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A vessel of capacity 286 litres, has “n” taps, numbered from 1 to N fitted to it. At the start of the nth minute, where, 1≤ n ≤ N, the tap numbered n, which fills the tank at the rate of n litres per minute, is opened. If it takes exactly N minutes to fill the tank in this manner, then, what is the value of N?
1. 10
2. 12
3. 9
4. 11

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Correct Answer - Option 4 : 11

During the 1st minute, only tap 1 would be opened, so 1 litre will be filled by it.

During the 2nd minute, taps 1 and 2 would be opened, so the taps 1 and 2 will fill 1 litre and 2 litres, respectively.

∴ (1 + 2) litres will be filled.

Proceeding in this manner, during the nth minute, n(n+1)/2 litre would be filled.

Now adding ∑n from 1 onwards to N –

⇒ 1 + (1 + 2) + (1+ 2 + 3) + ……… + (1 + 2 + 3 + 4 + …..N) = 286

Or, ∑n(n + 1)/2 = [N(N+1)(N +2)]/6

∴ [N(N + 1)(N +2)]/6 = 286

Now, from options –

Option (1):

If N = 10

[10 × 11 × 12]/6 = 220 ≠ 286

Option (2):

If N = 12

[12 × 13 × 14]/6 = 364 ≠ 286

Option (3):

If N = 9

[9 × 10 × 11]/6 = 165 ≠ 286

Option (4):

If N = 11

[11 × 12 × 13]/6 = 286

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