Practice focused topic-wise MCQs with answers and explanations for quick revision and exam preparation.
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Showing 391-400 of 588 questions.
If \(U=\mathbb{R}\) and (A=\(-\infty,2]\), what is (A')?
A. (\(2,\infty\))
(2) (A) में शामिल है, इसलिए पूरक में (2) नहीं आएगा। \(\mathbb{R}\) में पूरक (\(2,\infty\)) है। / (2) is included in (A), so it is not in the complement. In \(\mathbb{R}\), the complement is (\(2,\infty\)).
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A. (10)
(a) को हटाने पर (5) तत्व बचते हैं। उनमें से ठीक (3) चुनने के \(\binom{5}{3}=10\) तरीके हैं। / After excluding (a), (5) elements remain. Choosing exactly (3) of them gives \(\binom{5}{3}=10\) ways.
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If \(U=\{1,2,3,4,5,6,7,8\}\), (A'={1,4,7}) and (B'={2,4,8}), what is \(A\cap B\)?
A. ({3,5,6})
(A\cap B=\(A'\cup B'\)') है। \(A'\cup B'={1,2,4,7,8}\), इसलिए पूरक ({3,5,6}) है। / (A\cap B=\(A'\cup B'\)'). Since \(A'\cup B'={1,2,4,7,8}\), the complement is ({3,5,6}).
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If \(A=\{1,2,3,4,5\}\), how many elements of (\mathcal{P}(A)) have an even sum?
C. (16)
(A) में कम से कम एक विषम संख्या है, इसलिए उपसमुच्चयों के योग सम और विषम बराबर संख्या में होंगे। कुल (32) हैं, अतः सम योग वाले (16) हैं। / Since (A) has at least one odd number, subsets with even and odd sums are equal in number. Total subsets are (32), so even-sum subsets are (16).
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B. \(2^{45}\)
(10) के गुणज (10,20,30,40,50) हैं, इसलिए (|A|=5)। (A') में (45) तत्व हैं, अतः (|\mathcal{P}(A')|=2^{45})। / The multiples of (10) are (10,20,30,40,50), so (|A|=5). (A') has (45) elements, hence (|\mathcal{P}(A')|=2^{45}).
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B. (4)
({1,2,3}) के उपसमुच्चय में (1) नहीं होना चाहिए। केवल (2,3) से बनने वाले \(2^2=4\) उपसमुच्चय मिलेंगे। / The subset of ({1,2,3}) must not contain (1). Subsets formed only from (2,3) give \(2^2=4\) choices.
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A. (U) में (1) और संयुक्त संख्याएँ(1) and composite numbers in (U)
(A) में (U) की अभाज्य संख्याएँ हैं। इसलिए पूरक में (1) और संयुक्त संख्याएँ होंगी। / (A) contains the prime numbers of (U). Therefore the complement contains (1) and composite numbers.
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B. (4)
(A'={e,f,g,h}) और (B'={a,b,g,h}) हैं। केवल एक में आने वाले तत्व (a,b,e,f) हैं, इसलिए (4) तत्व हैं। / (A'={e,f,g,h}) and (B'={a,b,g,h}). Elements in exactly one are (a,b,e,f), so there are (4).
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C. (32)
(1,2,3) में से कम से कम दो चुनने के \(\binom{3}{2}+\binom{3}{3}=4\) तरीके हैं। शेष तीन तत्व स्वतंत्र हैं, इसलिए \(4\times2^3=32\) है। / Choosing at least two from (1,2,3) gives \(\binom{3}{2}+\binom{3}{3}=4\) ways. The remaining three elements are free, so the count is \(4\times2^3=32\).
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If (n(\mathcal{P}(A))=64) and (U) has (10) elements, how many elements are in (A')?
A. (4)
क्योंकि \(2^{n(A)}=64\), इसलिए (n(A)=6) और (n(A')=10-6=4)। परीक्षा में पहले आधार समुच्चय को पहचानें। / Since \(2^{n(A)}=64\), (n(A)=6), so (n(A')=10-6=4). In exams, identify the universal set first.
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