6. Two points \(A\) and \(B\) are in a vertical line, with \(A\) above \(B\) and \(A B = 4 a\). One end of a light elastic spring, of natural length \(a\) and modulus of elasticity \(3 m g\), is attached to \(A\). The other end of the spring is attached to a particle \(P\) of mass \(m\). Another light elastic spring, of natural length \(a\) and modulus of elasticity \(m g\), has one end attached to \(B\) and the other end attached to \(P\). The particle \(P\) hangs at rest in equilibrium.

1. Show that \(A P = \frac { 7 a } { 4 }\) The particle \(P\) is now pulled down vertically from its equilibrium position towards \(B\) and at time \(t = 0\) it is released from rest. At time \(t\), the particle \(P\) is moving with speed \(v\) and has displacement \(x\) from its equilibrium position. The particle \(P\) is subject to air resistance of magnitude \(m k v\), where \(k\) is a positive constant.
2. Show that $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + k \frac { \mathrm {~d} x } { \mathrm {~d} t } + \frac { 4 g } { a } x = 0$$
3. Find the range of values of \(k\) which would result in the motion of \(P\) being a damped oscillation.