October 2020 Paper 2 Q10

OCR MEICurrent spec9 marksNumerical Methods Radians

10 In this question you must show detailed reasoning.

The equation of a curve is

\(y = \dfrac{\sin 2x - x}{x\sin x}\).

(a) Use the small angle approximation given in the list of formulae on pages 2–3 of this question paper to show that
\(\displaystyle\int_{0.01}^{0.05} y\,\mathrm{d}x \approx \ln 5\). [4]

(b) Use the same small angle approximation to show that
\(\dfrac{\mathrm{d}y}{\mathrm{d}x} \approx -10000\) at the point where \(x = 0.01\). [2]

The equation \(y = 0\) has a root near \(x = 1\). Joan uses the Newton-Raphson method to find this root. The output from the spreadsheet she uses is shown in Fig. 10.1.

\(n\)	0	1	2	3	4	5	6	7
\(x_n\)	1	0.958509	0.950084	0.948261	0.94786	0.947772	0.947753	0.947748

Fig. 10.1

Joan carries out some analysis of this output. The results are shown in Fig. 10.2.

\(x\)	\(y\)
0.9477475	–7.79967E–07
0.9477485	–2.90821E–06
\(x\)	\(y\)
0.947745	4.54066E–06
0.947755	–1.67417E–05

Fig. 10.2

Write 4.54066E–06 in standard mathematical notation.
State the value of the root as accurately as you can, justifying your answer.

[3]

Mark scheme (a)

Scheme	Marks	AO
\(\sin 2x \approx 2x\) or \(\sin x \approx x\) used	M1	3.1a
\(\displaystyle\int \left(\frac{1}{x}\right)\mathrm{d}x\) or \(\displaystyle\int \left(\frac{1}{x} - x\right)\mathrm{d}x\) obtained oe nfww	A1	1.1
\(\mathrm{F}[x] = \ln x\) oe or \(\mathrm{F}[x] = \ln x - \frac{1}{2}x^2\) oe	A1	1.1
\(\ln(0.05) - \ln(0.01) = \ln 5\) oe or \(\ln(0.05) - \ln(0.01) - 0.0012 \approx \ln 5\) oe	A1	3.2a
	[4]

Notes

M1: may see \(\cos x \approx 1 - \frac{x^2}{2}\)

A1: intermediate step needed from here to earn final mark

(corrected from the printed mark scheme: the last line is printed as \(\ln(0.05) - \ln(0.01) + 0.0012 \approx \ln 5\); with \(\mathrm{F}[x] = \ln x - \frac{1}{2}x^2\) the 0.0012 is subtracted)

Mark scheme (b)

Scheme	Marks	AO
differentiation of their \(\frac{1}{x}\)	M1	2.1
substitution of 0.01 and \(-10\,000\) correctly obtained	A1	1.1
	[2]

Notes

M1: or differentiation of \(y\) using quotient rule and use of small angle approximation

A1: from \(-\frac{1}{x^2}\) or \(-\frac{1}{x^2} - 1\) oe

Mark scheme (c)

Scheme	Marks	AO
\(4.54066 \times 10^{-6}\) or 0.00000454066 cao	B1	2.5
(no sign change for 6 dp), but sign change for 5 dp or last two iterates agree to 5dp	E1	3.1a
0.94775	B1	3.2a
	[3]

Notes

E1: allow sign change between 0.947745 and 0.9477475