Diophantine equations

7 The points \(P \left( \frac { 1 } { 2 } , \frac { 13 } { 24 } \right)\) and \(Q \left( \frac { 3 } { 2 } , \frac { 31 } { 24 } \right)\) lie on the curve \(y = \frac { 1 } { 3 } x ^ { 3 } + \frac { 1 } { 4 x }\).
The area of the surface generated when arc \(P Q\) is rotated completely about the \(x\)-axis is denoted by \(A\).

1. Find the exact value of \(A\). Give your answer as a rational multiple of \(\pi\). Student X finds an approximation to \(A\) by modelling the arc \(P Q\) as the straight line segment \(P Q\), then rotating this line segment completely about the \(x\)-axis to form a surface.
2. Find the approximation to \(A\) obtained by student X . Give your answer as a rational multiple of \(\pi\). Student Y finds a second approximation to \(A\) by modelling the original curve as the line \(y = M\), where \(M\) is the mean value of the function \(\mathrm { f } ( x ) = \frac { 1 } { 3 } x ^ { 3 } + \frac { 1 } { 4 x }\), then rotating this line completely about the \(x\)-axis to form a surface.
3. Find the approximation to \(A\) obtained by student Y . Give your answer correct to four decimal places.

6 You are given that \(q \in \mathbb { Z }\) with \(q \geqslant 1\) and that
\(\mathrm { S } = \frac { 1 } { ( \mathrm { q } + 1 ) } + \frac { 1 } { ( \mathrm { q } + 1 ) ( \mathrm { q } + 2 ) } + \frac { 1 } { ( \mathrm { q } + 1 ) ( \mathrm { q } + 2 ) ( \mathrm { q } + 3 ) } + \ldots\).

1. By considering a suitable geometric series show that \(\mathrm { S } < \frac { 1 } { \mathrm { q } }\).
2. Deduce that \(S \notin \mathbb { Z }\). You are also given that \(\mathrm { e } = \sum _ { r = 0 } ^ { \infty } \frac { 1 } { r ! }\).
3. Assume that \(\mathrm { e } = \frac { \mathrm { p } } { \mathrm { q } }\), where \(p\) and \(q\) are positive integers. By writing the infinite series for e in a form using \(q\) and \(S\) and using the result from part (b), prove by contradiction that e is irrational.