8. The function \(\mathrm { f } ( x )\) is such that \(\mathrm { f } ( x ) = - x ^ { 3 } + 2 x ^ { 2 } + k x - 10\)
The graph of \(y = \mathrm { f } ( x )\) crosses the \(x\)-axis at the points with coordinates \(( a , 0 ) , ( 2,0 )\) and \(( b , 0 )\) where \(a < b\)
- Show that \(k = 5\)
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[1 mark] - Find the exact value of \(a\) and the exact value of \(b\)
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[3 marks] - The functions \(\mathrm { g } ( x )\) and \(\mathrm { h } ( x )\) are such that
$$\begin{aligned}
& g ( x ) = x ^ { 3 } + 2 x ^ { 2 } - 5 x - 10
& h ( x ) = - 8 x ^ { 3 } + 8 x ^ { 2 } + 10 x - 10
\end{aligned}$$ - Explain how the graph of \(y = \mathrm { f } ( x )\) can be transformed into the graph of \(y = \mathrm { g } ( x )\) Fully justify your answer.
- (ii) Explain how the graph of \(y = \mathrm { f } ( x )\) can be transformed into the graph of \(y = \mathrm { h } ( x )\)
Fully justify your answer.
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A geometric series has second term 16 and fourth term 8
All the terms of the series are positive.
The \(n\)th term of the series is \(u _ { n }\)
Find the exact value of \(\sum _ { n = 5 } ^ { \infty } u _ { n }\)
[0pt]
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