Edexcel AEA 2007 June — Question 1 9 marks

Exam BoardEdexcel
ModuleAEA (Advanced Extension Award)
Year2007
SessionJune
Marks9
PaperDownload PDF ↗
TopicGeneralised Binomial Theorem
TypeExpansion with algebraic manipulation
DifficultyChallenging +1.2 This is a structured multi-part question requiring binomial expansion, substitution of a trigonometric expression, and evaluation of specific series. While it involves the generalized binomial theorem and requires careful algebraic manipulation across multiple steps, each part follows logically from the previous one with clear guidance ('hence, or otherwise'). The techniques are standard for AEA level, though the trigonometric substitution and series evaluation require more sophistication than typical A-level questions.
Spec1.04c Extend binomial expansion: rational n, |x|<11.04d Binomial expansion validity: convergence conditions1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=1
1.(a)Write down the binomial expansion of \(\frac { 1 } { ( 1 - y ) ^ { 2 } } , | y | < 1\) ,in ascending powers of \(y\) up to and including the term in \(y ^ { 3 }\) .
(b)Hence,or otherwise,show that $$\frac { 1 } { 4 } \operatorname { cosec } ^ { 4 } \left( \frac { \theta } { 2 } \right) = 1 + 2 \cos \theta + 3 \cos ^ { 2 } \theta + 4 \cos ^ { 3 } \theta + \ldots + ( r + 1 ) \cos ^ { r } \theta + \ldots$$ and state the values of \(\theta\) for which this result is not valid.
(4)
Find
(c) $$\begin{aligned} & 1 + \frac { 2 } { 2 } + \frac { 3 } { 2 ^ { 2 } } + \frac { 4 } { 2 ^ { 3 } } + \ldots + \frac { ( r + 1 ) } { 2 ^ { r } } + \ldots \\ & 1 - \frac { 2 } { 2 } + \frac { 3 } { 2 ^ { 2 } } - \frac { 4 } { 2 ^ { 3 } } + \ldots + ( - 1 ) ^ { r } \frac { ( r + 1 ) } { 2 ^ { r } } + \ldots \end{aligned}$$ (d) 
1.(a)Write down the binomial expansion of $\frac { 1 } { ( 1 - y ) ^ { 2 } } , | y | < 1$ ,in ascending powers of $y$ up to and including the term in $y ^ { 3 }$ .\\
(b)Hence,or otherwise,show that

$$\frac { 1 } { 4 } \operatorname { cosec } ^ { 4 } \left( \frac { \theta } { 2 } \right) = 1 + 2 \cos \theta + 3 \cos ^ { 2 } \theta + 4 \cos ^ { 3 } \theta + \ldots + ( r + 1 ) \cos ^ { r } \theta + \ldots$$

and state the values of $\theta$ for which this result is not valid.\\
(4)\\
Find\\
(c)

$$\begin{aligned}
& 1 + \frac { 2 } { 2 } + \frac { 3 } { 2 ^ { 2 } } + \frac { 4 } { 2 ^ { 3 } } + \ldots + \frac { ( r + 1 ) } { 2 ^ { r } } + \ldots \\
& 1 - \frac { 2 } { 2 } + \frac { 3 } { 2 ^ { 2 } } - \frac { 4 } { 2 ^ { 3 } } + \ldots + ( - 1 ) ^ { r } \frac { ( r + 1 ) } { 2 ^ { r } } + \ldots
\end{aligned}$$

(d)

\hfill \mbox{\textit{Edexcel AEA 2007 Q1 [9]}}
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