A-LevelMathematicsNumerical methods9709/21 · May/June 2025 · Q37 Marks

Sketch, on a single diagram, the graphs of y = 3e^-2x and y = secx for values of x such that 0 < x < π/2.

(a)Sketch, on a single diagram, the graphs of y = 3e^-2x and y = secx for values of x such that 0 < x < π/2.[2]
(b)Show that the x-coordinate of the point of intersection of the two graphs satisfies the equation x = ½ ln(3cosx).[2]
(c)Use an iterative formula, based on the equation in part (b), to find the x-coordinate of the point of intersection correct to 3 decimal places. Give the result of each iteration to 5 decimal places.[3]

Mark scheme

Marking points from the official mark scheme (May/June 2025). The mark or mark type is shown in brackets.

[B1] Sketch decreasing positive curve for y = 3e^-2x.
[B1] Sketch curve for y = secx (Correctly placed with reference to first sketch or correct with '1' marked on y-axis).
[M1] Equate and arrange at least as far as e^-2x =... or e^2x =... with cosx present.
[A1] Confirm x = ½ ln(3cosx) (AG - necessary detail needed).
[M1] Use iterative process correctly at least once (Calculator must be in radian mode).
[A1] Obtain final answer 0.487 (Required to precisely 3 decimal places).
[A1] Show sufficient iterations to justify answer, or show a sign change in the interval [0.4865, 0.4875] (Allow iterations to greater accuracy).

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