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### Question

A tank is filled in 6 minutes by three taps running together. Times taken by the three taps independently are in an AP[Arithmetic Progression], whose first term is a and common difference d. Then, a and d satisfy the relation?

**A**

a^{3} - 18a^{2} - ad^{2} + 6d^{2} = 0.

**B**

a^{3} - 12a^{2} + ad^{2} + 6d^{2} = 0.

**C**

a^{3} - 6a^{2} - ad^{2} + 6d^{2} = 0.

**D**

a^{3} - 30a^{2} + ad^{2} + 6d^{2} = 0.

**Soln.**

**Ans: a**

Let the times taken by the three taps be a - d, a and a + d. Then 6 minutes work of all the taps should add to 1. So we have, $6 × 1/{a - d} + 6 × 1/a + 6 × 1/{a + d}$ = 1, which is same as a^{3} - 18a^{2} - ad^{2} + 6d^{2} = 0.