A2 October 2021 Paper 1 Q2

EdexcelCurrent spec7 marksDifferentiation & Maclaurin Integration

(a) Use the Maclaurin series expansion for \(\cos x\) to determine the series expansion of \(\cos^2\left(\dfrac{x}{3}\right)\) in ascending powers of \(x\), up to and including the term in \(x^4\)
Give each term in simplest form. (2)

(b) Use the answer to part (a) and calculus to find an approximation, to 5 decimal places, for\[\int_{\frac{\pi}{6}}^{\frac{\pi}{2}}\left(\frac{1}{x}\cos^2\left(\frac{x}{3}\right)\right)\mathrm{d}x\] (3)

(c) Use the integration function on your calculator to evaluate\[\int_{\frac{\pi}{6}}^{\frac{\pi}{2}}\left(\frac{1}{x}\cos^2\left(\frac{x}{3}\right)\right)\mathrm{d}x\]Give your answer to 5 decimal places. (1)

(d) Assuming that the calculator answer in part (c) is accurate to 5 decimal places, comment on the accuracy of the approximation found in part (b). (1)

Mark scheme (a)

Scheme	Marks	AO
\(\cos^2\dfrac{x}{3} = \left(1 - \dfrac{\left(\frac{x}{3}\right)^2}{2} + \dfrac{\left(\frac{x}{3}\right)^4}{24} - \ldots\right)^2 = \ldots\) or \(\left(1 - \dfrac{x^2}{18} + \dfrac{x^4}{1944} - \ldots\right)^2 = \ldots\) or \(\dfrac{1}{2}\left(1 \pm \cos\dfrac{2x}{3}\right) = \dfrac{1}{2}\left(1 \pm \left(1 - \dfrac{1}{2}\left(\dfrac{2x}{3}\right)^2 + \dfrac{1}{4!}\left(\dfrac{2x}{3}\right)^4 - \ldots\right)\right)\)	M1	2.2a
\(= 1 - \dfrac{x^2}{9} + \dfrac{1}{243}x^4\)	A1	1.1b
	(2)

Notes

M1: Deduces the required series by using the Maclaurin series for \(\cos x\), replacing \(x\) with \(\dfrac{x}{3}\) and squares, or first applying the double angle identity (allow sign error) and then applying the series for \(\cos x\) with \(\dfrac{2x}{3}\). Attempts at finding from differentiation score M0 as the cosine series is required.

A1: Correct series

Mark scheme (b)

Scheme	Marks	AO
\(\displaystyle\int\frac{1 - \frac{x^2}{9} + \frac{1}{243}x^4}{x} = \int\frac{1}{x} - \frac{x}{9} + \frac{1}{243}x^3 = A\ln x + Bx^2 + Cx^4\) where \(A\), \(B\) and \(C \neq 0\)	M1	3.1a
\(\ln x - \dfrac{x^2}{18} + \dfrac{1}{972}x^4\)	A1ft	1.1b
= awrt 0.98295	A1	2.2a
	(3)

Notes

M1: Divides their series in part (a) by \(x\) and integrates to the form \(A\ln x + Bx^2 + Cx^4\)

A1ft: Correct integration, follow through on their coefficients and need not be simplified.

A1: Deduces the definite integral awrt 0.98295

Scheme	Marks	AO
Calculator = awrt 0.98280	B1	1.1b
	(1)

Mark scheme (d)

Scheme	Marks	AO
E.g. the approximation is correct to 3 d.p.	B1	3.2b
	(1)
	(7 marks)

Notes

B1: Makes a quantitative statement about the accuracy, so e.g. how many decimal places or significant figures it is correct to, or calculates a percentage accuracy to deduce it is reasonable. Do not accept just “underestimate” or similar without quantitative evidence. Allow for a reasonable comment as long as (b) is correct to at least 2 s.f. but (c) must be the correct value.