समुच्चय \(A={x:x^2=16}\) और \(B=\{-4,4\}\) के बारे में सही कथन कौन सा है?
Which statement is correct about the sets \(A={x:x^2=16}\) and \(B=\{-4,4\}\)?
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#quadratic
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A (A=B)
B \(A\subset B\) पर \(A\ne B\) / \(A\subset B\) but \(A\ne B\)
C \(B\subset A\) पर \(A\ne B\) / \(B\subset A\) but \(A\ne B\)
D \(A\cap B=\varnothing\)
Explanation opens after your attempt
Step 1
Concept
The solutions of \(x^2=16\) are (-4) and (4) so both sets have the same elements. In exams compare elements not order.
Step 2
Why this answer is correct
The correct answer is A. (A=B). The solutions of \(x^2=16\) are (-4) and (4) so both sets have the same elements. In exams compare elements not order.
Step 3
Exam Tip
\(x^2=16\) के हल (-4) और (4) हैं इसलिए दोनों में वही अवयव हैं। परीक्षा में क्रम नहीं अवयव देखें।
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यदि \(A={X:X\) समुच्चय ({a,b,c}) का एक-अवयवी उपसमुच्चय है(}) और \(B=\{{a},{b},{c}\}\) हैं तो सही संबंध क्या है?
If \(A={X:X\) is a singleton subset of the set ({a,b,c})(}) and \(B=\{{a},{b},{c}\}\), what is the correct relation?
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#equal_sets
#singleton_subsets
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A (A=B)
B \(A\subset B\) पर \(A\ne B\) / \(A\subset B\) but \(A\ne B\)
C \(B\subset A\) पर \(A\ne B\) / \(B\subset A\) but \(A\ne B\)
D \(A=\{a,b,c\}\)
Explanation opens after your attempt
Step 1
Concept
The singleton subsets are ({a},{b},{c}), so the two sets are equal. In exams carefully check the level of elements and subsets.
Step 2
Why this answer is correct
The correct answer is A. (A=B). The singleton subsets are ({a},{b},{c}), so the two sets are equal. In exams carefully check the level of elements and subsets.
Step 3
Exam Tip
एक-अवयवी उपसमुच्चय ({a},{b},{c}) होते हैं इसलिए दोनों समुच्चय बराबर हैं। परीक्षा में अवयव और उपसमुच्चय के स्तर को ध्यान से देखें।
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यदि \(A=\{2,4,6,8\}\) और \(B={x:x\) दस से छोटे धनात्मक सम पूर्णांक हैं(}) तो कौन सा विकल्प सही है?
If \(A=\{2,4,6,8\}\) and \(B={x:x\) is a positive even integer less than ten(}), which option is correct?
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#roster_form
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A (A=B)
B \(A\subset B\) पर \(A\ne B\) / \(A\subset B\) but \(A\ne B\)
C \(B\subset A\) पर \(A\ne B\) / \(B\subset A\) but \(B\ne A\)
D \(A\not\subset B\)
Explanation opens after your attempt
Step 1
Concept
The positive even integers less than ten are (2,4,6,8). Convert set builder form to roster form before comparing.
Step 2
Why this answer is correct
The correct answer is A. (A=B). The positive even integers less than ten are (2,4,6,8). Convert set builder form to roster form before comparing.
Step 3
Exam Tip
दस से छोटे धनात्मक सम पूर्णांक (2,4,6,8) हैं। वर्णनात्मक रूप को रोस्टर रूप में बदलकर मिलान करें।
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यदि \(A={\varnothing,{\varnothing}}\) है तो निम्न में से कौन (A) का उपसमुच्चय नहीं है?
If \(A={\varnothing,{\varnothing}}\), which of the following is not a subset of (A)?
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#empty_set
#subset_test
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A \({\varnothing}\)
B \({{\varnothing}}\)
C \(\varnothing\)
D \({\varnothing,{{\varnothing}}}\)
Explanation opens after your attempt
Correct Answer
D. \({\varnothing,{{\varnothing}}}\)
Step 1
Concept
Option D contains \({{\varnothing}}\) as an element, which is not in (A). In empty set questions count the braces carefully.
Step 2
Why this answer is correct
The correct answer is D. \({\varnothing,{{\varnothing}}}\). Option D contains \({{\varnothing}}\) as an element, which is not in (A). In empty set questions count the braces carefully.
Step 3
Exam Tip
विकल्प D में \({{\varnothing}}\) अवयव है जो (A) में नहीं है। रिक्त समुच्चय वाले प्रश्नों में कोष्ठकों की संख्या गिनें।
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यदि \(A=\{1,2,{3}\}\) है तो कौन सा समुच्चय (A) के बराबर है?
If \(A=\{1,2,{3}\}\), which set is equal to (A)?
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#equal_sets
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A ({2,{3},1})
B ({1,2,3})
C ({1,{2,3}})
D ({{1},2,{3}})
Explanation opens after your attempt
Correct Answer
A. ({2,{3},1})
Step 1
Concept
Changing order does not change a set but ({3}) and (3) are different. Identify nested elements carefully.
Step 2
Why this answer is correct
The correct answer is A. ({2,{3},1}). Changing order does not change a set but ({3}) and (3) are different. Identify nested elements carefully.
Step 3
Exam Tip
समुच्चय में क्रम बदलने से समुच्चय नहीं बदलता पर ({3}) और (3) अलग हैं। नेस्टेड अवयव को अलग पहचानें।
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यदि \(A\subseteq B\), (n(A)=8) और (B) में (A) से बाहर कोई अवयव नहीं है तो कौन सा निष्कर्ष निश्चित है?
If \(A\subseteq B\), (n(A)=8), and (B) has no element outside (A), which conclusion is certain?
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#equal_sets
#cardinality_reasoning
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A (A=B)
B \(A\subset B\) और \(A\ne B\) / \(A\subset B\) and \(A\ne B\)
C (B) रिक्त समुच्चय है / (B) is the empty set
D (n(B)=16)
Explanation opens after your attempt
Step 1
Concept
Since \(A\subseteq B\) and (B) has no new element outside (A), both have the same elements. Thinking about inclusion from both sides helps prove equality.
Step 2
Why this answer is correct
The correct answer is A. (A=B). Since \(A\subseteq B\) and (B) has no new element outside (A), both have the same elements. Thinking about inclusion from both sides helps prove equality.
Step 3
Exam Tip
\(A\subseteq B\) है और (B) में (A) के अलावा कोई नया अवयव नहीं है इसलिए दोनों में वही अवयव हैं। बराबरी सिद्ध करने में दोनों ओर सम्मिलन की सोच उपयोगी है।
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यदि \(A=\{a,b,c\}\) है तो (A) के कुल उपसमुच्चयों की संख्या क्या होगी?
If \(A=\{a,b,c\}\), what is the total number of subsets of (A)?
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A (3)
B (6)
C (8)
D (9)
Explanation opens after your attempt
Step 1
Concept
A set with three elements has \(2^3=8\) subsets. Count the empty set and the set itself among all subsets.
Step 2
Why this answer is correct
The correct answer is C. (8). A set with three elements has \(2^3=8\) subsets. Count the empty set and the set itself among all subsets.
Step 3
Exam Tip
तीन अवयवों वाले समुच्चय के उपसमुच्चय \(2^3=8\) होते हैं। कुल उपसमुच्चय में रिक्त समुच्चय और वही समुच्चय भी गिनें।
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यदि \(A={x:x\in \mathbb{Z}\) और (|x-2|<2}) है तो निम्न में से कौन (A) का उचित उपसमुच्चय है?
If \(A={x:x\in \mathbb{Z}\) and (|x-2|<2}), which of the following is a proper subset of (A)?
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#integer_interval
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A ({1,2,3})
B ({1,3})
C ({0,1,2})
D ({2,4})
Explanation opens after your attempt
Correct Answer
B. ({1,3})
Step 1
Concept
The condition (|x-2|<2) gives integers (1,2,3), and ({1,3}) lies inside it but is not equal to it. A proper subset has all elements inside but is not the whole set.
Step 2
Why this answer is correct
The correct answer is B. ({1,3}). The condition (|x-2|<2) gives integers (1,2,3), and ({1,3}) lies inside it but is not equal to it. A proper subset has all elements inside but is not the whole set.
Step 3
Exam Tip
(|x-2|<2) से पूर्णांक (1,2,3) मिलते हैं और ({1,3}) इसके अंदर है पर बराबर नहीं है। उचित उपसमुच्चय में सभी अवयव अंदर होते हैं लेकिन पूरा समुच्चय नहीं होता।
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यदि किसी समुच्चय के उचित उपसमुच्चय (31) हैं तो उस समुच्चय में अवयवों की संख्या क्या है?
If a set has (31) proper subsets, how many elements does the set have?
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A (4)
B (5)
C (6)
D (31)
Explanation opens after your attempt
Step 1
Concept
The number of proper subsets is \(2^n-1\). From \(2^n-1=31\), we get (n=5).
Step 2
Why this answer is correct
The correct answer is B. (5). The number of proper subsets is \(2^n-1\). From \(2^n-1=31\), we get (n=5).
Step 3
Exam Tip
उचित उपसमुच्चय की संख्या \(2^n-1\) होती है। \(2^n-1=31\) से (n=5) मिलता है।
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यदि \(A=\varnothing\) और \(B=\{0\}\) हैं तो कौन सा विकल्प सही है?
If \(A=\varnothing\) and \(B=\{0\}\), which option is correct?
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A (A=B)
B \(A\subset B\) और \(A\ne B\) / \(A\subset B\) and \(A\ne B\)
C \(B\subset A\)
D \(0\in A\)
Explanation opens after your attempt
Correct Answer
B. \(A\subset B\) और \(A\ne B\) / \(A\subset B\) and \(A\ne B\)
Step 1
Concept
The empty set is a subset of every set and ({0}) is not empty. Keep \(\varnothing\) and (0) different.
Step 2
Why this answer is correct
The correct answer is B. \(A\subset B\) और \(A\ne B\) / \(A\subset B\) and \(A\ne B\). The empty set is a subset of every set and ({0}) is not empty. Keep \(\varnothing\) and (0) different.
Step 3
Exam Tip
रिक्त समुच्चय हर समुच्चय का उपसमुच्चय है और ({0}) रिक्त नहीं है। \(\varnothing\) और (0) को अलग रखें।
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यदि \(A={x:x\) अंग्रेजी शब्द गणित के अक्षर हैं(}) और \(B=\{m,a,t,h\}\) हैं तो सही कथन क्या है?
If \(A={x:x\) is a letter of the English word math(}) and \(B=\{m,a,t,h\}\), what is the correct statement?
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#equal_sets
#letters
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A (A=B)
B \(A\subset B\) पर \(A\ne B\) / \(A\subset B\) but \(A\ne B\)
C \(B\subset A\) पर \(A\ne B\) / \(B\subset A\) but \(A\ne B\)
D \(A\cap B=\varnothing\)
Explanation opens after your attempt
Step 1
Concept
Both sets contain the same four distinct letters. Repetition and order do not affect a set.
Step 2
Why this answer is correct
The correct answer is A. (A=B). Both sets contain the same four distinct letters. Repetition and order do not affect a set.
Step 3
Exam Tip
दोनों में वही चार अलग अक्षर हैं। समुच्चय में दोहराव और क्रम का प्रभाव नहीं होता।
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यदि \(A=\{1,3,5,7\}\) और \(B={x:x\) आठ से छोटे धनात्मक विषम पूर्णांक हैं(}) तो (A) और (B) का संबंध क्या है?
If \(A=\{1,3,5,7\}\) and \(B={x:x\) is a positive odd integer less than eight(}), what is the relation between (A) and (B)?
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#equal_sets
#odd_numbers
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A (A=B)
B \(A\subset B\) पर \(A\ne B\) / \(A\subset B\) but \(A\ne B\)
C \(B\subset A\) पर \(A\ne B\) / \(B\subset A\) but \(A\ne B\)
D \(A\not\subset B\)
Explanation opens after your attempt
Step 1
Concept
The positive odd integers less than eight are (1,3,5,7). Write the described set in elements before comparing.
Step 2
Why this answer is correct
The correct answer is A. (A=B). The positive odd integers less than eight are (1,3,5,7). Write the described set in elements before comparing.
Step 3
Exam Tip
आठ से छोटे धनात्मक विषम पूर्णांक (1,3,5,7) हैं। शब्दों में दिए समुच्चय को अवयवों में लिखें।
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यदि \(A=\{1,2,3,4\}\) और \(B=\{1,2\}\) है तो कौन सा कथन सही है?
If \(A=\{1,2,3,4\}\) and \(B=\{1,2\}\), which statement is correct?
#sets
#proper_subset
#basic_relation
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A \(A\subset B\)
B \(B\subset A\) और \(B\ne A\) / \(B\subset A\) and \(B\ne A\)
C (A=B)
D \(A\cap B=\varnothing\)
Explanation opens after your attempt
Correct Answer
B. \(B\subset A\) और \(B\ne A\) / \(B\subset A\) and \(B\ne A\)
Step 1
Concept
All elements of (B) are in (A), but (A) has extra elements. This is a proper subset relation.
Step 2
Why this answer is correct
The correct answer is B. \(B\subset A\) और \(B\ne A\) / \(B\subset A\) and \(B\ne A\). All elements of (B) are in (A), but (A) has extra elements. This is a proper subset relation.
Step 3
Exam Tip
(B) के सभी अवयव (A) में हैं लेकिन (A) में अतिरिक्त अवयव भी हैं। इसे उचित उपसमुच्चय कहते हैं।
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यदि \(A={x:x\) (12) का अभाज्य गुणनखंड है(}) और \(B=\{2,3\}\) हैं तो क्या सत्य है?
If \(A={x:x\) is a prime factor of (12)(}) and \(B=\{2,3\}\), what is true?
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#prime_factors
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A (A=B)
B \(A=\{2,2,3\}\) इसलिए \(A\ne B\) / \(A=\{2,2,3\}\) so \(A\ne B\)
C \(B\subset A\) पर \(A\ne B\) / \(B\subset A\) but \(A\ne B\)
D \(A\cap B={12}\)
Explanation opens after your attempt
Step 1
Concept
The distinct prime factors of (12) are (2) and (3). Repeated factors are not repeated in a set.
Step 2
Why this answer is correct
The correct answer is A. (A=B). The distinct prime factors of (12) are (2) and (3). Repeated factors are not repeated in a set.
Step 3
Exam Tip
(12) के अलग अभाज्य गुणनखंड (2) और (3) हैं। समुच्चय में दोहराव नहीं लिखा जाता।
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यदि \(A=\{1,2\}\) और \(B=\{{1,2}\}\) हैं तो कौन सा कथन सही है?
If \(A=\{1,2\}\) and \(B=\{{1,2}\}\), which statement is correct?
#sets
#element_vs_subset
#nested_sets
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A (A=B)
B \(A\in B\)
C \(B\subset A\)
D \(1\in B\)
Explanation opens after your attempt
Correct Answer
B. \(A\in B\)
Step 1
Concept
The only element of (B) is the whole set ({1,2}). Be careful with subset and element symbols.
Step 2
Why this answer is correct
The correct answer is B. \(A\in B\). The only element of (B) is the whole set ({1,2}). Be careful with subset and element symbols.
Step 3
Exam Tip
(B) का एकमात्र अवयव पूरा समुच्चय ({1,2}) है। उपसमुच्चय और अवयव के चिन्ह में सावधानी रखें।
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यदि \(A=\{p,q,r,s\}\) और \(B=\{p,r\}\) हैं तो (B) के कौन से उपसमुच्चय (A) के भी उपसमुच्चय हैं?
If \(A=\{p,q,r,s\}\) and \(B=\{p,r\}\), which subsets of (B) are also subsets of (A)?
#sets
#subset_transitivity
#reasoning
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A केवल ({p,r}) / Only ({p,r})
B केवल \(\varnothing\) / Only \(\varnothing\)
C (B) के सभी उपसमुच्चय / All subsets of (B)
D केवल एकल अवयव वाले उपसमुच्चय / Only singleton subsets
Explanation opens after your attempt
Correct Answer
C. (B) के सभी उपसमुच्चय / All subsets of (B)
Step 1
Concept
Because \(B\subset A\), every subset of (B) is also a subset of (A). This uses transitive reasoning.
Step 2
Why this answer is correct
The correct answer is C. (B) के सभी उपसमुच्चय / All subsets of (B). Because \(B\subset A\), every subset of (B) is also a subset of (A). This uses transitive reasoning.
Step 3
Exam Tip
क्योंकि \(B\subset A\), इसलिए (B) का हर उपसमुच्चय (A) का भी उपसमुच्चय है। यह ट्रांजिटिव सोच का प्रयोग है।
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यदि \(A=\{0,1,2\}\) है तो निम्न में से कौन (A) का उपसमुच्चय नहीं है?
If \(A=\{0,1,2\}\), which of the following is not a subset of (A)?
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#subset_test
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A \(\varnothing\)
B ({0,2})
C ({1})
D \({\varnothing}\)
Explanation opens after your attempt
Correct Answer
D. \({\varnothing}\)
Step 1
Concept
\(\varnothing\) is a subset, but \({\varnothing}\) has \(\varnothing\) as an element, which is not in (A). This is a common mistake.
Step 2
Why this answer is correct
The correct answer is D. \({\varnothing}\). \(\varnothing\) is a subset, but \({\varnothing}\) has \(\varnothing\) as an element, which is not in (A). This is a common mistake.
Step 3
Exam Tip
\(\varnothing\) उपसमुच्चय है लेकिन \({\varnothing}\) में अवयव \(\varnothing\) है जो (A) में नहीं है। यह सामान्य गलती है।
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यदि (P(A)) में (16) अवयव हैं तो (A) में कितने अवयव होंगे?
If (P(A)) has (16) elements, how many elements does (A) have?
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A (2)
B (3)
C (4)
D (16)
Explanation opens after your attempt
Step 1
Concept
If (A) has (n) elements, then (P(A)) has \(2^n\) elements. From \(2^n=16\), (n=4).
Step 2
Why this answer is correct
The correct answer is C. (4). If (A) has (n) elements, then (P(A)) has \(2^n\) elements. From \(2^n=16\), (n=4).
Step 3
Exam Tip
यदि (A) में (n) अवयव हों तो (P(A)) में \(2^n\) अवयव होते हैं। \(2^n=16\) से (n=4) है।
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कौन सा विकल्प \(A=\{1,2,3\}\) के पावर समुच्चय का अवयव है?
Which option is an element of the power set of \(A=\{1,2,3\}\)?
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A (2)
B ({2,3})
C ({4})
D ({{1}})
Explanation opens after your attempt
Correct Answer
B. ({2,3})
Step 1
Concept
Elements of a power set are subsets of the original set. ({2,3}) is a subset of (A).
Step 2
Why this answer is correct
The correct answer is B. ({2,3}). Elements of a power set are subsets of the original set. ({2,3}) is a subset of (A).
Step 3
Exam Tip
पावर समुच्चय के अवयव मूल समुच्चय के उपसमुच्चय होते हैं। ({2,3}) (A) का उपसमुच्चय है।
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यदि \(A\subset B\) और \(B\subset A\) हैं तो सही निष्कर्ष क्या है?
If \(A\subset B\) and \(B\subset A\), what is the correct conclusion?
#sets
#equal_sets
#proof_method
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A (A=B)
B \(A\cap B=\varnothing\)
C (A) और (B) अनंत हैं / (A) and (B) are infinite
D \(A\ne B\) हमेशा / \(A\ne B\) always
Explanation opens after your attempt
Step 1
Concept
Subset relation in both directions means both sets have exactly the same elements. This is a standard way to prove equality of sets.
Step 2
Why this answer is correct
The correct answer is A. (A=B). Subset relation in both directions means both sets have exactly the same elements. This is a standard way to prove equality of sets.
Step 3
Exam Tip
दोनों दिशाओं में उपसमुच्चय होने का अर्थ है कि दोनों में ठीक वही अवयव हैं। बराबर समुच्चय सिद्ध करने की यह मानक विधि है।
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यदि \(A={x:x\) (18) का धनात्मक भाजक है और (x) विषम है(}) तो कौन सा समुच्चय (A) के बराबर है?
If \(A={x:x\) is a positive divisor of (18) and (x) is odd(}), which set is equal to (A)?
#sets
#equal_sets
#divisors
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A ({1,3,9})
B ({2,6,18})
C ({1,2,3,6,9,18})
D ({3,6,9})
Explanation opens after your attempt
Correct Answer
A. ({1,3,9})
Step 1
Concept
The odd positive divisors of (18) are (1,3,9). Apply all conditions together.
Step 2
Why this answer is correct
The correct answer is A. ({1,3,9}). The odd positive divisors of (18) are (1,3,9). Apply all conditions together.
Step 3
Exam Tip
(18) के विषम धनात्मक भाजक (1,3,9) हैं। सभी शर्तों को साथ में लागू करें।
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कौन सा विकल्प \(A=\{1,{2},3\}\) का उपसमुच्चय है?
Which option is a subset of \(A=\{1,{2},3\}\)?
#sets
#nested_sets
#subset_test
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A ({1,2})
B ({{2},3})
C ({{1,2}})
D ({2,3})
Explanation opens after your attempt
Correct Answer
B. ({{2},3})
Step 1
Concept
The set (A) has ({2}) as an element but not (2). Therefore ({{2},3}) is a subset.
Step 2
Why this answer is correct
The correct answer is B. ({{2},3}). The set (A) has ({2}) as an element but not (2). Therefore ({{2},3}) is a subset.
Step 3
Exam Tip
(A) में ({2}) अवयव है पर (2) अवयव नहीं है। इसलिए ({{2},3}) उपसमुच्चय है।
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यदि \(A=\{1,2,3,4,5\}\) है तो (A) के दो अवयवों वाले उपसमुच्चयों की संख्या क्या है?
If \(A=\{1,2,3,4,5\}\), how many subsets of (A) have exactly two elements?
#sets
#subsets
#combinations
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A (5)
B (10)
C (20)
D (25)
Explanation opens after your attempt
Step 1
Concept
The number of ways to choose two elements is \(\binom{5}{2}=10\). Order is not counted in subsets.
Step 2
Why this answer is correct
The correct answer is B. (10). The number of ways to choose two elements is \(\binom{5}{2}=10\). Order is not counted in subsets.
Step 3
Exam Tip
दो अवयव चुनने की संख्या \(\binom{5}{2}=10\) है। उपसमुच्चय में क्रम नहीं गिना जाता।
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यदि \(A=\{a,b,c,d\}\) है तो (A) के कम से कम दो अवयवों वाले उपसमुच्चयों की संख्या क्या है?
If \(A=\{a,b,c,d\}\), how many subsets of (A) have at least two elements?
#sets
#subsets
#counting
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A (6)
B (10)
C (11)
D (12)
Explanation opens after your attempt
Step 1
Concept
There are (16) total subsets and (1+4=5) subsets with zero or one element. So the answer is (16-5=11).
Step 2
Why this answer is correct
The correct answer is C. (11). There are (16) total subsets and (1+4=5) subsets with zero or one element. So the answer is (16-5=11).
Step 3
Exam Tip
कुल उपसमुच्चय (16) हैं और शून्य या एक अवयव वाले (1+4=5) हैं। इसलिए उत्तर (16-5=11) है।
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यदि \(A=\{2,3,5,7\}\) और \(B={x:x\) दस से छोटे अभाज्य अंक हैं(}) हैं तो कौन सा विकल्प सही है?
If \(A=\{2,3,5,7\}\) and \(B={x:x\) is a prime digit less than ten(}), which option is correct?
#sets
#equal_sets
#prime_numbers
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A (A=B)
B \(A\subset B\) पर \(A\ne B\) / \(A\subset B\) but \(A\ne B\)
C \(B\subset A\) पर \(A\ne B\) / \(B\subset A\) but \(A\ne B\)
D \(A\cap B={1}\)
Explanation opens after your attempt
Step 1
Concept
The prime digits less than ten are (2,3,5,7). The number (1) is not prime.
Step 2
Why this answer is correct
The correct answer is A. (A=B). The prime digits less than ten are (2,3,5,7). The number (1) is not prime.
Step 3
Exam Tip
दस से छोटे अभाज्य अंक (2,3,5,7) हैं। (1) अभाज्य नहीं होता।
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यदि \(A={x:x\) \(x^2-5x+6=0\) का हल है(}) और \(B=\{2,3\}\) हैं तो क्या सत्य है?
If \(A={x:x\) is a solution of \(x^2-5x+6=0\)(}) and \(B=\{2,3\}\), what is true?
#sets
#equal_sets
#quadratic_roots
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A (A=B)
B \(A=\{6\}\)
C \(B\subset A\) पर \(A\ne B\) / \(B\subset A\) but \(A\ne B\)
D \(A\cap B=\varnothing\)
Explanation opens after your attempt
Step 1
Concept
Since (x-2 -5x+6=(x-2)(x-3)), the solutions are (2,3). Make the roots the elements of the set.
Step 2
Why this answer is correct
The correct answer is A. (A=B). Since (x-2 -5x+6=(x-2)(x-3)), the solutions are (2,3). Make the roots the elements of the set.
Step 3
Exam Tip
(x-2 -5x+6=(x-2)(x-3)) इसलिए हल (2,3) हैं। समीकरण के हलों को समुच्चय के अवयव बनाएं।
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यदि \(A=\{1,2,3\}\) और \(B=\{1,2,3,4\}\) हैं तो \(A\subset B\) के बारे में क्या सही है?
If \(A=\{1,2,3\}\) and \(B=\{1,2,3,4\}\), what is true about \(A\subset B\)?
#sets
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#definition
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A सत्य है और (A) उचित उपसमुच्चय है / It is true and (A) is a proper subset
B सत्य है पर (A=B) / It is true but (A=B)
C असत्य है क्योंकि \(4\notin A\) / It is false because \(4\notin A\)
D असत्य है क्योंकि (B) बड़ा है / It is false because (B) is larger
Explanation opens after your attempt
Correct Answer
A. सत्य है और (A) उचित उपसमुच्चय है / It is true and (A) is a proper subset
Step 1
Concept
All elements of (A) are in (B), and (B) has one extra element. Hence (A) is a proper subset.
Step 2
Why this answer is correct
The correct answer is A. सत्य है और (A) उचित उपसमुच्चय है / It is true and (A) is a proper subset. All elements of (A) are in (B), and (B) has one extra element. Hence (A) is a proper subset.
Step 3
Exam Tip
(A) के सभी अवयव (B) में हैं और (B) में एक अतिरिक्त अवयव है। इसलिए (A) उचित उपसमुच्चय है।
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यदि \(A={x:x\) (20) का धनात्मक भाजक है(}) और \(B=\{1,2,4,5,10,20\}\) हैं तो संबंध क्या है?
If \(A={x:x\) is a positive divisor of (20)(}) and \(B=\{1,2,4,5,10,20\}\), what is the relation?
#sets
#equal_sets
#divisors
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A (A=B)
B \(A\subset B\) पर \(A\ne B\) / \(A\subset B\) but \(A\ne B\)
C \(B\subset A\) पर \(A\ne B\) / \(B\subset A\) but \(A\ne B\)
D \(A=\{2,4,5,10\}\)
Explanation opens after your attempt
Step 1
Concept
The positive divisors of (20) are (1,2,4,5,10,20). In divisor questions do not forget (1) and the number itself.
Step 2
Why this answer is correct
The correct answer is A. (A=B). The positive divisors of (20) are (1,2,4,5,10,20). In divisor questions do not forget (1) and the number itself.
Step 3
Exam Tip
(20) के धनात्मक भाजक (1,2,4,5,10,20) हैं। भाजक वाले प्रश्न में (1) और संख्या स्वयं न भूलें।
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यदि \(A={x:x\in \mathbb{N},x\leq 4}\) और \(B=\{4,3,2,1\}\) हैं तो कौन सा कथन सही है?
If \(A={x:x\in \mathbb{N},x\leq 4}\) and \(B=\{4,3,2,1\}\), which statement is correct?
#sets
#equal_sets
#order_irrelevant
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A (A=B)
B \(A\ne B\) क्योंकि क्रम अलग है / \(A\ne B\) because the order is different
C \(A\subset B\) पर \(A\ne B\) / \(A\subset B\) but \(A\ne B\)
D \(B=\varnothing\)
Explanation opens after your attempt
Step 1
Concept
Both sets contain only (1,2,3,4). Order has no importance in a set.
Step 2
Why this answer is correct
The correct answer is A. (A=B). Both sets contain only (1,2,3,4). Order has no importance in a set.
Step 3
Exam Tip
दोनों समुच्चयों में (1,2,3,4) ही हैं। समुच्चय में क्रम का महत्व नहीं होता।
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यदि \(A=\{a,a,b,b,c\}\) और \(B=\{a,b,c\}\) हैं तो सही संबंध क्या है?
If \(A=\{a,a,b,b,c\}\) and \(B=\{a,b,c\}\), what is the correct relation?
#sets
#equal_sets
#repetition
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A (A=B)
B \(A\ne B\) क्योंकि (A) में पाँच स्थान हैं / \(A\ne B\) because (A) has five positions
C \(B\subset A\) पर \(A\ne B\) / \(B\subset A\) but \(A\ne B\)
D \(A\cap B=\varnothing\)
Explanation opens after your attempt
Step 1
Concept
Repeated elements are counted only once in a set. Therefore both have the same distinct elements.
Step 2
Why this answer is correct
The correct answer is A. (A=B). Repeated elements are counted only once in a set. Therefore both have the same distinct elements.
Step 3
Exam Tip
दोहराए गए अवयव समुच्चय में एक बार ही माने जाते हैं। इसलिए दोनों में वही अलग अवयव हैं।
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यदि \(A={1,2,\varnothing}\) है तो निम्न में से कौन (A) का उपसमुच्चय है?
If \(A={1,2,\varnothing}\), which of the following is a subset of (A)?
#sets
#empty_set
#subset_test
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A \({1,\varnothing}\)
B ({0})
C ({{1}})
D ({1,3})
Explanation opens after your attempt
Correct Answer
A. \({1,\varnothing}\)
Step 1
Concept
Both (1) and \(\varnothing\) are elements of (A). Every chosen element of a subset must belong to the original set.
Step 2
Why this answer is correct
The correct answer is A. \({1,\varnothing}\). Both (1) and \(\varnothing\) are elements of (A). Every chosen element of a subset must belong to the original set.
Step 3
Exam Tip
(1) और \(\varnothing\) दोनों (A) के अवयव हैं। उपसमुच्चय में हर चुना अवयव मूल समुच्चय में होना चाहिए।
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यदि \(A=\{1,2,3\}\) है तो (A) के सभी उचित उपसमुच्चयों की संख्या क्या है?
If \(A=\{1,2,3\}\), what is the number of all proper subsets of (A)?
#sets
#proper_subsets
#counting
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A (6)
B (7)
C (8)
D (9)
Explanation opens after your attempt
Step 1
Concept
A three element set has \(2^3=8\) total subsets. Removing the whole set (A) gives (7) proper subsets.
Step 2
Why this answer is correct
The correct answer is B. (7). A three element set has \(2^3=8\) total subsets. Removing the whole set (A) gives (7) proper subsets.
Step 3
Exam Tip
तीन अवयवों के कुल उपसमुच्चय \(2^3=8\) हैं। उचित उपसमुच्चय में पूरा (A) हटाने पर (7) बचते हैं।
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यदि \(A\subseteq B\), (n(A)=6), और (n(B)=6) है तो (A) और (B) के बारे में क्या निश्चित है?
If \(A\subseteq B\), (n(A)=6), and (n(B)=6), what is certain about (A) and (B)?
#sets
#equal_sets
#cardinality
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A (A=B)
B \(A\subset B\) और \(A\ne B\) / \(A\subset B\) and \(A\ne B\)
C \(A\cap B=\varnothing\)
D (B) में कम से कम सात अवयव हैं / (B) has at least seven elements
Explanation opens after your attempt
Step 1
Concept
When \(A\subseteq B\) and both have the same number of elements, no extra element remains. Hence (A=B).
Step 2
Why this answer is correct
The correct answer is A. (A=B). When \(A\subseteq B\) and both have the same number of elements, no extra element remains. Hence (A=B).
Step 3
Exam Tip
जब \(A\subseteq B\) और दोनों के अवयवों की संख्या समान हो तो कोई अतिरिक्त अवयव नहीं बचता। इसलिए (A=B)।
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यदि \(A={x:x\in \mathbb{Z},x^2=1}\) और \(B=\{-1,1\}\) हैं तो क्या सत्य है?
If \(A={x:x\in \mathbb{Z},x^2=1}\) and \(B=\{-1,1\}\), what is true?
#sets
#equal_sets
#integer_solutions
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A (A=B)
B \(A=\{1\}\)
C \(A=\{-1\}\)
D \(A=\varnothing\)
Explanation opens after your attempt
Step 1
Concept
In integers, the solutions of \(x^2=1\) are (-1) and (1). Do not miss the negative solution.
Step 2
Why this answer is correct
The correct answer is A. (A=B). In integers, the solutions of \(x^2=1\) are (-1) and (1). Do not miss the negative solution.
Step 3
Exam Tip
पूर्णांकों में \(x^2=1\) के हल (-1) और (1) हैं। ऋणात्मक हल को न छोड़ें।
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यदि \(A={x:x\) (36) का धनात्मक पूर्ण वर्ग भाजक है(}) तो (A) कौन सा है?
If \(A={x:x\) is a positive square divisor of (36)(}), which set is (A)?
#sets
#set_builder
#divisors
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A ({1,4,9,36})
B ({4,9})
C ({1,6,36})
D ({2,3,6})
Explanation opens after your attempt
Correct Answer
A. ({1,4,9,36})
Step 1
Concept
The square divisors of (36) are (1,4,9,36). Check both conditions square and divisor.
Step 2
Why this answer is correct
The correct answer is A. ({1,4,9,36}). The square divisors of (36) are (1,4,9,36). Check both conditions square and divisor.
Step 3
Exam Tip
(36) के वर्ग भाजक (1,4,9,36) हैं। वर्ग होने और भाजक होने दोनों शर्तें जांचें।
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यदि \(A=\{1,2,3,4\}\) और \(C=\{2,4\}\) हैं तो \(C\subset A\) क्यों सत्य है?
If \(A=\{1,2,3,4\}\) and \(C=\{2,4\}\), why is \(C\subset A\) true?
#sets
#proper_subset
#reasoning
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A क्योंकि (C) का हर अवयव (A) में है और \(A\ne C\) / Because every element of (C) is in (A) and \(A\ne C\)
B क्योंकि (A) का हर अवयव (C) में है / Because every element of (A) is in (C)
C क्योंकि दोनों का पहला अवयव समान है / Because the first element of both is same
D क्योंकि (C) में कम अवयव हैं इसलिए हमेशा उपसमुच्चय है / Because (C) has fewer elements so it is always a subset
Explanation opens after your attempt
Correct Answer
A. क्योंकि (C) का हर अवयव (A) में है और \(A\ne C\) / Because every element of (C) is in (A) and \(A\ne C\)
Step 1
Concept
A proper subset needs inclusion of every element and inequality. Having fewer elements alone is not enough.
Step 2
Why this answer is correct
The correct answer is A. क्योंकि (C) का हर अवयव (A) में है और \(A\ne C\) / Because every element of (C) is in (A) and \(A\ne C\). A proper subset needs inclusion of every element and inequality. Having fewer elements alone is not enough.
Step 3
Exam Tip
उचित उपसमुच्चय के लिए हर अवयव का सम्मिलन और असमानता दोनों चाहिए। केवल अवयवों की संख्या कम होना पर्याप्त नहीं है।
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यदि \(A={x:x\) (15) से छोटे (3) के धनात्मक गुणज हैं(}) और \(B=\{3,6,9,12\}\) हैं तो कौन सा निष्कर्ष सही है?
If \(A={x:x\) is a positive multiple of (3) less than (15)(}) and \(B=\{3,6,9,12\}\), which conclusion is correct?
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#equal_sets
#multiples
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A (A=B)
B \(A\subset B\) पर \(A\ne B\) / \(A\subset B\) but \(A\ne B\)
C \(B\subset A\) पर \(A\ne B\) / \(B\subset A\) but \(A\ne B\)
D \(A=\{3,6,9,12,15\}\)
Explanation opens after your attempt
Step 1
Concept
The positive multiples of (3) less than (15) are (3,6,9,12). The strict boundary excludes (15).
Step 2
Why this answer is correct
The correct answer is A. (A=B). The positive multiples of (3) less than (15) are (3,6,9,12). The strict boundary excludes (15).
Step 3
Exam Tip
(15) से छोटे (3) के धनात्मक गुणज (3,6,9,12) हैं। कठोर सीमा में (15) शामिल नहीं है।
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यदि \(A=\{1,2\}\), \(B=\{1,2,3\}\), और \(C=\{1,2,3,4\}\) हैं तो कौन सा कथन सही है?
If \(A=\{1,2\}\), \(B=\{1,2,3\}\), and \(C=\{1,2,3,4\}\), which statement is correct?
#sets
#subset_chain
#transitivity
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A \(A\subset B\subset C\)
B \(C\subset B\subset A\)
C (A=B=C)
D \(A\not\subset C\)
Explanation opens after your attempt
Correct Answer
A. \(A\subset B\subset C\)
Step 1
Concept
Each next set contains all elements of the previous one and one extra element. Hence it forms a chain of proper subsets.
Step 2
Why this answer is correct
The correct answer is A. \(A\subset B\subset C\). Each next set contains all elements of the previous one and one extra element. Hence it forms a chain of proper subsets.
Step 3
Exam Tip
हर अगले समुच्चय में पिछले के सभी अवयव और एक अतिरिक्त अवयव है। इसलिए उचित उपसमुच्चय की शृंखला बनती है।
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यदि \(A={x:x\in \mathbb{N},2<x\leq 6}\) तो निम्न में से कौन (A) का उचित उपसमुच्चय है?
If \(A={x:x\in \mathbb{N},2<x\leq 6}\), which of the following is a proper subset of (A)?
#sets
#proper_subset
#set_builder
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A ({3,4,5,6})
B ({3,5})
C ({2,3})
D ({6,7})
Explanation opens after your attempt
Correct Answer
B. ({3,5})
Step 1
Concept
Here \(A=\{3,4,5,6\}\), and ({3,5}) lies inside it but is not equal to it. So it is a proper subset.
Step 2
Why this answer is correct
The correct answer is B. ({3,5}). Here \(A=\{3,4,5,6\}\), and ({3,5}) lies inside it but is not equal to it. So it is a proper subset.
Step 3
Exam Tip
\(A=\{3,4,5,6\}\) है और ({3,5}) इसके अंदर है पर बराबर नहीं है। इसलिए यह उचित उपसमुच्चय है।
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यदि \(A=\{1,2,3\}\) और \(B=\{1,2,3,{1}\}\) हैं तो (A) और (B) का संबंध क्या है?
If \(A=\{1,2,3\}\) and \(B=\{1,2,3,{1}\}\), what is the relation between (A) and (B)?
#sets
#nested_sets
#proper_subset
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A (A=B)
B \(A\subset B\) और \(A\ne B\) / \(A\subset B\) and \(A\ne B\)
C \(B\subset A\) और \(A\ne B\) / \(B\subset A\) and \(A\ne B\)
D \(A\cap B=\varnothing\)
Explanation opens after your attempt
Correct Answer
B. \(A\subset B\) और \(A\ne B\) / \(A\subset B\) and \(A\ne B\)
Step 1
Concept
All elements of (A) are in (B), and (B) has the extra element ({1}). Treat ({1}) as different from (1).
Step 2
Why this answer is correct
The correct answer is B. \(A\subset B\) और \(A\ne B\) / \(A\subset B\) and \(A\ne B\). All elements of (A) are in (B), and (B) has the extra element ({1}). Treat ({1}) as different from (1).
Step 3
Exam Tip
(A) के सभी अवयव (B) में हैं और (B) में अतिरिक्त अवयव ({1}) है। ({1}) को (1) से अलग मानें।
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यदि \(A={x:x\) (50) से छोटे (10) के धनात्मक गुणज हैं(}) तो (A) के उचित उपसमुच्चयों की संख्या क्या है?
If \(A={x:x\) is a positive multiple of (10) less than (50)(}), how many proper subsets does (A) have?
#sets
#proper_subsets
#multiples
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A (7)
B (15)
C (16)
D (31)
Explanation opens after your attempt
Step 1
Concept
\(A=\{10,20,30,40\}\) has (4) elements. Proper subsets are \(2^4-1=15\).
Step 2
Why this answer is correct
The correct answer is B. (15). \(A=\{10,20,30,40\}\) has (4) elements. Proper subsets are \(2^4-1=15\).
Step 3
Exam Tip
\(A=\{10,20,30,40\}\) में (4) अवयव हैं। उचित उपसमुच्चय \(2^4-1=15\) होते हैं।
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यदि \(A={x:x\) \(x^2-4=0\) का प्राकृतिक हल है(}) और \(B=\{2\}\) हैं तो कौन सा विकल्प सही है?
If \(A={x:x\) is a natural solution of \(x^2-4=0\)(}) and \(B=\{2\}\), which option is correct?
#sets
#equal_sets
#natural_numbers
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A (A=B)
B \(A=\{-2,2\}\)
C \(B\subset A\) पर \(A\ne B\) / \(B\subset A\) but \(A\ne B\)
D \(A=\varnothing\)
Explanation opens after your attempt
Step 1
Concept
The equation has roots (-2,2), but the natural solution is only (2). Always check the given number set condition.
Step 2
Why this answer is correct
The correct answer is A. (A=B). The equation has roots (-2,2), but the natural solution is only (2). Always check the given number set condition.
Step 3
Exam Tip
समीकरण के हल (-2,2) हैं पर प्राकृतिक हल केवल (2) है। दिए हुए संख्या समुच्चय की शर्त अवश्य देखें।
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यदि \(A={x:x\) (24) का सम धनात्मक भाजक है(}) तो कौन सा विकल्प (A) का उपसमुच्चय है?
If \(A={x:x\) is an even positive divisor of (24)(}), which option is a subset of (A)?
#sets
#subset_test
#divisors
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A ({2,6,8})
B ({1,2,3})
C ({2,5,12})
D ({4,9})
Explanation opens after your attempt
Correct Answer
A. ({2,6,8})
Step 1
Concept
Here \(A=\{2,4,6,8,12,24\}\), and (2,6,8) all belong to (A). An option with an outside element is not a subset.
Step 2
Why this answer is correct
The correct answer is A. ({2,6,8}). Here \(A=\{2,4,6,8,12,24\}\), and (2,6,8) all belong to (A). An option with an outside element is not a subset.
Step 3
Exam Tip
\(A=\{2,4,6,8,12,24\}\) है और विकल्प में (2,6,8) सभी (A) में हैं। बाहरी अवयव वाला विकल्प उपसमुच्चय नहीं होता।
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यदि \(A=\{1,2,3,4\}\) है तो (A) के तीन अवयवों वाले उचित उपसमुच्चयों की संख्या क्या है?
If \(A=\{1,2,3,4\}\), how many proper subsets of (A) have exactly three elements?
#sets
#proper_subsets
#combinations
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A (3)
B (4)
C (6)
D (8)
Explanation opens after your attempt
Step 1
Concept
The number of ways to choose three elements is \(\binom{4}{3}=4\). All are proper because the whole set has four elements.
Step 2
Why this answer is correct
The correct answer is B. (4). The number of ways to choose three elements is \(\binom{4}{3}=4\). All are proper because the whole set has four elements.
Step 3
Exam Tip
तीन अवयव चुनने की संख्या \(\binom{4}{3}=4\) है। ये सभी उचित हैं क्योंकि पूरा समुच्चय चार अवयवों का है।
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यदि \(A={x:x\) \(0\leq x<5\) और (x) पूर्णांक है(}) तथा \(B=\{0,1,2,3,4\}\) हैं तो क्या सत्य है?
If \(A={x:x\) is an integer with \(0\leq x<5\)(}) and \(B=\{0,1,2,3,4\}\), what is true?
#sets
#equal_sets
#inequality
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A (A=B)
B \(A\subset B\) पर \(A\ne B\) / \(A\subset B\) but \(A\ne B\)
C \(B\subset A\) पर \(A\ne B\) / \(B\subset A\) but \(A\ne B\)
D \(5\in A\)
Explanation opens after your attempt
Step 1
Concept
The condition \(0\leq x<5\) includes (0) and excludes (5). Thus the elements are (0,1,2,3,4).
Step 2
Why this answer is correct
The correct answer is A. (A=B). The condition \(0\leq x<5\) includes (0) and excludes (5). Thus the elements are (0,1,2,3,4).
Step 3
Exam Tip
शर्त \(0\leq x<5\) में (0) शामिल और (5) बाहर है। इसलिए अवयव (0,1,2,3,4) हैं।
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यदि \(A=\{1,2,3\}\) है तो (P(A)) में ({1,3}) के बारे में कौन सा कथन सही है?
If \(A=\{1,2,3\}\), which statement about ({1,3}) in (P(A)) is correct?
#sets
#power_set
#subset_element
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A \({1,3}\in P(A)\)
B \({1,3}\notin P(A)\)
C \(1,3\notin A\)
D (P(A)=A)
Explanation opens after your attempt
Correct Answer
A. \({1,3}\in P(A)\)
Step 1
Concept
Since \({1,3}\subseteq A\), ({1,3}) is an element of the power set. In a power set, subsets become elements.
Step 2
Why this answer is correct
The correct answer is A. \({1,3}\in P(A)\). Since \({1,3}\subseteq A\), ({1,3}) is an element of the power set. In a power set, subsets become elements.
Step 3
Exam Tip
क्योंकि \({1,3}\subseteq A\), इसलिए ({1,3}) पावर समुच्चय का अवयव है। पावर समुच्चय में उपसमुच्चय अवयव बनते हैं।
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यदि \(A={x:x\) (9) से छोटे अऋण पूर्णांक हैं(}) और \(B=\{0,1,2,3,4,5,6,7,8\}\) हैं तो क्या सत्य है?
If \(A={x:x\) is a non-negative integer less than (9)(}) and \(B=\{0,1,2,3,4,5,6,7,8\}\), what is true?
#sets
#equal_sets
#nonnegative_integers
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A (A=B)
B \(A\ne B\) क्योंकि (0) प्राकृतिक नहीं है / \(A\ne B\) because (0) is not natural
C \(B\subset A\) पर \(A\ne B\) / \(B\subset A\) but \(A\ne B\)
D \(A=\{1,2,3,4,5,6,7,8,9\}\)
Explanation opens after your attempt
Step 1
Concept
Non-negative integers start from (0) and those less than (9) go up to (8). Hence the two sets are equal.
Step 2
Why this answer is correct
The correct answer is A. (A=B). Non-negative integers start from (0) and those less than (9) go up to (8). Hence the two sets are equal.
Step 3
Exam Tip
अऋण पूर्णांक (0) से शुरू होते हैं और (9) से छोटे होने पर (8) तक जाते हैं। इसलिए दोनों बराबर हैं।
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यदि \(A=\{2,4,6\}\) है तो कौन सा विकल्प (A) का उपसमुच्चय है पर उचित उपसमुच्चय नहीं है?
If \(A=\{2,4,6\}\), which option is a subset of (A) but not a proper subset?
#sets
#proper_subset
#self_subset
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A ({2,4})
B ({2,4,6})
C \(\varnothing\)
D ({6})
Explanation opens after your attempt
Correct Answer
B. ({2,4,6})
Step 1
Concept
A set is a subset of itself but not a proper subset of itself. Proper relation excludes equality.
Step 2
Why this answer is correct
The correct answer is B. ({2,4,6}). A set is a subset of itself but not a proper subset of itself. Proper relation excludes equality.
Step 3
Exam Tip
समुच्चय अपने आप का उपसमुच्चय है पर अपने आप का उचित उपसमुच्चय नहीं है। उचित संबंध में बराबरी हटती है।
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यदि \(A={x:x\) (30) का अभाज्य भाजक है(}) और \(B=\{2,3,5\}\) हैं तो संबंध क्या है?
If \(A={x:x\) is a prime divisor of (30)(}) and \(B=\{2,3,5\}\), what is the relation?
#sets
#equal_sets
#prime_divisors
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A (A=B)
B \(A\subset B\) पर \(A\ne B\) / \(A\subset B\) but \(A\ne B\)
C \(B\subset A\) पर \(A\ne B\) / \(B\subset A\) but \(A\ne B\)
D \(A=\{1,2,3,5,30\}\)
Explanation opens after your attempt
Step 1
Concept
The prime divisors of (30) are (2,3,5). The numbers (1) and (30) are not prime divisors.
Step 2
Why this answer is correct
The correct answer is A. (A=B). The prime divisors of (30) are (2,3,5). The numbers (1) and (30) are not prime divisors.
Step 3
Exam Tip
(30) के अभाज्य भाजक (2,3,5) हैं। (1) और (30) अभाज्य भाजक नहीं हैं।
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यदि \(A=\{1,2,3,4,5,6\}\), \(B=\{2,4,6\}\), और \(C={x:x\in A\) तथा (x) सम है(}) हैं तो कौन सा कथन सही है?
If \(A=\{1,2,3,4,5,6\}\), \(B=\{2,4,6\}\), and \(C={x:x\in A\) and (x) is even(}), which statement is correct?
#sets
#equal_sets
#proper_subset
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A (B=C) और \(B\subset A\) / (B=C) and \(B\subset A\)
B (B=C) और \(A\subset B\) / (B=C) and \(A\subset B\)
C \(B\ne C\)
D \(C=\varnothing\)
Explanation opens after your attempt
Correct Answer
A. (B=C) और \(B\subset A\) / (B=C) and \(B\subset A\)
Step 1
Concept
The even elements of (A) are (2,4,6), so (C=B) and (B) is a proper subset of (A). Apply the condition inside the original set.
Step 2
Why this answer is correct
The correct answer is A. (B=C) और \(B\subset A\) / (B=C) and \(B\subset A\). The even elements of (A) are (2,4,6), so (C=B) and (B) is a proper subset of (A). Apply the condition inside the original set.
Step 3
Exam Tip
(A) के सम अवयव (2,4,6) हैं इसलिए (C=B) है और (B) (A) का उचित उपसमुच्चय है। शर्त को मूल समुच्चय के अंदर लागू करें।
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यदि \(A={x:x\in\mathbb{Z}\) और \(x^2-4x+3=0}\) तथा \(B=\{1,3\}\) हैं तो सही संबंध कौन सा है?
If \(A={x:x\in\mathbb{Z}\) and \(x^2-4x+3=0}\) and \(B=\{1,3\}\), which relation is correct?
#sets
#equal_sets
#quadratic_roots
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A (A=B)
B (A) (B) का उचित उपसमुच्चय है / (A) is a proper subset of (B)
C (B) (A) का उचित उपसमुच्चय है / (B) is a proper subset of (A)
D \(A\cap B=\varnothing\)
Explanation opens after your attempt
Step 1
Concept
The equation has roots (1) and (3), so both sets have the same elements. In equal sets order is not considered.
Step 2
Why this answer is correct
The correct answer is A. (A=B). The equation has roots (1) and (3), so both sets have the same elements. In equal sets order is not considered.
Step 3
Exam Tip
समीकरण के हल (1) और (3) हैं इसलिए दोनों समुच्चयों में वही अवयव हैं। बराबर समुच्चय में क्रम नहीं देखा जाता।
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यदि \(A={X:X\subseteq{1,2,3,4},{1,4}\subseteq X\) और \(2\notin X}\) है तो (A) के बराबर कौन सा समुच्चय है?
If \(A={X:X\subseteq{1,2,3,4},{1,4}\subseteq X\) and \(2\notin X}\), which set is equal to (A)?
#sets
#equal_sets
#restricted_subsets
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A ({{1,4},{1,3,4}})
B ({{1,2,4},{1,3,4}})
C ({{1,4},{2,4},{1,2,3,4}})
D ({{1},{4},{1,4}})
Explanation opens after your attempt
Correct Answer
A. ({{1,4},{1,3,4}})
Step 1
Concept
Every (X) must contain (1) and (4), and must not contain (2), so only (3) is optional. In such questions, fix required and forbidden elements first.
Step 2
Why this answer is correct
The correct answer is A. ({{1,4},{1,3,4}}). Every (X) must contain (1) and (4), and must not contain (2), so only (3) is optional. In such questions, fix required and forbidden elements first.
Step 3
Exam Tip
हर (X) में (1) और (4) होना चाहिए तथा (2) नहीं होना चाहिए, इसलिए केवल (3) वैकल्पिक है। ऐसे प्रश्नों में अनिवार्य और निषिद्ध अवयव पहले तय करें।
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यदि \(A=\{1,2,{3,4}\}\) है तो निम्न में से कौन (A) का उपसमुच्चय है?
If \(A=\{1,2,{3,4}\}\), which of the following is a subset of (A)?
#sets
#subsets
#nested_sets
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A ({1,3})
B ({2,{3,4}})
C ({{1,2},3})
D ({4,{3}})
Explanation opens after your attempt
Correct Answer
B. ({2,{3,4}})
Step 1
Concept
Both (2) and ({3,4}) are elements of (A). Changing braces can change the element.
Step 2
Why this answer is correct
The correct answer is B. ({2,{3,4}}). Both (2) and ({3,4}) are elements of (A). Changing braces can change the element.
Step 3
Exam Tip
(2) और ({3,4}) दोनों (A) के अवयव हैं। कोष्ठक बदलने से अवयव बदल सकता है।
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यदि (P(A)) में (64) अवयव हैं तो (A) के उचित उपसमुच्चयों की संख्या क्या होगी?
If (P(A)) has (64) elements, what is the number of proper subsets of (A)?
#sets
#power_set
#proper_subsets
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A (31)
B (32)
C (63)
D (64)
Explanation opens after your attempt
Step 1
Concept
From \(2^n=64\), (n=6), and proper subsets are \(2^6-1=63\). Keep the formulas for power set and proper subsets separate.
Step 2
Why this answer is correct
The correct answer is C. (63). From \(2^n=64\), (n=6), and proper subsets are \(2^6-1=63\). Keep the formulas for power set and proper subsets separate.
Step 3
Exam Tip
\(2^n=64\) से (n=6) है और उचित उपसमुच्चय \(2^6-1=63\) होंगे। पावर समुच्चय और उचित उपसमुच्चय के सूत्र अलग रखें।
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यदि \(A={x:x\) (42) का अभाज्य भाजक है(}) और \(B=\{2,3,7\}\) हैं तो कौन सा कथन सही है?
If \(A={x:x\) is a prime divisor of (42)(}) and \(B=\{2,3,7\}\), which statement is correct?
#sets
#equal_sets
#prime_divisors
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A (A=B)
B \(A=\{1,2,3,6,7,14,21,42\}\)
C (B) (A) का उचित उपसमुच्चय है / (B) is a proper subset of (A)
D \(A=\varnothing\)
Explanation opens after your attempt
Step 1
Concept
The prime divisors of (42) are (2,3,7). Understand the difference between all divisors and prime divisors.
Step 2
Why this answer is correct
The correct answer is A. (A=B). The prime divisors of (42) are (2,3,7). Understand the difference between all divisors and prime divisors.
Step 3
Exam Tip
(42) के अभाज्य भाजक (2,3,7) हैं। सभी भाजकों और अभाज्य भाजकों में अंतर समझें।
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यदि \(A={x:x\in\mathbb{N}\) और \(x\mid 18}\) तथा \(B=\{1,2,3,6,9,18\}\) हैं तो सही विकल्प चुनिए।
If \(A={x:x\in\mathbb{N}\) and \(x\mid 18}\) and \(B=\{1,2,3,6,9,18\}\), choose the correct option.
#sets
#equal_sets
#divisibility
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A (A=B)
B (A) में केवल अभाज्य भाजक हैं / (A) contains only prime divisors
C \(B\subset A\) पर \(A\ne B\) / \(B\subset A\) but \(A\ne B\)
D \(18\notin A\)
Explanation opens after your attempt
Step 1
Concept
\(x\mid 18\) means (x) divides (18). A number itself is also its positive divisor.
Step 2
Why this answer is correct
The correct answer is A. (A=B). \(x\mid 18\) means (x) divides (18). A number itself is also its positive divisor.
Step 3
Exam Tip
\(x\mid 18\) का अर्थ है (x) (18) का भाजक है। संख्या स्वयं भी उसका धनात्मक भाजक होती है।
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यदि \(A=\{0,1,2,3\}\) है तो (A) के ऐसे उपसमुच्चयों की संख्या क्या है जिनमें (0) अवश्य हो?
If \(A=\{0,1,2,3\}\), how many subsets of (A) must contain (0)?
#sets
#subsets
#counting
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A (4)
B (6)
C (8)
D (16)
Explanation opens after your attempt
Step 1
Concept
Fixing (0), the remaining three elements are optional, so there are \(2^3=8\) subsets. For a fixed element, count choices for the remaining elements.
Step 2
Why this answer is correct
The correct answer is C. (8). Fixing (0), the remaining three elements are optional, so there are \(2^3=8\) subsets. For a fixed element, count choices for the remaining elements.
Step 3
Exam Tip
(0) को निश्चित रखने पर बाकी तीन अवयव स्वतंत्र हैं इसलिए \(2^3=8\) उपसमुच्चय मिलते हैं। निश्चित अवयव को चुनकर बाकी पर गिनती करें।
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यदि \(A=\{a,b,c,d,e\}\) है तो (A) के ऐसे उपसमुच्चयों की संख्या क्या है जिनमें (a) न हो?
If \(A=\{a,b,c,d,e\}\), how many subsets of (A) do not contain (a)?
#sets
#subsets
#counting_without_element
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A (8)
B (16)
C (24)
D (31)
Explanation opens after your attempt
Step 1
Concept
After excluding (a), four elements remain and they form \(2^4=16\) subsets. Remove the forbidden element first.
Step 2
Why this answer is correct
The correct answer is B. (16). After excluding (a), four elements remain and they form \(2^4=16\) subsets. Remove the forbidden element first.
Step 3
Exam Tip
(a) हटाने के बाद चार अवयव बचते हैं और उनसे \(2^4=16\) उपसमुच्चय बनते हैं। निषिद्ध अवयव को पहले बाहर करें।
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यदि \(A={x:x\in\mathbb{Z}\) और \(-3\leq x<2}\) तो (A) के बराबर कौन सा समुच्चय है?
If \(A={x:x\in\mathbb{Z}\) and \(-3\leq x<2}\), which set is equal to (A)?
#sets
#equal_sets
#integer_interval
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A ({-3,-2,-1,0,1})
B ({-2,-1,0,1,2})
C ({-3,-2,-1,0,1,2})
D ({-2,-1,0,1})
Explanation opens after your attempt
Correct Answer
A. ({-3,-2,-1,0,1})
Step 1
Concept
(-3) is included and (2) is not included. Read open and closed ends of inequalities carefully.
Step 2
Why this answer is correct
The correct answer is A. ({-3,-2,-1,0,1}). (-3) is included and (2) is not included. Read open and closed ends of inequalities carefully.
Step 3
Exam Tip
(-3) शामिल है और (2) शामिल नहीं है। असमानता में खुले और बंद सिरों को ध्यान से पढ़ें।
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यदि \(A\subseteq B\), \(B\subseteq C\), और \(A\not=C\) है तो कौन सा कथन हमेशा सत्य है?
If \(A\subseteq B\), \(B\subseteq C\), and \(A\not=C\), which statement is always true?
#sets
#subset_transitivity
#proper_subset
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A \(A\subset C\) उचित रूप से / (A) is a proper subset of (C)
B (B=C) अवश्य / (B=C) necessarily
C (A=B) अवश्य / (A=B) necessarily
D \(C\subset A\)
Explanation opens after your attempt
Correct Answer
A. \(A\subset C\) उचित रूप से / (A) is a proper subset of (C)
Step 1
Concept
By transitivity \(A\subseteq C\), and since \(A\ne C\), (A) is a proper subset. The middle set may or may not be equal.
Step 2
Why this answer is correct
The correct answer is A. \(A\subset C\) उचित रूप से / (A) is a proper subset of (C). By transitivity \(A\subseteq C\), and since \(A\ne C\), (A) is a proper subset. The middle set may or may not be equal.
Step 3
Exam Tip
ट्रांजिटिव नियम से \(A\subseteq C\) और \(A\not=C\) होने से (A) उचित उपसमुच्चय है। बीच वाला समुच्चय बराबर भी हो सकता है।
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यदि \(A={x:x\) (25) से छोटे पूर्ण वर्ग हैं और (x>0}) तो कौन सा विकल्प (A) के बराबर है?
If \(A={x:x\) is a positive perfect square less than (25)(}), which option is equal to (A)?
#sets
#equal_sets
#perfect_squares
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A ({1,4,9,16})
B ({0,1,4,9,16})
C ({1,4,9,16,25})
D ({2,3,4,5})
Explanation opens after your attempt
Correct Answer
A. ({1,4,9,16})
Step 1
Concept
The positive perfect squares are (1,4,9,16), and (25) is not included. Apply every condition together.
Step 2
Why this answer is correct
The correct answer is A. ({1,4,9,16}). The positive perfect squares are (1,4,9,16), and (25) is not included. Apply every condition together.
Step 3
Exam Tip
धनात्मक पूर्ण वर्ग (1,4,9,16) हैं और (25) सीमा में शामिल नहीं है। हर शर्त को साथ में लागू करें।
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यदि \(A=\{p,q,r,s\}\) है तो (A) के दो अवयवों वाले उपसमुच्चयों की संख्या क्या है?
If \(A=\{p,q,r,s\}\), how many subsets of (A) have exactly two elements?
#sets
#subsets
#combinations
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A (4)
B (6)
C (8)
D (12)
Explanation opens after your attempt
Step 1
Concept
The number of ways to choose two elements is \(\binom{4}{2}=6\). Order is not counted in subsets.
Step 2
Why this answer is correct
The correct answer is B. (6). The number of ways to choose two elements is \(\binom{4}{2}=6\). Order is not counted in subsets.
Step 3
Exam Tip
दो अवयव चुनने के तरीके \(\binom{4}{2}=6\) हैं। उपसमुच्चय में क्रम नहीं गिना जाता।
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यदि \(A={\varnothing,{\varnothing},1}\) है तो निम्न में से कौन (A) का उपसमुच्चय है?
If \(A={\varnothing,{\varnothing},1}\), which of the following is a subset of (A)?
#sets
#empty_set
#subset_test
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A \({\varnothing,1}\)
B ({0,1})
C \({{{\varnothing}}}\)
D \({2,\varnothing}\)
Explanation opens after your attempt
Correct Answer
A. \({\varnothing,1}\)
Step 1
Concept
Both \(\varnothing\) and (1) are elements of (A), so \({\varnothing,1}\) is a subset. (0) and \(\varnothing\) are different.
Step 2
Why this answer is correct
The correct answer is A. \({\varnothing,1}\). Both \(\varnothing\) and (1) are elements of (A), so \({\varnothing,1}\) is a subset. (0) and \(\varnothing\) are different.
Step 3
Exam Tip
\(\varnothing\) और (1) दोनों (A) के अवयव हैं इसलिए \({\varnothing,1}\) उपसमुच्चय है। (0) और \(\varnothing\) अलग हैं।
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यदि \(A={x:x\in\mathbb{N}\) और (x) (16) का गुणनखंड है पर (x) सम नहीं है(}) तो (A) क्या है?
If \(A={x:x\in\mathbb{N}\) and (x) is a factor of (16) but (x) is not even(}), what is (A)?
#sets
#set_builder
#odd_factor
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A ({1})
B ({2,4,8,16})
C ({1,2,4,8,16})
D \(\varnothing\)
Explanation opens after your attempt
Step 1
Concept
Among the factors of (16), only (1) is odd. In questions with a negative condition, pay attention to the excluded elements.
Step 2
Why this answer is correct
The correct answer is A. ({1}). Among the factors of (16), only (1) is odd. In questions with a negative condition, pay attention to the excluded elements.
Step 3
Exam Tip
(16) के भाजकों में केवल (1) विषम है। नकारात्मक शर्त वाले प्रश्न में हटाए गए अवयवों पर ध्यान दें।
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यदि \(A=\{1,2,3,4,5\}\) है तो (A) के तीन या अधिक अवयवों वाले उपसमुच्चयों की संख्या क्या है?
If \(A=\{1,2,3,4,5\}\), how many subsets of (A) have three or more elements?
#sets
#subsets
#combinations
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A (10)
B (16)
C (20)
D (26)
Explanation opens after your attempt
Step 1
Concept
The number is \(\binom{5}{3}+\binom{5}{4}+\binom{5}{5}=16\). For size based counting, add the combinations.
Step 2
Why this answer is correct
The correct answer is B. (16). The number is \(\binom{5}{3}+\binom{5}{4}+\binom{5}{5}=16\). For size based counting, add the combinations.
Step 3
Exam Tip
तीन या अधिक अवयवों की संख्या \(\binom{5}{3}+\binom{5}{4}+\binom{5}{5}=16\) है। आकार आधारित गिनती में संयोजन जोड़ें।
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यदि \(A={x:x\) अंग्रेजी शब्द (LEVEL) के अलग अक्षर हैं(}) और \(B=\{L,E,V\}\) हैं तो सही निष्कर्ष क्या है?
If \(A={x:x\) is a distinct letter of the English word (LEVEL)(}) and \(B=\{L,E,V\}\), what is the correct conclusion?
#sets
#equal_sets
#repeated_letters
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A (A=B)
B \(A\ne B\) क्योंकि (L) दो बार आता है / \(A\ne B\) because (L) occurs twice
C \(B\subset A\) पर \(A\ne B\) / \(B\subset A\) but \(A\ne B\)
D \(A=\{L,E,V,E,L\}\) और यह अलग है / \(A=\{L,E,V,E,L\}\) and it is different
Explanation opens after your attempt
Step 1
Concept
Repetition is not counted in a set, so the distinct letters are (L,E,V). In letter based questions count only distinct elements.
Step 2
Why this answer is correct
The correct answer is A. (A=B). Repetition is not counted in a set, so the distinct letters are (L,E,V). In letter based questions count only distinct elements.
Step 3
Exam Tip
समुच्चय में दोहराव नहीं गिना जाता इसलिए अलग अक्षर (L,E,V) हैं। अक्षरों वाले प्रश्न में केवल अलग अवयव गिनें।
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यदि \(A={x:x\in\mathbb{Z}\) और \(x^2\leq 4}\) है तो (A) का कौन सा उपसमुच्चय उचित नहीं है?
If \(A={x:x\in\mathbb{Z}\) and \(x^2\leq 4}\), which subset of (A) is not proper?
#sets
#proper_subset
#integer_squares
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A ({-2,-1,0,1,2})
B ({-1,0,1})
C ({-2,2})
D \(\varnothing\)
Explanation opens after your attempt
Correct Answer
A. ({-2,-1,0,1,2})
Step 1
Concept
Here \(A=\{-2,-1,0,1,2\}\), so the same set is a subset but not proper. A proper subset is not the whole set.
Step 2
Why this answer is correct
The correct answer is A. ({-2,-1,0,1,2}). Here \(A=\{-2,-1,0,1,2\}\), so the same set is a subset but not proper. A proper subset is not the whole set.
Step 3
Exam Tip
\(A=\{-2,-1,0,1,2\}\) है इसलिए वही समुच्चय उपसमुच्चय है पर उचित नहीं है। उचित उपसमुच्चय पूरा समुच्चय नहीं होता।
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यदि \(A=\{2,4,6,8\}\) और \(B={x:x=2n\) जहाँ \(n\in\mathbb{N}\) और \(1\leq n\leq 4}\) हैं तो कौन सा विकल्प सही है?
If \(A=\{2,4,6,8\}\) and \(B={x:x=2n\) where \(n\in\mathbb{N}\) and \(1\leq n\leq 4}\), which option is correct?
#sets
#equal_sets
#parametric_set
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A (A=B)
B \(A\subset B\) पर \(A\ne B\) / \(A\subset B\) but \(A\ne B\)
C \(B=\{1,2,3,4\}\)
D \(A\cap B=\varnothing\)
Explanation opens after your attempt
Step 1
Concept
Putting (n=1,2,3,4) gives (x=2,4,6,8). In parameter based sets, first find the values.
Step 2
Why this answer is correct
The correct answer is A. (A=B). Putting (n=1,2,3,4) gives (x=2,4,6,8). In parameter based sets, first find the values.
Step 3
Exam Tip
(n=1,2,3,4) रखने पर (x=2,4,6,8) मिलता है। पैरामीटर वाले समुच्चय में पहले मान निकालें।
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यदि \(A=\{1,2,3\}\) और (B=P(A)) है तो निम्न में से कौन (B) का अवयव नहीं है?
If \(A=\{1,2,3\}\) and (B=P(A)), which of the following is not an element of (B)?
#sets
#power_set
#element_vs_subset
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A ({1,2})
B \(\varnothing\)
C ({3})
D (2)
Explanation opens after your attempt
Step 1
Concept
The elements of (P(A)) are subsets of (A), and (2) itself is not a subset. ({2}) and (2) are different.
Step 2
Why this answer is correct
The correct answer is D. (2). The elements of (P(A)) are subsets of (A), and (2) itself is not a subset. ({2}) and (2) are different.
Step 3
Exam Tip
(P(A)) के अवयव (A) के उपसमुच्चय होते हैं और (2) स्वयं उपसमुच्चय नहीं है। ({2}) और (2) अलग हैं।
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यदि \(A={x:x\in\mathbb{N}\) और (x<10}), \(B={x:x\) एक अंक की प्राकृतिक संख्या है(}) हैं तो क्या सत्य है?
If \(A={x:x\in\mathbb{N}\) and (x<10}), \(B={x:x\) is a one digit natural number(}), what is true?
#sets
#equal_sets
#natural_numbers
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A (A=B)
B (A) में (0) है और (B) में नहीं है / (A) contains (0) but (B) does not
C \(B\subset A\) पर \(A\ne B\) / \(B\subset A\) but \(A\ne B\)
D \(A=\{0,1,2,3,4,5,6,7,8,9\}\)
Explanation opens after your attempt
Step 1
Concept
In Class (11), usually \(\mathbb{N}={1,2,3,\ldots}\), so both sets are (1) to (9). Read the given number system carefully.
Step 2
Why this answer is correct
The correct answer is A. (A=B). In Class (11), usually \(\mathbb{N}={1,2,3,\ldots}\), so both sets are (1) to (9). Read the given number system carefully.
Step 3
Exam Tip
कक्षा (11) में सामान्यतः \(\mathbb{N}={1,2,3,\ldots}\) लिया जाता है इसलिए दोनों (1) से (9) तक हैं। संकेत में दी गई संख्या पद्धति को ध्यान से देखें।
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यदि \(A={x:x\) (60) का धनात्मक भाजक है और (x) (15) का भाजक भी है(}) तो (A) कौन सा है?
If \(A={x:x\) is a positive divisor of (60) and also a divisor of (15)(}), what is (A)?
#sets
#set_builder
#common_divisors
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A ({1,3,5,15})
B ({2,4,6,10,12,20,30,60})
C ({15,30,45,60})
D ({3,5})
Explanation opens after your attempt
Correct Answer
A. ({1,3,5,15})
Step 1
Concept
The common positive divisors of both numbers are (1,3,5,15). For an and condition, choose common elements.
Step 2
Why this answer is correct
The correct answer is A. ({1,3,5,15}). The common positive divisors of both numbers are (1,3,5,15). For an and condition, choose common elements.
Step 3
Exam Tip
दोनों संख्याओं के सामान्य धनात्मक भाजक (1,3,5,15) हैं। और की शर्त में समान अवयव चुनें।
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यदि \(A=\{a,b\}\) है तो (P(A)) के बराबर कौन सा समुच्चय है?
If \(A=\{a,b\}\), which set is equal to (P(A))?
#sets
#power_set
#equal_sets
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A \({\varnothing,{a},{b},{a,b}}\)
B ({a,b,{a,b}})
C ({{a},{b}})
D \({\varnothing,a,b}\)
Explanation opens after your attempt
Correct Answer
A. \({\varnothing,{a},{b},{a,b}}\)
Step 1
Concept
A power set lists all subsets as elements. It contains ({a}), not (a) as the singleton subset.
Step 2
Why this answer is correct
The correct answer is A. \({\varnothing,{a},{b},{a,b}}\). A power set lists all subsets as elements. It contains ({a}), not (a) as the singleton subset.
Step 3
Exam Tip
पावर समुच्चय में सभी उपसमुच्चय अवयव के रूप में लिखे जाते हैं। (a) नहीं बल्कि ({a}) उसका अवयव होगा।
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यदि \(A={x:x\in\mathbb{Z}\) और \(|x|\leq 2}\) तथा \(B=\{-2,-1,0,1,2\}\) हैं तो कौन सा निष्कर्ष सही है?
If \(A={x:x\in\mathbb{Z}\) and \(|x|\leq 2}\) and \(B=\{-2,-1,0,1,2\}\), which conclusion is correct?
#sets
#equal_sets
#modulus
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A (A=B)
B \(A=\{0,1,2\}\)
C \(B\subset A\) पर \(A\ne B\) / \(B\subset A\) but \(A\ne B\)
D \(A=\{-2,2\}\)
Explanation opens after your attempt
Step 1
Concept
The condition \(|x|\leq 2\) includes all integers from (-2) to (2). In modulus, check both positive and negative sides.
Step 2
Why this answer is correct
The correct answer is A. (A=B). The condition \(|x|\leq 2\) includes all integers from (-2) to (2). In modulus, check both positive and negative sides.
Step 3
Exam Tip
\(|x|\leq 2\) में (-2) से (2) तक के सभी पूर्णांक आते हैं। मापांक में धन और ऋण दोनों पक्ष देखें।
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यदि \(A=\{1,2,3,4\}\) है तो (A) के ऐसे उपसमुच्चयों की संख्या क्या है जिनमें (1) हो और (4) न हो?
If \(A=\{1,2,3,4\}\), how many subsets of (A) contain (1) and do not contain (4)?
#sets
#subsets
#restricted_counting
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A (2)
B (4)
C (6)
D (8)
Explanation opens after your attempt
Step 1
Concept
(1) is fixed and (4) is forbidden, so (2,3) are free. Hence \(2^2=4\) subsets are possible.
Step 2
Why this answer is correct
The correct answer is B. (4). (1) is fixed and (4) is forbidden, so (2,3) are free. Hence \(2^2=4\) subsets are possible.
Step 3
Exam Tip
(1) निश्चित और (4) निषिद्ध है इसलिए (2,3) स्वतंत्र हैं। अतः \(2^2=4\) उपसमुच्चय बनते हैं।
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यदि \(A={x:x\) (10) से छोटे सम अभाज्य हैं(}) तो कौन सा समुच्चय (A) के बराबर है?
If \(A={x:x\) is an even prime less than (10)(}), which set is equal to (A)?
#sets
#equal_sets
#even_prime
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A ({2})
B ({2,4,6,8})
C ({2,3,5,7})
D \(\varnothing\)
Explanation opens after your attempt
Step 1
Concept
Only (2) is an even prime number. In combined conditions, both properties must hold together.
Step 2
Why this answer is correct
The correct answer is A. ({2}). Only (2) is an even prime number. In combined conditions, both properties must hold together.
Step 3
Exam Tip
केवल (2) ही सम अभाज्य संख्या है। संयुक्त शर्तों में दोनों गुण एक साथ चाहिए।
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यदि \(A=\{1,2,3\}\) और \(B=\{1,2,3,4,5\}\) हैं तो (B) के कितने उपसमुच्चय (A) के भी उपसमुच्चय हैं?
If \(A=\{1,2,3\}\) and \(B=\{1,2,3,4,5\}\), how many subsets of (B) are also subsets of (A)?
#sets
#subsets
#common_subsets
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A (3)
B (6)
C (8)
D (32)
Explanation opens after your attempt
Step 1
Concept
The subsets lying inside (A) are subsets of both, and their number is \(2^3=8\). Decide the common limit using the smaller set.
Step 2
Why this answer is correct
The correct answer is C. (8). The subsets lying inside (A) are subsets of both, and their number is \(2^3=8\). Decide the common limit using the smaller set.
Step 3
Exam Tip
जो उपसमुच्चय (A) के अंदर हैं वही दोनों के लिए उपसमुच्चय होंगे और उनकी संख्या \(2^3=8\) है। साझा सीमा को छोटे समुच्चय से तय करें।
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यदि \(A={x:x\in\mathbb{N}\) और \(x^2<50}\) तो (A) का कौन सा उपसमुच्चय नहीं है?
If \(A={x:x\in\mathbb{N}\) and \(x^2<50}\), which is not a subset of (A)?
#sets
#subset_test
#square_inequality
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A ({1,3,7})
B ({2,4,6})
C ({1,5,8})
D \(\varnothing\)
Explanation opens after your attempt
Correct Answer
C. ({1,5,8})
Step 1
Concept
Here \(A=\{1,2,3,4,5,6,7\}\), and (8) is not in it. One outside element makes the subset statement false.
Step 2
Why this answer is correct
The correct answer is C. ({1,5,8}). Here \(A=\{1,2,3,4,5,6,7\}\), and (8) is not in it. One outside element makes the subset statement false.
Step 3
Exam Tip
\(A=\{1,2,3,4,5,6,7\}\) है और (8) इसमें नहीं है। एक बाहरी अवयव पूरा उपसमुच्चय कथन गलत कर देता है।
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यदि \(A={x:x\) (24) और (36) दोनों के अभाज्य भाजक हैं(}) तो (A) क्या है?
If \(A={x:x\) is a prime divisor of both (24) and (36)(}), what is (A)?
#sets
#equal_sets
#common_prime_divisors
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A ({2,3})
B ({2,3,5})
C ({1,2,3,6,12})
D ({6,12})
Explanation opens after your attempt
Correct Answer
A. ({2,3})
Step 1
Concept
The common prime divisors of both numbers are (2) and (3). Common divisors and common prime divisors can be different.
Step 2
Why this answer is correct
The correct answer is A. ({2,3}). The common prime divisors of both numbers are (2) and (3). Common divisors and common prime divisors can be different.
Step 3
Exam Tip
दोनों संख्याओं के सामान्य अभाज्य भाजक (2) और (3) हैं। सामान्य भाजक और सामान्य अभाज्य भाजक अलग हो सकते हैं।
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यदि \(A=\{1,2,{1,2}\}\) है तो (A) के कुल अवयव कितने हैं?
If \(A=\{1,2,{1,2}\}\), how many elements does (A) have?
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#cardinality
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A (2)
B (3)
C (4)
D (5)
Explanation opens after your attempt
Step 1
Concept
Here (1), (2), and ({1,2}) are three distinct elements. A whole set can be an element of a larger set.
Step 2
Why this answer is correct
The correct answer is B. (3). Here (1), (2), and ({1,2}) are three distinct elements. A whole set can be an element of a larger set.
Step 3
Exam Tip
यहाँ (1), (2), और ({1,2}) तीन अलग अवयव हैं। पूरा समुच्चय भी किसी बड़े समुच्चय का अवयव हो सकता है।
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यदि \(A={x:x\in\mathbb{Z}\) और \(x^2=9}\) तथा \(B=\{-3,3,9\}\) हैं तो सही संबंध क्या है?
If \(A={x:x\in\mathbb{Z}\) and \(x^2=9}\) and \(B=\{-3,3,9\}\), what is the correct relation?
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#integer_roots
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A (A=B)
B (A) (B) का उचित उपसमुच्चय है / (A) is a proper subset of (B)
C (B) (A) का उचित उपसमुच्चय है / (B) is a proper subset of (A)
D \(A=\{9\}\)
Explanation opens after your attempt
Correct Answer
B. (A) (B) का उचित उपसमुच्चय है / (A) is a proper subset of (B)
Step 1
Concept
Here \(A=\{-3,3\}\), and (B) has the extra element (9). Distinguish roots of an equation from constants in it.
Step 2
Why this answer is correct
The correct answer is B. (A) (B) का उचित उपसमुच्चय है / (A) is a proper subset of (B). Here \(A=\{-3,3\}\), and (B) has the extra element (9). Distinguish roots of an equation from constants in it.
Step 3
Exam Tip
\(A=\{-3,3\}\) है और (B) में अतिरिक्त (9) है। समीकरण के हल और समीकरण में आए स्थिरांक को अलग समझें।
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यदि \(A={x:x\) (100) से छोटे (25) के धनात्मक गुणज हैं(}) तो (A) के उपसमुच्चयों की संख्या क्या है?
If \(A={x:x\) is a positive multiple of (25) less than (100)(}), how many subsets does (A) have?
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#multiples
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A (4)
B (6)
C (8)
D (16)
Explanation opens after your attempt
Step 1
Concept
\(A=\{25,50,75\}\) has (3) elements, so it has \(2^3=8\) subsets. The boundary (100) is not included.
Step 2
Why this answer is correct
The correct answer is C. (8). \(A=\{25,50,75\}\) has (3) elements, so it has \(2^3=8\) subsets. The boundary (100) is not included.
Step 3
Exam Tip
\(A=\{25,50,75\}\) में (3) अवयव हैं इसलिए \(2^3=8\) उपसमुच्चय होंगे। सीमा में (100) शामिल नहीं है।
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यदि \(A={x:x\) \(x^2-1=0\) का पूर्णांक हल है(}) और \(B={x:x\in\mathbb{Z}\) तथा (-2<x<2) और \(x\ne0}\) हैं तो क्या सत्य है?
If \(A={x:x\) is an integer solution of \(x^2-1=0\)(}) and \(B={x:x\in\mathbb{Z}\) with (-2<x<2) and \(x\ne0}\), what is true?
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#multiple_descriptions
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A (A=B)
B \(A=\{1\}\) और \(B=\{-1,1\}\) / \(A=\{1\}\) and \(B=\{-1,1\}\)
C \(B=\varnothing\)
D \(A\cap B=\varnothing\)
Explanation opens after your attempt
Step 1
Concept
Both sets contain only (-1) and (1). Different descriptions can give equal sets.
Step 2
Why this answer is correct
The correct answer is A. (A=B). Both sets contain only (-1) and (1). Different descriptions can give equal sets.
Step 3
Exam Tip
दोनों समुच्चयों में (-1) और (1) ही आते हैं। अलग वर्णन बराबर समुच्चय दे सकते हैं।
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यदि \(A=\{1,2,3,4,5,6\}\) और \(B=\{2,3,5\}\) हैं तो (B) के तीन अवयवों वाले उपसमुच्चय के बारे में क्या सही है?
If \(A=\{1,2,3,4,5,6\}\) and \(B=\{2,3,5\}\), what is true about the three element subset of (B)?
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#subsets
#reasoning
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A वह (B) ही है और (A) का उपसमुच्चय भी है / It is (B) itself and also a subset of (A)
B वह (A) के बराबर है / It is equal to (A)
C ऐसा कोई उपसमुच्चय नहीं है / No such subset exists
D वह उचित उपसमुच्चय नहीं हो सकता किसी भी समुच्चय का / It cannot be a proper subset of any set
Explanation opens after your attempt
Correct Answer
A. वह (B) ही है और (A) का उपसमुच्चय भी है / It is (B) itself and also a subset of (A)
Step 1
Concept
The only three element subset of (B) is (B) itself. Since \(B\subset A\), it is also a subset of (A).
Step 2
Why this answer is correct
The correct answer is A. वह (B) ही है और (A) का उपसमुच्चय भी है / It is (B) itself and also a subset of (A). The only three element subset of (B) is (B) itself. Since \(B\subset A\), it is also a subset of (A).
Step 3
Exam Tip
तीन अवयवों वाला (B) का केवल उपसमुच्चय (B) स्वयं है। क्योंकि \(B\subset A\) है इसलिए यह (A) का उपसमुच्चय भी है।
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यदि \(A={x:x\) (12) से छोटे (2) या (3) के धनात्मक गुणज हैं(}) तो (A) कौन सा है?
If \(A={x:x\) is a positive multiple of (2) or (3) less than (12)(}), what is (A)?
#sets
#set_builder
#multiples
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A ({2,3,4,6,8,9,10})
B ({6})
C ({2,4,6,8,10,12})
D ({3,6,9,12})
Explanation opens after your attempt
Correct Answer
A. ({2,3,4,6,8,9,10})
Step 1
Concept
Combining multiples of (2) or (3) gives (2,3,4,6,8,9,10). For an or condition, think like a union.
Step 2
Why this answer is correct
The correct answer is A. ({2,3,4,6,8,9,10}). Combining multiples of (2) or (3) gives (2,3,4,6,8,9,10). For an or condition, think like a union.
Step 3
Exam Tip
(2) या (3) के गुणजों को मिलाने पर (2,3,4,6,8,9,10) मिलते हैं। या की शर्त में संघ जैसी सोच रखें।
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यदि \(A=\{1,2,3\}\), \(B=\{1,2,3,4\}\), और \(C=\{1,2,3,4\}\) हैं तो कौन सा कथन सत्य है?
If \(A=\{1,2,3\}\), \(B=\{1,2,3,4\}\), and \(C=\{1,2,3,4\}\), which statement is true?
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#proper_subset
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A (B=C) और (A) (B) का उचित उपसमुच्चय है / (B=C) and (A) is a proper subset of (B)
B (A=B=C)
C (C) (A) का उचित उपसमुच्चय है / (C) is a proper subset of (A)
D \(A\not\subset C\)
Explanation opens after your attempt
Correct Answer
A. (B=C) और (A) (B) का उचित उपसमुच्चय है / (B=C) and (A) is a proper subset of (B)
Step 1
Concept
(B) and (C) have the same elements, and (A) does not contain (4). Check equality and proper subset separately.
Step 2
Why this answer is correct
The correct answer is A. (B=C) और (A) (B) का उचित उपसमुच्चय है / (B=C) and (A) is a proper subset of (B). (B) and (C) have the same elements, and (A) does not contain (4). Check equality and proper subset separately.
Step 3
Exam Tip
(B) और (C) में वही अवयव हैं और (A) में (4) नहीं है। समानता और उचित उपसमुच्चय दोनों अलग-अलग जांचें।
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यदि किसी समुच्चय के (15) उचित उपसमुच्चय हैं तो उसके पावर समुच्चय में कितने अवयव होंगे?
If a set has (15) proper subsets, how many elements are in its power set?
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#proper_subsets
#power_set
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A (15)
B (16)
C (30)
D (31)
Explanation opens after your attempt
Step 1
Concept
From proper subsets \(2^n-1=15\), we get \(2^n=16\). A power set has \(2^n\) elements.
Step 2
Why this answer is correct
The correct answer is B. (16). From proper subsets \(2^n-1=15\), we get \(2^n=16\). A power set has \(2^n\) elements.
Step 3
Exam Tip
उचित उपसमुच्चय \(2^n-1=15\) से \(2^n=16\) मिलता है। पावर समुच्चय में कुल \(2^n\) अवयव होते हैं।
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यदि \(A={x:x\in\mathbb{N}\) और \(x\leq 6}\) तथा \(B={x:x\) (6) का धनात्मक भाजक है(}) हैं तो संबंध क्या है?
If \(A={x:x\in\mathbb{N}\) and \(x\leq 6}\) and \(B={x:x\) is a positive divisor of (6)(}), what is the relation?
#sets
#proper_subset
#divisors
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A (A=B)
B (B) (A) का उचित उपसमुच्चय है / (B) is a proper subset of (A)
C (A) (B) का उचित उपसमुच्चय है / (A) is a proper subset of (B)
D \(A\cap B=\varnothing\)
Explanation opens after your attempt
Correct Answer
B. (B) (A) का उचित उपसमुच्चय है / (B) is a proper subset of (A)
Step 1
Concept
\(A=\{1,2,3,4,5,6\}\) and \(B=\{1,2,3,6\}\). All elements of (B) are in (A), but (A) is larger.
Step 2
Why this answer is correct
The correct answer is B. (B) (A) का उचित उपसमुच्चय है / (B) is a proper subset of (A). \(A=\{1,2,3,4,5,6\}\) and \(B=\{1,2,3,6\}\). All elements of (B) are in (A), but (A) is larger.
Step 3
Exam Tip
\(A=\{1,2,3,4,5,6\}\) और \(B=\{1,2,3,6\}\) है। (B) के सभी अवयव (A) में हैं पर (A) बड़ा है।
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यदि \(A=\{{1},{2},{1,2}\}\) है तो (A) के अवयवों में कौन सा शामिल है?
If \(A=\{{1},{2},{1,2}\}\), which one is included as an element of (A)?
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#nested_sets
#elements
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A ({1,2})
B (1)
C (2)
D ({{1},{2}})
Explanation opens after your attempt
Correct Answer
A. ({1,2})
Step 1
Concept
The elements of (A) are written as sets, and ({1,2}) is one of them. The inner (1) and (2) are not directly elements of (A).
Step 2
Why this answer is correct
The correct answer is A. ({1,2}). The elements of (A) are written as sets, and ({1,2}) is one of them. The inner (1) and (2) are not directly elements of (A).
Step 3
Exam Tip
(A) में अवयव समुच्चय के रूप में लिखे हैं और ({1,2}) उनमें से एक है। अंदर के (1) और (2) सीधे (A) के अवयव नहीं हैं।
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यदि \(A={x:x\) (40) का धनात्मक भाजक है और \(x\leq 10}\) तो कौन सा विकल्प (A) के बराबर है?
If \(A={x:x\) is a positive divisor of (40) and \(x\leq 10}\), which option is equal to (A)?
#sets
#equal_sets
#divisors
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A ({1,2,4,5,8,10})
B ({1,2,4,5,8,10,20,40})
C ({2,4,5,8})
D ({10,20,40})
Explanation opens after your attempt
Correct Answer
A. ({1,2,4,5,8,10})
Step 1
Concept
The positive divisors of (40) up to (10) are (1,2,4,5,8,10). Apply the boundary condition to the divisor list.
Step 2
Why this answer is correct
The correct answer is A. ({1,2,4,5,8,10}). The positive divisors of (40) up to (10) are (1,2,4,5,8,10). Apply the boundary condition to the divisor list.
Step 3
Exam Tip
(40) के (10) तक के धनात्मक भाजक (1,2,4,5,8,10) हैं। सीमा की शर्त को भाजक सूची पर लगाएं।
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यदि \(A=\{1,2,3,4\}\) है तो (A) के ऐसे उपसमुच्चयों की संख्या क्या है जिनमें (2) और (3) दोनों हों?
If \(A=\{1,2,3,4\}\), how many subsets of (A) contain both (2) and (3)?
#sets
#subsets
#restricted_counting
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A (2)
B (4)
C (6)
D (8)
Explanation opens after your attempt
Step 1
Concept
(2) and (3) are fixed, while (1,4) are optional, so \(2^2=4\) subsets are obtained. Include fixed elements first.
Step 2
Why this answer is correct
The correct answer is B. (4). (2) and (3) are fixed, while (1,4) are optional, so \(2^2=4\) subsets are obtained. Include fixed elements first.
Step 3
Exam Tip
(2) और (3) निश्चित हैं तथा (1,4) स्वतंत्र हैं इसलिए \(2^2=4\) उपसमुच्चय मिलते हैं। निश्चित अवयवों को पहले शामिल करें।
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यदि \(A={x:x\in\mathbb{Z}\) और \(x^2-2x=0}\) तथा \(B=\{0,2\}\) हैं तो क्या सत्य है?
If \(A={x:x\in\mathbb{Z}\) and \(x^2-2x=0}\) and \(B=\{0,2\}\), what is true?
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A (A=B)
B \(A=\{2\}\)
C (B) (A) का उचित उपसमुच्चय है / (B) is a proper subset of (A)
D \(A=\varnothing\)
Explanation opens after your attempt
Step 1
Concept
(x-2 -2x=x(x-2)), so the roots are (0) and (2). Do not forget zero as a root.
Step 2
Why this answer is correct
The correct answer is A. (A=B). (x-2 -2x=x(x-2)), so the roots are (0) and (2). Do not forget zero as a root.
Step 3
Exam Tip
(x-2 -2x=x(x-2)) इसलिए हल (0) और (2) हैं। शून्य को हल मानने में गलती न करें।
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यदि \(A={x:x\) (1) से (20) तक की अभाज्य संख्याएं हैं(}) और \(B=\{2,3,5,7,11,13,17,19\}\) हैं तो निष्कर्ष क्या है?
If \(A={x:x\) is a prime number from (1) to (20)(}) and \(B=\{2,3,5,7,11,13,17,19\}\), what is the conclusion?
#sets
#equal_sets
#prime_numbers
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A (A=B)
B \(A\ne B\) क्योंकि (1) छूट गया / \(A\ne B\) because (1) is missing
C \(B\subset A\) पर \(A\ne B\) / \(B\subset A\) but \(A\ne B\)
D \(A=\{1,2,3,5,7,11,13,17,19\}\)
Explanation opens after your attempt
Step 1
Concept
The number (1) is not prime, and the primes up to (20) are exactly those in (B). Remember the definition of a prime number.
Step 2
Why this answer is correct
The correct answer is A. (A=B). The number (1) is not prime, and the primes up to (20) are exactly those in (B). Remember the definition of a prime number.
Step 3
Exam Tip
(1) अभाज्य नहीं है और (20) तक अभाज्य संख्याएं वही हैं जो (B) में हैं। अभाज्य संख्या की परिभाषा याद रखें।
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यदि (A) में (5) अवयव हैं तो (A) के ऐसे उपसमुच्चयों की संख्या क्या है जो (A) के बराबर नहीं हैं?
If (A) has (5) elements, how many subsets of (A) are not equal to (A)?
#sets
#proper_subsets
#counting
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A (16)
B (30)
C (31)
D (32)
Explanation opens after your attempt
Step 1
Concept
Subsets not equal to (A) are proper subsets, and their number is \(2^5-1=31\). Only the whole set is removed once.
Step 2
Why this answer is correct
The correct answer is C. (31). Subsets not equal to (A) are proper subsets, and their number is \(2^5-1=31\). Only the whole set is removed once.
Step 3
Exam Tip
बराबर नहीं होने वाले उपसमुच्चय ही उचित उपसमुच्चय हैं और उनकी संख्या \(2^5-1=31\) है। पूरा समुच्चय केवल एक बार हटता है।
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यदि \(A={x:x\in\mathbb{R}\) और \(x^2+1=0}\) है तो वास्तविक संख्याओं में (A) क्या है?
If \(A={x:x\in\mathbb{R}\) and \(x^2+1=0}\), what is (A) in real numbers?
#sets
#empty_set
#real_numbers
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A \(\varnothing\)
B ({1,-1})
C ({0})
D ({i,-i})
Explanation opens after your attempt
Correct Answer
A. \(\varnothing\)
Step 1
Concept
For no real (x), \(x^2+1=0\) is possible. The number system condition can change the answer.
Step 2
Why this answer is correct
The correct answer is A. \(\varnothing\). For no real (x), \(x^2+1=0\) is possible. The number system condition can change the answer.
Step 3
Exam Tip
किसी वास्तविक (x) के लिए \(x^2+1=0\) संभव नहीं है। संख्या पद्धति की शर्त उत्तर बदल सकती है।
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यदि \(A={x:x\) (30) से छोटे (5) के धनात्मक गुणज हैं(}) और \(B=\{5,10,15,20,25\}\) हैं तो संबंध क्या है?
If \(A={x:x\) is a positive multiple of (5) less than (30)(}) and \(B=\{5,10,15,20,25\}\), what is the relation?
#sets
#equal_sets
#multiples
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A (A=B)
B \(A=\{5,10,15,20,25,30\}\)
C (B) (A) का उचित उपसमुच्चय है / (B) is a proper subset of (A)
D \(A=\varnothing\)
Explanation opens after your attempt
Step 1
Concept
Multiples less than (30) do not include (30). Therefore (A) and (B) are equal.
Step 2
Why this answer is correct
The correct answer is A. (A=B). Multiples less than (30) do not include (30). Therefore (A) and (B) are equal.
Step 3
Exam Tip
(30) से छोटे गुणजों में (30) शामिल नहीं होता। इसलिए (A) और (B) समान हैं।
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यदि \(A=\{{1,2},3,4\}\) है तो निम्न में से कौन (A) का उपसमुच्चय नहीं है?
If \(A=\{{1,2},3,4\}\), which of the following is not a subset of (A)?
#sets
#nested_sets
#subset_error
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A ({3,4})
B ({{1,2}})
C ({{1,2},4})
D ({1,2})
Explanation opens after your attempt
Correct Answer
D. ({1,2})
Step 1
Concept
In (A), ({1,2}) is one element, but (1) and (2) are not separate elements. So ({1,2}) is not a subset.
Step 2
Why this answer is correct
The correct answer is D. ({1,2}). In (A), ({1,2}) is one element, but (1) and (2) are not separate elements. So ({1,2}) is not a subset.
Step 3
Exam Tip
(A) में ({1,2}) एक अवयव है पर (1) और (2) अलग से अवयव नहीं हैं। इसलिए ({1,2}) उपसमुच्चय नहीं है।
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यदि \(A={x:x\in\mathbb{N}\) और \(x\leq 3}\) तो (P(A)) में कितने ऐसे अवयव हैं जो एक-अवयवी समुच्चय हैं?
If \(A={x:x\in\mathbb{N}\) and \(x\leq 3}\), how many elements of (P(A)) are singleton sets?
#sets
#power_set
#singleton_subsets
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A (1)
B (2)
C (3)
D (8)
Explanation opens after your attempt
Step 1
Concept
Here \(A=\{1,2,3\}\), and it has (3) singleton subsets. These subsets become elements of the power set.
Step 2
Why this answer is correct
The correct answer is C. (3). Here \(A=\{1,2,3\}\), and it has (3) singleton subsets. These subsets become elements of the power set.
Step 3
Exam Tip
\(A=\{1,2,3\}\) है और इसके एक-अवयवी उपसमुच्चय (3) हैं। पावर समुच्चय में ये उपसमुच्चय अवयव बनते हैं।
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यदि \(A=\{2,3,4\}\), \(B=\{2,3,4,5\}\), और \(C=\{3,4\}\) हैं तो कौन सा कथन सही है?
If \(A=\{2,3,4\}\), \(B=\{2,3,4,5\}\), and \(C=\{3,4\}\), which statement is correct?
#sets
#subset_chain
#proper_subset
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A \(C\subset A\subset B\) उचित रूप से / \(C\subset A\subset B\) properly
B \(B\subset A\subset C\) उचित रूप से / \(B\subset A\subset C\) properly
C (A=C)
D \(C\not\subset B\)
Explanation opens after your attempt
Correct Answer
A. \(C\subset A\subset B\) उचित रूप से / \(C\subset A\subset B\) properly
Step 1
Concept
All elements of (C) are in (A), and all elements of (A) are in (B). Each step is proper because there is an extra element.
Step 2
Why this answer is correct
The correct answer is A. \(C\subset A\subset B\) उचित रूप से / \(C\subset A\subset B\) properly. All elements of (C) are in (A), and all elements of (A) are in (B). Each step is proper because there is an extra element.
Step 3
Exam Tip
(C) के सभी अवयव (A) में हैं और (A) के सभी अवयव (B) में हैं। प्रत्येक चरण में अतिरिक्त अवयव होने से संबंध उचित है।
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यदि \(A={x:x\in\mathbb{Z}\) और \(0<x^2<10}\) तो (A) के बराबर कौन सा समुच्चय है?
If \(A={x:x\in\mathbb{Z}\) and \(0<x^2<10}\), which set is equal to (A)?
#sets
#equal_sets
#square_inequality
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A ({-3,-2,-1,1,2,3})
B ({0,1,2,3})
C ({-2,-1,0,1,2})
D ({-4,-3,-2,-1,1,2,3,4})
Explanation opens after your attempt
Correct Answer
A. ({-3,-2,-1,1,2,3})
Step 1
Concept
\(x^2\) can be (1,4,9), so \(x=\pm1,\pm2,\pm3\). Since \(0<x^2\), (0) is not included.
Step 2
Why this answer is correct
The correct answer is A. ({-3,-2,-1,1,2,3}). \(x^2\) can be (1,4,9), so \(x=\pm1,\pm2,\pm3\). Since \(0<x^2\), (0) is not included.
Step 3
Exam Tip
\(x^2\) (1,4,9) हो सकता है इसलिए \(x=\pm1,\pm2,\pm3\) हैं। \(0<x^2\) होने से (0) शामिल नहीं है।
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यदि \(A=\{1,2,3,4\}\) और \(B=\{1,3\}\) हैं तो (A) के कितने उपसमुच्चय (B) को उपसमुच्चय के रूप में रखते हैं?
If \(A=\{1,2,3,4\}\) and \(B=\{1,3\}\), how many subsets of (A) contain (B) as a subset?
#sets
#subsets
#contain_subset
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A (2)
B (4)
C (8)
D (16)
Explanation opens after your attempt
Step 1
Concept
(1) and (3) are fixed, while (2,4) are optional, so there are \(2^2=4\) subsets. When a subset must be contained, fix its elements first.
Step 2
Why this answer is correct
The correct answer is B. (4). (1) and (3) are fixed, while (2,4) are optional, so there are \(2^2=4\) subsets. When a subset must be contained, fix its elements first.
Step 3
Exam Tip
(1) और (3) निश्चित हैं तथा (2,4) स्वतंत्र हैं इसलिए \(2^2=4\) उपसमुच्चय होंगे। किसी उपसमुच्चय को शामिल रखना हो तो उसके अवयव पहले तय करें।
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यदि \(A=\{2,4,6\}\) और \(B={x:x\) संख्या 8 से छोटी धनात्मक सम संख्या है(}), तो सही कथन कौन सा है?
If \(A=\{2,4,6\}\) and \(B={x:x\) is a positive even number less than 8(}), which statement is correct?
#sets
#equal sets
#subset
#class 11
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A (A=B)
B \(A\subset B\) लेकिन \(A\ne B\) / \(A\subset B\) but \(A\ne B\)
C \(B\subset A\) लेकिन \(A\ne B\) / \(B\subset A\) but \(A\ne B\)
D \(A\cap B=\varnothing\)
Explanation opens after your attempt
Step 1
Concept
Both sets have exactly the elements 2, 4 and 6. Order or description does not change a set.
Step 2
Why this answer is correct
The correct answer is A. (A=B). Both sets have exactly the elements 2, 4 and 6. Order or description does not change a set.
Step 3
Exam Tip
दोनों समुच्चयों के सभी अवयव 2, 4 और 6 हैं। क्रम या वर्णन बदलने से समुच्चय नहीं बदलता।
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यदि \(A={x:x\in \mathbb{Z}, x^2-4x+3=0}\) और \(B=\{1,3\}\), तो सही संबंध कौन सा है?
If \(A={x:x\in \mathbb{Z}, x^2-4x+3=0}\) and \(B=\{1,3\}\), which relation is correct?
#sets
#equal sets
#quadratic
#level 9
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A (A=B)
B \(A\subset B\) लेकिन \(A\ne B\) / \(A\subset B\) but \(A\ne B\)
C \(B\subset A\) लेकिन \(A\ne B\) / \(B\subset A\) but \(A\ne B\)
D \(A=\varnothing\)
Explanation opens after your attempt
Step 1
Concept
The equation gives ((x-1)(x-3)=0), so \(A=\{1,3\}\). Equal sets have the same elements, order is not important.
Step 2
Why this answer is correct
The correct answer is A. (A=B). The equation gives ((x-1)(x-3)=0), so \(A=\{1,3\}\). Equal sets have the same elements, order is not important.
Step 3
Exam Tip
समीकरण से ((x-1)(x-3)=0), इसलिए \(A=\{1,3\}\)। बराबर समुच्चय में अवयव वही होते हैं, क्रम जरूरी नहीं।
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समुच्चय \(A=\{1,1,2,3\}\) और \(B=\{3,2,1\}\) के लिए कौन सा निष्कर्ष सही है?
For the sets \(A=\{1,1,2,3\}\) and \(B=\{3,2,1\}\), which conclusion is correct?
#sets
#duplicate elements
#equal sets
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A \(A\ne B\) क्योंकि 1 दो बार लिखा है / \(A\ne B\) because 1 is written twice
B (A=B)
C \(A\subset B\) लेकिन \(B\not\subset A\) / \(A\subset B\) but \(B\not\subset A\)
D \(B\subset A\) लेकिन \(A\not\subset B\) / \(B\subset A\) but \(A\not\subset B\)
Explanation opens after your attempt
Step 1
Concept
Repeated elements are not counted in a set. In exams, first list the distinct elements.
Step 2
Why this answer is correct
The correct answer is B. (A=B). Repeated elements are not counted in a set. In exams, first list the distinct elements.
Step 3
Exam Tip
समुच्चय में किसी अवयव की पुनरावृत्ति नहीं गिनी जाती। परीक्षा में पहले वास्तविक अलग-अलग अवयव लिखें।
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यदि \(A=\{a,b,c\}\) और \(B=\{a,b,c,d\}\), तो (A) और (B) के बारे में सही कथन कौन सा है?
If \(A=\{a,b,c\}\) and \(B=\{a,b,c,d\}\), which statement about (A) and (B) is correct?
#proper subset
#sets
#class 11
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A (A=B)
B \(A\subset B\)
C \(B\subset A\)
D \(A\not\subset B\)
Explanation opens after your attempt
Correct Answer
B. \(A\subset B\)
Step 1
Concept
Every element of (A) is in (B), and (B) has one extra element. So (A) is a proper subset of (B).
Step 2
Why this answer is correct
The correct answer is B. \(A\subset B\). Every element of (A) is in (B), and (B) has one extra element. So (A) is a proper subset of (B).
Step 3
Exam Tip
(A) का हर अवयव (B) में है और (B) में एक अतिरिक्त अवयव है। इसलिए (A), (B) का उचित उपसमुच्चय है।
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यदि \(A=\{a,b,c,d,e\}\), तो ऐसे उपसमुच्चयों की संख्या कितनी है जिनमें (a) हो, (c) न हो और ठीक 3 अवयव हों?
If \(A=\{a,b,c,d,e\}\), how many subsets contain (a), do not contain (c), and have exactly 3 elements?
#subsets
#counting
#combination
#level 9
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A 2
B 3
C 4
D 6
Explanation opens after your attempt
Step 1
Concept
(a) is fixed and (c) is excluded, so choose 2 from (b,d,e). The number is \(\binom{3}{2}=3\).
Step 2
Why this answer is correct
The correct answer is B. 3. (a) is fixed and (c) is excluded, so choose 2 from (b,d,e). The number is \(\binom{3}{2}=3\).
Step 3
Exam Tip
(a) निश्चित है और (c) हटाया गया है, इसलिए (b,d,e) में से 2 चुनने हैं। संख्या \(\binom{3}{2}=3\) है।
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यदि ({x,5}={3,y}), तो निम्न में से कौन सा युग्म सही हो सकता है?
If ({x,5}={3,y}), which pair can be correct?
#equal sets
#variables
#reasoning
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A (x=3,y=5)
B (x=5,y=3)
C (x=3,y=3)
D (x=5,y=5)
Explanation opens after your attempt
Correct Answer
A. (x=3,y=5)
Step 1
Concept
Equal sets must contain exactly the same elements, so 3 and 5 must appear on both sides. Do not be confused by order.
Step 2
Why this answer is correct
The correct answer is A. (x=3,y=5). Equal sets must contain exactly the same elements, so 3 and 5 must appear on both sides. Do not be confused by order.
Step 3
Exam Tip
बराबर समुच्चयों में वही अवयव होने चाहिए, इसलिए 3 और 5 दोनों ओर होने चाहिए। ऐसे प्रश्नों में क्रम से भ्रमित न हों।
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यदि ({m,m+2}={4,6}), तो (m) का मान क्या है?
If ({m,m+2}={4,6}), what is the value of (m)?
#equal sets
#variable
#reasoning
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A 2
B 4
C 6
D 8
Explanation opens after your attempt
Step 1
Concept
For equal sets, the two elements must be exactly 4 and 6. Substituting (m=4) gives ({m,m+2}={4,6}).
Step 2
Why this answer is correct
The correct answer is B. 4. For equal sets, the two elements must be exactly 4 and 6. Substituting (m=4) gives ({m,m+2}={4,6}).
Step 3
Exam Tip
बराबर समुच्चय के लिए दोनों अवयव 4 और 6 ही होने चाहिए। (m=4) रखने पर ({m,m+2}={4,6}) मिलता है।
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समुच्चय \(A={x:x^2=4,\ x\in \mathbb{Z}}\) और \(B=\{-2,2\}\) के लिए सही कथन चुनिए।
Choose the correct statement for \(A={x:x^2=4,\ x\in \mathbb{Z}}\) and \(B=\{-2,2\}\).
#equal sets
#roster form
#set builder
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A (A=B)
B \(A=\{2\}\) इसलिए \(A\ne B\) / \(A=\{2\}\) so \(A\ne B\)
C \(B\subset A\) लेकिन \(A\not\subset B\) / \(B\subset A\) but \(A\not\subset B\)
D \(A=\varnothing\)
Explanation opens after your attempt
Step 1
Concept
In integers, the solutions of \(x^2=4\) are (-2) and 2. Both sets are equal.
Step 2
Why this answer is correct
The correct answer is A. (A=B). In integers, the solutions of \(x^2=4\) are (-2) and 2. Both sets are equal.
Step 3
Exam Tip
पूर्णांकों में \(x^2=4\) के हल (-2) और 2 हैं। दोनों समुच्चय बराबर हैं।
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यदि \(A={\varnothing,{2},2}\), तो निम्न में से कौन सा कथन सही है?
If \(A={\varnothing,{2},2}\), which of the following statements is correct?
#element vs subset
#empty set
#notation
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A \({2}\in A\) और \({2}\subset A\) दोनों सत्य हैं / Both \({2}\in A\) and \({2}\subset A\) are true
B \(2\notin A\)
C \(\varnothing\not\subset A\)
D \(A=\varnothing\)
Explanation opens after your attempt
Correct Answer
A. \({2}\in A\) और \({2}\subset A\) दोनों सत्य हैं / Both \({2}\in A\) and \({2}\subset A\) are true
Step 1
Concept
({2}) is an element of (A), and 2 is also in (A), so \({2}\subset A\) is also true. Keep the difference between \(\in\) and \(\subset\) clear.
Step 2
Why this answer is correct
The correct answer is A. \({2}\in A\) और \({2}\subset A\) दोनों सत्य हैं / Both \({2}\in A\) and \({2}\subset A\) are true. ({2}) is an element of (A), and 2 is also in (A), so \({2}\subset A\) is also true. Keep the difference between \(\in\) and \(\subset\) clear.
Step 3
Exam Tip
({2}) (A) का अवयव है और 2 भी (A) में है, इसलिए \({2}\subset A\) भी सत्य है। \(\in\) और \(\subset\) का अंतर ध्यान रखें।
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यदि \(A={x:x\) 12 का धनात्मक गुणनखंड है(}) और \(B=\{1,2,3,4,6,12\}\), तो क्या सत्य है?
If \(A={x:x\) is a positive factor of 12(}) and \(B=\{1,2,3,4,6,12\}\), what is true?
#factors
#equal sets
#set builder
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A (A=B)
B \(A\subset B\) लेकिन \(A\ne B\) / \(A\subset B\) but \(A\ne B\)
C \(B\subset A\) लेकिन \(A\ne B\) / \(B\subset A\) but \(A\ne B\)
D (A) अनंत है / (A) is infinite
Explanation opens after your attempt
Step 1
Concept
These six numbers are exactly the positive factors of 12. Set-builder form and roster form give the same set.
Step 2
Why this answer is correct
The correct answer is A. (A=B). These six numbers are exactly the positive factors of 12. Set-builder form and roster form give the same set.
Step 3
Exam Tip
12 के सभी धनात्मक गुणनखंड यही छह हैं। वर्णनात्मक रूप और सूची रूप एक ही समुच्चय दे रहे हैं।
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यदि \(A\subseteq B\), (A) और (B) दोनों सीमित समुच्चय हैं और (n(A)=n(B)=7), तो निष्कर्ष क्या होगा?
If \(A\subseteq B\), (A) and (B) are finite sets, and (n(A)=n(B)=7), what will be the conclusion?
#finite sets
#subset
#equal sets
#proof
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A (A=B)
B \(A\subset B\) और \(A\ne B\) / \(A\subset B\) and \(A\ne B\)
C \(B\subset A\) लेकिन \(A\ne B\) / \(B\subset A\) but \(A\ne B\)
D \(A\cap B=\varnothing\)
Explanation opens after your attempt
Step 1
Concept
For finite sets, if one is a subset of the other and both have the same number of elements, they are equal. This is a useful rule for proving equality.
Step 2
Why this answer is correct
The correct answer is A. (A=B). For finite sets, if one is a subset of the other and both have the same number of elements, they are equal. This is a useful rule for proving equality.
Step 3
Exam Tip
सीमित समुच्चयों में यदि एक दूसरे का उपसमुच्चय हो और दोनों में समान संख्या में अवयव हों, तो वे बराबर होते हैं। यह बराबरी सिद्ध करने का उपयोगी नियम है।
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यदि \(A=\{1,2,{3}\}\), तो कौन सा कथन सही है?
If \(A=\{1,2,{3}\}\), which statement is correct?
#element
#subset
#common mistake
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A \(3\in A\)
B \({3}\in A\)
C ({1,2,3}=A)
D \({3}\subset A\)
Explanation opens after your attempt
Correct Answer
B. \({3}\in A\)
Step 1
Concept
Here ({3}) itself is an element of (A), not 3. Keep element and subset ideas separate.
Step 2
Why this answer is correct
The correct answer is B. \({3}\in A\). Here ({3}) itself is an element of (A), not 3. Keep element and subset ideas separate.
Step 3
Exam Tip
यहां ({3}) स्वयं (A) का अवयव है, 3 नहीं। अवयव और उपसमुच्चय में अंतर ध्यान रखें।
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यदि \(A=\{1,2,3\}\), तो निम्न में से कौन सा (A) का उपसमुच्चय नहीं है?
If \(A=\{1,2,3\}\), which of the following is not a subset of (A)?
#subset
#not subset
#class 11
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A \(\varnothing\)
B ({1,3})
C ({2,4})
D ({1,2,3})
Explanation opens after your attempt
Correct Answer
C. ({2,4})
Step 1
Concept
4 is not an element of (A), so ({2,4}) is not a subset. Look for any outside element in the option.
Step 2
Why this answer is correct
The correct answer is C. ({2,4}). 4 is not an element of (A), so ({2,4}) is not a subset. Look for any outside element in the option.
Step 3
Exam Tip
4, (A) का अवयव नहीं है, इसलिए ({2,4}) उपसमुच्चय नहीं है। किसी भी विकल्प में बाहरी अवयव खोजें।
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किसी 5 अवयव वाले समुच्चय के कुल उपसमुच्चयों की संख्या क्या होगी?
What is the total number of subsets of a set having 5 elements?
#number of subsets
#power set
#hard
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A 10
B 25
C 32
D 120
Explanation opens after your attempt
Step 1
Concept
A set with (n) elements has \(2^n\) subsets. Here \(2^5=32\).
Step 2
Why this answer is correct
The correct answer is C. 32. A set with (n) elements has \(2^n\) subsets. Here \(2^5=32\).
Step 3
Exam Tip
(n) अवयवों वाले समुच्चय के उपसमुच्चय \(2^n\) होते हैं। यहां \(2^5=32\) है।
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यदि (A) के कुल उपसमुच्चय 64 हैं, तो (A) में कितने अवयव हैं?
If (A) has 64 subsets, how many elements does (A) have?
#power set
#subset count
#exam
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A 5
B 6
C 7
D 8
Explanation opens after your attempt
Step 1
Concept
Total subsets are \(2^n=64\), so (n=6). Recognising powers is the quick method.
Step 2
Why this answer is correct
The correct answer is B. 6. Total subsets are \(2^n=64\), so (n=6). Recognising powers is the quick method.
Step 3
Exam Tip
कुल उपसमुच्चय \(2^n=64\) हैं, इसलिए (n=6)। घातों को पहचानना तेज तरीका है।
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4 अवयव वाले समुच्चय के उचित उपसमुच्चयों की संख्या क्या है?
What is the number of proper subsets of a 4-element set?
#proper subset
#subset count
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A 15
B 16
C 8
D 14
Explanation opens after your attempt
Step 1
Concept
Total subsets are \(2^4=16\), and the set itself is not a proper subset. So the number is 15.
Step 2
Why this answer is correct
The correct answer is A. 15. Total subsets are \(2^4=16\), and the set itself is not a proper subset. So the number is 15.
Step 3
Exam Tip
कुल उपसमुच्चय \(2^4=16\) हैं और समुच्चय स्वयं उचित उपसमुच्चय नहीं है। इसलिए संख्या 15 है।
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यदि \(A=\varnothing\), तो (A) के उपसमुच्चयों की संख्या कितनी है?
If \(A=\varnothing\), how many subsets does (A) have?
#empty set
#subset count
#power set
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A 0
B 1
C 2
D अनंत / Infinite
Explanation opens after your attempt
Step 1
Concept
The empty set has exactly one subset, the empty set itself. Remember \(2^0=1\).
Step 2
Why this answer is correct
The correct answer is B. 1. The empty set has exactly one subset, the empty set itself. Remember \(2^0=1\).
Step 3
Exam Tip
रिक्त समुच्चय का एक ही उपसमुच्चय है, वह स्वयं रिक्त समुच्चय है। \(2^0=1\) याद रखें।
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यदि \(A=\{p,q,r,s\}\), तो (A) के 2 अवयव वाले उपसमुच्चयों की संख्या कितनी है?
If \(A=\{p,q,r,s\}\), how many 2-element subsets of (A) are there?
#subset count
#combination
#hard
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A 4
B 5
C 6
D 8
Explanation opens after your attempt
Step 1
Concept
The number of ways to choose 2 elements is \(\binom{4}{2}=6\). Order is not counted in subsets.
Step 2
Why this answer is correct
The correct answer is C. 6. The number of ways to choose 2 elements is \(\binom{4}{2}=6\). Order is not counted in subsets.
Step 3
Exam Tip
2 अवयव चुनने के तरीके \(\binom{4}{2}=6\) हैं। उपसमुच्चयों में क्रम नहीं गिना जाता।
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यदि \(A=\{1,2,3,4,5\}\), तो ऐसे उपसमुच्चयों की संख्या कितनी है जिनमें 1 अवश्य हो?
If \(A=\{1,2,3,4,5\}\), how many subsets must contain 1?
#subsets
#contains element
#counting
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A 8
B 16
C 24
D 32
Explanation opens after your attempt
Step 1
Concept
Fix 1, and each of the remaining 4 elements may be chosen or not. So there are \(2^4=16\) subsets.
Step 2
Why this answer is correct
The correct answer is B. 16. Fix 1, and each of the remaining 4 elements may be chosen or not. So there are \(2^4=16\) subsets.
Step 3
Exam Tip
1 को निश्चित रखें और शेष 4 अवयवों को चुनना या न चुनना स्वतंत्र है। इसलिए \(2^4=16\) उपसमुच्चय मिलते हैं।
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यदि \(A=\{1,2,3,4,5,6\}\), तो ऐसे उपसमुच्चयों की संख्या कितनी है जिनमें 2 नहीं हो?
If \(A=\{1,2,3,4,5,6\}\), how many subsets do not contain 2?
#subset counting
#excluded element
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A 16
B 32
C 48
D 64
Explanation opens after your attempt
Step 1
Concept
After excluding 2, 5 elements remain. Their subsets are \(2^5=32\).
Step 2
Why this answer is correct
The correct answer is B. 32. After excluding 2, 5 elements remain. Their subsets are \(2^5=32\).
Step 3
Exam Tip
2 को हटाने के बाद 5 अवयव बचते हैं। इनके सभी उपसमुच्चय \(2^5=32\) हैं।
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यदि \(A=\{a,b,c,d,e\}\), तो ऐसे उपसमुच्चयों की संख्या कितनी है जिनमें (a) और (b) दोनों हों?
If \(A=\{a,b,c,d,e\}\), how many subsets contain both (a) and (b)?
#subset counting
#fixed elements
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A 4
B 8
C 16
D 32
Explanation opens after your attempt
Step 1
Concept
(a) and (b) are fixed, while the other 3 elements are optional. Hence \(2^3=8\) subsets.
Step 2
Why this answer is correct
The correct answer is B. 8. (a) and (b) are fixed, while the other 3 elements are optional. Hence \(2^3=8\) subsets.
Step 3
Exam Tip
(a) और (b) निश्चित हैं, बाकी 3 अवयव स्वतंत्र हैं। इसलिए \(2^3=8\) उपसमुच्चय होंगे।
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यदि \(A=\{1,2,3,4\}\), तो ऐसे उपसमुच्चयों की संख्या कितनी है जिनमें ठीक 3 अवयव हों?
If \(A=\{1,2,3,4\}\), how many subsets have exactly 3 elements?
#exact size subset
#combination
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A 3
B 4
C 6
D 8
Explanation opens after your attempt
Step 1
Concept
The number of ways to choose exactly 3 elements is \(\binom{4}{3}=4\). Changing order does not create a new subset.
Step 2
Why this answer is correct
The correct answer is B. 4. The number of ways to choose exactly 3 elements is \(\binom{4}{3}=4\). Changing order does not create a new subset.
Step 3
Exam Tip
ठीक 3 अवयव चुनने के तरीके \(\binom{4}{3}=4\) हैं। क्रम बदलने से नया उपसमुच्चय नहीं बनता।
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यदि \(A={x:x\) 10 से छोटी अभाज्य संख्या है(}) और \(B=\{2,3,5,7\}\), तो कौन सा कथन सही है?
If \(A={x:x\) is a prime number less than 10(}) and \(B=\{2,3,5,7\}\), which statement is correct?
#prime numbers
#equal sets
#common mistake
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A (A=B)
B \(A\subset B\) और \(A\ne B\) / \(A\subset B\) and \(A\ne B\)
C \(B\subset A\) और \(A\ne B\) / \(B\subset A\) and \(A\ne B\)
D \(A=\{1,2,3,5,7\}\)
Explanation opens after your attempt
Step 1
Concept
The prime numbers less than 10 are 2, 3, 5 and 7. The number 1 is not prime.
Step 2
Why this answer is correct
The correct answer is A. (A=B). The prime numbers less than 10 are 2, 3, 5 and 7. The number 1 is not prime.
Step 3
Exam Tip
10 से छोटी अभाज्य संख्याएं 2, 3, 5 और 7 हैं। 1 अभाज्य नहीं है।
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यदि \(A={x:x^2-5x+6=0}\) और \(B=\{2,3\}\), तो सही कथन कौन सा है?
If \(A={x:x^2-5x+6=0}\) and \(B=\{2,3\}\), which statement is correct?
#quadratic
#equal sets
#set builder
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A (A=B)
B \(A=\{6\}\)
C \(B\subset A\) लेकिन \(A\ne B\) / \(B\subset A\) but \(A\ne B\)
D \(A=\varnothing\)
Explanation opens after your attempt
Step 1
Concept
The equation gives ((x-2)(x-3)=0), so the solutions are 2 and 3. The solution set equals (B).
Step 2
Why this answer is correct
The correct answer is A. (A=B). The equation gives ((x-2)(x-3)=0), so the solutions are 2 and 3. The solution set equals (B).
Step 3
Exam Tip
समीकरण ((x-2)(x-3)=0) देता है, इसलिए हल 2 और 3 हैं। हलों का समुच्चय (B) के बराबर है।
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यदि \(A=\{0,1\}\) और \(B={x:x^2=x}\), तो सही संबंध क्या है?
If \(A=\{0,1\}\) and \(B={x:x^2=x}\), what is the correct relation?
#equal sets
#solution set
#algebra
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A (A=B)
B \(A\subset B\) लेकिन \(A\ne B\) / \(A\subset B\) but \(A\ne B\)
C \(B=\{1\}\)
D \(A\cap B=\varnothing\)
Explanation opens after your attempt
Step 1
Concept
From \(x^2=x\), (x(x-1)=0), so (x=0) or (x=1). Both sets are equal.
Step 2
Why this answer is correct
The correct answer is A. (A=B). From \(x^2=x\), (x(x-1)=0), so (x=0) or (x=1). Both sets are equal.
Step 3
Exam Tip
\(x^2=x\) से (x(x-1)=0), इसलिए (x=0) या (x=1)। दोनों समुच्चय समान हैं।
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यदि \(A=\{1,2,3\}\), तो (P(A)) में कितने अवयव होंगे?
If \(A=\{1,2,3\}\), how many elements will (P(A)) have?
#power set
#subset
#count
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A 3
B 6
C 8
D 9
Explanation opens after your attempt
Step 1
Concept
(P(A)) contains all subsets of (A). For 3 elements, the number is \(2^3=8\).
Step 2
Why this answer is correct
The correct answer is C. 8. (P(A)) contains all subsets of (A). For 3 elements, the number is \(2^3=8\).
Step 3
Exam Tip
(P(A)) में (A) के सभी उपसमुच्चय होते हैं। 3 अवयवों के लिए संख्या \(2^3=8\) है।
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यदि \(A\subseteq B\), तो \(A\cup B\) किसके बराबर होगा?
If \(A\subseteq B\), then \(A\cup B\) is equal to what?
#subset
#union
#concept
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A (A)
B (B)
C \(\varnothing\)
D \(A\cap B\) हमेशा रिक्त / \(A\cap B\) always empty
Explanation opens after your attempt
Step 1
Concept
When every element of (A) is already in (B), union adds nothing new. Hence \(A\cup B=B\).
Step 2
Why this answer is correct
The correct answer is B. (B). When every element of (A) is already in (B), union adds nothing new. Hence \(A\cup B=B\).
Step 3
Exam Tip
जब (A) का हर अवयव (B) में है, तो मिलाने पर नया अवयव नहीं आता। इसलिए \(A\cup B=B\)।
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यदि \(A\subseteq B\), तो \(A\cap B\) किसके बराबर होगा?
If \(A\subseteq B\), then \(A\cap B\) is equal to what?
#subset
#intersection
#concept
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A (A)
B (B)
C \(\varnothing\)
D \(A\cup B\)
Explanation opens after your attempt
Step 1
Concept
Every element of (A) is in (B), so the common elements are exactly (A). This is an important result of subset relation.
Step 2
Why this answer is correct
The correct answer is A. (A). Every element of (A) is in (B), so the common elements are exactly (A). This is an important result of subset relation.
Step 3
Exam Tip
(A) का हर अवयव (B) में है, इसलिए साझा अवयव पूरे (A) ही हैं। यह उपसमुच्चय का महत्वपूर्ण परिणाम है।
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यदि \(A=\{1,2\}\), तो (P(A)) कौन सा है?
If \(A=\{1,2\}\), which is (P(A))?
#power set
#subset notation
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A \({\varnothing,1,2}\)
B \({\varnothing,{1},{2},{1,2}}\)
C ({{1},{2}})
D ({1,2,{1,2}})
Explanation opens after your attempt
Correct Answer
B. \({\varnothing,{1},{2},{1,2}}\)
Step 1
Concept
In a power set, all subsets are written as elements. Thus single elements appear as ({1}) and ({2}).
Step 2
Why this answer is correct
The correct answer is B. \({\varnothing,{1},{2},{1,2}}\). In a power set, all subsets are written as elements. Thus single elements appear as ({1}) and ({2}).
Step 3
Exam Tip
पावर सेट में सभी उपसमुच्चय अवयव के रूप में लिखे जाते हैं। इसलिए एकल अवयव भी ({1}) और ({2}) रूप में होंगे।
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यदि \(A=\varnothing\) और \(B=\{0\}\), तो सही कथन कौन सा है?
If \(A=\varnothing\) and \(B=\{0\}\), which statement is correct?
#empty set
#proper subset
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A (A=B)
B \(A\subset B\)
C \(0\in A\)
D \(B\subset A\)
Explanation opens after your attempt
Correct Answer
B. \(A\subset B\)
Step 1
Concept
The empty set is a subset of every set, and (B) is not empty. Hence (A) is a proper subset.
Step 2
Why this answer is correct
The correct answer is B. \(A\subset B\). The empty set is a subset of every set, and (B) is not empty. Hence (A) is a proper subset.
Step 3
Exam Tip
रिक्त समुच्चय हर समुच्चय का उपसमुच्चय है और (B) रिक्त नहीं है। इसलिए (A) उचित उपसमुच्चय है।
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यदि \(A={x:x\) 6 का गुणज है और (x<20)(}) तथा \(B=\{6,12,18\}\), तो कौन सा संबंध सही है?
If \(A={x:x\) is a multiple of 6 and (x<20)(}) and \(B=\{6,12,18\}\), which relation is correct?
#multiples
#equal sets
#set builder
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A (A=B)
B \(A\subset B\) लेकिन \(A\ne B\) / \(A\subset B\) but \(A\ne B\)
C \(B\subset A\) लेकिन \(A\ne B\) / \(B\subset A\) but \(A\ne B\)
D (A) अनंत है / (A) is infinite
Explanation opens after your attempt
Step 1
Concept
The positive multiples of 6 less than 20 are 6, 12 and 18. So the two sets are equal.
Step 2
Why this answer is correct
The correct answer is A. (A=B). The positive multiples of 6 less than 20 are 6, 12 and 18. So the two sets are equal.
Step 3
Exam Tip
20 से छोटे धनात्मक 6 के गुणज 6, 12 और 18 हैं। इसलिए दोनों समुच्चय समान हैं।
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यदि \(A=\{1,2,3,4\}\) और \(B=\{2,4\}\), तो कौन सा कथन गलत है?
If \(A=\{1,2,3,4\}\) and \(B=\{2,4\}\), which statement is false?
#proper subset
#false statement
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A \(B\subset A\)
B \(B\subseteq A\)
C \(A\subseteq A\)
D \(A\subset B\)
Explanation opens after your attempt
Correct Answer
D. \(A\subset B\)
Step 1
Concept
(A) is a subset of itself, but it is not a proper subset of itself. Also \(A\subset B\) is false here.
Step 2
Why this answer is correct
The correct answer is D. \(A\subset B\). (A) is a subset of itself, but it is not a proper subset of itself. Also \(A\subset B\) is false here.
Step 3
Exam Tip
(A) स्वयं (A) का उपसमुच्चय है, लेकिन अपना उचित उपसमुच्चय नहीं है। इसलिए \(A\subset B\) भी असत्य है।
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यदि \(A=\{1,3,5,7\}\) और \(B={x:x\) 8 से छोटी धनात्मक विषम संख्या है(}), तो (A) और (B) कैसे संबंधित हैं?
If \(A=\{1,3,5,7\}\) and \(B={x:x\) is a positive odd number less than 8(}), how are (A) and (B) related?
#odd numbers
#equal sets
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A (A=B)
B \(A\subset B\) लेकिन \(A\ne B\) / \(A\subset B\) but \(A\ne B\)
C \(B\subset A\) लेकिन \(A\ne B\) / \(B\subset A\) but \(A\ne B\)
D \(A\cap B=\varnothing\)
Explanation opens after your attempt
Step 1
Concept
The positive odd numbers less than 8 are 1, 3, 5 and 7. The list and rule give the same set.
Step 2
Why this answer is correct
The correct answer is A. (A=B). The positive odd numbers less than 8 are 1, 3, 5 and 7. The list and rule give the same set.
Step 3
Exam Tip
8 से छोटी धनात्मक विषम संख्याएं 1, 3, 5 और 7 हैं। सूची और नियम दोनों समान समुच्चय देते हैं।
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यदि \(A=\{2,3,5,7,11\}\), तो (A) के ऐसे उपसमुच्चयों की संख्या कितनी है जिनमें 2 हो लेकिन 11 न हो?
If \(A=\{2,3,5,7,11\}\), how many subsets contain 2 but not 11?
#subset counting
#conditions
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A 4
B 8
C 16
D 32
Explanation opens after your attempt
Step 1
Concept
2 is fixed and 11 is excluded, while the remaining 3 elements are optional. Hence \(2^3=8\) subsets.
Step 2
Why this answer is correct
The correct answer is B. 8. 2 is fixed and 11 is excluded, while the remaining 3 elements are optional. Hence \(2^3=8\) subsets.
Step 3
Exam Tip
2 निश्चित है और 11 निषिद्ध है, शेष 3 अवयव स्वतंत्र हैं। इसलिए \(2^3=8\) उपसमुच्चय हैं।
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यदि \(A=\{a,b,c,d\}\), तो (A) के उन उपसमुच्चयों की संख्या कितनी है जिनमें कम से कम एक अवयव हो?
If \(A=\{a,b,c,d\}\), how many subsets of (A) have at least one element?
#nonempty subsets
#counting
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A 15
B 16
C 8
D 4
Explanation opens after your attempt
Step 1
Concept
Total subsets are \(2^4=16\). For at least one element, remove the empty set, giving 15.
Step 2
Why this answer is correct
The correct answer is A. 15. Total subsets are \(2^4=16\). For at least one element, remove the empty set, giving 15.
Step 3
Exam Tip
कुल उपसमुच्चय \(2^4=16\) हैं। कम से कम एक अवयव के लिए रिक्त समुच्चय हटाएं, इसलिए 15।
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यदि \(A=\{1,2,3,4,5\}\), तो (A) के ऐसे उपसमुच्चयों की संख्या कितनी है जिनमें ठीक 2 अवयव हों और 5 न हो?
If \(A=\{1,2,3,4,5\}\), how many subsets have exactly 2 elements and do not contain 5?
#exact subset
#counting
#combination
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A 4
B 6
C 8
D 10
Explanation opens after your attempt
Step 1
Concept
Exclude 5 and choose 2 elements from ({1,2,3,4}). The number is \(\binom{4}{2}=6\).
Step 2
Why this answer is correct
The correct answer is B. 6. Exclude 5 and choose 2 elements from ({1,2,3,4}). The number is \(\binom{4}{2}=6\).
Step 3
Exam Tip
5 को छोड़कर ({1,2,3,4}) से 2 अवयव चुनने हैं। संख्या \(\binom{4}{2}=6\) है।
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यदि \(A={x:x\in \mathbb{N}, 2x+1<10}\), तो (A) किसके बराबर है?
If \(A={x:x\in \mathbb{N}, 2x+1<10}\), what is (A) equal to?
#set builder
#inequality
#equal sets
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A ({1,2,3,4})
B ({0,1,2,3,4})
C ({1,2,3,4,5})
D ({2,3,4})
Explanation opens after your attempt
Correct Answer
A. ({1,2,3,4})
Step 1
Concept
From (2x+1<10), we get (x<4.5). The natural numbers are (1,2,3,4).
Step 2
Why this answer is correct
The correct answer is A. ({1,2,3,4}). From (2x+1<10), we get (x<4.5). The natural numbers are (1,2,3,4).
Step 3
Exam Tip
(2x+1<10) से (x<4.5) मिलता है। प्राकृतिक संख्याएं (1,2,3,4) हैं।
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यदि \(A=\{1,2,{1,2}\}\), तो निम्न में से कौन सा कथन सही है?
If \(A=\{1,2,{1,2}\}\), which of the following is correct?
#element
#subset
#advanced notation
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A \({1,2}\subset A\) और \({1,2}\in A\) दोनों सत्य हैं / Both \({1,2}\subset A\) and \({1,2}\in A\) are true
B \({1,2}\subset A\) सत्य है लेकिन \({1,2}\in A\) असत्य है / \({1,2}\subset A\) is true but \({1,2}\in A\) is false
C \({1,2}\in A\) सत्य है लेकिन \({1,2}\subset A\) असत्य है / \({1,2}\in A\) is true but \({1,2}\subset A\) is false
D दोनों असत्य हैं / Both are false
Explanation opens after your attempt
Correct Answer
A. \({1,2}\subset A\) और \({1,2}\in A\) दोनों सत्य हैं / Both \({1,2}\subset A\) and \({1,2}\in A\) are true
Step 1
Concept
Since (1) and (2) are in (A), \({1,2}\subset A\), and ({1,2}) is also an element. This is a good mixed notation example.
Step 2
Why this answer is correct
The correct answer is A. \({1,2}\subset A\) और \({1,2}\in A\) दोनों सत्य हैं / Both \({1,2}\subset A\) and \({1,2}\in A\) are true. Since (1) and (2) are in (A), \({1,2}\subset A\), and ({1,2}) is also an element. This is a good mixed notation example.
Step 3
Exam Tip
(1) और (2) (A) में हैं, इसलिए \({1,2}\subset A\), और ({1,2}) भी एक अवयव है। यह मिश्रित notation का अच्छा उदाहरण है।
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यदि \(A=\{1,2,3\}\), तो निम्न में से कौन सा (P(A)) का अवयव है?
If \(A=\{1,2,3\}\), which of the following is an element of (P(A))?
#power set
#element
#subset
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A 1
B ({1,2})
C ({4})
D ({1,4})
Explanation opens after your attempt
Correct Answer
B. ({1,2})
Step 1
Concept
Elements of (P(A)) are subsets of (A). ({1,2}) is a subset of (A).
Step 2
Why this answer is correct
The correct answer is B. ({1,2}). Elements of (P(A)) are subsets of (A). ({1,2}) is a subset of (A).
Step 3
Exam Tip
(P(A)) के अवयव (A) के उपसमुच्चय होते हैं। ({1,2}), (A) का उपसमुच्चय है।
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यदि \(A=\{1,2,3,4,5,6\}\), तो (A) के सम संख्या वाले सभी अवयवों का समुच्चय (B) क्या है?
If \(A=\{1,2,3,4,5,6\}\), what is the set (B) of all even elements of (A)?
#subset
#even numbers
#selection
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A ({1,3,5})
B ({2,4,6})
C ({2,4})
D ({1,2,3,4,5,6})
Explanation opens after your attempt
Correct Answer
B. ({2,4,6})
Step 1
Concept
The even elements in (A) are 2, 4 and 6. Thus \(B\subset A\) and \(B=\{2,4,6\}\).
Step 2
Why this answer is correct
The correct answer is B. ({2,4,6}). The even elements in (A) are 2, 4 and 6. Thus \(B\subset A\) and \(B=\{2,4,6\}\).
Step 3
Exam Tip
(A) में सम अवयव 2, 4 और 6 हैं। इसलिए \(B\subset A\) और \(B=\{2,4,6\}\)।
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यदि \(A={x:x\) 15 का धनात्मक भाजक है(}) और \(B=\{1,3,5,15\}\), तो कौन सा कथन सही है?
If \(A={x:x\) is a positive divisor of 15(}) and \(B=\{1,3,5,15\}\), which statement is correct?
#divisors
#equal sets
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A (A=B)
B \(A\subset B\) लेकिन \(A\ne B\) / \(A\subset B\) but \(A\ne B\)
C \(B\subset A\) लेकिन \(A\ne B\) / \(B\subset A\) but \(A\ne B\)
D \(A=\{3,5\}\)
Explanation opens after your attempt
Step 1
Concept
The positive divisors of 15 are 1, 3, 5 and 15. Therefore the sets are equal.
Step 2
Why this answer is correct
The correct answer is A. (A=B). The positive divisors of 15 are 1, 3, 5 and 15. Therefore the sets are equal.
Step 3
Exam Tip
15 के धनात्मक भाजक 1, 3, 5 और 15 हैं। इसलिए दोनों समुच्चय बराबर हैं।
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यदि \(A=\{a,b,c\}\) और \(B=\{b,c,d\}\), तो कौन सा कथन सही है?
If \(A=\{a,b,c\}\) and \(B=\{b,c,d\}\), which statement is correct?
#not subset
#equal sets
#comparison
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A (A=B)
B \(A\subset B\)
C \(B\subset A\)
D न तो \(A\subseteq B\) और न ही \(B\subseteq A\) / Neither \(A\subseteq B\) nor \(B\subseteq A\)
Explanation opens after your attempt
Correct Answer
D. न तो \(A\subseteq B\) और न ही \(B\subseteq A\) / Neither \(A\subseteq B\) nor \(B\subseteq A\)
Step 1
Concept
(a) is not in (B), and (d) is not in (A). Hence neither set is a subset of the other.
Step 2
Why this answer is correct
The correct answer is D. न तो \(A\subseteq B\) और न ही \(B\subseteq A\) / Neither \(A\subseteq B\) nor \(B\subseteq A\). (a) is not in (B), and (d) is not in (A). Hence neither set is a subset of the other.
Step 3
Exam Tip
(a), (B) में नहीं है और (d), (A) में नहीं है। इसलिए कोई भी दूसरे का उपसमुच्चय नहीं है।
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यदि \(A=\{1,2,3,4\}\) और \(B=\{2,3\}\), तो (A-B) के आधार पर कौन सा कथन सही है?
If \(A=\{1,2,3,4\}\) and \(B=\{2,3\}\), which statement based on (A-B) is correct?
#set difference
#subset
#application
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A (A-B={1,4}) और यह (A) का उपसमुच्चय है / (A-B={1,4}) and it is a subset of (A)
B (A-B={2,3}) और यह (B) का उपसमुच्चय है / (A-B={2,3}) and it is a subset of (B)
C (A-B=A)
D \(A-B=\varnothing\)
Explanation opens after your attempt
Correct Answer
A. (A-B={1,4}) और यह (A) का उपसमुच्चय है / (A-B={1,4}) and it is a subset of (A)
Step 1
Concept
(A-B) contains elements of (A) that are not in (B). Here they are 1 and 4.
Step 2
Why this answer is correct
The correct answer is A. (A-B={1,4}) और यह (A) का उपसमुच्चय है / (A-B={1,4}) and it is a subset of (A). (A-B) contains elements of (A) that are not in (B). Here they are 1 and 4.
Step 3
Exam Tip
(A-B) में (A) के वे अवयव हैं जो (B) में नहीं हैं। यहां वे 1 और 4 हैं।
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कथन पर विचार करें: हर समुच्चय स्वयं का उपसमुच्चय है। इसका सही मूल्य क्या है?
Consider the statement: Every set is a subset of itself. What is its truth value?
#subset property
#reflexive
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A सत्य / True
B असत्य / False
C केवल रिक्त समुच्चय के लिए सत्य / True only for empty set
D केवल सीमित समुच्चय के लिए सत्य / True only for finite sets
Explanation opens after your attempt
Correct Answer
A. सत्य / True
Step 1
Concept
Every element of a set is in the same set, so \(A\subseteq A\) is true. Remember it like a reflexive property.
Step 2
Why this answer is correct
The correct answer is A. सत्य / True. Every element of a set is in the same set, so \(A\subseteq A\) is true. Remember it like a reflexive property.
Step 3
Exam Tip
हर अवयव अपने ही समुच्चय में होता है, इसलिए \(A\subseteq A\) सत्य है। इसे reflexive property की तरह याद रखें।
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कथन पर विचार करें: यदि \(A\subset B\), तो \(A\ne B\)। यह कथन कैसा है?
Consider the statement: If \(A\subset B\), then \(A\ne B\). What is the nature of this statement?
#proper subset
#notation
#true false
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A सत्य / True
B असत्य / False
C केवल अनंत समुच्चयों के लिए सत्य / True only for infinite sets
D केवल रिक्त समुच्चय के लिए असत्य / False only for empty set
Explanation opens after your attempt
Correct Answer
A. सत्य / True
Step 1
Concept
\(A\subset B\) denotes a proper subset, so equality is not possible. Distinguish \(\subset\) from \(\subseteq\).
Step 2
Why this answer is correct
The correct answer is A. सत्य / True. \(A\subset B\) denotes a proper subset, so equality is not possible. Distinguish \(\subset\) from \(\subseteq\).
Step 3
Exam Tip
\(A\subset B\) उचित उपसमुच्चय को दर्शाता है, इसलिए बराबरी संभव नहीं। प्रतीक \(\subset\) और \(\subseteq\) में अंतर रखें।
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यदि \(A=\{1,2,3,4\}\), तो (A) के ऐसे उपसमुच्चयों की संख्या कितनी है जिनमें 1 हो या 2 हो?
If \(A=\{1,2,3,4\}\), how many subsets contain 1 or 2?
#subset counting
#or condition
#hard
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A 8
B 12
C 14
D 16
Explanation opens after your attempt
Step 1
Concept
There are 16 total subsets, and those containing neither 1 nor 2 are the \(2^2=4\) subsets of ({3,4}). Hence (16-4=12).
Step 2
Why this answer is correct
The correct answer is B. 12. There are 16 total subsets, and those containing neither 1 nor 2 are the \(2^2=4\) subsets of ({3,4}). Hence (16-4=12).
Step 3
Exam Tip
कुल 16 उपसमुच्चय हैं और जिनमें न 1 है न 2, वे ({3,4}) के \(2^2=4\) उपसमुच्चय हैं। इसलिए (16-4=12)।
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यदि \(A=\{1,2,3,4,5\}\), तो (A) के ऐसे उपसमुच्चयों की संख्या कितनी है जिनमें 1 हो और 2 न हो?
If \(A=\{1,2,3,4,5\}\), how many subsets contain 1 and do not contain 2?
#subset counting
#and condition
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A 4
B 8
C 16
D 24
Explanation opens after your attempt
Step 1
Concept
1 is fixed and 2 is excluded, while the remaining 3 elements are optional. Thus \(2^3=8\) subsets.
Step 2
Why this answer is correct
The correct answer is B. 8. 1 is fixed and 2 is excluded, while the remaining 3 elements are optional. Thus \(2^3=8\) subsets.
Step 3
Exam Tip
1 निश्चित है और 2 नहीं लेना है, शेष 3 अवयव स्वतंत्र हैं। इसलिए \(2^3=8\) उपसमुच्चय होंगे।
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यदि \(A={x:x\in \mathbb{N}, x\le 4}\) और \(B={x:x\) 5 से छोटी प्राकृतिक संख्या है(}), तो कौन सा कथन सही है?
If \(A={x:x\in \mathbb{N}, x\le 4}\) and \(B={x:x\) is a natural number less than 5(}), which statement is correct?
#equal sets
#natural numbers
#inequality
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A (A=B)
B \(A\subset B\) लेकिन \(A\ne B\) / \(A\subset B\) but \(A\ne B\)
C \(B\subset A\) लेकिन \(A\ne B\) / \(B\subset A\) but \(A\ne B\)
D \(A=\varnothing\)
Explanation opens after your attempt
Step 1
Concept
Both descriptions give ({1,2,3,4}). Do not be confused by different forms of inequalities.
Step 2
Why this answer is correct
The correct answer is A. (A=B). Both descriptions give ({1,2,3,4}). Do not be confused by different forms of inequalities.
Step 3
Exam Tip
दोनों वर्णन ({1,2,3,4}) देते हैं। असमानता के अलग रूपों से भ्रमित न हों।
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यदि \(A=\{1,2,3\}\) और \(B=\{1,2,3,4,5\}\), तो (B) के ऐसे उपसमुच्चयों की संख्या कितनी है जो (A) को अवश्य समाहित करते हैं?
If \(A=\{1,2,3\}\) and \(B=\{1,2,3,4,5\}\), how many subsets of (B) must contain (A)?
#subset containing set
#counting
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A 2
B 4
C 8
D 16
Explanation opens after your attempt
Step 1
Concept
1, 2 and 3 are fixed, while 4 and 5 are optional. Hence \(2^2=4\) subsets.
Step 2
Why this answer is correct
The correct answer is B. 4. 1, 2 and 3 are fixed, while 4 and 5 are optional. Hence \(2^2=4\) subsets.
Step 3
Exam Tip
1, 2 और 3 निश्चित हैं, जबकि 4 और 5 वैकल्पिक हैं। इसलिए \(2^2=4\) उपसमुच्चय होंगे।
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