11 The displacement of a door from its equilibrium (closed) position is measured by the angle, \(\theta\) radians, which the door makes with its closed position. The door can swing either side of the equilibrium position so that \(\theta\) can take positive and negative values. The door is released from rest from an open position at time \(t = 0\). A proposed differential equation to model the motion of the door for \(t \geqslant 0\) is \(\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } + \lambda \frac { \mathrm { d } \theta } { \mathrm { d } t } + 3 \theta = 0\) where \(\lambda\) is a constant and \(\lambda \geqslant 0\).

1. 1. According to the model, for what value of \(\lambda\) will the motion of the door be simple harmonic?
   2. Explain briefly why modelling the motion of the door as simple harmonic is unlikely to be realistic.
2. Find the range of values of \(\lambda\) for which the model predicts that the door will never pass through the equilibrium position.
3. Sketch a possible graph of \(\theta\) against \(t\) when \(\lambda\) lies outside the range found in part (b) but the motion is not simple harmonic.