यदि \(A=\{2,4,6\}\) और \(B=\{1,3\}\) हैं, तो \(A\times B\) में कुल कितने क्रमित युग्म होंगे?
If \(A=\{2,4,6\}\) and \(B=\{1,3\}\), 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)=3\times2=6). In exams, first count the 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)=3\times2=6). In exams, first count the elements of both sets.
Step 3
Exam Tip
(n\(A\times B\)=n(A)n(B)=3\times2=6) होता है। परीक्षा में पहले दोनों समुच्चयों के अवयव गिनें।
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यदि \(A=\{a,b,c\}\) और \(B=\{0,1\}\) हैं, तो \(A\times B\) कौन सा है?
If \(A=\{a,b,c\}\) and \(B=\{0,1\}\), which one is \(A\times B\)?
#cartesian-product
#listing
#definition
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A ({(a,0),(a,1),(b,0),(b,1),(c,0),(c,1)})
B ({(0,a),(1,a),(0,b),(1,b),(0,c),(1,c)})
C ({(a,b),(b,c),(c,a)})
D ({(a,0),(b,1),(c,0)})
Explanation opens after your attempt
Correct Answer
A. ({(a,0),(a,1),(b,0),(b,1),(c,0),(c,1)})
Step 1
Concept
In \(A\times B\), the first component comes from (A) and the second from (B). All possible pairs must be written.
Step 2
Why this answer is correct
The correct answer is A. ({(a,0),(a,1),(b,0),(b,1),(c,0),(c,1)}). In \(A\times B\), the first component comes from (A) and the second from (B). All possible pairs must be written.
Step 3
Exam Tip
\(A\times B\) में पहला घटक (A) से और दूसरा घटक (B) से आता है। सभी संभव युग्म लिखना जरूरी है।
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यदि \(A=\{1,3,5\}\) और \(B=\{2,4\}\) हैं, तो कौन सा युग्म \(B\times A\) में है?
If \(A=\{1,3,5\}\) and \(B=\{2,4\}\), which pair belongs to \(B\times A\)?
#cartesian-product
#membership
#order
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A ((4,3))
B ((3,4))
C ((5,2))
D ((1,4))
Explanation opens after your attempt
Correct Answer
A. ((4,3))
Step 1
Concept
In \(B\times A\), the first element must be from (B) and the second from (A). Hence ((4,3)) is correct.
Step 2
Why this answer is correct
The correct answer is A. ((4,3)). In \(B\times A\), the first element must be from (B) and the second from (A). Hence ((4,3)) is correct.
Step 3
Exam Tip
\(B\times A\) में पहला अवयव (B) से और दूसरा (A) से होना चाहिए। इसलिए ((4,3)) सही है।
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यदि \(A=\{0,2,4\}\) और \(B=\{1,2,3\}\) हैं, तो \((2,3)\in A\times B\) का सत्य मान क्या है?
If \(A=\{0,2,4\}\) and \(B=\{1,2,3\}\), what is the truth value of \((2,3)\in A\times B\)?
#cartesian-product
#membership
#truth-value
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A सत्य / true
B असत्य / false
C केवल जब (A=B) / only when (A=B)
D निर्धारित नहीं / not determined
Explanation opens after your attempt
Correct Answer
A. सत्य / true
Step 1
Concept
Since \(2\in A\) and \(3\in B\), \((2,3)\in A\times B\) is true. Check both positions separately in an ordered pair.
Step 2
Why this answer is correct
The correct answer is A. सत्य / true. Since \(2\in A\) and \(3\in B\), \((2,3)\in A\times B\) is true. Check both positions separately in an ordered pair.
Step 3
Exam Tip
क्योंकि \(2\in A\) और \(3\in B\), इसलिए \((2,3)\in A\times B\) सत्य है। क्रमित युग्म में दोनों स्थान अलग-अलग जांचें।
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यदि \(A=\{1,2,3,4\}\), \(B=\{2,4,6\}\) और \(C=\{4,6,8\}\) हैं, तो (A\times\(B\cap C\)) में कितने अवयव होंगे?
If \(A=\{1,2,3,4\}\), \(B=\{2,4,6\}\) and \(C=\{4,6,8\}\), how many elements are in (A\times\(B\cap C\))?
#cartesian-product
#intersection
#cardinality
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A (8)
B (12)
C (6)
D (4)
Explanation opens after your attempt
Step 1
Concept
\(B\cap C={4,6}\), so (n(A\times\(B\cap C\))=4\times2=8). First find the intersection and then multiply.
Step 2
Why this answer is correct
The correct answer is A. (8). \(B\cap C={4,6}\), so (n(A\times\(B\cap C\))=4\times2=8). First find the intersection and then multiply.
Step 3
Exam Tip
\(B\cap C={4,6}\), इसलिए (n(A\times\(B\cap C\))=4\times2=8)। पहले प्रतिच्छेद निकालें फिर गुणन करें।
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यदि \(A=\{1,2\}\), \(B=\{3,4\}\) और \(C=\{4,5,6\}\) हैं, तो (A\times\(B\cup C\)) में कितने अवयव होंगे?
If \(A=\{1,2\}\), \(B=\{3,4\}\) and \(C=\{4,5,6\}\), how many elements are in (A\times\(B\cup C\))?
#cartesian-product
#union
#counting
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A (8)
B (10)
C (6)
D (12)
Explanation opens after your attempt
Step 1
Concept
\(B\cup C={3,4,5,6}\), so total pairs are \(2\times4=8\). Do not count a common element twice in union.
Step 2
Why this answer is correct
The correct answer is A. (8). \(B\cup C={3,4,5,6}\), so total pairs are \(2\times4=8\). Do not count a common element twice in union.
Step 3
Exam Tip
\(B\cup C={3,4,5,6}\), इसलिए कुल युग्म \(2\times4=8\) हैं। संघ में समान अवयव को दो बार न गिनें।
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यदि \(A=\{1,2,3,4,5\}\), \(B=\{1,3,5\}\) और \(C=\{0,2\}\) हैं, तो \((A-B)\times C\) में कितने अवयव होंगे?
If \(A=\{1,2,3,4,5\}\), \(B=\{1,3,5\}\) and \(C=\{0,2\}\), how many elements are there in \((A-B)\times C\)?
#cartesian-product
#set-difference
#cardinality
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A (4)
B (6)
C (8)
D (10)
Explanation opens after your attempt
Step 1
Concept
(A-B={2,4}), so (n\((A-B)\times C\)=2\times2=4). It is necessary to find the set difference first.
Step 2
Why this answer is correct
The correct answer is A. (4). (A-B={2,4}), so (n\((A-B)\times C\)=2\times2=4). It is necessary to find the set difference first.
Step 3
Exam Tip
(A-B={2,4}), इसलिए (n\((A-B)\times C\)=2\times2=4)। पहले समुच्चय अंतर निकालना जरूरी है।
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यदि (n(A)=5) और (n\(A\times B\)=35) है, तो (n(B)) कितना होगा?
If (n(A)=5) and (n\(A\times B\)=35), what is (n(B))?
#cartesian-product
#cardinality
#numerical
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A (7)
B (30)
C (40)
D (5)
Explanation opens after your attempt
Step 1
Concept
(n\(A\times B\)=n(A)n(B)), so (35=5n(B)) and (n(B)=7). In such questions, use the formula in reverse too.
Step 2
Why this answer is correct
The correct answer is A. (7). (n\(A\times B\)=n(A)n(B)), so (35=5n(B)) and (n(B)=7). In such questions, use the formula in reverse too.
Step 3
Exam Tip
(n\(A\times B\)=n(A)n(B)), इसलिए (35=5n(B)) और (n(B)=7)। ऐसे प्रश्नों में सूत्र को उल्टा भी लगाएं।
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यदि (n(A)=3), (n(B)=4) और (A,B) दोनों अरिक्त हैं, तो \(A\times B\) और \(B\times A\) में अवयवों की संख्या का संबंध क्या है?
If (n(A)=3), (n(B)=4) and both (A,B) are non-empty, what is the relation between the numbers of elements in \(A\times B\) and \(B\times A\)?
#cartesian-product
#comparison
#noncommutative
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A दोनों में (12) अवयव होंगे / both have (12) elements
B \(A\times B\) में (7) अवयव होंगे / \(A\times B\) has (7) elements
C \(B\times A\) में (1) अवयव होगा / \(B\times A\) has (1) element
D दोनों हमेशा समान समुच्चय होंगे / both are always equal sets
Explanation opens after your attempt
Correct Answer
A. दोनों में (12) अवयव होंगे / both have (12) elements
Step 1
Concept
Both have \(3\times4=12\) elements. The number can be the same, but the order of pairs is generally different.
Step 2
Why this answer is correct
The correct answer is A. दोनों में (12) अवयव होंगे / both have (12) elements. Both have \(3\times4=12\) elements. The number can be the same, but the order of pairs is generally different.
Step 3
Exam Tip
दोनों में अवयवों की संख्या \(3\times4=12\) होती है। संख्या समान हो सकती है पर युग्मों का क्रम सामान्यतः अलग होता है।
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यदि \(A={x:x\in\mathbb{N},x\le4}\) और \(B={y:y\in\mathbb{N},y<3}\) हैं, तो (n\(A\times B\)) कितना है?
If \(A={x:x\in\mathbb{N},x\le4}\) and \(B={y:y\in\mathbb{N},y<3}\), what is (n\(A\times B\))?
#cartesian-product
#set-builder
#natural-numbers
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A (8)
B (6)
C (12)
D (4)
Explanation opens after your attempt
Step 1
Concept
\(A=\{1,2,3,4\}\) and \(B=\{1,2\}\), so (n\(A\times B\)=4\times2=8). Convert set-builder form into roster form first.
Step 2
Why this answer is correct
The correct answer is A. (8). \(A=\{1,2,3,4\}\) and \(B=\{1,2\}\), so (n\(A\times B\)=4\times2=8). Convert set-builder form into roster form first.
Step 3
Exam Tip
\(A=\{1,2,3,4\}\) और \(B=\{1,2\}\), इसलिए (n\(A\times B\)=4\times2=8)। सेट-बिल्डर रूप को पहले सूची रूप में बदलें।
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यदि \(A={x:x\in\mathbb{Z},-2\le x\le1}\) और \(B=\{0,2\}\) हैं, तो \(A\times B\) में कितने युग्म होंगे?
If \(A={x:x\in\mathbb{Z},-2\le x\le1}\) and \(B=\{0,2\}\), how many pairs are in \(A\times B\)?
#cartesian-product
#set-builder
#integers
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A (8)
B (6)
C (4)
D (10)
Explanation opens after your attempt
Step 1
Concept
\(A=\{-2,-1,0,1\}\) has (4) elements and (B) has (2) elements. Therefore total pairs are \(4\times2=8\).
Step 2
Why this answer is correct
The correct answer is A. (8). \(A=\{-2,-1,0,1\}\) has (4) elements and (B) has (2) elements. Therefore total pairs are \(4\times2=8\).
Step 3
Exam Tip
\(A=\{-2,-1,0,1\}\) में (4) अवयव हैं और (B) में (2) अवयव हैं। इसलिए कुल युग्म \(4\times2=8\) होंगे।
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यदि \(A=\{1,2,3\}\) और \(B=\varnothing\) हैं, तो \(B\times A\) क्या होगा?
If \(A=\{1,2,3\}\) and \(B=\varnothing\), what is \(B\times A\)?
#cartesian-product
#empty-set
#concept
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A \(\varnothing\)
B ({1,2,3})
C ({\(\varnothing,1\),\(\varnothing,2\),\(\varnothing,3\)})
D ({\(1,\varnothing\),\(2,\varnothing\),\(3,\varnothing\)})
Explanation opens after your attempt
Correct Answer
A. \(\varnothing\)
Step 1
Concept
No first component can be taken from the empty set, so no ordered pair is formed. If either side is empty, the Cartesian product is \(\varnothing\).
Step 2
Why this answer is correct
The correct answer is A. \(\varnothing\). No first component can be taken from the empty set, so no ordered pair is formed. If either side is empty, the Cartesian product is \(\varnothing\).
Step 3
Exam Tip
रिक्त समुच्चय से पहला घटक नहीं लिया जा सकता, इसलिए कोई क्रमित युग्म नहीं बनेगा। किसी भी तरफ रिक्त समुच्चय हो तो कार्तीय गुणन \(\varnothing\) होता है।
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यदि \(A=\{p,q\}\) है, तो \(A\times A\) में कितने क्रमित युग्म होंगे?
If \(A=\{p,q\}\), how many ordered pairs are in \(A\times A\)?
#cartesian-product
#self-product
#cardinality
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A (4)
B (2)
C (3)
D (1)
Explanation opens after your attempt
Step 1
Concept
(n\(A\times A\)=n(A)2 =22 =4). Pairs with equal components such as ((p,p)) and ((q,q)) are also included.
Step 2
Why this answer is correct
The correct answer is A. (4). (n\(A\times A\)=n(A)2 =22 =4). Pairs with equal components such as ((p,p)) and ((q,q)) are also included.
Step 3
Exam Tip
(n\(A\times A\)=n(A)2 =22 =4) होता है। ((p,p)) और ((q,q)) जैसे समान घटक वाले युग्म भी शामिल होते हैं।
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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<y)?
If \(A=\{1,2,3,4\}\) and \(B=\{2,4,6\}\), how many pairs ((x,y)) in \(A\times B\) satisfy (x<y)?
#cartesian-product
#inequality
#counting
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A (9)
B (8)
C (7)
D (6)
Explanation opens after your attempt
Step 1
Concept
The valid pairs are ((1,2),(1,4),(2,4),(3,4),(1,6),(2,6),(3,6),(4,6)), so the count is (8). List carefully to avoid overcounting.
Step 2
Why this answer is correct
The correct answer is A. (9). The valid pairs are ((1,2),(1,4),(2,4),(3,4),(1,6),(2,6),(3,6),(4,6)), so the count is (8). List carefully to avoid overcounting.
Step 3
Exam Tip
(y=2) के लिए (1) युग्म, (y=4) के लिए (3) युग्म और (y=6) के लिए (4) युग्म मिलते हैं। कुल (1+3+4=8) नहीं, सही गिनती ((1,2),(1,4),(2,4),(3,4),(1,6),(2,6),(3,6),(4,6)) यानी (8) है।
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यदि \(A=\{0,1,2,3\}\) और \(B=\{1,2,3\}\) हैं, तो \(A\times B\) में कितने युग्म ((x,y)) ऐसे हैं कि (x+y=4)?
If \(A=\{0,1,2,3\}\) and \(B=\{1,2,3\}\), how many pairs ((x,y)) in \(A\times B\) satisfy (x+y=4)?
#cartesian-product
#sum-condition
#application
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A (3)
B (4)
C (2)
D (1)
Explanation opens after your attempt
Step 1
Concept
The valid pairs are ((1,3),(2,2),(3,1)). The positions in an ordered pair remain fixed.
Step 2
Why this answer is correct
The correct answer is A. (3). The valid pairs are ((1,3),(2,2),(3,1)). The positions in an ordered pair remain fixed.
Step 3
Exam Tip
सही युग्म ((1,3),(2,2),(3,1)) हैं। क्रमित युग्म में स्थान निश्चित रहते हैं।
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यदि \(A=\{1,2,4\}\) और \(B=\{1,2,4,8\}\) हैं, तो \(A\times B\) में कितने युग्म ((x,y)) ऐसे हैं कि (y=2x)?
If \(A=\{1,2,4\}\) and \(B=\{1,2,4,8\}\), how many pairs ((x,y)) in \(A\times B\) satisfy (y=2x)?
#cartesian-product
#equation-condition
#function-pattern
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A (3)
B (2)
C (4)
D (6)
Explanation opens after your attempt
Step 1
Concept
The valid pairs are ((1,2),(2,4),(4,8)). A simple method is to choose (x) first and then find (y).
Step 2
Why this answer is correct
The correct answer is A. (3). The valid pairs are ((1,2),(2,4),(4,8)). A simple method is to choose (x) first and then find (y).
Step 3
Exam Tip
सही युग्म ((1,2),(2,4),(4,8)) हैं। पहले (x) चुनकर (y) निकालना आसान तरीका है।
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यदि \(A=\{2,3,6\}\) और \(B=\{1,2,3\}\) हैं, तो \(A\times B\) में कितने युग्म ((x,y)) ऐसे हैं कि (y) संख्या (x) को विभाजित करती है?
If \(A=\{2,3,6\}\) and \(B=\{1,2,3\}\), how many pairs ((x,y)) in \(A\times B\) satisfy that (y) divides (x)?
#cartesian-product
#divisibility
#counting
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A (7)
B (6)
C (5)
D (8)
Explanation opens after your attempt
Step 1
Concept
The valid pairs are ((2,1),(2,2),(3,1),(3,3),(6,1),(6,2),(6,3)). In divisibility, note which component is the divisor.
Step 2
Why this answer is correct
The correct answer is A. (7). The valid pairs are ((2,1),(2,2),(3,1),(3,3),(6,1),(6,2),(6,3)). In divisibility, note which component is the divisor.
Step 3
Exam Tip
सही युग्म ((2,1),(2,2),(3,1),(3,3),(6,1),(6,2),(6,3)) हैं। विभाज्यता में कौन सा घटक भाजक है यह ध्यान रखें।
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यदि \(A=\{1,2\}\) और \(B=\{3,4,5\}\) हैं, तो \(A\times B\) में पहले घटक (2) वाले कितने युग्म होंगे?
If \(A=\{1,2\}\) and \(B=\{3,4,5\}\), how many pairs in \(A\times B\) have first component (2)?
#cartesian-product
#components
#counting
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A (3)
B (2)
C (1)
D (6)
Explanation opens after your attempt
Step 1
Concept
When the first component (2) is fixed, the second component can be any of the (3) elements of (B). Hence there are (3) pairs.
Step 2
Why this answer is correct
The correct answer is A. (3). When the first component (2) is fixed, the second component can be any of the (3) elements of (B). Hence there are (3) pairs.
Step 3
Exam Tip
पहला घटक (2) तय करने पर दूसरा घटक (B) के (3) अवयवों में से कोई भी हो सकता है। इसलिए (3) युग्म मिलेंगे।
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यदि \(A=\{r,s,t,u\}\) और \(B=\{5,6\}\) हैं, तो \(A\times B\) में दूसरा घटक (6) वाले कितने युग्म होंगे?
If \(A=\{r,s,t,u\}\) and \(B=\{5,6\}\), how many pairs in \(A\times B\) have second component (6)?
#cartesian-product
#components
#counting
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A (4)
B (2)
C (6)
D (8)
Explanation opens after your attempt
Step 1
Concept
When the second component (6) is fixed, the first component can be any of the (4) elements of (A). So there are (4) pairs.
Step 2
Why this answer is correct
The correct answer is A. (4). When the second component (6) is fixed, the first component can be any of the (4) elements of (A). So there are (4) pairs.
Step 3
Exam Tip
दूसरा घटक (6) तय होने पर पहला घटक (A) के (4) अवयवों में से कोई भी हो सकता है। इसलिए (4) युग्म होंगे।
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यदि \(A=\{1,2,3\}\) और \(B=\{4,5\}\) हैं, तो (A) से (B) तक कुल कितने संबंध संभव हैं?
If \(A=\{1,2,3\}\) and \(B=\{4,5\}\), how many relations are possible from (A) to (B)?
#cartesian-product
#relations
#subsets
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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 total 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 total 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 होता है। इसलिए कुल संबंध \(2^6\) हैं।
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यदि \(A=\{1,2,3\}\) और \(B=\{4,5\}\) हैं, तो \(A\times B\) का कौन सा उपसमुच्चय (A) से (B) तक संबंध है?
If \(A=\{1,2,3\}\) and \(B=\{4,5\}\), which subset of \(A\times B\) is a relation from (A) to (B)?
#cartesian-product
#relation
#subset
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A ({(1,4),(3,5)})
B ({(4,1),(5,3)})
C ({(1,6)})
D ({(0,4),(2,5)})
Explanation opens after your attempt
Correct Answer
A. ({(1,4),(3,5)})
Step 1
Concept
A relation is a subset of \(A\times B\). Only all pairs of ({(1,4),(3,5)}) belong to \(A\times B\).
Step 2
Why this answer is correct
The correct answer is A. ({(1,4),(3,5)}). A relation is a subset of \(A\times B\). Only all pairs of ({(1,4),(3,5)}) belong to \(A\times B\).
Step 3
Exam Tip
संबंध \(A\times B\) का उपसमुच्चय होता है। केवल ({(1,4),(3,5)}) के सभी युग्म \(A\times B\) में हैं।
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यदि \(A=\{0,1\}\) और \(B=\{1,2,3\}\) हैं, तो \(A\times B\) के कितने उपसमुच्चय होंगे?
If \(A=\{0,1\}\) and \(B=\{1,2,3\}\), how many subsets does \(A\times B\) have?
#cartesian-product
#subsets
#counting
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A \(2^6\)
B (6)
C \(2^5\)
D \(3^2\)
Explanation opens after your attempt
Correct Answer
A. \(2^6\)
Step 1
Concept
(n\(A\times B\)=2\times3=6), so the number of subsets is \(2^6\). Use the formula \(2^n\) for counting subsets.
Step 2
Why this answer is correct
The correct answer is A. \(2^6\). (n\(A\times B\)=2\times3=6), so the number of subsets is \(2^6\). Use the formula \(2^n\) for counting subsets.
Step 3
Exam Tip
(n\(A\times B\)=2\times3=6), इसलिए उपसमुच्चयों की संख्या \(2^6\) है। उपसमुच्चय गिनने में \(2^n\) सूत्र लगाएं।
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यदि \(A=\{1,2\}\) और \(B=\{2,3\}\) हैं, तो \(A\times B\cap B\times A\) में कितने युग्म होंगे?
If \(A=\{1,2\}\) and \(B=\{2,3\}\), how many pairs are in \(A\times B\cap B\times A\)?
#cartesian-product
#intersection
#ordered-pairs
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A (1)
B (2)
C (3)
D (4)
Explanation opens after your attempt
Step 1
Concept
Only ((2,2)) is common to both. For intersection, the whole ordered pair must be identical.
Step 2
Why this answer is correct
The correct answer is A. (1). Only ((2,2)) is common to both. For intersection, the whole ordered pair must be identical.
Step 3
Exam Tip
दोनों में केवल ((2,2)) समान युग्म है। प्रतिच्छेद के लिए पूरा क्रमित युग्म समान होना चाहिए।
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यदि (A=[0,2]) और \(B=\{1\}\) हैं, तो \(A\times B\) का ज्यामितीय रूप क्या है?
If (A=[0,2]) and \(B=\{1\}\), what is the geometric form of \(A\times B\)?
#cartesian-product
#interval
#geometry
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A रेखा (y=1) पर \(0\le x\le2\) वाला रेखाखंड / line segment on (y=1) with \(0\le x\le2\)
B रेखा (x=1) पर \(0\le y\le2\) वाला रेखाखंड / line segment on (x=1) with \(0\le y\le2\)
C पूरा तल \(\mathbb{R}^2\) / whole plane \(\mathbb{R}^2\)
D केवल बिंदु ((1,0)) / only point ((1,0))
Explanation opens after your attempt
Correct Answer
A. रेखा (y=1) पर \(0\le x\le2\) वाला रेखाखंड / line segment on (y=1) with \(0\le x\le2\)
Step 1
Concept
\(A\times B={(x,1):0\le x\le2}\). Hence it is a horizontal line segment on (y=1).
Step 2
Why this answer is correct
The correct answer is A. रेखा (y=1) पर \(0\le x\le2\) वाला रेखाखंड / line segment on (y=1) with \(0\le x\le2\). \(A\times B={(x,1):0\le x\le2}\). Hence it is a horizontal line segment on (y=1).
Step 3
Exam Tip
\(A\times B={(x,1):0\le x\le2}\) है। इसलिए यह (y=1) पर एक क्षैतिज रेखाखंड है।
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यदि \(A=\{-1\}\) और (B=[2,5]) हैं, तो \(A\times B\) कौन सा है?
If \(A=\{-1\}\) and (B=[2,5]), which one is \(A\times B\)?
#cartesian-product
#interval
#set-builder
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A ({(-1,y):2\le y\le5})
B ({(x,-1):2\le x\le5})
C ({(2,y):-1\le y\le5})
D ({(-1,2)})
Explanation opens after your attempt
Correct Answer
A. ({(-1,y):2\le y\le5})
Step 1
Concept
The first component is always (-1), and the second varies in ([2,5]). This gives a vertical line segment.
Step 2
Why this answer is correct
The correct answer is A. ({(-1,y):2\le y\le5}). The first component is always (-1), and the second varies in ([2,5]). This gives a vertical line segment.
Step 3
Exam Tip
पहला घटक हमेशा (-1) है और दूसरा ([2,5]) में बदलता है। यह ऊर्ध्वाधर रेखाखंड का रूप देता है।
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यदि \(A=\{1,2\}\), \(B=\{3\}\) और \(C=\{4,5\}\) हैं, तो (A\times\(B\times C\)) में कितने अवयव होंगे?
If \(A=\{1,2\}\), \(B=\{3\}\) and \(C=\{4,5\}\), how many elements are in (A\times\(B\times C\))?
#cartesian-product
#nested-product
#cardinality
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A (4)
B (5)
C (6)
D (2)
Explanation opens after your attempt
Step 1
Concept
(n\(B\times C\)=1\times2=2), so (n(A\times\(B\times C\))=2\times2=4). Count the inner Cartesian product first.
Step 2
Why this answer is correct
The correct answer is A. (4). (n\(B\times C\)=1\times2=2), so (n(A\times\(B\times C\))=2\times2=4). Count the inner Cartesian product first.
Step 3
Exam Tip
(n\(B\times C\)=1\times2=2), इसलिए (n(A\times\(B\times C\))=2\times2=4)। अंदर का कार्तीय गुणन पहले गिनें।
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यदि \(A=\{m,n\}\) और \(B=\{7,8,9\}\) हैं, तो \(A\times B\) में दूसरे घटकों का समुच्चय क्या है?
If \(A=\{m,n\}\) and \(B=\{7,8,9\}\), what is the set of second components in \(A\times B\)?
#cartesian-product
#projection
#components
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A ({7,8,9})
B ({m,n})
C ({m,n,7,8,9})
D ({(m,7),(n,9)})
Explanation opens after your attempt
Correct Answer
A. ({7,8,9})
Step 1
Concept
Second components always come from (B). Since (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. ({7,8,9}). Second components always come from (B). Since (A) is non-empty, every element of (B) appears as a second component.
Step 3
Exam Tip
दूसरे घटक हमेशा (B) से आते हैं। क्योंकि (A) अरिक्त है, (B) का हर अवयव दूसरे घटक में आएगा।
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यदि \(A=\{1,2,3\}\) और \(B=\{2,3,4\}\) हैं, तो \(A\times B\) में कितने युग्म ((x,y)) ऐसे हैं कि (x+y) विषम है?
If \(A=\{1,2,3\}\) and \(B=\{2,3,4\}\), how many pairs ((x,y)) in \(A\times B\) have (x+y) odd?
#cartesian-product
#parity
#counting
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A (5)
B (4)
C (6)
D (3)
Explanation opens after your attempt
Step 1
Concept
A sum is odd when one component is odd and the other is even. Here (5) such pairs are formed.
Step 2
Why this answer is correct
The correct answer is A. (5). A sum is odd when one component is odd and the other is even. Here (5) such pairs are formed.
Step 3
Exam Tip
योग विषम तब होता है जब एक घटक विषम और दूसरा सम हो। यहां ऐसे (5) युग्म बनते हैं।
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यदि \(A=\{0,1,2\}\) और \(B=\{0,2,4\}\) हैं, तो \(A\times B\) में कितने युग्म ((x,y)) ऐसे हैं कि (x+y=2)?
If \(A=\{0,1,2\}\) and \(B=\{0,2,4\}\), how many pairs ((x,y)) in \(A\times B\) satisfy (x+y=2)?
#cartesian-product
#sum-condition
#membership
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A (2)
B (1)
C (3)
D (4)
Explanation opens after your attempt
Step 1
Concept
The valid pairs are ((0,2)) and ((2,0)). ((1,1)) will not appear because \(1\notin B\).
Step 2
Why this answer is correct
The correct answer is A. (2). The valid pairs are ((0,2)) and ((2,0)). ((1,1)) will not appear because \(1\notin B\).
Step 3
Exam Tip
सही युग्म ((0,2)) और ((2,0)) हैं। ((1,1)) नहीं आएगा क्योंकि \(1\notin B\)।
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यदि \(A=\{1,2,4\}\) और \(B=\{2,3,5\}\) हैं, तो \(A\times B\) में सबसे बड़े घटक-योग वाला युग्म कौन सा है?
If \(A=\{1,2,4\}\) and \(B=\{2,3,5\}\), which pair in \(A\times B\) has the greatest sum of components?
#cartesian-product
#maximum
#application
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A ((4,5))
B ((5,4))
C ((2,5))
D ((4,3))
Explanation opens after your attempt
Correct Answer
A. ((4,5))
Step 1
Concept
The greatest first component is (4) and the greatest second component is (5). So ((4,5)) has the greatest sum.
Step 2
Why this answer is correct
The correct answer is A. ((4,5)). The greatest first component is (4) and the greatest second component is (5). So ((4,5)) has the greatest sum.
Step 3
Exam Tip
सबसे बड़ा पहला घटक (4) और सबसे बड़ा दूसरा घटक (5) है। इसलिए ((4,5)) का योग सबसे बड़ा है।
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यदि \(A=\{1,2,3,4\}\) और \(B=\{1,2\}\) हैं, तो \(A\times B\) में कितने युग्म ((x,y)) ऐसे हैं कि (x-y=2)?
If \(A=\{1,2,3,4\}\) and \(B=\{1,2\}\), how many pairs ((x,y)) in \(A\times B\) satisfy (x-y=2)?
#cartesian-product
#difference-condition
#application
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A (2)
B (1)
C (3)
D (4)
Explanation opens after your attempt
Step 1
Concept
The valid pairs are ((3,1)) and ((4,2)). A quick method is to choose (y) and find (x=y+2).
Step 2
Why this answer is correct
The correct answer is A. (2). The valid pairs are ((3,1)) and ((4,2)). A quick method is to choose (y) and find (x=y+2).
Step 3
Exam Tip
सही युग्म ((3,1)) और ((4,2)) हैं। (y) चुनकर (x=y+2) निकालना तेज तरीका है।
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यदि \(A=\{1,2,3,4\}\) और \(B=\{2,4,6,8\}\) हैं, तो \(A\times B\) में कितने युग्म ((x,y)) ऐसे हैं कि (xy=8)?
If \(A=\{1,2,3,4\}\) and \(B=\{2,4,6,8\}\), how many pairs ((x,y)) in \(A\times B\) satisfy (xy=8)?
#cartesian-product
#product-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 ((1,8),(2,4),(4,2)). In a product condition, always check membership of both components.
Step 2
Why this answer is correct
The correct answer is A. (3). The valid pairs are ((1,8),(2,4),(4,2)). In a product condition, always check membership of both components.
Step 3
Exam Tip
सही युग्म ((1,8),(2,4),(4,2)) हैं। उत्पाद शर्त में दोनों घटकों की सदस्यता जरूर जांचें।
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यदि \(A=\{1,2,3\}\), \(B=\{3,4\}\) और \(C=\{5,6\}\) हैं, तो \(A\times(B-C)\) क्या होगा?
If \(A=\{1,2,3\}\), \(B=\{3,4\}\) and \(C=\{5,6\}\), what is \(A\times(B-C)\)?
#cartesian-product
#set-difference
#listing
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A ({(1,3),(1,4),(2,3),(2,4),(3,3),(3,4)})
B \(\varnothing\)
C ({(1,5),(2,6)})
D ({(3,1),(4,1),(3,2),(4,2),(3,3),(4,3)})
Explanation opens after your attempt
Correct Answer
A. ({(1,3),(1,4),(2,3),(2,4),(3,3),(3,4)})
Step 1
Concept
Since (B-C={3,4}), each element of (A) pairs with (3) and (4). Find the difference first and then write the pairs.
Step 2
Why this answer is correct
The correct answer is A. ({(1,3),(1,4),(2,3),(2,4),(3,3),(3,4)}). Since (B-C={3,4}), each element of (A) pairs with (3) and (4). Find the difference first and then write the pairs.
Step 3
Exam Tip
क्योंकि (B-C={3,4}), इसलिए (A) के हर अवयव के साथ (3) और (4) जुड़ेंगे। पहले अंतर निकालें फिर युग्म लिखें।
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यदि \(A=\{1,2\}\) और \(B=\{3,4,5\}\) हैं, तो \(A\times B\) का सही सेट-बिल्डर रूप कौन सा है?
If \(A=\{1,2\}\) and \(B=\{3,4,5\}\), 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
Cartesian product is a set of ordered pairs. Therefore the correct form is \({(x,y):x\in A,,y\in B}\).
Step 2
Why this answer is correct
The correct answer is A. \({(x,y):x\in A,,y\in B}\). Cartesian product is a set of ordered pairs. Therefore the correct form is \({(x,y):x\in A,,y\in B}\).
Step 3
Exam Tip
कार्तीय गुणन क्रमित युग्मों का समुच्चय है। इसलिए सही रूप \({(x,y):x\in A,,y\in B}\) 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 \(x\le y\)?
If \(A=\{1,2,3\}\) and \(B=\{1,2,3\}\), how many pairs ((x,y)) in \(A\times B\) satisfy \(x\le y\)?
#cartesian-product
#inequality
#equality-included
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A (6)
B (3)
C (9)
D (5)
Explanation opens after your attempt
Step 1
Concept
The valid pairs are ((1,1),(1,2),(1,3),(2,2),(2,3),(3,3)). Include equal pairs when the condition is \(\le\).
Step 2
Why this answer is correct
The correct answer is A. (6). The valid pairs are ((1,1),(1,2),(1,3),(2,2),(2,3),(3,3)). Include equal pairs when the condition is \(\le\).
Step 3
Exam Tip
सही युग्म ((1,1),(1,2),(1,3),(2,2),(2,3),(3,3)) हैं। \(\le\) में बराबरी वाले युग्म भी शामिल करें।
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यदि \(A=\{1,2,3\}\) और \(B=\{2,3,4\}\) हैं, तो \(A\times B\) में कितने युग्म ((x,y)) ऐसे हैं कि (x+y>5)?
If \(A=\{1,2,3\}\) and \(B=\{2,3,4\}\), how many pairs ((x,y)) in \(A\times B\) satisfy (x+y>5)?
#cartesian-product
#inequality
#sum-condition
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A (3)
B (4)
C (2)
D (5)
Explanation opens after your attempt
Step 1
Concept
The valid pairs are ((2,4),(3,3),(3,4)). Pairs with (x+y=5) are not included.
Step 2
Why this answer is correct
The correct answer is A. (3). The valid pairs are ((2,4),(3,3),(3,4)). Pairs with (x+y=5) are not included.
Step 3
Exam Tip
सही युग्म ((2,4),(3,3),(3,4)) हैं। (x+y=5) वाले युग्म शामिल नहीं होंगे।
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कथन: यदि \(A\ne\varnothing\) और \(B\ne\varnothing\), तो \(A\times B=\varnothing\)। यह कथन कैसा है?
Statement: If \(A\ne\varnothing\) and \(B\ne\varnothing\), then \(A\times B=\varnothing\). What is this statement?
#cartesian-product
#assertion
#empty-set
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A असत्य / false
B सत्य / true
C कभी सत्य कभी असत्य / sometimes true sometimes false
D निर्धारित नहीं / not determined
Explanation opens after your attempt
Correct Answer
A. असत्य / false
Step 1
Concept
If both sets are non-empty, at least one ordered pair is formed. Therefore \(A\times B=\varnothing\) cannot happen.
Step 2
Why this answer is correct
The correct answer is A. असत्य / false. If both sets are non-empty, at least one ordered pair is formed. Therefore \(A\times B=\varnothing\) cannot happen.
Step 3
Exam Tip
यदि दोनों समुच्चय अरिक्त हैं तो कम से कम एक क्रमित युग्म बनता है। इसलिए \(A\times B=\varnothing\) नहीं हो सकता।
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कथन: \(A\times B=B\times A\) हमेशा सत्य है। सही विकल्प चुनिए।
Statement: \(A\times B=B\times A\) is always true. Choose the correct option.
#cartesian-product
#assertion
#noncommutative
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A कथन असत्य है / the statement is false
B कथन सदैव सत्य है / the statement is always true
C कथन केवल (n(A)=n(B)) से सत्य है / the statement is true only because (n(A)=n(B))
D कथन सभी अरिक्त समुच्चयों के लिए सत्य है / the statement is true for all non-empty sets
Explanation opens after your attempt
Correct Answer
A. कथन असत्य है / the statement is false
Step 1
Concept
Order matters in Cartesian product, so it is generally not commutative. It may be equal in special cases such as (A=B) or because of an empty set.
Step 2
Why this answer is correct
The correct answer is A. कथन असत्य है / the statement is false. Order matters in Cartesian product, so it is generally not commutative. It may be equal in special cases such as (A=B) or because of an empty set.
Step 3
Exam Tip
कार्तीय गुणन में क्रम महत्वपूर्ण है, इसलिए यह सामान्यतः क्रमविनिमेय नहीं होता। यह विशेष स्थिति में (A=B) या रिक्त समुच्चय के कारण समान हो सकता है।
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\(यदि (A={1,2,3}) और (B={a,b}) हैं, तो (A\times B) में कौन सा युग्म उस संबंध (R={(x,y):x\in A,,y\in B,,x\) विषम है}) में होगा?
\(If (A={1,2,3}) and (B={a,b}), which pair belongs to the relation (R={(x,y):x\in A,,y\in B,,x\) is odd})?
#cartesian-product
#relation
#condition
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A ((3,b))
B ((2,a))
C ((a,3))
D ((b,1))
Explanation opens after your attempt
Correct Answer
A. ((3,b))
Step 1
Concept
\(3\in A\) is odd and \(b\in B\), so \((3,b)\in R\). Apply the given condition to the Cartesian product.
Step 2
Why this answer is correct
The correct answer is A. ((3,b)). \(3\in A\) is odd and \(b\in B\), so \((3,b)\in R\). Apply the given condition to the Cartesian product.
Step 3
Exam Tip
\(3\in A\) विषम है और \(b\in B\), इसलिए \((3,b)\in R\)। संबंध में दी गई शर्त को कार्तीय गुणन पर लागू करें।
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यदि \(A=\{1,2\}\), \(B=\{3,4\}\) और \(R=\{(1,3),(2,4)\}\) है, तो (R) किसका उपसमुच्चय है?
If \(A=\{1,2\}\), \(B=\{3,4\}\) and \(R=\{(1,3),(2,4)\}\), then (R) is a subset of which set?
#cartesian-product
#subset
#relation
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A \(A\times B\)
B \(B\times A\)
C \(A\cap B\)
D \(A\cup B\)
Explanation opens after your attempt
Correct Answer
A. \(A\times B\)
Step 1
Concept
In every pair of (R), the first component is from (A) and the second is from (B). Therefore \(R\subseteq A\times B\).
Step 2
Why this answer is correct
The correct answer is A. \(A\times B\). In every pair of (R), the first component is from (A) and the second is from (B). Therefore \(R\subseteq A\times B\).
Step 3
Exam Tip
(R) के हर युग्म में पहला घटक (A) से और दूसरा (B) से है। इसलिए \(R\subseteq A\times B\)।
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यदि \(A=\{0,1\}\) और \(B=\{2,3\}\) हैं, तो \(A\times B\) में सभी युग्मों के घटक-योगों का समुच्चय क्या है?
If \(A=\{0,1\}\) and \(B=\{2,3\}\), what is the set of sums of components of all pairs in \(A\times B\)?
#cartesian-product
#sum-set
#application
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A ({2,3,4})
B ({2,3})
C ({0,1,2,3})
D ({1,5})
Explanation opens after your attempt
Correct Answer
A. ({2,3,4})
Step 1
Concept
The sums are (0+2=2), (0+3=3), (1+2=3) and (1+3=4). In a set, (3) is not written twice.
Step 2
Why this answer is correct
The correct answer is A. ({2,3,4}). The sums are (0+2=2), (0+3=3), (1+2=3) and (1+3=4). In a set, (3) is not written twice.
Step 3
Exam Tip
युग्मों से योग (0+2=2), (0+3=3), (1+2=3) और (1+3=4) मिलते हैं। समुच्चय में (3) को दो बार नहीं लिखते।
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यदि \(A=\{1,2,3\}\) और \(B=\{2,4,6\}\) हैं, तो \(A\times B\) में कितने युग्म ((x,y)) ऐसे हैं कि (y) सम संख्या है?
If \(A=\{1,2,3\}\) and \(B=\{2,4,6\}\), how many pairs ((x,y)) in \(A\times B\) have (y) even?
#cartesian-product
#parity
#components
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A (9)
B (6)
C (3)
D (0)
Explanation opens after your attempt
Step 1
Concept
All (3) elements of (B) are even and pair with (3) elements of (A). Hence \(3\times3=9\) pairs are obtained.
Step 2
Why this answer is correct
The correct answer is A. (9). All (3) elements of (B) are even and pair with (3) elements of (A). Hence \(3\times3=9\) pairs are obtained.
Step 3
Exam Tip
(B) के सभी (3) अवयव सम हैं और (A) के (3) अवयवों से जुड़ते हैं। इसलिए \(3\times3=9\) युग्म मिलते हैं।
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यदि \(A=\{1,2,3,4\}\) और \(B=\{1,3\}\) हैं, तो \(A\times B\) में कितने युग्म ((x,y)) ऐसे हैं कि (x) विषम है?
If \(A=\{1,2,3,4\}\) and \(B=\{1,3\}\), how many pairs ((x,y)) in \(A\times B\) have (x) odd?
#cartesian-product
#parity
#counting
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A (4)
B (2)
C (6)
D (8)
Explanation opens after your attempt
Step 1
Concept
The odd elements in (A) are (1) and (3), and each pairs with (2) elements of (B). Therefore \(2\times2=4\) pairs are formed.
Step 2
Why this answer is correct
The correct answer is A. (4). The odd elements in (A) are (1) and (3), and each pairs with (2) elements of (B). Therefore \(2\times2=4\) pairs are formed.
Step 3
Exam Tip
(A) में विषम अवयव (1) और (3) हैं, और हर एक (B) के (2) अवयवों से जुड़ेगा। इसलिए \(2\times2=4\) युग्म होंगे।
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यदि \(A=\{1,2\}\), \(B=\{2,3\}\) और \(C=\{3,4\}\) हैं, तो (\(A\cap B\)\times C) में कौन सा युग्म होगा?
If \(A=\{1,2\}\), \(B=\{2,3\}\) and \(C=\{3,4\}\), which pair belongs to (\(A\cap B\)\times C)?
#cartesian-product
#intersection
#membership
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A ((2,4))
B ((1,3))
C ((3,2))
D ((4,2))
Explanation opens after your attempt
Correct Answer
A. ((2,4))
Step 1
Concept
\(A\cap B={2}\), so the first component can only be (2) and the second must be from (C). Hence ((2,4)) is correct.
Step 2
Why this answer is correct
The correct answer is A. ((2,4)). \(A\cap B={2}\), so the first component can only be (2) and the second must be from (C). Hence ((2,4)) is correct.
Step 3
Exam Tip
\(A\cap B={2}\), इसलिए पहला घटक केवल (2) हो सकता है और दूसरा (C) से होगा। इसलिए ((2,4)) सही है।
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यदि \(A=\{1,2,5\}\), \(B=\{2,3,5\}\) और \(C=\{5,7\}\) हैं, तो (A\times\(B\cap C\)) में कितने अवयव होंगे?
If \(A=\{1,2,5\}\), \(B=\{2,3,5\}\) and \(C=\{5,7\}\), how many elements are there in (A\times\(B\cap C\))?
#cartesian-product
#intersection
#cardinality
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A (3)
B (6)
C (9)
D (1)
Explanation opens after your attempt
Step 1
Concept
\(B\cap C={5}\), so (n(A\times\(B\cap C\))=3\times1=3). First find the intersection and then count the Cartesian product.
Step 2
Why this answer is correct
The correct answer is A. (3). \(B\cap C={5}\), so (n(A\times\(B\cap C\))=3\times1=3). First find the intersection and then count the Cartesian product.
Step 3
Exam Tip
\(B\cap C={5}\), इसलिए (n(A\times\(B\cap C\))=3\times1=3)। पहले प्रतिच्छेद निकालें फिर कार्तीय गुणन गिनें।
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यदि \(A=\{0,2,4\}\) और \(B=\{1,3,5\}\) हैं, तो \(A\times B\) में कितने युग्म ((x,y)) ऐसे हैं कि (x+y=5)?
If \(A=\{0,2,4\}\) and \(B=\{1,3,5\}\), how many pairs ((x,y)) in \(A\times B\) satisfy (x+y=5)?
#cartesian-product
#sum-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 ((0,5),(2,3),(4,1)). In a sum condition, always check membership of both components.
Step 2
Why this answer is correct
The correct answer is A. (3). The valid pairs are ((0,5),(2,3),(4,1)). In a sum condition, always check membership of both components.
Step 3
Exam Tip
सही युग्म ((0,5),(2,3),(4,1)) हैं। योग की शर्त में दोनों घटकों की सदस्यता जरूर जांचें।
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यदि \(A=\{1,2,3,4\}\) और \(B=\{1,2,4\}\) हैं, तो \(A\times B\) में कितने युग्म ((x,y)) ऐसे हैं कि (x) संख्या (y) से बड़ा या बराबर है?
If \(A=\{1,2,3,4\}\) and \(B=\{1,2,4\}\), how many pairs ((x,y)) in \(A\times B\) satisfy \(x\ge y\)?
#cartesian-product
#inequality
#counting
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A (8)
B (7)
C (6)
D (9)
Explanation opens after your attempt
Step 1
Concept
The correct count is (4) for (y=1), (3) for (y=2), and (1) for (y=4). Thus total pairs are (4+3+1=8).
Step 2
Why this answer is correct
The correct answer is A. (8). The correct count is (4) for (y=1), (3) for (y=2), and (1) for (y=4). Thus total pairs are (4+3+1=8).
Step 3
Exam Tip
सही गिनती (y=1) के लिए (4), (y=2) के लिए (3) और (y=4) के लिए (1) है। इसलिए कुल (4+3+1=8) युग्म हैं।
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यदि \(A=\{2,4,8\}\) और \(B=\{1,2,4\}\) हैं, तो \(A\times B\) में कितने युग्म ((x,y)) ऐसे हैं कि \(\frac{x}{y}=2\)?
If \(A=\{2,4,8\}\) and \(B=\{1,2,4\}\), how many pairs ((x,y)) in \(A\times B\) satisfy \(\frac{x}{y}=2\)?
#cartesian-product
#fraction-condition
#application
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A (3)
B (2)
C (4)
D (1)
Explanation opens after your attempt
Step 1
Concept
The valid pairs are ((2,1),(4,2),(8,4)). In a fraction condition, take (y) only from set (B).
Step 2
Why this answer is correct
The correct answer is A. (3). The valid pairs are ((2,1),(4,2),(8,4)). In a fraction condition, take (y) only from set (B).
Step 3
Exam Tip
सही युग्म ((2,1),(4,2),(8,4)) हैं। भिन्न वाली शर्त में (y) का मान समुच्चय (B) से ही लें।
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यदि \(A=\{1,2,3\}\) और \(B=\{2,4,6\}\) हैं, तो \(A\times B\) में कौन सा युग्म उस संबंध \(R=\{(x,y):y=2x\}\) में नहीं होगा?
If \(A=\{1,2,3\}\) and \(B=\{2,4,6\}\), which pair in \(A\times B\) will not belong to the relation \(R=\{(x,y):y=2x\}\)?
#cartesian-product
#relation
#condition
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A ((1,4))
B ((1,2))
C ((2,4))
D ((3,6))
Explanation opens after your attempt
Correct Answer
A. ((1,4))
Step 1
Concept
In ((1,4)), \(4\ne2\times1\), so it is not in (R). Apply the given relation condition to each option.
Step 2
Why this answer is correct
The correct answer is A. ((1,4)). In ((1,4)), \(4\ne2\times1\), so it is not in (R). Apply the given relation condition to each option.
Step 3
Exam Tip
((1,4)) में \(4\ne2\times1\), इसलिए यह (R) में नहीं है। संबंध में दी गई शर्त को हर विकल्प पर लगाएं।
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यदि \(A=\{1,3\}\) और \(B=\{2,4,6\}\) हैं, तो \(A\times B\) में सभी युग्मों के घटक-गुणनफलों का समुच्चय क्या होगा?
If \(A=\{1,3\}\) and \(B=\{2,4,6\}\), what is the set of products of components of all pairs in \(A\times B\)?
#cartesian-product
#product-set
#application
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A ({2,4,6,12,18})
B ({2,4,6})
C ({3,6,9,12,18})
D ({1,2,3,4,6})
Explanation opens after your attempt
Correct Answer
A. ({2,4,6,12,18})
Step 1
Concept
The products are \(1\cdot2=2\), \(1\cdot4=4\), \(1\cdot6=6\), \(3\cdot2=6\), \(3\cdot4=12\), \(3\cdot6=18\). In a set, (6) is not written twice.
Step 2
Why this answer is correct
The correct answer is A. ({2,4,6,12,18}). The products are \(1\cdot2=2\), \(1\cdot4=4\), \(1\cdot6=6\), \(3\cdot2=6\), \(3\cdot4=12\), \(3\cdot6=18\). In a set, (6) is not written twice.
Step 3
Exam Tip
गुणनफल \(1\cdot2=2\), \(1\cdot4=4\), \(1\cdot6=6\), \(3\cdot2=6\), \(3\cdot4=12\), \(3\cdot6=18\) मिलते हैं। समुच्चय में (6) को दो बार नहीं लिखते।
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