Moderate -0.8 This is a straightforward multi-part question requiring routine binomial expansion of (2+y)³, simple substitution and algebraic manipulation to show terms cancel, then direct integration of a polynomial. All steps are standard textbook exercises with no problem-solving insight required, making it easier than average.
2. (a) (i) Using the binomial expansion, or otherwise, express \(( 2 + y ) ^ { 3 }\) in the form \(a + b y + c y ^ { 2 } + y ^ { 3 }\), where \(a , b\) and \(c\) are integers.
(ii) Hence show that \(\left( 2 + x ^ { - 2 } \right) ^ { 3 } + \left( 2 - x ^ { - 2 } \right) ^ { 3 }\) can be expressed in the form \(p + q x ^ { - 4 }\), where \(p\) and \(q\) are integers.
(b) (i) Hence find \(\int \left[ \left( 2 + x ^ { - 2 } \right) ^ { 3 } + \left( 2 - x ^ { - 2 } \right) ^ { 3 } \right] \mathrm { d } x\).
2. (a) (i) Using the binomial expansion, or otherwise, express $( 2 + y ) ^ { 3 }$ in the form $a + b y + c y ^ { 2 } + y ^ { 3 }$, where $a , b$ and $c$ are integers.\\
(ii) Hence show that $\left( 2 + x ^ { - 2 } \right) ^ { 3 } + \left( 2 - x ^ { - 2 } \right) ^ { 3 }$ can be expressed in the form $p + q x ^ { - 4 }$, where $p$ and $q$ are integers.\\
(b) (i) Hence find $\int \left[ \left( 2 + x ^ { - 2 } \right) ^ { 3 } + \left( 2 - x ^ { - 2 } \right) ^ { 3 } \right] \mathrm { d } x$.\\
\hfill \mbox{\textit{SPS SPS SM Pure 2021 Q2 [7]}}