Explanation opens after your attempt
Correct Answer
A. ({(0,y):-2\le y\le2})
Step 1
Concept
The first component is always (0), and the second varies in ([-2,2]). So it is a segment on the (y)-axis.
Step 2
Why this answer is correct
The correct answer is A. ({(0,y):-2\le y\le2}). The first component is always (0), and the second varies in ([-2,2]). So it is a segment on the (y)-axis.
Step 3
Exam Tip
पहला घटक हमेशा (0) होगा और दूसरा ([-2,2]) से बदलता रहेगा। इसलिए यह (y)-अक्ष का रेखाखंड है।
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यदि \(A={x:x\in\mathbb{N},x\le2}\) और \(B={y:y\in\mathbb{N},y\le3}\) हैं, तो \(A\times B\) में कितने युग्म होंगे?
If \(A={x:x\in\mathbb{N},x\le2}\) and \(B={y:y\in\mathbb{N},y\le3}\), how many pairs are in \(A\times B\)?
#cartesian-product
#set-builder
#natural-numbers
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A (6)
B (5)
C (3)
D (2)
Explanation opens after your attempt
Step 1
Concept
\(A=\{1,2\}\) and \(B=\{1,2,3\}\). Hence (n\(A\times B\)=2\times3=6).
Step 2
Why this answer is correct
The correct answer is A. (6). \(A=\{1,2\}\) and \(B=\{1,2,3\}\). Hence (n\(A\times B\)=2\times3=6).
Step 3
Exam Tip
\(A=\{1,2\}\) और \(B=\{1,2,3\}\) हैं। अतः (n\(A\times B\)=2\times3=6)।
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यदि \(A=\{1,2\}\) और \(B=\{3,4\}\) हैं, तो \(A\times B\) का कौन सा सही सेट-बिल्डर रूप है?
If \(A=\{1,2\}\) and \(B=\{3,4\}\), which is the correct set-builder form of \(A\times B\)?
#cartesian-product
#set-builder
#definition
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A \({(x,y):x\in A,,y\in B}\)
B \({(x,y):x\in B,,y\in A}\)
C \({x+y:x\in A,,y\in B}\)
D \({xy:x\in A,,y\in B}\)
Explanation opens after your attempt
Correct Answer
A. \({(x,y):x\in A,,y\in B}\)
Step 1
Concept
\(A\times B\) is a set of ordered pairs, so its form is \({(x,y):x\in A,,y\in B}\). Sum or product of entries is not Cartesian product.
Step 2
Why this answer is correct
The correct answer is A. \({(x,y):x\in A,,y\in B}\). \(A\times B\) is a set of ordered pairs, so its form is \({(x,y):x\in A,,y\in B}\). Sum or product of entries is not Cartesian product.
Step 3
Exam Tip
\(A\times B\) क्रमित युग्मों का समुच्चय है, इसलिए रूप \({(x,y):x\in A,,y\in B}\) है। योग या गुणन कार्तीय गुणन नहीं है।
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यदि \(A=\{1,2,3\}\) और \(B=\{4,5\}\) हैं, तो \(A\times B\) में पहले घटकों का समुच्चय क्या होगा?
If \(A=\{1,2,3\}\) and \(B=\{4,5\}\), what is the set of first components in \(A\times B\)?
#cartesian-product
#components
#projection
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A ({1,2,3})
B ({4,5})
C ({1,2,3,4,5})
D \(\varnothing\)
Explanation opens after your attempt
Correct Answer
A. ({1,2,3})
Step 1
Concept
In \(A\times B\), all first components come from (A), and (B) is non-empty. Therefore the set of first components is (A).
Step 2
Why this answer is correct
The correct answer is A. ({1,2,3}). In \(A\times B\), all first components come from (A), and (B) is non-empty. Therefore the set of first components is (A).
Step 3
Exam Tip
\(A\times B\) में सभी पहले घटक (A) से आते हैं और (B) रिक्त नहीं है। इसलिए पहले घटकों का समुच्चय (A) ही है।
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यदि \(A=\{r,s\}\) और \(B=\{u,v,w\}\) हैं, तो \(A\times B\) में दूसरे घटकों का समुच्चय क्या होगा?
If \(A=\{r,s\}\) and \(B=\{u,v,w\}\), what is the set of second components in \(A\times B\)?
#cartesian-product
#components
#projection
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A ({u,v,w})
B ({r,s})
C ({r,s,u,v,w})
D ({(r,u),(s,w)})
Explanation opens after your attempt
Correct Answer
A. ({u,v,w})
Step 1
Concept
Second components always come from (B). If (A) is non-empty, every element of (B) appears as a second component.
Step 2
Why this answer is correct
The correct answer is A. ({u,v,w}). Second components always come from (B). If (A) is non-empty, every element of (B) appears as a second component.
Step 3
Exam Tip
दूसरे घटक हमेशा (B) से आते हैं। यदि (A) रिक्त नहीं है तो (B) का हर अवयव दूसरे घटक के रूप में आएगा।
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यदि \(A=\{1,2\}\) और \(B=\{3,4,5\}\) हैं, तो \(A\times B\) का कौन सा कथन सही है?
If \(A=\{1,2\}\) and \(B=\{3,4,5\}\), which statement about \(A\times B\) is correct?
#cartesian-product
#concept
#definition
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A हर \(a\in A\) के लिए (B) के सभी अवयवों से युग्म बनते हैं / for every \(a\in A\), pairs are formed with all elements of (B)
B हर \(a\in A\) के लिए केवल एक युग्म बनता है / for every \(a\in A\), only one pair is formed
C केवल समान अवयवों से युग्म बनते हैं / only equal elements form pairs
D केवल \(A\cap B\) से युग्म बनते हैं / pairs are formed only from \(A\cap B\)
Explanation opens after your attempt
Correct Answer
A. हर \(a\in A\) के लिए (B) के सभी अवयवों से युग्म बनते हैं / for every \(a\in A\), pairs are formed with all elements of (B)
Step 1
Concept
In Cartesian product, every element of (A) pairs with each element of (B). That is why total pairs are (n(A)n(B)).
Step 2
Why this answer is correct
The correct answer is A. हर \(a\in A\) के लिए (B) के सभी अवयवों से युग्म बनते हैं / for every \(a\in A\), pairs are formed with all elements of (B). In Cartesian product, every element of (A) pairs with each element of (B). That is why total pairs are (n(A)n(B)).
Step 3
Exam Tip
कार्तीय गुणन में (A) का हर अवयव (B) के प्रत्येक अवयव के साथ जुड़ता है। यही कारण है कि कुल युग्म (n(A)n(B)) होते हैं।
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यदि \(A=\{1,2,3\}\) और \(B=\{2,4\}\) हैं, तो \(A\times B\) में कितने युग्म ((x,y)) ऐसे हैं कि (x) संख्या (y) से छोटा है?
If \(A=\{1,2,3\}\) and \(B=\{2,4\}\), how many pairs ((x,y)) in \(A\times B\) satisfy (x<y)?
#cartesian-product
#inequality
#common-error
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A (5)
B (4)
C (6)
D (3)
Explanation opens after your attempt
Step 1
Concept
The listed valid distinct pairs are ((1,2),(1,4),(2,4),(3,4)), so the count is (4). Avoid counting the same pair twice.
Step 2
Why this answer is correct
The correct answer is A. (5). The listed valid distinct pairs are ((1,2),(1,4),(2,4),(3,4)), so the count is (4). Avoid counting the same pair twice.
Step 3
Exam Tip
सही युग्म ((1,2),(1,4),(2,4),(3,4),(2,4)) नहीं दोहराएं; वास्तविक अलग युग्म ((1,2),(1,4),(2,4),(3,4)) और ((2,4)) की पुनरावृत्ति हटाने पर (4) होता है।
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