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Six numbers a, b, c, d, e, f are such that ab=1, bc=1/2, cd=6, de=2 and ef=1/2. What is the value of (ad:be:cf)?
1. 4:3:27
2. 6:1:9
3. 8:9:9
4. 72:1:9
छह संख्याएँ a, b, c, d, e, f इस प्रकार हैं कि ab=1, bc=1/2, cd=6, de=2 और ef=1/2. (ad:be:cf) का मान क्या है?
1. 4:3:27
2. 6:1:9
3. 8:9:9
4. 72:1:9
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UGC-NET-Sociology-16-January-2025-Shift-1-Q13
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We are given the equations:
ab = 1
bc = 1/2
cd = 6
de = 2
ef = 1/2
We need to find the ratio (ad : be : cf).
Step 1: Express each variable in terms of a
From ab = 1, we get b = 1/a
From bc = 1/2, substituting b: (1/a) * c = 1/2 → c = a/2
From cd = 6, substituting c: (a/2) * d = 6 → d = 12/a
From de = 2, substituting d: (12/a) * e = 2 → e = a/6
From ef = 1/2, substituting e: (a/6) * f = 1/2 → f = 3/a
Step 2: Compute ad, be, cf
ad = a * (12/a) = 12
be = (1/a) * (a/6) = 1/6
cf = (a/2) * (3/a) = 3/2
Step 3: Find the ratio (ad : be : cf)
(12 : 1/6 : 3/2)
Multiply all terms by 6 to eliminate fractions:
(72 : 1 : 9)
हमें समीकरण दिए गए हैं:
ab = 1
bc = 1/2
cd = 6
de = 2
ef = 1/2
हमें अनुपात (ad : be : cf) ज्ञात करना है।
चरण 1: प्रत्येक चर को a के पदों में व्यक्त करें
ab = 1 से, हमें b = 1/a मिलता है
bc = 1/2 से, b प्रतिस्थापित करने पर: (1/a) * c = 1/2 → c = a/2
cd = 6 से, c प्रतिस्थापित करने पर: (a/2) * d = 6 → d = 12/a
de = 2 से, d प्रतिस्थापित करने पर: (12/a) * e = 2 → e = a/6
ef = 1/2 से, e प्रतिस्थापित करने पर: (a/6) * f = 1/2 → f = 3/a
चरण 2: ad, be, cf की गणना करें
ad = a * (12/a) = 12
be = (1/a) * (a/6) = 1/6
cf = (a/2) * (3/a) = 3/2
चरण 3: अनुपात ज्ञात करें (ad : be : cf)
(12 : 1/6 : 3/2)
भिन्नों को हटाने के लिए सभी पदों को 6 से गुणा करें:
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