यदि \(|A\times B|=24\), (|A|=4) और (|C|=7) हैं, तो \(|B\times C|\) का मान क्या होगा?
If \(|A\times B|=24\), (|A|=4), and (|C|=7), what is the value of \(|B\times C|\)?
#cartesian-product
#cardinality
#expert
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A (28)
B (35)
C (42)
D (56)
Explanation opens after your attempt
Step 1
Concept
Since \(|B|=\frac{24}{4}=6\), \(|B\times C|=6\cdot7=42\). First find the unknown cardinality.
Step 2
Why this answer is correct
The correct answer is C. (42). Since \(|B|=\frac{24}{4}=6\), \(|B\times C|=6\cdot7=42\). First find the unknown cardinality.
Step 3
Exam Tip
\(|B|=\frac{24}{4}=6\), इसलिए \(|B\times C|=6\cdot7=42\)। पहले अज्ञात समुच्चय की कार्डिनलिटी निकालें।
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यदि \(A=\{1,2,3,4\}\) और \(B=\{5,6,7\}\) हैं, तो \(A\times B\) के कितने उपसमुच्चय ((1,5)) और ((4,7)) को रखते हैं पर ((2,6)) को नहीं रखते हैं?
If \(A=\{1,2,3,4\}\) and \(B=\{5,6,7\}\), how many subsets of \(A\times B\) contain ((1,5)) and ((4,7)) but do not contain ((2,6))?
#power-set
#fixed-pairs
#counting
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A (256)
B (512)
C (1024)
D (2048)
Explanation opens after your attempt
Step 1
Concept
There are (12) pairs, and after forcing (2) in and (1) out, (9) pairs remain free. Hence the number is \(2^9=512\).
Step 2
Why this answer is correct
The correct answer is B. (512). There are (12) pairs, and after forcing (2) in and (1) out, (9) pairs remain free. Hence the number is \(2^9=512\).
Step 3
Exam Tip
कुल (12) युग्म हैं, (2) युग्म रखने और (1) युग्म हटाने के बाद (9) स्वतंत्र युग्म बचते हैं। इसलिए संख्या \(2^9=512\) है।
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यदि (|A|=2) और (|B|=5) हैं, तो \(A\times B\) के ठीक (5) अवयवों वाले उपसमुच्चयों की संख्या क्या है?
If (|A|=2) and (|B|=5), what is the number of subsets of \(A\times B\) having exactly (5) elements?
#combinations
#subset-counting
#relations
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A (120)
B (210)
C (252)
D (512)
Explanation opens after your attempt
Step 1
Concept
\(|A\times B|=10\), so choosing exactly (5) pairs gives \(\binom{10}{5}=252\). Use combinations for exact-size subsets.
Step 2
Why this answer is correct
The correct answer is C. (252). \(|A\times B|=10\), so choosing exactly (5) pairs gives \(\binom{10}{5}=252\). Use combinations for exact-size subsets.
Step 3
Exam Tip
\(|A\times B|=10\), इसलिए ठीक (5) युग्म चुनने के तरीके \(\binom{10}{5}=252\) हैं। ठीक संख्या के लिए संयोजन लगाएँ।
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यदि \(A=\{2,3,4,5,6,7,8\}\), \(B=\{1,2,3,4,5\}\) और \(R=\{(a,b):a+3b=17\}\) है, तो (|R|) क्या है?
If \(A=\{2,3,4,5,6,7,8\}\), \(B=\{1,2,3,4,5\}\), and \(R=\{(a,b):a+3b=17\}\), what is (|R|)?
#linear-condition
#relation-subset
#counting
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A (1)
B (2)
C (3)
D (4)
Explanation opens after your attempt
Step 1
Concept
The condition gives ((8,3)) and ((5,4)), so (|R|=2). Check possible (b)-values systematically.
Step 2
Why this answer is correct
The correct answer is B. (2). The condition gives ((8,3)) and ((5,4)), so (|R|=2). Check possible (b)-values systematically.
Step 3
Exam Tip
शर्त से ((8,3)) और ((5,4)) मिलते हैं, इसलिए (|R|=2)। समीकरण में (b) के संभावित मान व्यवस्थित रूप से जाँचें।
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यदि \(A=\{0,1,2,3,4,5\}\) और \(B=\{0,1,2,3,4,5\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें \(a^2+b^2=25\) है?
If \(A=\{0,1,2,3,4,5\}\) and \(B=\{0,1,2,3,4,5\}\), how many pairs ((a,b)) in \(A\times B\) satisfy \(a^2+b^2=25\)?
#quadratic-condition
#ordered-pair
#counting
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A (2)
B (3)
C (4)
D (5)
Explanation opens after your attempt
Step 1
Concept
The pairs are ((0,5),(3,4),(4,3),(5,0)), so the count is (4). Reversed ordered pairs are counted separately.
Step 2
Why this answer is correct
The correct answer is C. (4). The pairs are ((0,5),(3,4),(4,3),(5,0)), so the count is (4). Reversed ordered pairs are counted separately.
Step 3
Exam Tip
युग्म ((0,5),(3,4),(4,3),(5,0)) हैं, इसलिए संख्या (4) है। क्रमित युग्मों में उलटे युग्म अलग गिने जाते हैं।
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यदि \(A=\{1,2,3,4,5,6\}\) और \(B=\{1,2,3,4,5,6\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें \(a\mid b\) और (a<b) दोनों हैं?
If \(A=\{1,2,3,4,5,6\}\) and \(B=\{1,2,3,4,5,6\}\), how many pairs ((a,b)) in \(A\times B\) satisfy both \(a\mid b\) and (a<b)?
#divisibility
#strict-inequality
#counting
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A (6)
B (8)
C (9)
D (10)
Explanation opens after your attempt
Step 1
Concept
For (a=1), there are (5) pairs; for (a=2), (2); and for (a=3), (1), totaling (8). Do not include (a=b).
Step 2
Why this answer is correct
The correct answer is B. (8). For (a=1), there are (5) pairs; for (a=2), (2); and for (a=3), (1), totaling (8). Do not include (a=b).
Step 3
Exam Tip
(a=1) पर (5), (a=2) पर (2), और (a=3) पर (1) युग्म मिलते हैं, कुल (8)। समानता (a=b) को शामिल न करें।
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यदि \(A=\{1,2,3,4,5\}\) और \(B=\{2,3,4,5,6,7\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (a+b) अभाज्य और (a<b) है?
If \(A=\{1,2,3,4,5\}\) and \(B=\{2,3,4,5,6,7\}\), how many pairs ((a,b)) in \(A\times B\) have (a+b) prime and (a<b)?
#prime-sum
#combined-condition
#counting
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A (6)
B (8)
C (10)
D (12)
Explanation opens after your attempt
Step 1
Concept
Applying both conditions gives (8) pairs. First filter by (a<b), then test prime sums.
Step 2
Why this answer is correct
The correct answer is B. (8). Applying both conditions gives (8) pairs. First filter by (a<b), then test prime sums.
Step 3
Exam Tip
दोनों शर्तें लगाने पर (8) युग्म मिलते हैं। पहले (a<b) से छाँटें और फिर अभाज्य योग जाँचें।
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\(यदि (A={1,2,3,4}), (B={1,2,3,4}) और (S={(a,b):a+b\) is even\(}) है, तो (S) के पूरक में (A\times B) के सापेक्ष कितने युग्म होंगे\)?
\(If (A={1,2,3,4}), (B={1,2,3,4}), and (S={(a,b):a+b\) is even\(}), how many pairs are in the complement of (S) relative to (A\times B)\)?
#complement
#parity
#cartesian-product
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A (6)
B (8)
C (10)
D (12)
Explanation opens after your attempt
Step 1
Concept
The complement contains pairs with odd sum, and the count is \(2\cdot2+2\cdot2=8\). For complements, write the opposite condition.
Step 2
Why this answer is correct
The correct answer is B. (8). The complement contains pairs with odd sum, and the count is \(2\cdot2+2\cdot2=8\). For complements, write the opposite condition.
Step 3
Exam Tip
पूरक में विषम योग वाले युग्म होंगे और उनकी संख्या \(2\cdot2+2\cdot2=8\) है। पूरक गिनती में विपरीत शर्त लिखें।
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यदि \(A=\{0,1,2,3\}\), \(B=\{0,1,2,3,4\}\) और \(R=\{(a,b):b\ge a^2-1\}\) है, तो (|R|) क्या है?
If \(A=\{0,1,2,3\}\), \(B=\{0,1,2,3,4\}\), and \(R=\{(a,b):b\ge a^2-1\}\), what is (|R|)?
#quadratic-inequality
#relation-subset
#counting
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A (11)
B (12)
C (13)
D (14)
Explanation opens after your attempt
Step 1
Concept
For (a=0,1,2,3), the counts of (b) are (5,5,2,0), totaling (12). Check the boundary separately for each (a).
Step 2
Why this answer is correct
The correct answer is B. (12). For (a=0,1,2,3), the counts of (b) are (5,5,2,0), totaling (12). Check the boundary separately for each (a).
Step 3
Exam Tip
(a=0,1,2,3) के लिए (b) के क्रमशः (5,5,2,0) मान मिलते हैं, कुल (12)। हर (a) पर सीमा अलग जाँचें।
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यदि \(A=\{1,2,3,4,5\}\) और \(B=\{1,2,3,4,5\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (\gcd(a,b)=1) और (a+b) सम है?
If \(A=\{1,2,3,4,5\}\) and \(B=\{1,2,3,4,5\}\), how many pairs ((a,b)) in \(A\times B\) satisfy (\gcd(a,b)=1) and (a+b) even?
#gcd
#parity
#combined-condition
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A (6)
B (7)
C (8)
D (9)
Explanation opens after your attempt
Step 1
Concept
The components must have the same parity and also (\gcd(a,b)=1), giving (7) pairs. Check \(\gcd\) carefully among same-parity pairs.
Step 2
Why this answer is correct
The correct answer is B. (7). The components must have the same parity and also (\gcd(a,b)=1), giving (7) pairs. Check \(\gcd\) carefully among same-parity pairs.
Step 3
Exam Tip
दोनों अवयवों की समता समान होनी चाहिए और (\gcd(a,b)=1) भी चाहिए, ऐसे (7) युग्म हैं। सम-सम युग्मों में \(\gcd\) जल्दी जाँचें।
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यदि \(A=\{1,2,3,4\}\), \(B=\{2,4,6,8\}\) और \(R=\{(a,b):\frac{b}{a}\in\mathbb{N}\}\) है, तो (|R|) क्या है?
If \(A=\{1,2,3,4\}\), \(B=\{2,4,6,8\}\), and \(R=\{(a,b):\frac{b}{a}\in\mathbb{N}\}\), what is (|R|)?
#divisibility
#fraction-condition
#counting
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A (9)
B (10)
C (11)
D (12)
Explanation opens after your attempt
Step 1
Concept
This is the condition \(a\mid b\); for (a=1,2,3,4), the counts are (4,4,1,2). The total is (11) pairs.
Step 2
Why this answer is correct
The correct answer is C. (11). This is the condition \(a\mid b\); for (a=1,2,3,4), the counts are (4,4,1,2). The total is (11) pairs.
Step 3
Exam Tip
यह \(a\mid b\) की शर्त है; (a=1,2,3,4) के लिए गिनती (4,4,1,2) है। कुल (11) युग्म मिलते हैं।
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यदि \(A=\{1,2,3\}\), \(B=\{1,2,3,4\}\) और \(C=\{2,3,5\}\) हैं, तो (|A\times(B-C)|+|\(A\times B\)\cap\(A\times C\)|) क्या है?
If \(A=\{1,2,3\}\), \(B=\{1,2,3,4\}\), and \(C=\{2,3,5\}\), what is (|A\times(B-C)|+|\(A\times B\)\cap\(A\times C\)|)?
#set-operations
#cardinality
#cartesian-product
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A (9)
B (12)
C (15)
D (18)
Explanation opens after your attempt
Step 1
Concept
(B-C={1,4}) gives (6) pairs, and \(B\cap C={2,3}\) gives (6) pairs. The sum is (12).
Step 2
Why this answer is correct
The correct answer is B. (12). (B-C={1,4}) gives (6) pairs, and \(B\cap C={2,3}\) gives (6) pairs. The sum is (12).
Step 3
Exam Tip
(B-C={1,4}) से (6) और \(B\cap C={2,3}\) से (6) युग्म मिलते हैं। योग (12) है।
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यदि \(A=\{1,2,3,4,5\}\) और \(B=\{1,2,3,4,5\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (a+b=6) या (a=b) है?
If \(A=\{1,2,3,4,5\}\) and \(B=\{1,2,3,4,5\}\), how many pairs ((a,b)) in \(A\times B\) satisfy (a+b=6) or (a=b)?
#inclusion-exclusion
#diagonal
#sum-condition
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A (8)
B (9)
C (10)
D (11)
Explanation opens after your attempt
Step 1
Concept
There are (5) pairs with (a+b=6) and (5) with (a=b), but ((3,3)) is counted twice. Hence (5+5-1=9).
Step 2
Why this answer is correct
The correct answer is B. (9). There are (5) pairs with (a+b=6) and (5) with (a=b), but ((3,3)) is counted twice. Hence (5+5-1=9).
Step 3
Exam Tip
(a+b=6) वाले (5) और (a=b) वाले (5) युग्म हैं, पर ((3,3)) दो बार गिना गया। इसलिए (5+5-1=9)।
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यदि \(A=\{1,2,3,4\}\) और \(B=\{1,2,3,4,5,6\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें \(ab\le12\) और (a+b) विषम है?
If \(A=\{1,2,3,4\}\) and \(B=\{1,2,3,4,5,6\}\), how many pairs ((a,b)) in \(A\times B\) satisfy \(ab\le12\) and (a+b) is odd?
#product-inequality
#parity
#counting
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A (9)
B (10)
C (11)
D (12)
Explanation opens after your attempt
Step 1
Concept
Counting opposite-parity pairs under the product bound gives (10). For combined conditions, check the bound first and then parity.
Step 2
Why this answer is correct
The correct answer is B. (10). Counting opposite-parity pairs under the product bound gives (10). For combined conditions, check the bound first and then parity.
Step 3
Exam Tip
पहली शर्त के अंदर विपरीत समता वाले युग्म गिनने पर (10) मिलते हैं। संयुक्त शर्तों में पहले सीमा और फिर समता जाँचें।
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यदि \(A=\{1,2,3,4\}\) और \(B=\{2,3,5,7,11\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (b-a) अभाज्य है?
If \(A=\{1,2,3,4\}\) and \(B=\{2,3,5,7,11\}\), how many pairs ((a,b)) in \(A\times B\) have (b-a) prime?
#prime-difference
#counting
#ordered-pair
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A (5)
B (6)
C (7)
D (8)
Explanation opens after your attempt
Step 1
Concept
Checking possible differences gives (6) pairs with prime difference. Remember that (1) is not prime.
Step 2
Why this answer is correct
The correct answer is B. (6). Checking possible differences gives (6) pairs with prime difference. Remember that (1) is not prime.
Step 3
Exam Tip
संभव अंतरों को जाँचने पर अभाज्य अंतर वाले (6) युग्म मिलते हैं। ध्यान रखें कि (1) अभाज्य नहीं है।
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यदि \(A=\{1,2,3,4\}\), \(B=\{1,2,3\}\) और \(C=\{0,1\}\) हैं, तो \(A\times B\times C\) में कितने क्रमित त्रिक ((a,b,c)) ऐसे हैं जिनमें (a=b+c) है?
If \(A=\{1,2,3,4\}\), \(B=\{1,2,3\}\), and \(C=\{0,1\}\), how many ordered triples ((a,b,c)) in \(A\times B\times C\) satisfy (a=b+c)?
#ordered-triple
#linear-rule
#counting
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A (4)
B (5)
C (6)
D (7)
Explanation opens after your attempt
Step 1
Concept
Each ((b,c)) gives (a=b+c), and all (6) values lie in (A). Hence there are (6) triples.
Step 2
Why this answer is correct
The correct answer is C. (6). Each ((b,c)) gives (a=b+c), and all (6) values lie in (A). Hence there are (6) triples.
Step 3
Exam Tip
हर ((b,c)) से (a=b+c) बनता है और सभी (6) मान (A) में हैं। इसलिए (6) त्रिक मिलते हैं।
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यदि \(A=\{1,2,3,4,5,6\}\) और \(B=\{1,2,3,4,5,6\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (\operatorname{lcm}(a,b)=6) है?
If \(A=\{1,2,3,4,5,6\}\) and \(B=\{1,2,3,4,5,6\}\), how many pairs ((a,b)) in \(A\times B\) satisfy (\operatorname{lcm}(a,b)=6)?
#lcm
#divisors
#counting
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A (7)
B (8)
C (9)
D (10)
Explanation opens after your attempt
Step 1
Concept
For (\operatorname{lcm}(a,b)=6), both values must divide (6), and there are (9) pairs. Check using the list of divisors.
Step 2
Why this answer is correct
The correct answer is C. (9). For (\operatorname{lcm}(a,b)=6), both values must divide (6), and there are (9) pairs. Check using the list of divisors.
Step 3
Exam Tip
(\operatorname{lcm}(a,b)=6) के लिए (a,b) के मान (6) के भाजक होने चाहिए और कुल (9) युग्म मिलते हैं। भाजकों की सूची से जाँचें।
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यदि \(A=\{1,2,3,4,5\}\), \(B=\{1,2,3,4,5\}\) और \(T=\{(a,b):|a-b|\le2\}\) है, तो (|T|) क्या है?
If \(A=\{1,2,3,4,5\}\), \(B=\{1,2,3,4,5\}\), and \(T=\{(a,b):|a-b|\le2\}\), what is (|T|)?
#absolute-inequality
#complement
#counting
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A (17)
B (19)
C (21)
D (23)
Explanation opens after your attempt
Step 1
Concept
Among (25) total pairs, (6) pairs have (|a-b|>2), so (25-6=19). Complement counting is quicker.
Step 2
Why this answer is correct
The correct answer is B. (19). Among (25) total pairs, (6) pairs have (|a-b|>2), so (25-6=19). Complement counting is quicker.
Step 3
Exam Tip
कुल (25) युग्मों में (|a-b|>2) वाले (6) युग्म हैं, इसलिए (25-6=19)। पूरक गिनती तेज रहती है।
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यदि \(A=\{1,2,3,4\}\), \(B=\{1,2,3,4\}\) और \(U=A\times B\) है, तो (U) में कितने युग्म (a<b) या (a+b=5) को संतुष्ट करते हैं?
If \(A=\{1,2,3,4\}\), \(B=\{1,2,3,4\}\), and \(U=A\times B\), how many pairs in (U) satisfy (a<b) or (a+b=5)?
#inclusion-exclusion
#inequality
#sum-condition
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A (8)
B (9)
C (10)
D (11)
Explanation opens after your attempt
Step 1
Concept
There are (6) pairs with (a<b) and (4) with (a+b=5), with (2) common. Thus (6+4-2=8).
Step 2
Why this answer is correct
The correct answer is A. (8). There are (6) pairs with (a<b) and (4) with (a+b=5), with (2) common. Thus (6+4-2=8).
Step 3
Exam Tip
(a<b) वाले (6) और (a+b=5) वाले (4) युग्म हैं, साझा (2) हैं। इसलिए (6+4-2=8)।
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यदि \(A=\{0,1,2,3,4\}\) और \(B=\{0,1,2,3,4\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें \(a+b\le4\) और \(a\ne b\) है?
If \(A=\{0,1,2,3,4\}\) and \(B=\{0,1,2,3,4\}\), how many pairs ((a,b)) in \(A\times B\) satisfy \(a+b\le4\) and \(a\ne b\)?
#sum-inequality
#not-equal
#counting
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A (9)
B (10)
C (11)
D (12)
Explanation opens after your attempt
Step 1
Concept
There are (15) pairs with \(a+b\le4\), and equal pairs among them are ((0,0),(1,1),(2,2)). Hence (15-3=12).
Step 2
Why this answer is correct
The correct answer is D. (12). There are (15) pairs with \(a+b\le4\), and equal pairs among them are ((0,0),(1,1),(2,2)). Hence (15-3=12).
Step 3
Exam Tip
\(a+b\le4\) वाले कुल (15) युग्म हैं और उनमें बराबर युग्म ((0,0),(1,1),(2,2)) हैं। इसलिए (15-3=12)।
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यदि \(A=\{1,2,3,4,5,6\}\) और \(B=\{1,2,3,4,5,6\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (a+b) (3) से विभाज्य और \(a\ne b\) है?
If \(A=\{1,2,3,4,5,6\}\) and \(B=\{1,2,3,4,5,6\}\), how many pairs ((a,b)) in \(A\times B\) have (a+b) divisible by (3) and \(a\ne b\)?
#modular-counting
#not-equal
#divisibility
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A (8)
B (10)
C (12)
D (14)
Explanation opens after your attempt
Step 1
Concept
There are (12) pairs with sum divisible by (3), and equal pairs among them are ((3,3),(6,6)). Hence (12-2=10).
Step 2
Why this answer is correct
The correct answer is B. (10). There are (12) pairs with sum divisible by (3), and equal pairs among them are ((3,3),(6,6)). Hence (12-2=10).
Step 3
Exam Tip
(3) से विभाज्य योग वाले (12) युग्म हैं और उनमें ((3,3),(6,6)) बराबर युग्म हैं। इसलिए (12-2=10)।
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यदि \(A=\{1,2,3,4\}\), \(B=\{1,2,3,4,5\}\) और \(R=\{(a,b):a^2<b\}\) है, तो (|R|) क्या है?
If \(A=\{1,2,3,4\}\), \(B=\{1,2,3,4,5\}\), and \(R=\{(a,b):a^2<b\}\), what is (|R|)?
#quadratic-inequality
#counting
#relation
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A (5)
B (6)
C (7)
D (8)
Explanation opens after your attempt
Step 1
Concept
For (a=1), there are (4) values, and for (a=2), there is (1); none for the rest. Total is (5).
Step 2
Why this answer is correct
The correct answer is A. (5). For (a=1), there are (4) values, and for (a=2), there is (1); none for the rest. Total is (5).
Step 3
Exam Tip
(a=1) पर (4) और (a=2) पर (1) मान मिलते हैं, बाकी पर कोई नहीं। कुल (5) युग्म हैं।
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यदि \(A=\{1,2,3,4,5\}\), \(B=\{1,2,3,4,5\}\) और \(R=\{(a,b):a+b=7\}\) है, तो \(R^{-1}\) में कौन सा युग्म अवश्य होगा?
If \(A=\{1,2,3,4,5\}\), \(B=\{1,2,3,4,5\}\), and \(R=\{(a,b):a+b=7\}\), which pair must be in \(R^{-1}\)?
#inverse-relation
#ordered-pair
#cartesian-product
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A ((1,5))
B ((2,5))
C ((5,2))
D ((4,4))
Explanation opens after your attempt
Correct Answer
C. ((5,2))
Step 1
Concept
Since \((2,5)\in R\), we have \((5,2)\in R^{-1}\). In an inverse relation, the order reverses.
Step 2
Why this answer is correct
The correct answer is C. ((5,2)). Since \((2,5)\in R\), we have \((5,2)\in R^{-1}\). In an inverse relation, the order reverses.
Step 3
Exam Tip
क्योंकि \((2,5)\in R\), इसलिए \((5,2)\in R^{-1}\)। प्रतिलोम संबंध में क्रम उलटता है।
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यदि \(A=\{1,2,3\}\) और \(B=\{1,2,3,4,5,6\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (b) (a) का गुणज है पर \(b\ne a\) है?
If \(A=\{1,2,3\}\) and \(B=\{1,2,3,4,5,6\}\), how many pairs ((a,b)) in \(A\times B\) have (b) as a multiple of (a) but \(b\ne a\)?
#multiples
#not-equal
#counting
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A (8)
B (9)
C (10)
D (11)
Explanation opens after your attempt
Step 1
Concept
The total multiple count is (6+3+2=11), and ((1,1),(2,2),(3,3)) are removed. Thus (11-3=8).
Step 2
Why this answer is correct
The correct answer is A. (8). The total multiple count is (6+3+2=11), and ((1,1),(2,2),(3,3)) are removed. Thus (11-3=8).
Step 3
Exam Tip
गुणजों की कुल गिनती (6+3+2=11) है और ((1,1),(2,2),(3,3)) हटते हैं। इसलिए (11-3=8)।
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यदि \(A=\{1,2,3,4\}\) और \(B=\{0,1,2,3,4,5\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें \(b\equiv a^2 \pmod{5}\) है?
If \(A=\{1,2,3,4\}\) and \(B=\{0,1,2,3,4,5\}\), how many pairs ((a,b)) in \(A\times B\) satisfy \(b\equiv a^2 \pmod{5}\)?
#modular-arithmetic
#square-residue
#counting
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A (4)
B (5)
C (6)
D (8)
Explanation opens after your attempt
Step 1
Concept
The residues of \(a^2\) are (1,4,4,1), and each such residue has one value in (B). Therefore there are (4) pairs.
Step 2
Why this answer is correct
The correct answer is A. (4). The residues of \(a^2\) are (1,4,4,1), and each such residue has one value in (B). Therefore there are (4) pairs.
Step 3
Exam Tip
\(a^2\) के अवशेष (1,4,4,1) हैं और (B) में हर ऐसे अवशेष का एक-एक मान है। इसलिए कुल (4) युग्म हैं।
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यदि \(A=\{1,2,3,4,5\}\), \(B=\{1,2,3,4,5\}\) और \(R=\{(a,b):a+b\le6\}\) है, तो (R) में ऐसे कितने युग्म हैं जिनमें (a) सम है?
If \(A=\{1,2,3,4,5\}\), \(B=\{1,2,3,4,5\}\), and \(R=\{(a,b):a+b\le6\}\), how many pairs in (R) have (a) even?
#restricted-relation
#parity
#sum-inequality
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A (5)
B (6)
C (7)
D (8)
Explanation opens after your attempt
Step 1
Concept
For even (a), (a=2) gives (4) values and (a=4) gives (2). Total is (6) pairs.
Step 2
Why this answer is correct
The correct answer is B. (6). For even (a), (a=2) gives (4) values and (a=4) gives (2). Total is (6) pairs.
Step 3
Exam Tip
सम (a) के लिए (a=2) पर (4) और (a=4) पर (2) मान मिलते हैं। कुल (6) युग्म हैं।
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यदि \(A=\{1,2,3,4,5,6\}\) और \(B=\{1,2,3,4,5,6\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (a+b=ab) है?
If \(A=\{1,2,3,4,5,6\}\) and \(B=\{1,2,3,4,5,6\}\), how many pairs ((a,b)) in \(A\times B\) satisfy (a+b=ab)?
#equation-condition
#factorization
#counting
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A (0)
B (1)
C (2)
D (3)
Explanation opens after your attempt
Step 1
Concept
The equation gives ((a-1)(b-1)=1), so only ((2,2)) works. Factor transformation makes counting easier.
Step 2
Why this answer is correct
The correct answer is B. (1). The equation gives ((a-1)(b-1)=1), so only ((2,2)) works. Factor transformation makes counting easier.
Step 3
Exam Tip
समीकरण ((a-1)(b-1)=1) देता है, इसलिए केवल ((2,2)) मिलता है। रूपांतरण से गिनती आसान होती है।
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\(यदि (A={1,2,3,4}), (B={1,2,3,4}) और (R={(a,b):a+b\) is odd}) है, तो (R) के उपसमुच्चयों की संख्या क्या है?
\(If (A={1,2,3,4}), (B={1,2,3,4}), and (R={(a,b):a+b\) is odd}), what is the number of subsets of (R)?
#power-set
#parity
#relation-subset
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A (128)
B (256)
C (512)
D (1024)
Explanation opens after your attempt
Step 1
Concept
There are (8) pairs with odd sum, so subsets of (R) are \(2^8=256\). First find the cardinality of the relation.
Step 2
Why this answer is correct
The correct answer is B. (256). There are (8) pairs with odd sum, so subsets of (R) are \(2^8=256\). First find the cardinality of the relation.
Step 3
Exam Tip
विषम योग वाले युग्म (8) हैं, इसलिए (R) के उपसमुच्चय \(2^8=256\) हैं। पहले संबंध की कार्डिनलिटी निकालें।
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यदि \(A=\{1,2,3,4,5\}\) और \(B=\{1,2,3,4,5\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (a<b) और (a+b) सम है?
If \(A=\{1,2,3,4,5\}\) and \(B=\{1,2,3,4,5\}\), how many pairs ((a,b)) in \(A\times B\) satisfy (a<b) and (a+b) is even?
#parity
#strict-inequality
#counting
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A (3)
B (4)
C (5)
D (6)
Explanation opens after your attempt
Step 1
Concept
Even sum requires same parity, along with (a<b). Odd values give (3) pairs and even values give (1), totaling (4).
Step 2
Why this answer is correct
The correct answer is B. (4). Even sum requires same parity, along with (a<b). Odd values give (3) pairs and even values give (1), totaling (4).
Step 3
Exam Tip
योग सम के लिए दोनों की समता समान चाहिए और (a<b) भी चाहिए। विषमों से (3) और समों से (1) युग्म मिलते हैं, कुल (4)।
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\(यदि (A={1,2,3,4}), (B={1,2,3,4,5,6}) और (R={(a,b):b=2a\) or \(b=3a}) है, तो (|R|) क्या है\)?
\(If (A={1,2,3,4}), (B={1,2,3,4,5,6}), and (R={(a,b):b=2a\) or \(b=3a}), what is (|R|)\)?
#union-of-rules
#linear-rule
#counting
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A (5)
B (6)
C (7)
D (8)
Explanation opens after your attempt
Step 1
Concept
The rule (b=2a) gives (3) pairs and (b=3a) gives (2), with no overlap. Total is (5).
Step 2
Why this answer is correct
The correct answer is A. (5). The rule (b=2a) gives (3) pairs and (b=3a) gives (2), with no overlap. Total is (5).
Step 3
Exam Tip
(b=2a) से (3) और (b=3a) से (2) युग्म मिलते हैं, कोई साझा युग्म नहीं है। कुल (5) युग्म हैं।
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यदि \(A=\{1,2,3,4,5,6\}\) और \(B=\{1,2,3,4,5,6\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (a) और (b) दोनों अभाज्य हैं या दोनों भाज्य हैं?
If \(A=\{1,2,3,4,5,6\}\) and \(B=\{1,2,3,4,5,6\}\), how many pairs ((a,b)) in \(A\times B\) have both (a) and (b) prime or both composite?
#prime-composite
#counting
#classification
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A (12)
B (13)
C (14)
D (18)
Explanation opens after your attempt
Step 1
Concept
There are (3) primes and (2) composites, so the count is \(3^2+2^2=13\). The number (1) is neither prime nor composite.
Step 2
Why this answer is correct
The correct answer is B. (13). There are (3) primes and (2) composites, so the count is \(3^2+2^2=13\). The number (1) is neither prime nor composite.
Step 3
Exam Tip
अभाज्य संख्याएँ (3) और भाज्य संख्याएँ (2) हैं, इसलिए गिनती \(3^2+2^2=13\) है। (1) न अभाज्य है न भाज्य।
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यदि \(A=\{1,2,3,4,5\}\), \(B=\{0,1,2,3,4\}\) और \(R=\{(a,b):a-b\in{1,3}\}\) है, तो (|R|) क्या है?
If \(A=\{1,2,3,4,5\}\), \(B=\{0,1,2,3,4\}\), and \(R=\{(a,b):a-b\in{1,3}\}\), what is (|R|)?
#difference-condition
#set-membership
#counting
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A (6)
B (7)
C (8)
D (9)
Explanation opens after your attempt
Step 1
Concept
The condition (a-b=1) gives (5) pairs and (a-b=3) gives (3), totaling (8). Add the counts for separate differences.
Step 2
Why this answer is correct
The correct answer is C. (8). The condition (a-b=1) gives (5) pairs and (a-b=3) gives (3), totaling (8). Add the counts for separate differences.
Step 3
Exam Tip
(a-b=1) से (5) और (a-b=3) से (3) युग्म मिलते हैं, कुल (8)। अलग-अलग अंतरों को जोड़ें।
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यदि \(A=\{1,2,3,4\}\), \(B=\{1,2,3,4\}\) और \(R=A\times B\) है, तो (R) से कम से कम कितने युग्म हटाने होंगे ताकि कोई भी युग्म ((a,a)) न बचे?
If \(A=\{1,2,3,4\}\), \(B=\{1,2,3,4\}\), and \(R=A\times B\), what is the minimum number of pairs to remove from (R) so that no pair ((a,a)) remains?
#diagonal-pairs
#removal
#cartesian-product
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A (2)
B (3)
C (4)
D (8)
Explanation opens after your attempt
Step 1
Concept
The diagonal pairs are ((1,1),(2,2),(3,3),(4,4)), so (4) must be removed. The number of diagonal pairs is (|A|).
Step 2
Why this answer is correct
The correct answer is C. (4). The diagonal pairs are ((1,1),(2,2),(3,3),(4,4)), so (4) must be removed. The number of diagonal pairs is (|A|).
Step 3
Exam Tip
विकर्ण युग्म ((1,1),(2,2),(3,3),(4,4)) हैं, इसलिए (4) हटाने होंगे। विकर्ण की संख्या (|A|) होती है।
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यदि \(A=\{1,2,3\}\), \(B=\{1,2,3\}\) और \(C=\{1,2,3\}\) हैं, तो \(A\times B\times C\) में कितने त्रिक ((a,b,c)) ऐसे हैं जिनमें (a+b+c=6) है?
If \(A=\{1,2,3\}\), \(B=\{1,2,3\}\), and \(C=\{1,2,3\}\), how many triples ((a,b,c)) in \(A\times B\times C\) satisfy (a+b+c=6)?
#ordered-triple
#sum-condition
#counting
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A (6)
B (7)
C (8)
D (9)
Explanation opens after your attempt
Step 1
Concept
There are (7) ordered triples from (1) to (3) with sum (6). In ordered triples, order is counted separately.
Step 2
Why this answer is correct
The correct answer is B. (7). There are (7) ordered triples from (1) to (3) with sum (6). In ordered triples, order is counted separately.
Step 3
Exam Tip
तीन संख्याओं (1) से (3) के बीच योग (6) के कुल (7) क्रमित त्रिक हैं। क्रमित त्रिक में क्रम अलग गिना जाता है।
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यदि \(A=\{1,2,3,4,5\}\) और \(B=\{1,2,3,4,5\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (a+b) (4) से विभाज्य है?
If \(A=\{1,2,3,4,5\}\) and \(B=\{1,2,3,4,5\}\), how many pairs ((a,b)) in \(A\times B\) have (a+b) divisible by (4)?
#modular-counting
#divisibility
#sum-condition
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A (5)
B (6)
C (7)
D (8)
Explanation opens after your attempt
Step 1
Concept
The sum can be (4) or (8), and the total number of valid pairs is (6). Count using remainders or a sum list.
Step 2
Why this answer is correct
The correct answer is B. (6). The sum can be (4) or (8), and the total number of valid pairs is (6). Count using remainders or a sum list.
Step 3
Exam Tip
योग (4) या (8) हो सकता है और अनुकूल युग्मों की कुल संख्या (6) है। शेषफल या योग-सूची से गिनें।
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यदि \(A=\{1,2,3,4,5,6\}\) और \(B=\{1,2,3,4,5,6\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (a+b) (5) से विभाज्य है?
If \(A=\{1,2,3,4,5,6\}\) and \(B=\{1,2,3,4,5,6\}\), how many pairs ((a,b)) in \(A\times B\) have (a+b) divisible by (5)?
#modular-counting
#divisibility
#ordered-pair
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A (6)
B (7)
C (8)
D (9)
Explanation opens after your attempt
Step 1
Concept
The number of ordered pairs with sum (5) or (10) is (7). Complementary remainders give a quick count.
Step 2
Why this answer is correct
The correct answer is B. (7). The number of ordered pairs with sum (5) or (10) is (7). Complementary remainders give a quick count.
Step 3
Exam Tip
योग (5) या (10) बनने वाले क्रमित युग्मों की संख्या (7) है। शेषफलों के पूरक जोड़ से तेज गिनती होती है।
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यदि \(A=\{0,1,2,3,4\}\), \(B=\{0,1,2,3,4\}\) और (R={(a,b):\(a^2\equiv b \pmod{5}\)}) है, तो (|R|) क्या है?
If \(A=\{0,1,2,3,4\}\), \(B=\{0,1,2,3,4\}\), and (R={(a,b):\(a^2\equiv b \pmod{5}\)}), what is (|R|)?
#modular-arithmetic
#square-residue
#relation
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A (3)
B (4)
C (5)
D (6)
Explanation opens after your attempt
Step 1
Concept
For each (a), exactly one residue for (b) appears in (B). Therefore there are (5) pairs.
Step 2
Why this answer is correct
The correct answer is C. (5). For each (a), exactly one residue for (b) appears in (B). Therefore there are (5) pairs.
Step 3
Exam Tip
प्रत्येक (a) के लिए (b) का ठीक एक अवशेष (B) में मिलता है। इसलिए कुल (5) युग्म हैं।
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यदि \(A=\{1,2,3,4\}\) और \(B=\{1,2,3,4,5,6,7,8\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें \(a\mid b\) पर \(b\mid a\) नहीं है?
If \(A=\{1,2,3,4\}\) and \(B=\{1,2,3,4,5,6,7,8\}\), how many pairs ((a,b)) in \(A\times B\) satisfy \(a\mid b\) but not \(b\mid a\)?
#divisibility
#asymmetric-condition
#counting
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A (10)
B (11)
C (12)
D (13)
Explanation opens after your attempt
Step 1
Concept
There are (16) pairs with \(a\mid b\), and pairs where \(b\mid a\) also holds are ((1,1),(2,2),(3,3),(4,4)). Hence (16-4=12).
Step 2
Why this answer is correct
The correct answer is C. (12). There are (16) pairs with \(a\mid b\), and pairs where \(b\mid a\) also holds are ((1,1),(2,2),(3,3),(4,4)). Hence (16-4=12).
Step 3
Exam Tip
\(a\mid b\) वाले (16) युग्म हैं और \(b\mid a\) भी होने वाले ((1,1),(2,2),(3,3),(4,4)) हैं। इसलिए (16-4=12)।
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यदि \(A=\{1,2,3,4,5\}\), \(B=\{1,2,3,4,5\}\) और \(R=\{(a,b):a+b=6\}\) है, तो \(R\circ R\) में कितने युग्म होंगे?
If \(A=\{1,2,3,4,5\}\), \(B=\{1,2,3,4,5\}\), and \(R=\{(a,b):a+b=6\}\), how many pairs are in \(R\circ R\)?
#composition
#relation
#cartesian-product
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A (3)
B (5)
C (7)
D (9)
Explanation opens after your attempt
Step 1
Concept
For each (x), (R) maps it to (6-x) and then back to (x). Thus \(R\circ R={(x,x):x\in A}\) has (5) pairs.
Step 2
Why this answer is correct
The correct answer is B. (5). For each (x), (R) maps it to (6-x) and then back to (x). Thus \(R\circ R={(x,x):x\in A}\) has (5) pairs.
Step 3
Exam Tip
हर (x) के लिए (R) उसे (6-x) से और फिर वापस (x) से जोड़ता है। इसलिए \(R\circ R={(x,x):x\in A}\) में (5) युग्म हैं।
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यदि \(A=\{1,2,3,4,5,6\}\) और \(B=\{1,2,3,4,5,6\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (\gcd(a,b)=2) है?
If \(A=\{1,2,3,4,5,6\}\) and \(B=\{1,2,3,4,5,6\}\), how many pairs ((a,b)) in \(A\times B\) satisfy (\gcd(a,b)=2)?
#gcd
#counting
#common-factor
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A (5)
B (6)
C (7)
D (8)
Explanation opens after your attempt
Step 1
Concept
Both numbers must be multiples of (2), and after division the reduced pair must be coprime, giving (7) pairs. Factor out the common divisor in \(\gcd\) questions.
Step 2
Why this answer is correct
The correct answer is C. (7). Both numbers must be multiples of (2), and after division the reduced pair must be coprime, giving (7) pairs. Factor out the common divisor in \(\gcd\) questions.
Step 3
Exam Tip
दोनों संख्याएँ (2) से गुणित हों और भाग देने पर परस्पर अभाज्य रहें, ऐसे (7) युग्म हैं। \(\gcd\) प्रश्न में सामान्य गुणनखंड निकालें।
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यदि \(A=\{1,2,3,4\}\), \(B=\{1,2,3,4,5\}\) और \(R=\{(a,b):a+b\ge6\}\) है, तो (R) के पूरक में \(A\times B\) के सापेक्ष कितने युग्म होंगे?
If \(A=\{1,2,3,4\}\), \(B=\{1,2,3,4,5\}\), and \(R=\{(a,b):a+b\ge6\}\), how many pairs are in the complement of (R) relative to \(A\times B\)?
#complement
#sum-inequality
#counting
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A (8)
B (9)
C (10)
D (11)
Explanation opens after your attempt
Step 1
Concept
The complement has pairs with (a+b<6), whose row counts are (4,3,2,1). Hence there are (10) pairs.
Step 2
Why this answer is correct
The correct answer is C. (10). The complement has pairs with (a+b<6), whose row counts are (4,3,2,1). Hence there are (10) pairs.
Step 3
Exam Tip
पूरक में (a+b<6) वाले युग्म होंगे, जिनकी पंक्ति-गिनती (4,3,2,1) है। इसलिए कुल (10) युग्म हैं।
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यदि \(A=\{1,2,3,4,5\}\) और \(B=\{1,2,3,4,5\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (a+b) सम और (ab) (3) से विभाज्य है?
If \(A=\{1,2,3,4,5\}\) and \(B=\{1,2,3,4,5\}\), how many pairs ((a,b)) in \(A\times B\) have (a+b) even and (ab) divisible by (3)?
#parity
#divisibility
#combined-condition
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A (4)
B (5)
C (6)
D (7)
Explanation opens after your attempt
Step 1
Concept
Even sum needs same parity, and (ab) must contain a factor (3). This gives (5) pairs.
Step 2
Why this answer is correct
The correct answer is B. (5). Even sum needs same parity, and (ab) must contain a factor (3). This gives (5) pairs.
Step 3
Exam Tip
योग सम के लिए समता समान चाहिए और (ab) में (3) का गुणनखंड चाहिए। ऐसे (5) युग्म मिलते हैं।
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यदि \(A=\{1,2,3,4,5,6\}\) और \(B=\{1,2,3,4,5,6\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (a+b=9) या (ab=12) है?
If \(A=\{1,2,3,4,5,6\}\) and \(B=\{1,2,3,4,5,6\}\), how many pairs ((a,b)) in \(A\times B\) satisfy (a+b=9) or (ab=12)?
#union-condition
#sum-product
#counting
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A (7)
B (8)
C (9)
D (10)
Explanation opens after your attempt
Step 1
Concept
There are (4) pairs with (a+b=9) and (4) with (ab=12), with no overlap. Hence the total is (8).
Step 2
Why this answer is correct
The correct answer is B. (8). There are (4) pairs with (a+b=9) and (4) with (ab=12), with no overlap. Hence the total is (8).
Step 3
Exam Tip
(a+b=9) वाले (4) और (ab=12) वाले (4) युग्म हैं, साझा कोई नहीं है। इसलिए कुल (8) युग्म हैं।
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यदि \(A=\{1,2,3,4,5\}\), \(B=\{1,2,3,4,5\}\) और \(R=\{(a,b):a<b\}\) है, तो \(R^{-1}\cap R\) में कितने युग्म होंगे?
If \(A=\{1,2,3,4,5\}\), \(B=\{1,2,3,4,5\}\), and \(R=\{(a,b):a<b\}\), how many pairs are in \(R^{-1}\cap R\)?
#inverse-relation
#intersection
#inequality
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A (0)
B (5)
C (10)
D (20)
Explanation opens after your attempt
Step 1
Concept
\(R^{-1}\) has the condition (a>b), which cannot hold together with (a<b). Therefore the intersection is empty.
Step 2
Why this answer is correct
The correct answer is A. (0). \(R^{-1}\) has the condition (a>b), which cannot hold together with (a<b). Therefore the intersection is empty.
Step 3
Exam Tip
\(R^{-1}\) में (a>b) वाली शर्त होगी, जो (a<b) के साथ एक साथ संभव नहीं है। इसलिए प्रतिच्छेद रिक्त है।
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यदि \(A=\{1,2,3,4\}\), \(B=\{1,2,3,4,5\}\) और \(R=\{(a,b):b-a\in{0,2}\}\) है, तो (|R|) क्या है?
If \(A=\{1,2,3,4\}\), \(B=\{1,2,3,4,5\}\), and \(R=\{(a,b):b-a\in{0,2}\}\), what is (|R|)?
#difference-condition
#relation-subset
#counting
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A (6)
B (7)
C (8)
D (9)
Explanation opens after your attempt
Step 1
Concept
The condition (b-a=0) gives (4) pairs and (b-a=2) gives (3). The total is (7) pairs.
Step 2
Why this answer is correct
The correct answer is B. (7). The condition (b-a=0) gives (4) pairs and (b-a=2) gives (3). The total is (7) pairs.
Step 3
Exam Tip
(b-a=0) से (4) और (b-a=2) से (3) युग्म मिलते हैं। कुल (7) युग्म हैं।
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यदि \(A=\{1,2,3\}\), \(B=\{1,2,3\}\) और \(C=\{1,2\}\) हैं, तो \(A\times B\times C\) में कितने त्रिक ((a,b,c)) ऐसे हैं जिनमें (a+b=2c+1) है?
If \(A=\{1,2,3\}\), \(B=\{1,2,3\}\), and \(C=\{1,2\}\), how many triples ((a,b,c)) in \(A\times B\times C\) satisfy (a+b=2c+1)?
#ordered-triple
#linear-condition
#counting
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A (4)
B (5)
C (6)
D (7)
Explanation opens after your attempt
Step 1
Concept
For (c=1), sum (3) gives (2) triples, and for (c=2), sum (5) gives (2). Total is (4).
Step 2
Why this answer is correct
The correct answer is A. (4). For (c=1), sum (3) gives (2) triples, and for (c=2), sum (5) gives (2). Total is (4).
Step 3
Exam Tip
(c=1) पर योग (3) के (2) त्रिक और (c=2) पर योग (5) के (2) त्रिक मिलते हैं। कुल (4) हैं।
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यदि \(A=\{1,2,3,4,5\}\) और \(B=\{1,2,3,4,5,6\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें (a+b) विषम और (b>a) है?
If \(A=\{1,2,3,4,5\}\) and \(B=\{1,2,3,4,5,6\}\), how many pairs ((a,b)) in \(A\times B\) have (a+b) odd and (b>a)?
#parity
#strict-inequality
#counting
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A (6)
B (7)
C (8)
D (9)
Explanation opens after your attempt
Step 1
Concept
For (a=1,2,3,4,5), the counts of valid (b) are (3,2,2,1,1), totaling (9). Odd sum requires opposite parity.
Step 2
Why this answer is correct
The correct answer is D. (9). For (a=1,2,3,4,5), the counts of valid (b) are (3,2,2,1,1), totaling (9). Odd sum requires opposite parity.
Step 3
Exam Tip
(a=1,2,3,4,5) के लिए योग्य (b) की गिनती (3,2,2,1,1) है, कुल (9)। विषम योग के लिए समता अलग होनी चाहिए।
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यदि \(A=\{1,2,3,4\}\), \(B=\{1,2,3,4\}\) और \(R=\{(a,b):a+b\le5\}\) है, तो (R) में कितने युग्म ऐसे हैं जिनका प्रतिलोम युग्म भी (R) में है?
If \(A=\{1,2,3,4\}\), \(B=\{1,2,3,4\}\), and \(R=\{(a,b):a+b\le5\}\), how many pairs in (R) have their inverse pair also in (R)?
#inverse-pair
#symmetric-condition
#relation
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A (8)
B (9)
C (10)
D (11)
Explanation opens after your attempt
Step 1
Concept
The condition \(a+b\le5\) remains the same after reversal, so all (10) pairs of (R) qualify. Recognize symmetric conditions.
Step 2
Why this answer is correct
The correct answer is C. (10). The condition \(a+b\le5\) remains the same after reversal, so all (10) pairs of (R) qualify. Recognize symmetric conditions.
Step 3
Exam Tip
शर्त \(a+b\le5\) उलटने पर भी वही रहती है, इसलिए (R) के सभी (10) युग्म योग्य हैं। सममित शर्त को पहचानें।
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यदि \(A=\{1,2,3,4,5\}\) और \(B=\{1,2,3,4,5\}\) हैं, तो \(A\times B\) में कितने युग्म ((a,b)) ऐसे हैं जिनमें \(a^2-b^2=0\) है?
If \(A=\{1,2,3,4,5\}\) and \(B=\{1,2,3,4,5\}\), how many pairs ((a,b)) in \(A\times B\) satisfy \(a^2-b^2=0\)?
#difference-of-squares
#diagonal
#counting
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A (4)
B (5)
C (10)
D (25)
Explanation opens after your attempt
Step 1
Concept
For positive numbers, \(a^2=b^2\) gives (a=b). Therefore there are (5) diagonal pairs.
Step 2
Why this answer is correct
The correct answer is B. (5). For positive numbers, \(a^2=b^2\) gives (a=b). Therefore there are (5) diagonal pairs.
Step 3
Exam Tip
धनात्मक संख्याओं के लिए \(a^2=b^2\) से (a=b) मिलता है। इसलिए (5) विकर्ण युग्म हैं।
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यदि \(A=\{1,2,3,4,5,6\}\), \(B=\{1,2,3,4,5,6\}\) और \(R=\{(a,b):a+b=7\}\) है, तो (R) के कितने उपसमुच्चय कम से कम एक युग्म रखते हैं जिसका पहला अवयव (1) है?
If \(A=\{1,2,3,4,5,6\}\), \(B=\{1,2,3,4,5,6\}\), and \(R=\{(a,b):a+b=7\}\), how many subsets of (R) contain at least one pair whose first component is (1)?
#subset-counting
#fixed-condition
#relation
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A (16)
B (32)
C (48)
D (64)
Explanation opens after your attempt
Step 1
Concept
There are (6) pairs in (R), and only ((1,6)) has first component (1). It must be included, so the other (5) pairs are free, giving \(2^5=32\).
Step 2
Why this answer is correct
The correct answer is B. (32). There are (6) pairs in (R), and only ((1,6)) has first component (1). It must be included, so the other (5) pairs are free, giving \(2^5=32\).
Step 3
Exam Tip
(R) में (6) युग्म हैं और पहला अवयव (1) वाला केवल ((1,6)) है। उसे रखना होगा, इसलिए बाकी (5) युग्म स्वतंत्र हैं और संख्या \(2^5=32\) है।
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