Easy -1.2 This is a routine C1 exercise testing basic index laws (converting surds to fractional powers), algebraic manipulation, and arithmetic series formulas. All parts require only direct application of standard techniques with no problem-solving insight needed, making it easier than average.
\begin{enumerate}
\item Given that $2 ^ { x } = \frac { 1 } { \sqrt { 2 } }$ and $2 ^ { y } = 4 \sqrt { } 2$,\\
(a) find the exact value of $x$ and the exact value of $y$,\\
(b) calculate the exact value of $2 ^ { y - x }$.
\item $f ( x ) = \frac { \left( x ^ { 2 } - 3 \right) ^ { 2 } } { x ^ { 3 } } , x \neq 0$.\\
(a) Show that $\mathrm { f } ( x ) \equiv x - 6 x ^ { - 1 } + 9 x ^ { - 3 }$.\\
(b) Hence, or otherwise, differentiate $\mathrm { f } ( x )$ with respect to $x$.
\item The sum of an arithmetic series is $\sum _ { r = 1 } ^ { n } ( 80 - 3 r )$.\\
(a) Write down the first two terms of the series.\\
(b) Find the common difference of the series.
\end{enumerate}
Given that $n = 50$,\\
(c) find the sum of the series.\\
\hfill \mbox{\textit{Edexcel C1 Q1 [5]}}