A-LevelMathematicsNumerical methods9709/31 · Oct/Nov 2020 · Q55 Marks
By sketching a suitable pair of graphs, show that the equation cosec x = 1 + e¯½ˣ has exactly two roots in the interval 0 < x < π.
- (a)By sketching a suitable pair of graphs, show that the equation cosec x = 1 + e¯½ˣ has exactly two roots in the interval 0 < x < π.[2]
- (b)The sequence of values given by the iterative formula Xn+1 = π - sin⁻¹(1/(e^(x_n/2) + 1)), with initial value x₁ = 2, converges to one of these roots. Use the formula to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.[3]
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Mark scheme
Marking points from the official mark scheme (Oct/Nov 2020). The mark or mark type is shown in brackets.
- Sketch a relevant graph, e.g. y = cosec x
- Sketch a second relevant graph, e.g. y=1+ e^(-x/2), and justify the given statement
- Use the iterative formula correctly at least twice
- Obtain final answer 2.28
- Show sufficient iterations to at least 4 d.p. to justify 2.28 to 2 d.p., or show there is a sign change in the interval (2.275, 2.285)
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