October 2020 Paper 3 Q5

OCR MEICurrent spec11 marksNumerical MethodsTrigonometry

5 Fig. 5 shows part of the curve \(y = \operatorname{cosec} x\) together with the \(x\)- and \(y\)-axes.

Fig. 5: one U-shaped branch of y = cosec x lying above the x-axis to the right of the y-axis, with a minimum point between two vertical asymptotes
Fig. 5
(a) For the section of the curve which is shown in Fig. 5, write down
(i) the equations of the two vertical asymptotes, [2]
(ii) the coordinates of the minimum point. [1]
(b) Show that the equation \(x = \operatorname{cosec} x\) has a root which lies between \(x = 1\) and \(x = 2\). [2]
(c) Use the iteration \(x_{n+1} = \operatorname{cosec}(x_n)\), with \(x_0 = 1\), to find
(i) the values of \(x_1\) and \(x_2\), correct to 5 decimal places, [1]
(ii) this root of the equation, correct to 3 decimal places. [1]
(d) There is another root of \(x = \operatorname{cosec} x\) which lies between \(x = 2\) and \(x = 3\).
Determine whether the iteration \(x_{n+1} = \operatorname{cosec}(x_n)\) with \(x_0 = 2.5\) converges to this root. [1]
(e) Sketch the staircase or cobweb diagram for the iteration, starting with \(x_0 = 2.5\), on the diagram in the Printed Answer Booklet. [3]

Diagram from the Printed Answer Booklet:

Printed Answer Booklet diagram: the branch of y = cosec x and the line y = x through O, which cuts the curve twice, once to the left of the minimum and once on the steep right-hand side
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