A-LevelMathematicsSeries9709/12 · May/June 2016 · Q99 Marks

A water tank holds 2000 litres when full. A small hole in the base is gradually getting bigger so that each day a greater amount of water is lost.

(i)(a)On the first day after filling, 10 litres of water are lost and this increases by 2 litres each day. How many litres will be lost on the 30th day after filling?[2]
(i)(b)The tank becomes empty during the nth day after filling. Find the value of n.[3]
(ii)Assume instead that 10 litres of water are lost on the first day and that the amount of water lost increases by 10% on each succeeding day. Find what percentage of the original 2000 litres is left in the tank at the end of the 30th day after filling.[4]

Mark scheme

Marking points from the official mark scheme (May/June 2016). The mark or mark type is shown in brackets.

[M1] a+(n-1)d = 10 + 29×2 Use of nth term of an AP with a=±10, d=±2, n=30 or 29
[A1] = 68 Condone – 68 → 68
[M1] ½n(20+ 2(n-1)) = 2000 or 0 Use of S_n formula for an AP with a=±10, d=±2 and equated to either 0 or 2000.
[A1] → 2n² + 18n - 4000 = 0 oe Correct 3 term quadratic = 0.
[A1] (n=) 41
[B1] r = 1.1, oe e.g. 11/10, 110%
[M1] Uses S₃₀ = 10(1.1³⁰-1) / (1.1-1) (= 1645) Use of S_n formula for a GP, a=±10, n=30.
[DM1] Percentage lost = (2000-1645)/2000 * 100 Fully correct method for % left with "their 1645"
[A1] = 17.75 allow 17.7 or 17.8.

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