Practice focused topic-wise MCQs with answers and explanations for quick revision and exam preparation.
Difficulty select karke sirf is topic ke focused MCQs practice karein. Score, timer aur explanations student-friendly flow me milenge.
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C. (6)
\(2^n=2^{12-n}\), इसलिए (n=12-n) और (n=6)। परीक्षा में समान base के exponents बराबर करें। / \(2^n=2^{12-n}\), so (n=12-n) and (n=6). In exams, equate exponents when the bases are equal.
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If (|U|=15) and (|\mathcal{P}(A)|=32), where \(A\subseteq U\), what is (|\mathcal{P}(A')|)?
B. \(2^{10}\)
(|\mathcal{P}(A)|=32=25), इसलिए (|A|=5) और (|A'|=10)। अतः (|\mathcal{P}(A')|=2^{10})। / (|\mathcal{P}(A)|=32=25), so (|A|=5) and (|A'|=10). Hence (|\mathcal{P}(A')|=2^{10}).
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If \(A=\{1,2,3,4,5\}\), how many subsets in (\mathcal{P}(A)) have cardinality greater than (2)?
A. (16)
कुल \(2^5=32\) subsets हैं और sizes (0,1,2) वाले (1+5+10=16) हैं, इसलिए उत्तर (16) है। परीक्षा में complement counting करें। / Total subsets are \(2^5=32\), and those of sizes (0,1,2) are (1+5+10=16), so the answer is (16). In exams, use complement counting.
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If \(A=\{1,2,3,4,5,6,7\}\), how many subsets in (\mathcal{P}(A)) have at least (5) elements?
B. (29)
संख्या \(\binom{7}{5}+\binom{7}{6}+\binom{7}{7}=21+7+1=29\) है। परीक्षा में at least का अर्थ सभी बड़े sizes जोड़ना है। / The number is \(\binom{7}{5}+\binom{7}{6}+\binom{7}{7}=21+7+1=29\). In exams, at least means adding all larger sizes.
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If \(A=\{1,2,3,4,5,6\}\), among the (3)-element subsets in (\mathcal{P}(A)), how many contain (1)?
B. (10)
(1) fixed है, इसलिए बाकी (2) तत्व (5) में से चुने जाएंगे: \(\binom{5}{2}=10\)। परीक्षा में compulsory element के बाद remaining choice गिनें। / (1) is fixed, so the remaining (2) elements are chosen from (5): \(\binom{5}{2}=10\). In exams, count remaining choices after fixing the compulsory element.
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A. (10)
(1) हटाने पर (5) तत्व बचते हैं, जिनमें से (3) चुनने हैं: \(\binom{5}{3}=10\)। परीक्षा में excluded element को पहले हटा दें। / After removing (1), (5) elements remain and (3) must be chosen: \(\binom{5}{3}=10\). In exams, remove the excluded element first.
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B. (16)
(A) fixed है और (U-A) के (4) तत्व optional हैं, इसलिए \(2^4=16\)। परीक्षा में containing (A) का अर्थ \(A\subseteq S\) लें। / (A) is fixed and the (4) elements of (U-A) are optional, so \(2^4=16\). In exams, containing (A) means \(A\subseteq S\).
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B. (16)
(A') से disjoint subsets केवल (A) के subsets होंगे, इसलिए संख्या \(2^4=16\) है। परीक्षा में disjoint from complement का अर्थ subset of original set समझें। / Subsets disjoint from (A') are exactly subsets of (A), so the number is \(2^4=16\). In exams, understand disjoint from complement as subset of the original set.
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If \(U=\{1,2,3,4,5,6,7,8,9,10\}\) and \(A=\{2,3,5,7\}\), what is (|\mathcal{P}(A')|)?
C. (64)
यहां (A') में (6) तत्व हैं, इसलिए (|\mathcal{P}(A')|=26=64)। परीक्षा में complement हमेशा (U) के सापेक्ष निकालें। / Here (A') has (6) elements, so (|\mathcal{P}(A')|=26=64). In exams, always find complement with respect to (U).
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If (|U|=14), \(A\subseteq U\), and (|\mathcal{P}(A)|=64), what is the value of (|\mathcal{P}(A')|)?
C. \(2^8\)
(|\mathcal{P}(A)|=64=26), इसलिए (|A|=6) और (|A'|=8)। अतः (|\mathcal{P}(A')|=28)। / Since (|\mathcal{P}(A)|=64=26), (|A|=6) and (|A'|=8). Hence (|\mathcal{P}(A')|=28).
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