A-LevelMathematicsNumerical methods9709/31 · Oct/Nov 2021 · Q810 Marks

The constant a is such that ∫ᵃ₁ (lnx / √x) dx = 6. [exp(x) is an alternative notation for eˣ.]

(a)Show that a = exp( (1/√a) + 2).[5]
(b)Verify by calculation that a lies between 9 and 11.[2]
(c)Use an iterative formula based on the equation in part (a) 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 (Oct/Nov 2021). The mark or mark type is shown in brackets.

[*M1] Commence integration and reach `a√x lnx + b√x . 1/x dx`, or equivalent
[A1] Obtain `2√x lnx - 2√x . 1/x dx`, or equivalent
[A1] Obtain integral `2√x lnx - 4√x`, or equivalent
[DM1] Substitute limits and equate result to 6
[A1] Rearrange and obtain `a = exp( (1/√a) + 2)`
[M1] Calculate the values of a relevant expression or pair of expressions at a = 9 and a = 11
[A1] Complete the argument correctly with correct values
[M1] Use the iterative process `a_n+1 = exp((1/√a_n) + 2)` correctly at least once
[A1] Obtain answer 10.12
[A1] Show sufficient iterations to 4dp to justify 10.12 to 2dp, or show there is a sign change in the interval (10.115, 10.125)

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