2015 IJC P1 Q9

At a robotic exhibition, a miniature robotic mouse is programmed to walk \(2\) different simulation paths from a starting point \(O\) to and from a series of points, \({{P}_{1}},\ {{P}_{2}},\ {{P}_{3}},\ \ldots ,\)increasingly far away in a straight line. The robotic mouse starts at \(O\) and walks stage \(1\) from \(O\) to \({{P}_{1}}\) and back to \(O\), then stage \(2\) from \(O\) to \({{P}_{2}}\) and back to \(O\), and so on. The miniature robotic mouse can walk a maximum distance of \(100\) m before its battery runs out.

1. In simulation \(A\), the distances between adjacent points are all \(3\) cm (see Fig. 1).
   1. Find the distance travelled by the robotic mouse that completes the first \(12\) stages of simulation \(A\).[2]
   2. Write down an expression for the distance travelled by the robotic mouse that completes \(n\) stages of simulation \(A\). Hence find the greatest number of stages that the robotic mouse can complete before its battery runs out.[4]

2. In simulation B, the distances between the points are such that \(O{{P}_{1}}=3\) cm,
   \({{P}_{1}}{{P}_{2}}=3\) cm, \({{P}_{2}}{{P}_{3}}=6\) cm and \({{P}_{n}}{{P}_{n+1}}=2{{P}_{n-1}}{{P}_{n}}\) (see Fig. 2). Write down an expression for the distance travelled by the robotic mouse that completes \(N\) stages of simulation \(B\). Hence find the distance from \(O\), and the direction of travel, of the robotic mouse at the instance when the battery runs out in simulation \(B\).[5]