A-LevelMathematicsNumerical methods9709/22 · May/June 2018 · Q611 Marks

It is given that ∫ (1 + e^(x/2))² dx from 0 to a = 10, where a is a positive constant.

(i)Show that a = 2ln((15-a)/(4 + e^(a/2)))[6]
(ii)Use the equation in part (i) to show by calculation that 1.5 < a < 1.6.[2]
(iii)Use an iterative formula based on the equation in part (i) to find the value of a correct to 3 significant figures. Give the result of each iteration to 5 significant figures.[3]

Mark scheme

Marking points from the official mark scheme (May/June 2018). The mark or mark type is shown in brackets.

[B1] Rewrite integrand as 1 + 2e^(x/2) + e^x
[M1] Integrate to obtain form x + k₁e^(x/2) + k₂e^x
[A1] Obtain x + 4e^(x/2) + e^x
[A1] Use limits to obtain a + 4e^(a/2) + e^a - 5 = 10
[M1] Rearrange as far as e^(a/2) = ... including use of 4e^(a/2) + e^a = e^(a/2)(4 + e^(a/2))
[A1] AG; necessary detail needed Confirm a = 2ln((15-a)/(4+e^(a/2)))
[M1] Consider sign of a - 2ln((15-a)/(4+e^(a/2))) for 1.5 and 1.6 or equivalent
[A1] Obtain -0.08... and 0.06... or equivalents and justify conclusion
[M1] Use iterative process correctly at least once
[A1] Obtain final answer 1.56
[A1] Show sufficient iterations to 5 sf to justify answer or show sign change in interval (1.555,1.565)

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