A-LevelMathematicsNumerical methods9709/22 · Feb/Mar 2022 · Q712 Marks

A curve has equation e^(2xy) – e^y = 100.

(a)Show that dy/dx = (2e^(2xy) y) / (e^y – e^(2x)).[3]
(b)Show that the curve has no stationary points.[2]
(c)Show that the x-coordinate of P satisfies the equation x = ln 10 – ½ ln(2x – 1).[4]
(d)Use an iterative formula, based on the equation in part (c), to find the x-coordinate of P correct to 3 significant figures. Use an initial value of 2 and give the result of each iteration to 5 significant figures.[3]

Mark scheme

Marking points from the official mark scheme (Feb/Mar 2022). The mark or mark type is shown in brackets.

[M1] Use product rule to differentiate e^(2xy)
[A1] Obtain 2e^(2xy) y + e^(2xy) dy/dx - e^y dy/dx = 0 and rearrange to confirm given result
[A1] AG – necessary detail needed
[M1] Consider 2e^(2x)y = 0 and either state e^(2x) ≠ 0 or substitute y = 0 in equation of curve
[A1] Complete argument with e^(2x) ≠ 0 or e^(2x) > 0 and substitution to show y cannot be zero [AG – necessary detail needed]
[B1] State or imply e^y – e^(2x) = 0 and hence y = 2x
[M1] Substitute for y in equation of curve and attempt rearrangement as far as e^(2x) = ...
[M1] Use relevant logarithm properties
[A1] Confirm equation x = ln 10 - ½ ln(2x – 1) [AG – necessary detail needed]
[M1] Use iteration process correctly at least once
[A1] Obtain final answer 1.82 [Answer required to exactly 3 sf]
[A1] Show sufficient iterations to 5 sf to justify answer or show sign change in interval [1.815, 1.825]

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