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A sonometer wire of total length 114 cm is fixed at both ends. Two bridges divide it into three segments whose fundamental frequencies are in the ratio 1: 3: 4. If the length of the largest segment is 12x cm, find the value of x.

1. 4
2. 5
3. 6
4. 8

Correct answer: 6

Solution

For a string under uniform tension, f is proportional to 1/L. So if f1: f2: f3 = 1: 3: 4, then L1: L2: L3 = 1: 1/3: 1/4. Multiplying by 12: L1: L2: L3 = 12: 4: 3. Sum = 19 parts = 114 cm, so 1 part = 6 cm. L1 = 72 cm, L2 = 24 cm, L3 = 18 cm. The largest segment is 72 cm. Given largest = 12x, so 12x = 72, giving x = 6.

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