OCR MEI Further Pure Core 2023 June — Question 12 7 marks

Exam BoardOCR MEI
ModuleFurther Pure Core (Further Pure Core)
Year2023
SessionJune
Marks7
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicAddition & Double Angle Formulae
TypeDerive triple angle formula only
DifficultyChallenging +1.2 This is a Further Maths question requiring systematic application of multiple angle formulae and binomial expansion of sin⁡θ using complex numbers or De Moivre's theorem. While it involves several steps and algebraic manipulation, it's a standard technique in Further Pure with a clear methodβ€”express sin ΞΈ in terms of exponentials, expand, and collect terms. More routine than typical Further Pure proof questions but harder than standard A-level due to the algebraic complexity.
Spec4.02q De Moivre's theorem: multiple angle formulae
12 Show that \(\sin ^ { 5 } \theta = \operatorname { asin } 5 \theta + \mathrm { b } \sin 3 \theta + \mathrm { csin } \theta\), where \(a , b\) and \(c\) are constants to be determined.
Question 12:
AnswerMarks
12(cid:2873)
(cid:2869)
Considering (cid:4672)π‘§βˆ’ (cid:4673)
(cid:3053)
32isin(cid:2873)πœƒ
1 1 1
= 𝑧(cid:2873)βˆ’5𝑧(cid:2871)+10π‘§βˆ’10 +5 βˆ’
𝑧 𝑧(cid:2871) 𝑧(cid:2873)
1 1 1
= 𝑧(cid:2873)βˆ’ βˆ’5(cid:3436)𝑧(cid:2871)βˆ’ (cid:3440)+10(cid:3436)π‘§βˆ’ (cid:3440)
𝑧(cid:2873) 𝑧(cid:2871) 𝑧
Using 𝑧(cid:3041)βˆ’ (cid:2869) =2isinπ‘›πœƒ with sum of terms
(cid:3053)(cid:3289)
= 2isin5πœƒβˆ’10isin3πœƒ+20isinπœƒ
οƒž sin(cid:2873)πœƒ = (cid:2869) sin5πœƒβˆ’ (cid:2873) sin3πœƒ+ (cid:2873) sinπœƒ
AnswerMarks
(cid:2869)(cid:2874) (cid:2869)(cid:2874) (cid:2876)M1
B1
M1
A1
M1
A1
AnswerMarks
A13.1a
1.1
1.1
2.1
2.1
1.1
AnswerMarks
2.2aseen at any stage
(cid:2873)
(cid:2869)
expansion of (cid:4672)π‘§βˆ’ (cid:4673)
(cid:3053)
correct expansion with 𝑧(cid:3041), (cid:2869) terms paired and factorised
(cid:3053)(cid:3289)
FT their expansion
i must appear in each term
www
Alternative method 1
AnswerMarks Guidance
Considering (cid:3435)e(cid:2919)(cid:3087) βˆ’e(cid:2879)(cid:2919)(cid:3087)(cid:3439) (cid:2873)M1
32isin(cid:2873)πœƒB1
= e(cid:2873)(cid:2919)(cid:3087)βˆ’5e(cid:2871)(cid:2919)(cid:3087)+10e(cid:2919)(cid:3087)βˆ’10e(cid:2879)(cid:2919)(cid:3087) +5e(cid:2879)(cid:2871)(cid:2919)(cid:3087)βˆ’e(cid:2879)(cid:2873)(cid:2919)(cid:3087)M1 M1
expansion of (cid:3435)e(cid:2919)(cid:3087) βˆ’e(cid:2879)(cid:2919)(cid:3087)(cid:3439)
AnswerMarks Guidance
= e(cid:2873)(cid:2919)(cid:3087)βˆ’e(cid:2879)(cid:2873)(cid:2919)(cid:3087) βˆ’5(cid:3435)e(cid:2871)(cid:2919)(cid:3087) βˆ’e(cid:2879)(cid:2871)(cid:2919)(cid:3087)(cid:3439)+10(e(cid:2919)(cid:3087) βˆ’e(cid:2879)(cid:2919)(cid:3087))A1 correct expansion with e(cid:2919)(cid:3087), e(cid:2879)(cid:2919)(cid:3087) terms paired and factorised
Using e(cid:2919)(cid:3087) βˆ’e(cid:2879)(cid:2919)(cid:3087) = 2isinπ‘›πœƒ with sum of termsM1 FT their expansion
= 2isin5πœƒβˆ’10isin3πœƒ+20isinπœƒA1 i must appear in each term
οƒž sin(cid:2873)πœƒ = (cid:2869) sin5πœƒβˆ’ (cid:2873) sin3πœƒ+ (cid:2873) sinπœƒ
AnswerMarks Guidance
(cid:2869)(cid:2874) (cid:2869)(cid:2874) (cid:2876)A1 A1
Alternative method 2
AnswerMarks Guidance
Equating Im components of (cosπœƒ+isinπœƒ)(cid:2873)using binomial expansion and de Moivre’s theorem
sin5πœƒ =5cos(cid:2872)πœƒsinπœƒβˆ’10cos(cid:2870)πœƒsin(cid:2871)πœƒ+sin(cid:2873) πœƒB1
=5(1βˆ’sin(cid:2870)πœƒ)(cid:2870)sinΞΈβˆ’10(1βˆ’sin(cid:2870)πœƒ)sin(cid:2871)πœƒ+sin(cid:2873)πœƒ=5(1βˆ’sin(cid:2870)πœƒ)(cid:2870)sinΞΈβˆ’10(1βˆ’sin(cid:2870)πœƒ)sin(cid:2871)πœƒ+sin(cid:2873)πœƒ M1*
sin(cid:2870)πœƒ
AnswerMarks Guidance
sin5πœƒ =16sin(cid:2873)πœƒβˆ’20sin(cid:2871)πœƒ+5sinπœƒA1 or 16sin(cid:2873)πœƒ =sin5πœƒ+20sin(cid:2871)πœƒβˆ’5sinπœƒ or any correct
rearrangement
AnswerMarks Guidance
Equating Im components of (cosπœƒ+isinπœƒ)(cid:2871)using binomial expansion and de Moivre’s theorem
sin3πœƒ = 3cos(cid:2870)πœƒsinπœƒβˆ’sin(cid:2871)πœƒM1* correct expression for sin3πœƒ in terms of sinπœƒ and cosπœƒ
3 1
sin(cid:2871)πœƒ = sinπœƒβˆ’ sin3πœƒ
AnswerMarks Guidance
4 4A1 A1
οƒž16sin(cid:2873)πœƒ =sin5πœƒ+20(cid:4672) (cid:2871) sinπœƒβˆ’ (cid:2869) sin3πœƒ(cid:4673)βˆ’5sinπœƒ
AnswerMarks Guidance
(cid:2872) (cid:2872)M1dep substituting expression for sin(cid:2871)πœƒ into sin(cid:2873)πœƒ.
οƒž sin(cid:2873)πœƒ = (cid:2869) sin5πœƒβˆ’ (cid:2873) sin3πœƒ+ (cid:2873) sinπœƒ
AnswerMarks Guidance
(cid:2869)(cid:2874) (cid:2869)(cid:2874) (cid:2876)A1 www
[7]
Question 12:
12 | (cid:2873)
(cid:2869)
Considering (cid:4672)π‘§βˆ’ (cid:4673)
(cid:3053)
32isin(cid:2873)πœƒ
1 1 1
= 𝑧(cid:2873)βˆ’5𝑧(cid:2871)+10π‘§βˆ’10 +5 βˆ’
𝑧 𝑧(cid:2871) 𝑧(cid:2873)
1 1 1
= 𝑧(cid:2873)βˆ’ βˆ’5(cid:3436)𝑧(cid:2871)βˆ’ (cid:3440)+10(cid:3436)π‘§βˆ’ (cid:3440)
𝑧(cid:2873) 𝑧(cid:2871) 𝑧
Using 𝑧(cid:3041)βˆ’ (cid:2869) =2isinπ‘›πœƒ with sum of terms
(cid:3053)(cid:3289)
= 2isin5πœƒβˆ’10isin3πœƒ+20isinπœƒ
οƒž sin(cid:2873)πœƒ = (cid:2869) sin5πœƒβˆ’ (cid:2873) sin3πœƒ+ (cid:2873) sinπœƒ
(cid:2869)(cid:2874) (cid:2869)(cid:2874) (cid:2876) | M1
B1
M1
A1
M1
A1
A1 | 3.1a
1.1
1.1
2.1
2.1
1.1
2.2a | seen at any stage
(cid:2873)
(cid:2869)
expansion of (cid:4672)π‘§βˆ’ (cid:4673)
(cid:3053)
correct expansion with 𝑧(cid:3041), (cid:2869) terms paired and factorised
(cid:3053)(cid:3289)
FT their expansion
i must appear in each term
www
Alternative method 1
Considering (cid:3435)e(cid:2919)(cid:3087) βˆ’e(cid:2879)(cid:2919)(cid:3087)(cid:3439) (cid:2873) | M1
32isin(cid:2873)πœƒ | B1
= e(cid:2873)(cid:2919)(cid:3087)βˆ’5e(cid:2871)(cid:2919)(cid:3087)+10e(cid:2919)(cid:3087)βˆ’10e(cid:2879)(cid:2919)(cid:3087) +5e(cid:2879)(cid:2871)(cid:2919)(cid:3087)βˆ’e(cid:2879)(cid:2873)(cid:2919)(cid:3087) | M1 | M1 | (cid:2873)
expansion of (cid:3435)e(cid:2919)(cid:3087) βˆ’e(cid:2879)(cid:2919)(cid:3087)(cid:3439)
= e(cid:2873)(cid:2919)(cid:3087)βˆ’e(cid:2879)(cid:2873)(cid:2919)(cid:3087) βˆ’5(cid:3435)e(cid:2871)(cid:2919)(cid:3087) βˆ’e(cid:2879)(cid:2871)(cid:2919)(cid:3087)(cid:3439)+10(e(cid:2919)(cid:3087) βˆ’e(cid:2879)(cid:2919)(cid:3087)) | A1 | correct expansion with e(cid:2919)(cid:3087), e(cid:2879)(cid:2919)(cid:3087) terms paired and factorised
Using e(cid:2919)(cid:3087) βˆ’e(cid:2879)(cid:2919)(cid:3087) = 2isinπ‘›πœƒ with sum of terms | M1 | FT their expansion
= 2isin5πœƒβˆ’10isin3πœƒ+20isinπœƒ | A1 | i must appear in each term
οƒž sin(cid:2873)πœƒ = (cid:2869) sin5πœƒβˆ’ (cid:2873) sin3πœƒ+ (cid:2873) sinπœƒ
(cid:2869)(cid:2874) (cid:2869)(cid:2874) (cid:2876) | A1 | A1 | www | www
Alternative method 2
Equating Im components of (cosπœƒ+isinπœƒ)(cid:2873) | using binomial expansion and de Moivre’s theorem
sin5πœƒ =5cos(cid:2872)πœƒsinπœƒβˆ’10cos(cid:2870)πœƒsin(cid:2871)πœƒ+sin(cid:2873) πœƒ | B1
=5(1βˆ’sin(cid:2870)πœƒ)(cid:2870)sinΞΈβˆ’10(1βˆ’sin(cid:2870)πœƒ)sin(cid:2871)πœƒ+sin(cid:2873)πœƒ | =5(1βˆ’sin(cid:2870)πœƒ)(cid:2870)sinΞΈβˆ’10(1βˆ’sin(cid:2870)πœƒ)sin(cid:2871)πœƒ+sin(cid:2873)πœƒ | M1* | correct expression for sin5πœƒ and substituting in cos(cid:2870)πœƒ =1βˆ’
sin(cid:2870)πœƒ
sin5πœƒ =16sin(cid:2873)πœƒβˆ’20sin(cid:2871)πœƒ+5sinπœƒ | A1 | or 16sin(cid:2873)πœƒ =sin5πœƒ+20sin(cid:2871)πœƒβˆ’5sinπœƒ or any correct
rearrangement
Equating Im components of (cosπœƒ+isinπœƒ)(cid:2871) | using binomial expansion and de Moivre’s theorem
sin3πœƒ = 3cos(cid:2870)πœƒsinπœƒβˆ’sin(cid:2871)πœƒ | M1* | correct expression for sin3πœƒ in terms of sinπœƒ and cosπœƒ
3 1
sin(cid:2871)πœƒ = sinπœƒβˆ’ sin3πœƒ
4 4 | A1 | A1 | or sin3πœƒ =3sinπœƒβˆ’4sin(cid:2871)πœƒ or any correct rearrangement | or sin3πœƒ =3sinπœƒβˆ’4sin(cid:2871)πœƒ or any correct rearrangement
οƒž16sin(cid:2873)πœƒ =sin5πœƒ+20(cid:4672) (cid:2871) sinπœƒβˆ’ (cid:2869) sin3πœƒ(cid:4673)βˆ’5sinπœƒ
(cid:2872) (cid:2872) | M1dep | substituting expression for sin(cid:2871)πœƒ into sin(cid:2873)πœƒ.
οƒž sin(cid:2873)πœƒ = (cid:2869) sin5πœƒβˆ’ (cid:2873) sin3πœƒ+ (cid:2873) sinπœƒ
(cid:2869)(cid:2874) (cid:2869)(cid:2874) (cid:2876) | A1 | www
[7]
12 Show that $\sin ^ { 5 } \theta = \operatorname { asin } 5 \theta + \mathrm { b } \sin 3 \theta + \mathrm { csin } \theta$, where $a , b$ and $c$ are constants to be determined.

\hfill \mbox{\textit{OCR MEI Further Pure Core 2023 Q12 [7]}}
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