Question 8 - A-Level Maths

Exam Board	OCR MEI
Module	Paper 3 (Paper 3)
Year	2020
Session	November
Topic	Integration by Substitution

The curve \(y = \frac { 1 } { \left( 1 + x ^ { 2 } \right) ^ { 2 } }\) is shown in Fig. 8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a13f7a05-e2d3-4354-a0c7-ef7283eff514-08_495_1058_1105_315} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure}
1. Show that \(\frac { d ^ { 2 } y } { d x ^ { 2 } } = \frac { 20 x ^ { 2 } - 4 } { \left( 1 + x ^ { 2 } \right) ^ { 4 } }\).
2. In this question you must show detailed reasoning. Find the set of values of \(x\) for which the curve is concave downwards.
Use the substitution \(x = \tan \theta\) to find the exact value of \(\int _ { - 1 } ^ { 1 } \frac { 1 } { \left( 1 + x ^ { 2 } \right) ^ { 2 } } d x\). Answer all the questions.
Section B (15 marks) The questions in this section refer to the article on the Insert. You should read the article before attempting the questions.