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\includegraphics[max width=\textwidth, alt={}, center]{dc8515fb-7104-4d3d-a8ea-ada2b75c70a2-8_442_1134_255_244} A conical shell, of semi-vertical angle \(\alpha\), is fixed with its axis vertical and its vertex V upwards. A light inextensible string passes through a small smooth hole at V and a particle P of mass 4 kg hangs in equilibrium at one end of the string. The other end of the string is attached to a particle Q of mass 2 kg which moves in a horizontal circle at a constant angular speed \(2.8 \mathrm { rads } ^ { - 1 }\) on the smooth outer surface of the shell at a vertical depth \(h \mathrm {~m}\) below V (see diagram).

1. Show that \(\mathrm { k } _ { 1 } \mathrm {~h} \sin ^ { 2 } \alpha + \mathrm { k } _ { 2 } \cos ^ { 2 } \alpha = \mathrm { k } _ { 3 } \cos \alpha\), where \(k _ { 1 } , k _ { 2 }\) and \(k _ { 3 }\) are integers to be determined.
2. Determine the greatest value of \(h\) for which Q remains in contact with the shell. \section*{END OF QUESTION PAPER}