A-LevelMathematicsNumerical methods9709/31 · May/June 2024 · Q69 Marks

By sketching a suitable pair of graphs, show that the equation cosec(1/2)x = eˣ - 3 has exactly one root, denoted by α, in the interval 0 < x < π.

(a)By sketching a suitable pair of graphs, show that the equation cosec(1/2)x = eˣ - 3 has exactly one root, denoted by α, in the interval 0 < x < π.[2]
(b)Verify by calculation that α lies between 1 and 2.[2]
(c)Show that if a sequence of values in the interval 0 < x < π given by the iterative formula x_(n+1) = ln(cosec(1/2)x_n + 3) converges, then it converges to α.[1]
(d)Use this iterative formula with an initial value of 1.4 to determine α correct to 2 decimal places. Give the result of each iteration to 4 decimal places.[3]
(e)State the minimum number of calculated iterations needed with this initial value to determine α correct to 2 decimal places.[1]

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Mark scheme

Marking points from the official mark scheme (May/June 2024). The mark or mark type is shown in brackets.

[B1] Sketch a relevant graph, e.g. y = eˣ – 3. Correct shape, correct vertical intercept.
[B1] Sketch a second relevant graph, e.g. y = cosec(x/2) (correct shape, minimum above the axis) and justify the given statement. Need to mark intersection with a dot, a cross, or say root at points of intersection, or equivalent
[M1] Calculate the values of a relevant expression or pair of expressions at x = 1 and x = 2
[A1] Complete the argument correctly with correct calculated values
[B1] State x = ln(cosec(x/2)+3) and rearrange to the given equation cosec(x/2) = eˣ – 3. Or vice versa and obtain the iterative formula.
[M1] Use the iterative formula correctly at least twice
[A1] Obtain final answer 1.50
[A1] Show sufficient iterations to 4 dp to justify 1.50 to 4 dp or show there is a sign change in the interval (1.495, 1.505)
[B1] 4

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