यदि (n\(A\times B\)=72) और (n(A)=8) है, तो (n(B)) कितना होगा?
If (n\(A\times B\)=72) and (n(A)=8), then what is (n(B))?
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A (9)
B (8)
C (64)
D (80)
Explanation opens after your attempt
Step 1
Concept
Since (n\(A\times B\)=n(A)n(B)), (72=8n(B)). In exams, apply the product rule first.
Step 2
Why this answer is correct
The correct answer is A. (9). Since (n\(A\times B\)=n(A)n(B)), (72=8n(B)). In exams, apply the product rule first.
Step 3
Exam Tip
क्योंकि (n\(A\times B\)=n(A)n(B)), इसलिए (72=8n(B))। परीक्षा में पहले गुणन नियम लगाएं।
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यदि \(A=\{1,2,3,4\}\), \(B=\{2,4,6\}\), \(C=\{3,4,5\}\) और \(D=\{4,6,8\}\) है, तो (\(A\times B\)\setminus\(C\times D\)) में कितने अवयव होंगे?
If \(A=\{1,2,3,4\}\), \(B=\{2,4,6\}\), \(C=\{3,4,5\}\) and \(D=\{4,6,8\}\), how many elements are in (\(A\times B\)\setminus\(C\times D\))?
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A (6)
B (7)
C (8)
D (9)
Explanation opens after your attempt
Step 1
Concept
(n\(A\times B\)=12), and (\(A\times B\)\cap\(C\times D\)=\(A\cap C\)\times\(B\cap D\)) has (4) elements. So the difference has (12-4=8) elements.
Step 2
Why this answer is correct
The correct answer is C. (8). (n\(A\times B\)=12), and (\(A\times B\)\cap\(C\times D\)=\(A\cap C\)\times\(B\cap D\)) has (4) elements. So the difference has (12-4=8) elements.
Step 3
Exam Tip
(n\(A\times B\)=12) और (\(A\times B\)\cap\(C\times D\)=\(A\cap C\)\times\(B\cap D\)) में (4) अवयव हैं। इसलिए अंतर में (12-4=8) अवयव होंगे।
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यदि \(A\neq \varnothing\), \(B\neq \varnothing\) और \(A\times B=B\times A\) है, तो कौन सा निष्कर्ष सही है?
If \(A\neq \varnothing\), \(B\neq \varnothing\) and \(A\times B=B\times A\), which conclusion is correct?
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A (A=B)
B \(A\subset B\)
C \(B\subset A\)
D \(A\cap B=\varnothing\)
Explanation opens after your attempt
Step 1
Concept
For non-empty sets, \(A\times B=B\times A\) only when (A=B). In exams, always check the empty-set condition.
Step 2
Why this answer is correct
The correct answer is A. (A=B). For non-empty sets, \(A\times B=B\times A\) only when (A=B). In exams, always check the empty-set condition.
Step 3
Exam Tip
अशून्य समुच्चयों के लिए \(A\times B=B\times A\) तभी होता है जब (A=B)। परीक्षा में खाली समुच्चय की शर्त जरूर देखें।
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यदि \(A=\{1,2,3\}\), \(B=\{1,2,3,4\}\) और \(R=\{(x,y)\in A\times B:y=x+2\}\) है, तो (R) क्या है?
If \(A=\{1,2,3\}\), \(B=\{1,2,3,4\}\), and \(R=\{(x,y)\in A\times B:y=x+2\}\), what is (R)?
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A ( {(1,3),(2,4)} )
B ( {(1,2),(2,3),(3,4)} )
C ( {(3,1),(4,2)} )
D ( {(1,3),(2,4),(3,5)} )
Explanation opens after your attempt
Correct Answer
A. ( {(1,3),(2,4)} )
Step 1
Concept
The condition (y=x+2) gives \(y\in B\) only for (x=1,2). For (x=3), (y=5), which is not in (B).
Step 2
Why this answer is correct
The correct answer is A. ( {(1,3),(2,4)} ). The condition (y=x+2) gives \(y\in B\) only for (x=1,2). For (x=3), (y=5), which is not in (B).
Step 3
Exam Tip
शर्त (y=x+2) से (x=1,2) पर ही \(y\in B\) मिलता है। (x=3) पर (y=5) है जो (B) में नहीं है।
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यदि \(A={x:x^2=4}\) और \(B={y:y^2-5y+6=0}\) है, तो \(A\times B\) में कितने अवयव होंगे?
If \(A={x:x^2=4}\) and \(B={y:y^2-5y+6=0}\), how many elements are in \(A\times B\)?
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A (2)
B (4)
C (6)
D (8)
Explanation opens after your attempt
Step 1
Concept
Here \(A=\{-2,2\}\) and \(B=\{2,3\}\), so (n\(A\times B\)=2\cdot 2=4). In exams, first list each set.
Step 2
Why this answer is correct
The correct answer is B. (4). Here \(A=\{-2,2\}\) and \(B=\{2,3\}\), so (n\(A\times B\)=2\cdot 2=4). In exams, first list each set.
Step 3
Exam Tip
यहां \(A=\{-2,2\}\) और \(B=\{2,3\}\), इसलिए (n\(A\times B\)=2\cdot 2=4)। परीक्षा में पहले प्रत्येक समुच्चय के अवयव निकालें।
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यदि \(A=\{1,2,3\}\) और \(B=\{2,4,6\}\) है, तो \(A\times B\) का कौन सा अवयव नहीं है?
If \(A=\{1,2,3\}\) and \(B=\{2,4,6\}\), which is not an element of \(A\times B\)?
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A ( (1,2) )
B ( (3,6) )
C ( (4,2) )
D ( (2,4) )
Explanation opens after your attempt
Correct Answer
C. ( (4,2) )
Step 1
Concept
In ( (4,2) ), the first element (4) is not in (A). In exams, take the first coordinate from (A) and the second from (B).
Step 2
Why this answer is correct
The correct answer is C. ( (4,2) ). In ( (4,2) ), the first element (4) is not in (A). In exams, take the first coordinate from (A) and the second from (B).
Step 3
Exam Tip
( (4,2) ) में पहला अवयव (4) है जो (A) में नहीं है। परीक्षा में क्रमबद्ध युग्म में पहला स्थान (A) से और दूसरा स्थान (B) से लें।
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यदि \(A=\{1,2,3,4\}\), \(B=\{a,b\}\), \(C=\{2,3,5\}\) और \(D=\{b,c\}\) है, तो (\(A\times B\)\cap\(C\times D\)) में कितने अवयव होंगे?
If \(A=\{1,2,3,4\}\), \(B=\{a,b\}\), \(C=\{2,3,5\}\) and \(D=\{b,c\}\), how many elements are in (\(A\times B\)\cap\(C\times D\))?
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A (1)
B (2)
C (3)
D (4)
Explanation opens after your attempt
Step 1
Concept
(\(A\times B\)\cap\(C\times D\)=\(A\cap C\)\times\(B\cap D\)), where \(A\cap C={2,3}\) and \(B\cap D={b}\). So there are \(2\cdot1=2\) elements.
Step 2
Why this answer is correct
The correct answer is B. (2). (\(A\times B\)\cap\(C\times D\)=\(A\cap C\)\times\(B\cap D\)), where \(A\cap C={2,3}\) and \(B\cap D={b}\). So there are \(2\cdot1=2\) elements.
Step 3
Exam Tip
(\(A\times B\)\cap\(C\times D\)=\(A\cap C\)\times\(B\cap D\)), जहां \(A\cap C={2,3}\) और \(B\cap D={b}\)। इसलिए कुल \(2\cdot1=2\) अवयव हैं।
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यदि \(A\subseteq C\) और \(B\subseteq D\) है, तो कौन सा संबंध निश्चित रूप से सही है?
If \(A\subseteq C\) and \(B\subseteq D\), which relation is definitely true?
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A \(A\times B\subseteq C\times D\)
B \(C\times D\subseteq A\times B\)
C \(A\times D\subseteq B\times C\)
D \(A\times B=C\times D\)
Explanation opens after your attempt
Correct Answer
A. \(A\times B\subseteq C\times D\)
Step 1
Concept
Every \(a\in A\) lies in (C), and every \(b\in B\) lies in (D), so \((a,b)\in C\times D\). In exams, check membership of both coordinates separately.
Step 2
Why this answer is correct
The correct answer is A. \(A\times B\subseteq C\times D\). Every \(a\in A\) lies in (C), and every \(b\in B\) lies in (D), so \((a,b)\in C\times D\). In exams, check membership of both coordinates separately.
Step 3
Exam Tip
हर \(a\in A\) के लिए \(a\in C\) और हर \(b\in B\) के लिए \(b\in D\), इसलिए \((a,b)\in C\times D\)। परीक्षा में दोनों निर्देशांकों की सदस्यता अलग जांचें।
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यदि (A=[1,3]) और (B=(2,5)) है, तो कौन सा बिंदु \(A\times B\) में है?
If (A=[1,3]) and (B=(2,5)), which point is in \(A\times B\)?
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A ( (3,5) )
B ( (1,2) )
C ( (2,4) )
D ( (0,4) )
Explanation opens after your attempt
Correct Answer
C. ( (2,4) )
Step 1
Concept
\(2\in[1,3]\) and \(4\in(2,5)\), so \((2,4)\in A\times B\). In exams, check endpoints of closed and open intervals carefully.
Step 2
Why this answer is correct
The correct answer is C. ( (2,4) ). \(2\in[1,3]\) and \(4\in(2,5)\), so \((2,4)\in A\times B\). In exams, check endpoints of closed and open intervals carefully.
Step 3
Exam Tip
\(2\in[1,3]\) और \(4\in(2,5)\), इसलिए \((2,4)\in A\times B\)। परीक्षा में बंद और खुले अंतराल के सिरों को ध्यान से देखें।
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यदि (n(A)=6) है, तो \(A\times A\) में ऐसे कितने युग्म होंगे जिनमें दोनों निर्देशांक बराबर हैं?
If (n(A)=6), how many pairs in \(A\times A\) have equal coordinates?
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A (6)
B (12)
C (30)
D (36)
Explanation opens after your attempt
Step 1
Concept
Pairs with equal coordinates are of the form ((a,a)), and their number is (n(A)). In exams, treat them as diagonal pairs.
Step 2
Why this answer is correct
The correct answer is A. (6). Pairs with equal coordinates are of the form ((a,a)), and their number is (n(A)). In exams, treat them as diagonal pairs.
Step 3
Exam Tip
बराबर निर्देशांक वाले युग्म ((a,a)) रूप के होते हैं और ऐसे युग्मों की संख्या (n(A)) होती है। परीक्षा में इन्हें विकर्ण युग्म समझें।
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यदि (n(A)=7) है, तो \(A\times A\) में ऐसे कितने युग्म होंगे जिनमें दोनों निर्देशांक अलग हैं?
If (n(A)=7), how many pairs in \(A\times A\) have distinct coordinates?
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A (7)
B (14)
C (42)
D (49)
Explanation opens after your attempt
Step 1
Concept
Total pairs are \(7^2=49\), and equal-coordinate pairs are (7). Hence distinct-coordinate pairs are (49-7=42).
Step 2
Why this answer is correct
The correct answer is C. (42). Total pairs are \(7^2=49\), and equal-coordinate pairs are (7). Hence distinct-coordinate pairs are (49-7=42).
Step 3
Exam Tip
कुल युग्म \(7^2=49\) और बराबर निर्देशांक वाले (7) हैं। इसलिए अलग निर्देशांक वाले (49-7=42) हैं।
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यदि (n(A)=4) और (n(B)=5) है, तो (A) से (B) तक कुल कितने संबंध संभव हैं?
If (n(A)=4) and (n(B)=5), how many relations from (A) to (B) are possible?
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A (20)
B \(2^{20}\)
C \(4^5\)
D \(5^4\)
Explanation opens after your attempt
Correct Answer
B. \(2^{20}\)
Step 1
Concept
A relation is any subset of \(A\times B\), and (n\(A\times B\)=20). Hence the number of relations is \(2^{20}\).
Step 2
Why this answer is correct
The correct answer is B. \(2^{20}\). A relation is any subset of \(A\times B\), and (n\(A\times B\)=20). Hence the number of relations is \(2^{20}\).
Step 3
Exam Tip
संबंध \(A\times B\) का कोई भी उपसमुच्चय होता है और (n\(A\times B\)=20)। इसलिए कुल संबंधों की संख्या \(2^{20}\) है।
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यदि \(A=\{1,2,3\}\) और \(B=\{4,5\}\) है, तो \(A\times B\) के ऐसे उपसमुच्चयों की संख्या क्या है जिनमें ठीक (2) क्रमबद्ध युग्म हों?
If \(A=\{1,2,3\}\) and \(B=\{4,5\}\), what is the number of subsets of \(A\times B\) having exactly (2) ordered pairs?
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A (6)
B (12)
C (15)
D (36)
Explanation opens after your attempt
Step 1
Concept
Here (n\(A\times B\)=3\cdot2=6). The number of ways to choose exactly (2) pairs is \(\binom{6}{2}=15\).
Step 2
Why this answer is correct
The correct answer is C. (15). Here (n\(A\times B\)=3\cdot2=6). The number of ways to choose exactly (2) pairs is \(\binom{6}{2}=15\).
Step 3
Exam Tip
यहां (n\(A\times B\)=3\cdot2=6)। ठीक (2) युग्म चुनने की संख्या \(\binom{6}{2}=15\) है।
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यदि ((2x-1,y+3)=(5,10)) है, तो ((x,y)) क्या होगा?
If ((2x-1,y+3)=(5,10)), what is ((x,y))?
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A ( (2,7) )
B ( (3,7) )
C ( (3,13) )
D ( (4,7) )
Explanation opens after your attempt
Correct Answer
B. ( (3,7) )
Step 1
Concept
Equal ordered pairs give (2x-1=5) and (y+3=10). Thus (x=3) and (y=7).
Step 2
Why this answer is correct
The correct answer is B. ( (3,7) ). Equal ordered pairs give (2x-1=5) and (y+3=10). Thus (x=3) and (y=7).
Step 3
Exam Tip
क्रमबद्ध युग्म बराबर होने पर (2x-1=5) और (y+3=10) होते हैं। इससे (x=3) और (y=7) मिलता है।
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यदि ((a+b,a-b)=(9,1)) है, तो \(a^2-b^2\) का मान क्या है?
If ((a+b,a-b)=(9,1)), what is the value of \(a^2-b^2\)?
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A (8)
B (9)
C (10)
D (45)
Explanation opens after your attempt
Step 1
Concept
Since (a-2 -b-2 =(a+b)(a-b)), the value is \(9\cdot1=9\). In exams, match coordinates directly from pair equality.
Step 2
Why this answer is correct
The correct answer is B. (9). Since (a-2 -b-2 =(a+b)(a-b)), the value is \(9\cdot1=9\). In exams, match coordinates directly from pair equality.
Step 3
Exam Tip
क्योंकि (a-2 -b-2 =(a+b)(a-b)), इसलिए मान \(9\cdot1=9\) है। परीक्षा में युग्म बराबरी से सीधे निर्देशांक मिलाएं।
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यदि \(A\times B={(1,3),(1,4),(2,3),(2,4)}\) है, तो (A) और (B) क्या हैं?
If \(A\times B={(1,3),(1,4),(2,3),(2,4)}\), what are (A) and (B)?
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A \(A=\{1,2},,B={3,4\}\)
B \(A=\{3,4},,B={1,2\}\)
C \(A=\{1,3},,B={2,4\}\)
D \(A=\{1,4},,B={2,3\}\)
Explanation opens after your attempt
Correct Answer
A. \(A=\{1,2},,B={3,4\}\)
Step 1
Concept
The first coordinates are (1,2), and the second coordinates are (3,4). Hence \(A=\{1,2\}\) and \(B=\{3,4\}\).
Step 2
Why this answer is correct
The correct answer is A. \(A=\{1,2},,B={3,4\}\). The first coordinates are (1,2), and the second coordinates are (3,4). Hence \(A=\{1,2\}\) and \(B=\{3,4\}\).
Step 3
Exam Tip
पहले निर्देशांक (1,2) हैं और दूसरे निर्देशांक (3,4) हैं। इसलिए \(A=\{1,2\}\) और \(B=\{3,4\}\)।
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यदि \(A\times B=\varnothing\) है, तो कौन सा निष्कर्ष हमेशा सत्य है?
If \(A\times B=\varnothing\), which conclusion is always true?
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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
A Cartesian product is empty when at least one set is empty. In exams, 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 \(B=\varnothing\). A Cartesian product is empty when at least one set is empty. In exams, both sets need not be empty.
Step 3
Exam Tip
कार्तीय गुणन खाली तभी होता है जब कम से कम एक समुच्चय खाली हो। परीक्षा में दोनों के खाली होने की जरूरत नहीं होती।
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यदि \(A=\{1,2\}\), \(B=\{3\}\) और \(C=\{4,5\}\) है, तो (A\times\(B\cup C\)) में कितने अवयव होंगे?
If \(A=\{1,2\}\), \(B=\{3\}\) and \(C=\{4,5\}\), how many elements are in (A\times\(B\cup C\))?
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A (3)
B (4)
C (5)
D (6)
Explanation opens after your attempt
Step 1
Concept
\(B\cup C={3,4,5}\), so (n(A\times\(B\cup C\))=2\cdot3=6). In exams, simplify the inner set first.
Step 2
Why this answer is correct
The correct answer is D. (6). \(B\cup C={3,4,5}\), so (n(A\times\(B\cup C\))=2\cdot3=6). In exams, simplify the inner set first.
Step 3
Exam Tip
\(B\cup C={3,4,5}\), इसलिए (n(A\times\(B\cup C\))=2\cdot3=6)। परीक्षा में पहले अंदर का समुच्चय सरल करें।
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यदि \(A=\{0,1,2\}\) और \(B=\{1,2,3\}\) है, तो \(A\times B\) में ऐसे कितने युग्म ((x,y)) हैं जिनके लिए (x<y) है?
If \(A=\{0,1,2\}\) and \(B=\{1,2,3\}\), how many pairs ((x,y)) in \(A\times B\) satisfy (x<y)?
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A (3)
B (5)
C (6)
D (9)
Explanation opens after your attempt
Step 1
Concept
For (x=0) there are (3) choices, for (x=1) there are (2), and for (x=2) there is (1). Total pairs are (3+2+1=6).
Step 2
Why this answer is correct
The correct answer is C. (6). For (x=0) there are (3) choices, for (x=1) there are (2), and for (x=2) there is (1). Total pairs are (3+2+1=6).
Step 3
Exam Tip
(x=0) पर (3), (x=1) पर (2), और (x=2) पर (1) विकल्प हैं। कुल (3+2+1=6) युग्म मिलते हैं।
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यदि \(A=\{-1,0,1,2\}\) और \(B=\{0,1,2\}\) है, तो \(A\times B\) में ऐसे कितने युग्म ((x,y)) हैं जिनके लिए (x+y=1) है?
If \(A=\{-1,0,1,2\}\) and \(B=\{0,1,2\}\), how many pairs ((x,y)) in \(A\times B\) satisfy (x+y=1)?
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A (2)
B (3)
C (4)
D (5)
Explanation opens after your attempt
Step 1
Concept
The valid pairs are ((-1,2)), ((0,1)), and ((1,0)). In exams, choose (x) and check (y=1-x).
Step 2
Why this answer is correct
The correct answer is B. (3). The valid pairs are ((-1,2)), ((0,1)), and ((1,0)). In exams, choose (x) and check (y=1-x).
Step 3
Exam Tip
सही युग्म ((-1,2)), ((0,1)) और ((1,0)) हैं। परीक्षा में (x) चुनकर (y=1-x) जांचें।
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यदि \(A=\{1,2,3,4\}\) और \(B=\{1,2,3,4\}\) है, तो \(A\times B\) में ऐसे कितने युग्म ((x,y)) हैं जिनके लिए \(x\mid y\) है?
If \(A=\{1,2,3,4\}\) and \(B=\{1,2,3,4\}\), how many pairs ((x,y)) in \(A\times B\) satisfy \(x\mid y\)?
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A (6)
B (7)
C (8)
D (10)
Explanation opens after your attempt
Step 1
Concept
For (x=1) there are (4), for (x=2) there are (2), for (x=3) there is (1), and for (x=4) there is (1). Total is (4+2+1+1=8).
Step 2
Why this answer is correct
The correct answer is C. (8). For (x=1) there are (4), for (x=2) there are (2), for (x=3) there is (1), and for (x=4) there is (1). Total is (4+2+1+1=8).
Step 3
Exam Tip
(x=1) से (4), (x=2) से (2), (x=3) से (1), (x=4) से (1) युग्म मिलते हैं। कुल (4+2+1+1=8) है।
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यदि \(A=\{1,2,3\}\) और \(B=\{2,3,4\}\) है, तो \(A\times B\) में ऐसे कितने युग्म ((x,y)) हैं जिनके लिए (xy) सम है?
If \(A=\{1,2,3\}\) and \(B=\{2,3,4\}\), how many pairs ((x,y)) in \(A\times B\) have (xy) even?
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A (4)
B (5)
C (6)
D (7)
Explanation opens after your attempt
Step 1
Concept
There are (9) total pairs, and the product is odd only for ((1,3)) and ((3,3)). Hence even-product pairs are (9-2=7).
Step 2
Why this answer is correct
The correct answer is D. (7). There are (9) total pairs, and the product is odd only for ((1,3)) and ((3,3)). Hence even-product pairs are (9-2=7).
Step 3
Exam Tip
कुल (9) युग्म हैं और गुणनफल विषम केवल ((1,3)) तथा ((3,3)) में है। इसलिए सम गुणनफल वाले (9-2=7) हैं।
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यदि \(A=\{2,4,6\}\) और \(B=\{1,2,3,4\}\) है, तो \(A\times B\) में ऐसे कितने युग्म ((x,y)) हैं जिनके लिए (x-y) धनात्मक है?
If \(A=\{2,4,6\}\) and \(B=\{1,2,3,4\}\), how many pairs ((x,y)) in \(A\times B\) satisfy (x-y) is positive?
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A (6)
B (7)
C (8)
D (9)
Explanation opens after your attempt
Step 1
Concept
For (x=2) there is (1) choice, for (x=4) there are (3), and for (x=6) there are (4). Hence the total is (1+3+4=8), so choose the option (8).
Step 2
Why this answer is correct
The correct answer is D. (9). For (x=2) there is (1) choice, for (x=4) there are (3), and for (x=6) there are (4). Hence the total is (1+3+4=8), so choose the option (8).
Step 3
Exam Tip
(x=2) पर (1), (x=4) पर (3), और (x=6) पर (4) विकल्प हैं। कुल (1+3+4=8) नहीं बल्कि (x=6) के लिए चारों (y) सही हैं, अतः कुल (8) है।
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यदि \(A=\{1,3,5\}\) और \(B=\{2,4,6,8\}\) है, तो \(A\times B\) में ऐसे कितने युग्म ((x,y)) हैं जिनके लिए (x+y>9) है?
If \(A=\{1,3,5\}\) and \(B=\{2,4,6,8\}\), how many pairs ((x,y)) in \(A\times B\) satisfy (x+y>9)?
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A (2)
B (3)
C (4)
D (5)
Explanation opens after your attempt
Step 1
Concept
With (x=3), only (8) works, and with (x=5), (6,8) work. So there are (1+2=3) pairs.
Step 2
Why this answer is correct
The correct answer is B. (3). With (x=3), only (8) works, and with (x=5), (6,8) work. So there are (1+2=3) pairs.
Step 3
Exam Tip
(x=3) के साथ केवल (8) और (x=5) के साथ (6,8) सही हैं। इसलिए कुल (1+2=3) युग्म हैं।
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यदि \(A=\{1,2,3,4,5\}\) और \(B=\{1,2,3\}\) है, तो \(A\times B\) में ऐसे कितने युग्म ((x,y)) हैं जिनके लिए (x=2y-1) है?
If \(A=\{1,2,3,4,5\}\) and \(B=\{1,2,3\}\), how many pairs ((x,y)) in \(A\times B\) satisfy (x=2y-1)?
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A (1)
B (2)
C (3)
D (4)
Explanation opens after your attempt
Step 1
Concept
For (y=1,2,3), we get (x=1,3,5), all of which lie in (A). Hence there are (3) pairs.
Step 2
Why this answer is correct
The correct answer is C. (3). For (y=1,2,3), we get (x=1,3,5), all of which lie in (A). Hence there are (3) pairs.
Step 3
Exam Tip
(y=1,2,3) देने पर (x=1,3,5) मिलते हैं और ये सभी (A) में हैं। अतः कुल (3) युग्म हैं।
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यदि \(A=\{1,2,4,8\}\) और \(B=\{2,4,8,16\}\) है, तो \(A\times B\) में ऐसे कितने युग्म ((x,y)) हैं जिनके लिए (y=2x) है?
If \(A=\{1,2,4,8\}\) and \(B=\{2,4,8,16\}\), how many pairs ((x,y)) in \(A\times B\) satisfy (y=2x)?
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A (2)
B (3)
C (4)
D (5)
Explanation opens after your attempt
Step 1
Concept
For every \(x\in A\), \(2x\in B\), giving ((1,2),(2,4),(4,8),(8,16)). There are (4) pairs.
Step 2
Why this answer is correct
The correct answer is C. (4). For every \(x\in A\), \(2x\in B\), giving ((1,2),(2,4),(4,8),(8,16)). There are (4) pairs.
Step 3
Exam Tip
हर \(x\in A\) के लिए \(2x\in B\), इसलिए ((1,2),(2,4),(4,8),(8,16)) मिलते हैं। कुल (4) युग्म हैं।
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यदि \(A=\{0,1,2,3\}\) और \(B=\{0,1,4,9\}\) है, तो \(A\times B\) में ऐसे कितने युग्म ((x,y)) हैं जिनके लिए \(y=x^2\) है?
If \(A=\{0,1,2,3\}\) and \(B=\{0,1,4,9\}\), how many pairs ((x,y)) in \(A\times B\) satisfy \(y=x^2\)?
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A (2)
B (3)
C (4)
D (5)
Explanation opens after your attempt
Step 1
Concept
For (x=0,1,2,3), (y=0,1,4,9), all in (B). Hence there are (4) pairs.
Step 2
Why this answer is correct
The correct answer is C. (4). For (x=0,1,2,3), (y=0,1,4,9), all in (B). Hence there are (4) pairs.
Step 3
Exam Tip
(x=0,1,2,3) पर (y=0,1,4,9) मिलते हैं और सभी (B) में हैं। इसलिए कुल (4) युग्म हैं।
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\(यदि (A={1,2,3}), (B={1,2,3}) और (R={(x,y)\in A\times B:x+y\) is even}), तो (n(R)) क्या है?
\(If (A={1,2,3}), (B={1,2,3}), and (R={(x,y)\in A\times B:x+y\) is even}), what is (n(R))?
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A (3)
B (4)
C (5)
D (6)
Explanation opens after your attempt
Step 1
Concept
The sum is even when both numbers have the same parity. Odd-odd gives (4) pairs and even-even gives (1), so total is (5).
Step 2
Why this answer is correct
The correct answer is C. (5). The sum is even when both numbers have the same parity. Odd-odd gives (4) pairs and even-even gives (1), so total is (5).
Step 3
Exam Tip
योग सम तब होगा जब दोनों संख्याएं समान समता की हों। विषम-विषम (4) और सम-सम (1) मिलकर कुल (5) युग्म देते हैं।
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यदि \(A=\{1,2,3,4\}\), \(B=\{1,2,3,4\}\) और \(S=\{(x,y)\in A\times B:x+y=5\}\), तो (S) क्या है?
If \(A=\{1,2,3,4\}\), \(B=\{1,2,3,4\}\), and \(S=\{(x,y)\in A\times B:x+y=5\}\), what is (S)?
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A ( {(1,4),(2,3),(3,2),(4,1)} )
B ( {(1,5),(2,3),(3,2),(5,1)} )
C ( {(1,4),(4,1)} )
D ( {(2,2),(3,3)} )
Explanation opens after your attempt
Correct Answer
A. ( {(1,4),(2,3),(3,2),(4,1)} )
Step 1
Concept
(x+y=5), and both coordinates must lie from (1) to (4). Thus the four pairs are ((1,4),(2,3),(3,2),(4,1)).
Step 2
Why this answer is correct
The correct answer is A. ( {(1,4),(2,3),(3,2),(4,1)} ). (x+y=5), and both coordinates must lie from (1) to (4). Thus the four pairs are ((1,4),(2,3),(3,2),(4,1)).
Step 3
Exam Tip
(x+y=5) और दोनों निर्देशांक (1) से (4) में होने चाहिए। इसलिए चार युग्म ((1,4),(2,3),(3,2),(4,1)) मिलते हैं।
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यदि \(A=\{a,b,c\}\) और \(B=\{1,2\}\) है, तो \(B\times A\) में कौन सा युग्म होगा?
If \(A=\{a,b,c\}\) and \(B=\{1,2\}\), which pair belongs to \(B\times A\)?
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A ( (a,1) )
B ( (1,a) )
C ( (b,2) )
D ( (c,1) )
Explanation opens after your attempt
Correct Answer
B. ( (1,a) )
Step 1
Concept
In \(B\times A\), the first coordinate comes from (B) and the second from (A). Hence ((1,a)) is correct.
Step 2
Why this answer is correct
The correct answer is B. ( (1,a) ). In \(B\times A\), the first coordinate comes from (B) and the second from (A). Hence ((1,a)) is correct.
Step 3
Exam Tip
\(B\times A\) में पहला निर्देशांक (B) से और दूसरा (A) से आता है। इसलिए ((1,a)) सही है।
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यदि \(A=\{1,2\}\), \(B=\{3,4\}\) और \(C=\{5\}\) है, तो (\(A\times B\)\times C) में कौन सा अवयव होगा?
If \(A=\{1,2\}\), \(B=\{3,4\}\), and \(C=\{5\}\), which element belongs to (\(A\times B\)\times C)?
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A ( (1,3,5) )
B ( ((1,3),5) )
C ( (1,(3,5)) )
D ( ((5,1),3) )
Explanation opens after your attempt
Correct Answer
B. ( ((1,3),5) )
Step 1
Concept
An element of (\(A\times B\)\times C) has the form (((a,b),c)). Therefore (((1,3),5)) has the correct structure.
Step 2
Why this answer is correct
The correct answer is B. ( ((1,3),5) ). An element of (\(A\times B\)\times C) has the form (((a,b),c)). Therefore (((1,3),5)) has the correct structure.
Step 3
Exam Tip
(\(A\times B\)\times C) का अवयव (((a,b),c)) रूप का होता है। इसलिए (((1,3),5)) सही संरचना है।
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कौन सा कथन सामान्यतः सत्य नहीं है?
Which statement is generally not true?
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A (A\times\(B\cap C\)=\(A\times B\)\cap\(A\times C\))
B (A\times\(B\cup C\)=\(A\times B\)\cup\(A\times C\))
C (A\times\(B\setminus C\)=\(A\times B\)\setminus\(A\times C\))
D \(A\times B=B\times A\)
Explanation opens after your attempt
Correct Answer
D. \(A\times B=B\times A\)
Step 1
Concept
Generally \(A\times B\neq B\times A\) because the order changes. In exams, treat ((a,b)) and ((b,a)) as different.
Step 2
Why this answer is correct
The correct answer is D. \(A\times B=B\times A\). Generally \(A\times B\neq B\times A\) because the order changes. In exams, treat ((a,b)) and ((b,a)) as different.
Step 3
Exam Tip
सामान्यतः \(A\times B\neq B\times A\) क्योंकि क्रम बदल जाता है। परीक्षा में ((a,b)) और ((b,a)) को अलग समझें।
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यदि \(A=\{1,2,3\}\), \(B=\{3,4\}\) और \(C=\{2,3,4\}\) है, तो (\(A\cap C\)\times\(B\cap C\)) में कितने अवयव हैं?
If \(A=\{1,2,3\}\), \(B=\{3,4\}\), and \(C=\{2,3,4\}\), how many elements are in (\(A\cap C\)\times\(B\cap C\))?
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A (2)
B (4)
C (6)
D (8)
Explanation opens after your attempt
Step 1
Concept
\(A\cap C={2,3}\) and \(B\cap C={3,4}\). Therefore the number of elements is \(2\cdot2=4\).
Step 2
Why this answer is correct
The correct answer is B. (4). \(A\cap C={2,3}\) and \(B\cap C={3,4}\). Therefore the number of elements is \(2\cdot2=4\).
Step 3
Exam Tip
\(A\cap C={2,3}\) और \(B\cap C={3,4}\)। इसलिए अवयवों की संख्या \(2\cdot2=4\) है।
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यदि \(A\times B\) में (24) अवयव हैं और (A) में (B) से (2) अधिक अवयव हैं, तो (n(A)) क्या है?
If \(A\times B\) has (24) elements and (A) has (2) more elements than (B), what is (n(A))?
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A (4)
B (6)
C (8)
D (12)
Explanation opens after your attempt
Step 1
Concept
Let (n(B)=m), then (n(A)=m+2) and (m(m+2)=24). This gives (m=4), so (n(A)=6).
Step 2
Why this answer is correct
The correct answer is B. (6). Let (n(B)=m), then (n(A)=m+2) and (m(m+2)=24). This gives (m=4), so (n(A)=6).
Step 3
Exam Tip
मान लें (n(B)=m), तब (n(A)=m+2) और (m(m+2)=24)। इससे (m=4), इसलिए (n(A)=6)।
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यदि (n\(A\times B\)=n\(B\times C\)=30), (n(B)=5), और (A,B,C) अशून्य सीमित समुच्चय हैं, तो (n\(A\times C\)) कितना होगा?
If (n\(A\times B\)=n\(B\times C\)=30), (n(B)=5), and (A,B,C) are non-empty finite sets, what is (n\(A\times C\))?
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A (25)
B (30)
C (36)
D (60)
Explanation opens after your attempt
Step 1
Concept
(n(A)=30/5=6) and (n(C)=30/5=6). Therefore (n\(A\times C\)=6\cdot6=36).
Step 2
Why this answer is correct
The correct answer is C. (36). (n(A)=30/5=6) and (n(C)=30/5=6). Therefore (n\(A\times C\)=6\cdot6=36).
Step 3
Exam Tip
(n(A)=30/5=6) और (n(C)=30/5=6)। इसलिए (n\(A\times C\)=6\cdot6=36)।
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यदि \(A=\{1,2,3\}\) और \(B=\{1,2,3,4\}\) है, तो \(A\times B\) में ऐसे युग्मों की संख्या कितनी है जिनमें पहला निर्देशांक दूसरे से बड़ा है?
If \(A=\{1,2,3\}\) and \(B=\{1,2,3,4\}\), how many pairs in \(A\times B\) have first coordinate greater than the second?
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A (2)
B (3)
C (4)
D (5)
Explanation opens after your attempt
Step 1
Concept
For (x=2), (y=1) works, and for (x=3), (y=1,2) work. Total is (1+2=3) pairs.
Step 2
Why this answer is correct
The correct answer is B. (3). For (x=2), (y=1) works, and for (x=3), (y=1,2) work. Total is (1+2=3) pairs.
Step 3
Exam Tip
(x=2) पर (y=1) और (x=3) पर (y=1,2) सही हैं। कुल (1+2=3) युग्म हैं।
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यदि \(A=\{1,2,3,4\}\) और \(B=\{1,2,3,4\}\) है, तो \(A\times B\) में ऐसे युग्म ((x,y)) कितने हैं जिनके लिए \(x+y\le 5\) है?
If \(A=\{1,2,3,4\}\) and \(B=\{1,2,3,4\}\), how many pairs ((x,y)) in \(A\times B\) satisfy \(x+y\le 5\)?
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A (8)
B (9)
C (10)
D (11)
Explanation opens after your attempt
Step 1
Concept
For (x=1,2,3,4), the numbers of choices are (4,3,2,1). Total pairs are (4+3+2+1=10).
Step 2
Why this answer is correct
The correct answer is C. (10). For (x=1,2,3,4), the numbers of choices are (4,3,2,1). Total pairs are (4+3+2+1=10).
Step 3
Exam Tip
(x=1,2,3,4) के लिए क्रमशः (4,3,2,1) विकल्प हैं। कुल (4+3+2+1=10) युग्म मिलते हैं।
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यदि \(A=\{0,2,4\}\) और \(B=\{1,3,5\}\) है, तो \(A\times B\) में ऐसे युग्म कितने हैं जिनका दूसरा निर्देशांक पहले से (1) अधिक है?
If \(A=\{0,2,4\}\) and \(B=\{1,3,5\}\), how many pairs in \(A\times B\) have the second coordinate (1) greater than the first?
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A (1)
B (2)
C (3)
D (4)
Explanation opens after your attempt
Step 1
Concept
The condition is (y=x+1), and the pairs are ((0,1),(2,3),(4,5)). Therefore there are (3) pairs.
Step 2
Why this answer is correct
The correct answer is C. (3). The condition is (y=x+1), and the pairs are ((0,1),(2,3),(4,5)). Therefore there are (3) pairs.
Step 3
Exam Tip
शर्त (y=x+1) है और युग्म ((0,1),(2,3),(4,5)) हैं। इसलिए कुल (3) युग्म हैं।
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यदि \(A=\{1,2,3,4,5\}\) और \(B=\{1,4,9,16,25\}\) है, तो \(A\times B\) में ऐसे युग्म ((x,y)) कितने हैं जिनके लिए \(y=x^2\) और (x) विषम है?
If \(A=\{1,2,3,4,5\}\) and \(B=\{1,4,9,16,25\}\), how many pairs ((x,y)) in \(A\times B\) satisfy \(y=x^2\) and (x) is odd?
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A (2)
B (3)
C (4)
D (5)
Explanation opens after your attempt
Step 1
Concept
The odd values of (x) are (1,3,5), and their squares (1,9,25) are in (B). Hence there are (3) pairs.
Step 2
Why this answer is correct
The correct answer is B. (3). The odd values of (x) are (1,3,5), and their squares (1,9,25) are in (B). Hence there are (3) pairs.
Step 3
Exam Tip
विषम (x) मान (1,3,5) हैं और इनके वर्ग (1,9,25) (B) में हैं। इसलिए कुल (3) युग्म हैं।
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\(यदि (A={1,2,3}), (B={2,3,4}) और (T={(x,y)\in A\times B:x+y\) is prime}), तो (n(T)) क्या है?
\(If (A={1,2,3}), (B={2,3,4}), and (T={(x,y)\in A\times B:x+y\) is prime}), what is (n(T))?
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A (3)
B (4)
C (5)
D (6)
Explanation opens after your attempt
Step 1
Concept
The prime sums possible are (3,5,7). The valid pairs are ((1,2),(1,4),(2,3),(3,2),(3,4)), so there are (5) pairs.
Step 2
Why this answer is correct
The correct answer is C. (5). The prime sums possible are (3,5,7). The valid pairs are ((1,2),(1,4),(2,3),(3,2),(3,4)), so there are (5) pairs.
Step 3
Exam Tip
योग (3,5,7) अभाज्य हो सकते हैं। सही युग्म ((1,2),(1,4),(2,3),(3,2),(3,4)) हैं, इसलिए (5) युग्म हैं।
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यदि \(A=\{1,2,3,4\}\) और \(B=\{2,4,6,8\}\) है, तो \(A\times B\) में ऐसे कितने युग्म ((x,y)) हैं जिनके लिए (y) (x) का गुणज है?
If \(A=\{1,2,3,4\}\) and \(B=\{2,4,6,8\}\), how many pairs ((x,y)) in \(A\times B\) have (y) as a multiple of (x)?
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A (8)
B (9)
C (10)
D (11)
Explanation opens after your attempt
Step 1
Concept
For (x=1) there are (4), for (x=2) there are (4), for (x=3) there is (1), and for (x=4) there are (2). Total is (11).
Step 2
Why this answer is correct
The correct answer is D. (11). For (x=1) there are (4), for (x=2) there are (4), for (x=3) there is (1), and for (x=4) there are (2). Total is (11).
Step 3
Exam Tip
(x=1) से (4), (x=2) से (4), (x=3) से (1), और (x=4) से (2) युग्म मिलते हैं। कुल (4+4+1+2=11) है।
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यदि \(A=\{1,2,3\}\), \(B=\{1,2,3\}\) और \(U=A\times B\), तो (U) में ऐसे युग्म कितने हैं जो \(x\neq y\) को संतुष्ट करते हैं?
If \(A=\{1,2,3\}\), \(B=\{1,2,3\}\), and \(U=A\times B\), how many pairs in (U) satisfy \(x\neq y\)?
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A (3)
B (6)
C (8)
D (9)
Explanation opens after your attempt
Step 1
Concept
There are (9) total pairs and (3) equal-coordinate pairs. Hence pairs with \(x\neq y\) are (9-3=6).
Step 2
Why this answer is correct
The correct answer is B. (6). There are (9) total pairs and (3) equal-coordinate pairs. Hence pairs with \(x\neq y\) are (9-3=6).
Step 3
Exam Tip
कुल (9) युग्म हैं और बराबर निर्देशांक वाले (3) हैं। इसलिए \(x\neq y\) वाले (9-3=6) युग्म हैं।
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यदि \(A=\{2,3,5\}\) और \(B=\{4,6,10,15\}\) है, तो \(A\times B\) में ऐसे कितने युग्म ((x,y)) हैं जिनके लिए (\gcd(x,y)>1) है?
If \(A=\{2,3,5\}\) and \(B=\{4,6,10,15\}\), how many pairs ((x,y)) in \(A\times B\) satisfy (\gcd(x,y)>1)?
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A (6)
B (7)
C (8)
D (9)
Explanation opens after your attempt
Step 1
Concept
For (x=2), (4,6,10) work; for (x=3), (6,15) work; for (x=5), (10,15) work. The total is (3+2+2=7), so the correct option is (7).
Step 2
Why this answer is correct
The correct answer is C. (8). For (x=2), (4,6,10) work; for (x=3), (6,15) work; for (x=5), (10,15) work. The total is (3+2+2=7), so the correct option is (7).
Step 3
Exam Tip
(x=2) पर (4,6,10), (x=3) पर (6,15), और (x=5) पर (10,15) सही हैं। कुल (3+2+2=7) नहीं बल्कि ((5,15)) सहित (7) है, इसलिए सही विकल्प (7) होना चाहिए।
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यदि \(A=\{1,2,3\}\) और \(B=\{1,2,3,4,5\}\) है, तो \(A\times B\) में ऐसे कितने युग्म ((x,y)) हैं जिनके लिए \(y-x\ge 2\) है?
If \(A=\{1,2,3\}\) and \(B=\{1,2,3,4,5\}\), how many pairs ((x,y)) in \(A\times B\) satisfy \(y-x\ge 2\)?
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A (6)
B (7)
C (8)
D (9)
Explanation opens after your attempt
Step 1
Concept
For (x=1), (3,4,5) work; for (x=2), (4,5) work; for (x=3), (5) works. Total is (3+2+1=6).
Step 2
Why this answer is correct
The correct answer is A. (6). For (x=1), (3,4,5) work; for (x=2), (4,5) work; for (x=3), (5) works. Total is (3+2+1=6).
Step 3
Exam Tip
(x=1) पर (3,4,5), (x=2) पर (4,5), और (x=3) पर (5) सही हैं। कुल (3+2+1=6) युग्म हैं।
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यदि \(A=\{1,2,3,4\}\) और \(B=\{1,2,3,4\}\) है, तो \(A\times B\) में ऐसे कितने युग्म ((x,y)) हैं जिनके लिए (x+y) (3) से विभाज्य है?
If \(A=\{1,2,3,4\}\) and \(B=\{1,2,3,4\}\), how many pairs ((x,y)) in \(A\times B\) have (x+y) divisible by (3)?
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A (4)
B (5)
C (6)
D (7)
Explanation opens after your attempt
Step 1
Concept
The sum can be (3) or (6). Valid pairs are ((1,2),(2,1),(2,4),(3,3),(4,2)), so there are (5) pairs.
Step 2
Why this answer is correct
The correct answer is B. (5). The sum can be (3) or (6). Valid pairs are ((1,2),(2,1),(2,4),(3,3),(4,2)), so there are (5) pairs.
Step 3
Exam Tip
योग (3) या (6) हो सकता है। सही युग्म ((1,2),(2,1),(2,4),(3,3),(4,2)) हैं, इसलिए (5) युग्म हैं।
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यदि \(A=\{0,1,2\}\) और \(B=\{-1,0,1\}\) है, तो \(A\times B\) में ऐसे कितने युग्म ((x,y)) हैं जिनके लिए \(x^2+y^2=1\) है?
If \(A=\{0,1,2\}\) and \(B=\{-1,0,1\}\), how many pairs ((x,y)) in \(A\times B\) satisfy \(x^2+y^2=1\)?
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A (2)
B (3)
C (4)
D (5)
Explanation opens after your attempt
Step 1
Concept
The valid pairs are ((0,-1),(0,1),(1,0)). Hence there are (3) pairs.
Step 2
Why this answer is correct
The correct answer is B. (3). The valid pairs are ((0,-1),(0,1),(1,0)). Hence there are (3) pairs.
Step 3
Exam Tip
सही युग्म ((0,-1),(0,1),(1,0)) हैं। इसलिए कुल (3) युग्म मिलते हैं।
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यदि \(A=\{-2,-1,0,1,2\}\) और \(B=\{-1,0,1\}\) है, तो \(A\times B\) में ऐसे कितने युग्म ((x,y)) हैं जिनके लिए (x+y=0) है?
If \(A=\{-2,-1,0,1,2\}\) and \(B=\{-1,0,1\}\), how many pairs ((x,y)) in \(A\times B\) satisfy (x+y=0)?
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A (2)
B (3)
C (4)
D (5)
Explanation opens after your attempt
Step 1
Concept
The valid pairs are ((-1,1),(0,0),(1,-1)). Therefore there are (3) pairs.
Step 2
Why this answer is correct
The correct answer is B. (3). The valid pairs are ((-1,1),(0,0),(1,-1)). Therefore there are (3) pairs.
Step 3
Exam Tip
सही युग्म ((-1,1),(0,0),(1,-1)) हैं। इसलिए कुल (3) युग्म हैं।
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यदि \(A=\{1,2,3,4\}\) और \(B=\{1,3,5,7\}\) है, तो \(A\times B\) में ऐसे कितने युग्म ((x,y)) हैं जिनके लिए (y=2x-1) है?
If \(A=\{1,2,3,4\}\) and \(B=\{1,3,5,7\}\), how many pairs ((x,y)) in \(A\times B\) satisfy (y=2x-1)?
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A (2)
B (3)
C (4)
D (5)
Explanation opens after your attempt
Step 1
Concept
For (x=1,2,3,4), (y=1,3,5,7), all in (B). Hence there are (4) pairs.
Step 2
Why this answer is correct
The correct answer is C. (4). For (x=1,2,3,4), (y=1,3,5,7), all in (B). Hence there are (4) pairs.
Step 3
Exam Tip
(x=1,2,3,4) पर (y=1,3,5,7) मिलते हैं और ये सभी (B) में हैं। अतः कुल (4) युग्म हैं।
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यदि \(A=\{1,2,3\}\) और \(B=\{2,4,6,8\}\) है, तो \(A\times B\) में ऐसे कितने युग्म ((x,y)) हैं जिनके लिए (y/x) पूर्णांक है?
If \(A=\{1,2,3\}\) and \(B=\{2,4,6,8\}\), how many pairs ((x,y)) in \(A\times B\) have (y/x) as an integer?
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A (7)
B (8)
C (9)
D (10)
Explanation opens after your attempt
Step 1
Concept
For (x=1), (4) values work; for (x=2), (4) values work; and for (x=3), only (6) works. Total is (9).
Step 2
Why this answer is correct
The correct answer is C. (9). For (x=1), (4) values work; for (x=2), (4) values work; and for (x=3), only (6) works. Total is (9).
Step 3
Exam Tip
(x=1) पर (4), (x=2) पर (4), और (x=3) पर केवल (6) सही है। कुल (4+4+1=9) है।
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यदि \(A=\{1,2,3,4,5\}\) और \(B=\{1,2,3,4,5\}\) है, तो \(A\times B\) में ऐसे कितने युग्म ((x,y)) हैं जिनके लिए (x+y=xy) है?
If \(A=\{1,2,3,4,5\}\) and \(B=\{1,2,3,4,5\}\), how many pairs ((x,y)) in \(A\times B\) satisfy (x+y=xy)?
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A (0)
B (1)
C (2)
D (3)
Explanation opens after your attempt
Step 1
Concept
The equation (xy-x-y=0) gives ((x-1)(y-1)=1). Among positive integers, only ((2,2)) works, so the count is (1).
Step 2
Why this answer is correct
The correct answer is B. (1). The equation (xy-x-y=0) gives ((x-1)(y-1)=1). Among positive integers, only ((2,2)) works, so the count is (1).
Step 3
Exam Tip
समीकरण (xy-x-y=0) से ((x-1)(y-1)=1) मिलता है। धनात्मक पूर्णांकों में केवल ((2,2)) सही है, इसलिए संख्या (1) है।
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