A-LevelMathematicsNumerical methods9709/33 · Oct/Nov 2019 · Q57 Marks

By sketching a suitable pair of graphs, show that the equation ln(x + 2) = 4e⁻ˣ has exactly one real root.

(i)By sketching a suitable pair of graphs, show that the equation ln(x + 2) = 4e⁻ˣ has exactly one real root.[2]
(ii)Show by calculation that this root lies between x = 1 and x = 1.5.[2]
(iii)Use the iterative formula x_(n+1) = ln(4 / ln(x_n + 2)) to determine the root 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 2019). The mark or mark type is shown in brackets.

[B1] Sketch a relevant graph, e.g. y = ln(x+2)
[B1] Sketch a second relevant graph, e.g. y = 4e⁻ˣ, and justify the given statement
[M1] Calculate the values of a relevant expression or pair of expressions at x = 1 and x = 1.5
[A1] Complete the argument correctly with correct calculated values
[M1] Use the iterative formula correctly at least twice using output from a previous iteration
[A1] Obtain final answer 1.23
[A1] Show sufficient iterations to 4 d.p. to justify 1.23 to 2 d.p., or show there is a sign change in the interval (1.225, 1.235)

On Oracle Prep you can write or photograph your own answer and have it marked by AI against this mark scheme, with feedback on what you missed.