A-LevelMathematicsNumerical methods9709/31 · May/June 2018 · Q810 Marks

The positive constant a is such that ∫(from 0 to a) x e^(-x/2) dx = 2.

(i)Show that a satisfies the equation a = 2ln(a + 2).[5]
(ii)Verify by calculation that a lies between 3 and 3.5.[2]
(iii)Use an iteration based on the equation in part (i) to determine a correct to 2 decimal places. Give the result of each iteration to 4 decimal places.[3]

Mark scheme

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

[M1*] Integrate by parts and reach [xe^(-x/2)] + m ∫e^(-x/2) dx
[A1] Obtain -2xe^(-x/2) + 2∫e^(-x/2) dx
[A1] Complete the integration and obtain -2xe^(-x/2) - 4e^(-x/2), or equivalent
[M1(dep*)] Having integrated twice, use limits and equate result to 2
[A1] Obtain the given equation correctly
[M1] Calculate values of a relevant expression or pair of expressions at a = 3 and a = 3.5
[A1] Complete the argument correctly with correct calculated values
[M1] Use the iterative formula a_(n+1) = 2ln(a_n + 2) correctly at least once
[A1] Obtain final answer 3.36
[A1] Show sufficient iterations to 4 d.p. to justify 3.36 to 2 d.p., or show there is a sign change in the interval (3.355, 3.365)

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