Match List-I with List-II
| List-I | List-II | ||
|---|---|---|---|
| A. | The fourth proportional to 4, 9, 12 | I. | 0.12 |
| B. | The mean proportional between 0.08 and 0.18 | II. | \(2\sqrt{3}\) |
| C. | The third proportional to 16 and 36 | III. | 27 |
| D. | The mean proportional between \(6+3\sqrt{3}\) and \(8-4\sqrt{3}\) | IV. | 81 |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → A-III, B-I, C-IV, D-II
The three kinds of proportional come from the same idea, equal ratios:
Fourth proportional to \(a,b,c\): the \(x\) with \(a:b=c:x\), so \(x=\frac{bc}{a}\)
Mean proportional between \(a\) and \(b\): the \(x\) with \(a:x=x:b\), so \(x=\sqrt{ab}\)
Third proportional to \(a\) and \(b\): the \(x\) with \(a:b=b:x\), so \(x=\frac{b^2}{a}\)
(A) The fourth proportional to 4, 9, 12 — this matches (III) 27.
\(\displaystyle x=\frac{9\times12}{4}=27\)
(B) The mean proportional between 0.08 and 0.18 — this matches (I) 0.12.
\(x=\sqrt{0.08\times0.18}=\sqrt{0.0144}=0.12\)
(C) The third proportional to 16 and 36 — this matches (IV) 81.
\(\displaystyle x=\frac{36^2}{16}=\frac{1296}{16}=81\)
(D) The mean proportional between \(6+3\sqrt{3}\) and \(8-4\sqrt{3}\) — this matches (II) \(2\sqrt{3}\).
Product \(=(6+3\sqrt{3})(8-4\sqrt{3})=48-24\sqrt{3}+24\sqrt{3}-36=12\)
\(x=\sqrt{12}=2\sqrt{3}\)