By sketching a suitable pair of graphs, show that the equation √x = e^x - 3 has only one root.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2023). The mark or mark type is shown in brackets.

1. [B1] (a) Sketch a relevant graph, e.g. y = e^x – 3 Should cut vertical axis at (0, -2) and have increasing gradient.
2. [B1] (a) Sketch a second relevant graph, e.g. y = √x and justify the given statement y = √x should start at (0, 0) and have reducing grading For second B1 need to mark intersection with a dot, a cross, or say root at point of intersection, or equivalent.
3. [M1] (b) Calculate the values of a relevant expression or pair of expressions at x = 1 and x = 2
4. [A1] (b) Complete the argument correctly with correct calculated values e.g. 1 -0.28.., 1.41 < 4.39.. 1.28 > 0, -2.98 < 0.
5. [B1] (c) State x = ln(3+√x) and rearrange to the given equation √x = e^x - 3 AG Or rearrange √x = e^x - 3 to x = ln(3 + √x) and state iterative formula of x_(n+1) = ln(3+√x_n).
6. [M1] (d) Use the iterative process correctly at least once
7. [A1] (d) Obtain final answer 1.43
8. [A1] (d) Show sufficient iterations to at least 4 d.p. to justify 1.43 to 2 d.p. or show there is a sign change in the interval (1.425, 1.435) Condone recovery and small differences in the final figure in the iteration e.g. 1, 1.3864, 1.4297, 1.4341, ... 1.5, 1.4210, 1.4332, 14.344, 1.4345, ... 2, 1.4848, 1.4395, 1.4350, 1.4346, 1.4345, ...