Standard +0.8 This question requires combining two standard series expansions (binomial and Maclaurin) with careful algebraic manipulation and composition. While the individual series are given, students must substitute sin θ into the binomial expansion, then substitute the Maclaurin series for sin θ, and collect terms up to θ³—requiring multiple steps of careful bookkeeping. This is moderately challenging for Further Maths, testing technique rather than deep insight, but the composition and term collection make it above average difficulty.
In this question you may use results from the formulae booklet without proof.
Use the binomial series for \((1 + x)^n\) and the Maclaurin's series for \(\sin x\) to find the series expansion for \(\frac{1}{(1 + \sin \theta)^4}\) up to and including the term in \(\theta^3\)
[4 marks]
Question 18:
18 | Substitutes
sin θ for x in
the first four
terms of the
correct
binomial series
with n=−4.
Condone sign
errors. | 1.1a | M1 | (−4)(−5) (−4)(−5)(−6)
(1+sinθ) −4 =1+(−4)sinθ+ sin2θ+ sin3θ+...
2 6
2
θ3 θ3
=1−4θ− +...+10θ− +...
6 6
3
θ3
−20θ− +... +...
6
2θ3
( )
=1−4θ+ +10θ2 −20θ3 +o θ4
3
1 58
=1−4θ+10θ2 − θ3 +...
(1+sinθ)4 3
Substitutes in
their binomial
series the first
two terms of
the sine series
in the first
bracket, and
the first term of
the sine series
in subsequent
brackets. | 3.1a | M1
Obtains the
correct series.
ISW
Condone x
u sed for θ | 1.1b | A1
Discards
higher powers
of θ in a
correct, fully
simplified,
series | 2.2a | A1
Question total | 4
Q | Marking
instructions | AO | Marks | Typical solution
In this question you may use results from the formulae booklet without proof.
Use the binomial series for $(1 + x)^n$ and the Maclaurin's series for $\sin x$ to find the series expansion for $\frac{1}{(1 + \sin \theta)^4}$ up to and including the term in $\theta^3$
[4 marks]
\hfill \mbox{\textit{AQA Further Paper 2 2024 Q18 [4]}}