Integration using De Moivre identities

Edexcel CP1 2024 June Q4

10 marks Standard +0.8

The complex number $z = \mathrm { e } ^ { \mathrm { i } \theta }$, where $\theta$ is real.
1. Show that
$$z ^ { n } + \frac { 1 } { z ^ { n } } \equiv 2 \cos n \theta$$ where $n$ is a positive integer.
Show that $$\cos ^ { 5 } \theta = \frac { 1 } { 16 } ( \cos 5 \theta + 5 \cos 3 \theta + 10 \cos \theta )$$
Hence, making your reasoning clear, determine all the solutions of $$\cos 5 \theta + 5 \cos 3 \theta + 12 \cos \theta = 0$$ in the interval $0 \leqslant \theta < 2 \pi$

OCR Further Pure Core 1 2023 June Q9

14 marks Challenging +1.8

9 In this question you must show detailed reasoning.}

Use de Moivre's theorem to determine constants $A$, $B$ and $C$ such that $$\sin ^ { 4 } \theta \equiv A \cos 4 \theta + B \cos 2 \theta + C .$$ The function f is defined by
$\mathrm { f } ( x ) = \sin \left( 4 \sin ^ { - 1 } \left( x ^ { \frac { 1 } { 5 } } \right) \right) - 8 \sin \left( 2 \sin ^ { - 1 } \left( x ^ { \frac { 1 } { 5 } } \right) \right) + 12 \sin ^ { - 1 } \left( x ^ { \frac { 1 } { 5 } } \right) , \quad x \in \mathbb { R } , 0 \leqslant x < 1$.
Show that $\mathrm { f } ^ { \prime } ( x ) = \frac { 32 } { 5 \sqrt { 1 - x ^ { \frac { 2 } { 5 } } } }$.
\includegraphics[max width=\textwidth, alt={}, center]{478c66d2-16a0-41ef-9444-25cfcd47d11d-7_894_842_1000_260} The diagram shows the curve with equation $\mathrm { y } = \frac { 1 } { \sqrt { 1 - x ^ { \frac { 2 } { 5 } } } }$ for $0 \leqslant x < 1$ and the asymptote $x = 1$. The region $R$ is the unbounded region between the curve, the $x$-axis, the line $x = 0$ and the line $x = 1$. You are given that the area of $R$ is finite.
Determine the exact area of $R$.

AQA Further Paper 1 2019 June Q8

10 marks Challenging +1.2

8

If $z = \cos \theta + \mathrm { i } \sin \theta$, use de Moivre's theorem to prove that $$z ^ { n } - \frac { 1 } { z ^ { n } } = 2 \mathrm { i } \sin n \theta$$ 8
Express $\sin ^ { 5 } \theta$ in terms of $\sin 5 \theta , \sin 3 \theta$ and $\sin \theta$
8
Hence show that $$\int _ { 0 } ^ { \frac { \pi } { 3 } } \sin ^ { 5 } \theta d \theta = \frac { 53 } { 480 }$$

OCR Further Pure Core 2 2021 June Q4

8 marks

4 In this question you must show detailed reasoning.

By writing $\sin \theta$ in terms of $\mathrm { e } ^ { \mathrm { i } \theta }$ and $\mathrm { e } ^ { - \mathrm { i } \theta }$ show that $$\sin ^ { 6 } \theta = \frac { 1 } { 32 } ( 10 - 15 \cos 2 \theta + 6 \cos 4 \theta - \cos 6 \theta ) .$$
Hence show that $\sin \frac { 1 } { 8 } \pi = \frac { 1 } { 2 } \sqrt [ 6 ] { 20 - 14 \sqrt { 2 } }$.
Use differentiation to find the first two non-zero terms of the Maclaurin expansion of $\ln \left( \frac { 1 } { 2 } + \cos x \right)$.

By considering the root of the equation $\ln \left( \frac { 1 } { 2 } + \cos x \right) = 0$ deduce that $\pi \approx 3 \sqrt { 3 \ln \left( \frac { 3 } { 2 } \right) }$. \section*{Total Marks for Question Set 3: 38} \section*{Mark scheme} \section*{Marking Instructions} a An element of professional judgement is required in the marking of any written paper. Remember that the mark scheme is designed to assist in marking incorrect solutions. Correct solutions leading to correct answers are awarded full marks but work must not always be judged on the answer alone, and answers that are given in the question, especially, must be validly obtained; key steps in the working must always be looked at and anything unfamiliar must be investigated thoroughly. Correct but unfamiliar or unexpected methods are often signalled by a correct result following an apparently incorrect method. Such work must be carefully assessed.
b The following types of marks are available. \section*{M} A suitable method has been selected and applied in a manner which shows that the method is essentially understood. Method marks are not usually lost for numerical errors, algebraic slips or errors in units. However, it is not usually sufficient for a candidate just to indicate an intention of using some method or just to quote a formula; the formula or idea must be applied to the specific problem in hand, e.g. by substituting the relevant quantities into the formula. In some cases the nature of the errors allowed for the award of an M mark may be specified.
A method mark may usually be implied by a correct answer unless the question includes the DR statement, the command words "Determine" or "Show that", or some other indication that the method must be given explicitly. \section*{A} Accuracy mark, awarded for a correct answer or intermediate step correctly obtained. Accuracy marks cannot be given unless the associated Method mark is earned (or implied). Therefore M0 A1 cannot ever be awarded. \section*{B} Mark for a correct result or statement independent of Method marks. \section*{E} A given result is to be established or a result has to be explained. This usually requires more working or explanation than the establishment of an unknown result. Unless otherwise indicated, marks once gained cannot subsequently be lost, e.g. wrong working following a correct form of answer is ignored. Sometimes this is reinforced in the mark scheme by the abbreviation isw. However, this would not apply to a case where a candidate passes through the correct answer as part of a wrong argument.
c When a part of a question has two or more 'method' steps, the M marks are in principle independent unless the scheme specifically says otherwise; and similarly where there are several B marks allocated. (The notation 'dep*' is used to indicate that a particular mark is dependent on an earlier, asterisked, mark in the scheme.) Of course, in practice it may happen that when a candidate has once gone wrong in a part of a question, the work from there on is worthless so that no more marks can sensibly be given. On the other hand, when two or more steps are successfully run together by the candidate, the earlier marks are implied and full credit must be given.
d The abbreviation FT implies that the A or B mark indicated is allowed for work correctly following on from previously incorrect results. Otherwise, A and B marks are given for correct work only - differences in notation are of course permitted. A (accuracy) marks are not given for answers obtained from incorrect working. When A or B marks are awarded for work at an intermediate stage of a solution, there may be various alternatives that are equally acceptable. In such cases, what is acceptable will be detailed in the mark scheme. Sometimes the answer to one part of a question is used in a later part of the same question. In this case, A marks will often be 'follow through'.
e We are usually quite flexible about the accuracy to which the final answer is expressed; over-specification is usually only penalised where the scheme explicitly says so.

When a value is given in the paper only accept an answer correct to at least as many significant figures as the given value.
When a value is not given in the paper accept any answer that agrees with the correct value to $\mathbf { 3 ~ s } . \mathbf { f }$. unless a different level of accuracy has been asked for in the question, or the mark scheme specifies an acceptable range.

Follow through should be used so that only one mark in any question is lost for each distinct accuracy error.
Candidates using a value of $9.80,9.81$ or 10 for $g$ should usually be penalised for any final accuracy marks which do not agree to the value found with 9.8 which is given in the rubric.
f Rules for replaced work and multiple attempts:

If one attempt is clearly indicated as the one to mark, or only one is left uncrossed out, then mark that attempt and ignore the others.
If more than one attempt is left not crossed out, then mark the last attempt unless it only repeats part of the first attempt or is substantially less complete.
if a candidate crosses out all of their attempts, the assessor should attempt to mark the crossed out answer(s) as above and award marks appropriately.

For a genuine misreading (of numbers or symbols) which is such that the object and the difficulty of the question remain unaltered, mark according to the scheme but following through from the candidate's data. A penalty is then applied; 1 mark is generally appropriate, though this may differ for some units. This is achieved by withholding one A or B mark in the question. Marks designated as cao may be awarded as long as there are no other errors.
If a candidate corrects the misread in a later part, do not continue to follow through. Note that a miscopy of the candidate's own working is not a misread but an accuracy error.
h If a calculator is used, some answers may be obtained with little or no working visible. Allow full marks for correct answers, provided that there is nothing in the wording of the question specifying that analytical methods are required such as the bold "In this question you must show detailed reasoning", or the command words "Show" or "Determine". Where an answer is wrong but there is some evidence of method, allow appropriate method marks. Wrong answers with no supporting method score zero. \section*{Abbreviations}

Question

Answer

Marks

AO

Guidance

4

(a)

DR \(\begin{aligned}

\sin \theta = \frac { \mathrm { e } ^ { \mathrm { i } \theta } - \mathrm { e } ^ { - \mathrm { i } \theta } } { 2 \mathrm { i } }

\sin ^ { 6 } \theta = \left( \frac { \mathrm { e } ^ { \mathrm { i } \theta } - \mathrm { e } ^ { - \mathrm { i } \theta } } { 2 i } \right) ^ { 6 } = - \frac { 1 } { 64 } \left( \mathrm { e } ^ { \mathrm { i } \theta } - \mathrm { e } ^ { - \mathrm { i } \theta } \right) ^ { 6 }

\left( e ^ { i \theta } - e ^ { - i \theta } \right) ^ { 6 } =

\mathrm { e } ^ { 6 \mathrm { i } \theta } - 6 \mathrm { e } ^ { 4 \mathrm { i } \theta } + 15 \mathrm { e } ^ { 2 \mathrm { i } \theta } - 20 + 15 \mathrm { e } ^ { - 2 \mathrm { i } \theta } - 6 \mathrm { e } ^ { - 4 \mathrm { i } \theta } + \mathrm { e } ^ { - 6 \mathrm { i } \theta } \end{aligned}\) \(\begin{aligned}

\mathrm { e } ^ { 6 \mathrm { i } \theta } + \mathrm { e } ^ { - 6 \mathrm { i } \theta } - 6 \left( \mathrm { e } ^ { 4 \mathrm { i } \theta } + \mathrm { e } ^ { - 4 \mathrm { i } \theta } \right) + 15 \left( \mathrm { e } ^ { 2 \mathrm { i } \theta } + \mathrm { e } ^ { - 2 \mathrm { i } \theta } \right) - 20

= 2 \cos 6 \theta - 6 \times 2 \cos 4 \theta + 15 \times 2 \cos 2 \theta - 20

\therefore \sin ^ { 6 } \theta =

- \frac { 1 } { 64 } ( 2 \cos 6 \theta - 12 \cos 4 \theta + 30 \cos 2 \theta - 20 )

= \frac { 1 } { 32 } ( 10 - 15 \cos 2 \theta + 6 \cos 4 \theta - \cos 6 \theta ) \end{aligned}\)

M1

dep*A1

[5]

1.1a

2.1

Genuine attempt to use binomial expansion with correct evaluated binomial coefficients. Condone sign errors

Collecting terms and using $\mathrm { e } ^ { \mathrm { i } \phi } + \mathrm { e } ^ { - \mathrm { i } \phi } = 2 \cos \phi$ at least once.

AG. Fully correct argument

Condone $2 \mathrm { i } \sin \theta = \mathrm { e } ^ { \mathrm { i } \theta } - \mathrm { e } ^ { - \mathrm { i } \theta }$

Allow use of $\sin \theta = \frac { e ^ { i \theta } + e ^ { - i \theta } } { 2 i }$ for $1 ^ { \text {st } }$ two M marks only

If i omitted from denominator their expression for $\sin \theta$ then only this M mark can still be awarded

(b)

DR $\theta = \frac { \pi } { 8 } \text { and } \mathrm { eg } \cos 2 \theta = \frac { \sqrt { 2 } } { 2 }$

$\sin ^ { 6 } \frac { \pi } { 8 } = \frac { 1 } { 32 } \left( 10 - 15 \times \frac { \sqrt { 2 } } { 2 } - \frac { - \sqrt { 2 } } { 2 } ( + 6 ( 0 ) ) \right)$

$\sin \frac { \pi } { 8 } = \sqrt [ 6 ] { \frac { 1 } { 64 } ( 20 - 15 \sqrt { 2 } + \sqrt { 2 } ) }$

$= \frac { 1 } { 2 } \sqrt [ 6 ] { 20 - 14 \sqrt { 2 } }$

*M1

dep*M1

2.2a

Choice of $\theta$ soi and calculation of at least one cos term.

Substitution and calculation of all cos terms

Terms must be shown distinct either in this line or in the form of $\cos n \frac { \pi } { 8 }$

Question

Answer

Marks

AO

Guidance

5

(a)

$f ( 0 ) = \ln \left( \frac { 1 } { 2 } + \cos 0 \right) = \ln \left( \frac { 3 } { 2 } \right)$ $\frac { \mathrm { d } ^ { 2 } \ln \left( \frac { 1 } { 2 } + \cos x \right) } { \mathrm { d } x ^ { 2 } } = \frac { - \cos x \left( \frac { 1 } { 2 } + \cos x \right) + \sin x ( - \sin x ) } { \left( \frac { 1 } { 2 } + \cos x \right) ^ { 2 } }$ $\ldots \Rightarrow f ^ { \prime \prime } ( 0 ) = - \frac { 2 } { 3 }$

$\ln \left( \frac { 1 } { 2 } + \cos x \right) = \ln \left( \frac { 3 } { 2 } \right) - \frac { x ^ { 2 } } { 3 } + \ldots$

B1

3.1a

Differentiating using chain rule (or rule for $\ln ( \mathrm { f } ( x ) )$ and evaluating when $x = 0$

Allow sign error in numerator

A1

1.1

Differentiating again using quotient (or product/chain) rule.

\multirow[t]{3}{*}{

NB Simplifies to $- \frac { \frac { 1 } { 2 } \cos x + 1 } { \left( \frac { 1 } { 2 } + \cos x \right) ^ { 2 } }$

If zero scored then SC1 for correct expansion

}

[4]

\multirow{4}{*}{}

(b)

\multirow{4}{*}{}

B1

1.1

Finding either $\pm \pi / 3$ as a root. Allow $60 ^ { \circ }$ for B 1 . Ignore other roots

\multirow{4}{*}{Or equating their expression (approximately) to 0 and rearranging for $x$ : $\ln \left( \frac { 3 } { 2 } \right) - \frac { x ^ { 2 } } { 3 } \approx 0 \Rightarrow x \approx \sqrt { 3 \ln \left( \frac { 3 } { 2 } \right) }$}

\(\begin{aligned}

\ln \left( \frac { 1 } { 2 } + \cos x \right) = 0 \Rightarrow x = \frac { \pi } { 3 } \left( \text { or } - \frac { \pi } { 3 } \right)

\therefore \ln \left( \frac { 3 } { 2 } \right) - \frac { \left( \frac { \pi } { 3 } \right) ^ { 2 } } { 3 } \approx 0 \end{aligned}\)

M1

3.1a

Substituting their root, in radians, into their Maclaurin series and equating (approximately) to 0 .

$\ln \left( \frac { 3 } { 2 } \right) - \frac { \pi ^ { 2 } } { 27 } \approx 0 \Rightarrow \pi \approx \sqrt { 27 \ln \left( \frac { 3 } { 2 } \right) } = 3 \sqrt { 3 \ln \left( \frac { 3 } { 2 } \right) }$

A1

3.2a

Could see ± but must be removed by final conclusion.

Must use approximately equals symbol (not just equals symbol)