Edexcel AEA (Advanced Extension Award) 2002 Specimen

1．（a）By considering the series $$1 + t + t ^ { 2 } + t ^ { 3 } + \ldots + t ^ { n }$$ or otherwise，sum the series $$1 + 2 t + 3 t ^ { 2 } + 4 t ^ { 3 } + \ldots + n t ^ { n - 1 }$$ for \(t \neq 1\) ．
（b）Hence find and simplify an expression for $$1 + 2 \times 3 + 3 \times 3 ^ { 2 } + 4 \times 3 ^ { 3 } + \ldots + 2001 \times 3 ^ { 2000 }$$ （c）Write down an expression for both the sums of the series in part（a）for the case where \(t = 1\) ．

2．Given that \(S = \int _ { 0 } ^ { \frac { \pi } { 2 } } \mathrm { e } ^ { 2 x } \sin x \mathrm {~d} x\) and \(C = \int _ { 0 } ^ { \frac { \pi } { 2 } } \mathrm { e } ^ { 2 x } \cos x \mathrm {~d} x\) ，
（a）show that \(S = 1 + 2 C\) ，
（b）find the exact value of \(S\) ．

3．Solve for values of \(\theta\) ，in degrees，in the range \(0 \leq \theta \leq 360\) ， $$\sqrt { } 2 \times ( \sin 2 \theta + \cos \theta ) + \cos 3 \theta = \sin 2 \theta + \cos \theta$$

4．A curve \(C\) has equation \(y = \mathrm { f } ( x )\) with \(\mathrm { f } ^ { \prime } ( x ) > 0\) ．The \(x\)－coordinate of the point \(P\) on the curve is \(a\) ．The tangent and the normal to \(C\) are drawn at \(P\) ．The tangent cuts the \(x\)－axis at the point \(A\) and the normal cuts the \(x\)－axis at the point \(B\) ．
（a）Show that the area of \(\triangle A P B\) is $$\frac { 1 } { 2 } [ \mathrm { f } ( a ) ] ^ { 2 } \left( \frac { \left[ \mathrm { f } ^ { \prime } ( a ) \right] ^ { 2 } + 1 } { \mathrm { f } ^ { \prime } ( a ) } \right)$$ （b）Given that \(\mathrm { f } ( x ) = \mathrm { e } ^ { 5 x }\) and the area of \(\triangle A P B\) is \(\mathrm { e } ^ { 5 a }\) ，find and simplify the exact value of \(a\) ．

5．The function f is defined on the domain \([ - 2,2 ]\) by： $$f ( x ) = \left\{ \begin{array} { r l r } - k x ( 2 + x ) & \text { if } & - 2 \leq x < 0 ,
k x ( 2 - x ) & \text { if } & 0 \leq x \leq 2 , \end{array} \right.$$ where \(k\) is a positive constant．
The function g is defined on the domain \([ - 2,2 ]\) by \(\mathrm { g } ( x ) = ( 2.5 ) ^ { 2 } - x ^ { 2 }\) ．
（a）Prove that there is a value of \(k\) such that the graph of f touches the graph of g ．
（b）For this value of \(k\) sketch the graphs of the functions f and g on the same axes，stating clearly where the graphs touch．
（c）Find the exact area of the region bounded by the two graphs．

6．Given that the coefficients of \(x , x ^ { 2 }\) and \(x ^ { 4 }\) in the expansion of \(( 1 + k x ) ^ { n }\) ，where \(n \geq 4\) and \(k\) is a positive constant，are the consecutive terms of a geometric series，
（a）show that \(k = \frac { 6 ( n - 1 ) } { ( n - 2 ) ( n - 3 ) }\) ．
（b）Given further that both \(n\) and \(k\) are positive integers，find all possible pairs of values for \(n\) and \(k\) ．You should show clearly how you know that you have found all possible pairs of values．
（c）For the case where \(k = 1.4\) ，find the value of the positive integer \(n\) ．
（d）Given that \(k = 1.4 , n\) is a positive integer and that the first term of the geometric series is the coefficient of \(x\) ，estimate how many terms are required for the sum of the geometric series to exceed \(1.12 \times 10 ^ { 12 }\) ．［You may assume that \(\log _ { 10 } 4 \approx 0.6\) and \(\log _ { 10 } 5 \approx 0.7\) ．］

7．The variable \(y\) is defined by $$y = \ln \left( \sec ^ { 2 } x + \operatorname { cosec } ^ { 2 } x \right) \text { for } 0 < x < \frac { \pi } { 2 } .$$ A student was asked to prove that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = - 4 \cot 2 x .$$ The attempted proof was as follows： $$\begin{aligned} y & = \ln \left( \sec ^ { 2 } x + \operatorname { cosec } ^ { 2 } x \right)
& = \ln \left( \sec ^ { 2 } x \right) + \ln \left( \operatorname { cosec } ^ { 2 } x \right)
& = 2 \ln \sec x + 2 \ln \operatorname { cosec } x
\frac { \mathrm {~d} y } { \mathrm {~d} x } & = 2 \tan x - 2 \cot x
& = \frac { 2 \left( \sin ^ { 2 } x - \cos ^ { 2 } x \right) } { \sin x \cos x }
& = \frac { - 2 \cos 2 x } { \frac { 1 } { 2 } \sin 2 x }
& = - 4 \cot 2 x \end{aligned}$$ （a）Identify the error in this attempt at a proof．
（b）Give a correct version of the proof．
（c）Find and simplify a general relationship between \(p\) and \(q\) ，where \(p\) and \(q\) are variables that depend on \(x\) ，such that the student would obtain the correct result when differentiating \(\ln ( p + q )\) with respect to \(x\) by the above incorrect method．
（d）Given that \(p ( x ) = k \sec r x\) and \(q ( x ) = \operatorname { cosec } ^ { 2 } x\) ，where \(k\) and \(r\) are positive integers，find the values of \(k\) and \(r\) such that \(p\) and \(q\) satisfy the relationship found in part（c）． \section*{END} Marks for presentation： 7
TOTAL MARKS： 100